A Vauchez

G Barruol

A Tommasit

Why do continents break-up parallel to ancient orogenic belts?

The frequently observed parallelism between rifts and the pre existing orogenic fabric of continents suggests that the inherited tectonic fabric of the lithosphere influences the rupture of continents. We propose that the existence of a pervasive fabric in the lithospheric mantle induces an anisotropie strength in the lithosphere, that guides the propagation of continental rifts. Subcrustal mantle mechanical anisotropy is supported by (i) the anisotropie strength of olivine, (ii) an ubiquitous tectonic fabric in exposed mantle rocks, and (iii) measurements of seismic and electrical anisotropy. During major episodes of continent

Rifting parallel to orogenic belts

Ocean-opening through rifting and continent break-up is frequently related to the occurrence ofhotspots. There is, howevcr, a discrepancy between hot spots acting as pin point sources of heat and the linear extent of rifts over thou sands ofkilometres. Moreover rifts tend to parallel pre-existing orogenic fab rics, e.g. North Atlantic rift [START_REF]Did the Atlantic close and then re-open•)[END_REF] , Rio Grande rift [START_REF] Olsen | Rio Grande Rift: An overview[END_REF], North-east China rift [START_REF] Ma | Ccnozoic extensional tectonics in China[END_REF], Baïkal rift [START_REF] Delvaux | Paleostress reconstructions and geodynamics of the Baikal region, Central Asia. Part II: Paleozoic and Cenozoic pre rift evolution[END_REF] , East African rift [START_REF] Ring | The influence of preex1sting structure on the evolution of the Cenozoic Malawi rift (East African rift system)[END_REF] , West African rift (e.g. [START_REF] Fairhead | Differential opening of the Central and South Atlantic oceans and the opening of the West African rift system[END_REF], Cape Graben [START_REF] Burke | Development of graben associa ted with the initial ruptures of the Atlantic ocean[END_REF] , and Eastern Brazilian rift [START_REF] Chang | Tectonics and stratigraphy of the East Brazil Rift system: an overview[END_REF]. From these observations it was suggested that structural inheri tance influences rift propagation. This paper propounds that the source of structural inheritance lies in the fabric of the lithospheric mantle developed during major orogenic episodes.

The East African rift and the North and South Atlantic oceans provide spectacular examples of parallelism be tween rifts and older orogenic belts. The East African rift system (Fig. 1 ) , > 2000 km long and still active , formed mostly during the Cenozoic in a shield stable since the Neoproterozoic [START_REF] Ring | The influence of preex1sting structure on the evolution of the Cenozoic Malawi rift (East African rift system)[END_REF] . The southern part of the system trends almost N-S, parallel to the Mo zambique belt. Northward, the rift sys tem splits into two branches that wrap around the Tanzania craton. The east ern branch remains parallel to the From Florida to the Barents Sea, the North Atlantic ocean opening occurred parallel to the Hercynian and Caledo nian orogens ( Fig. 2 ) . In eastcrn North America, parallelism between the Ap palachians , Triassic rifts. the continent margin, and the main magnetic anoma lies that mark the Atlantic opening indicates that the initial rift propagated along the internai domain of the bclt. South of the West African craton, the Atlantic rift formed paralle! to Neopro terozoic ( southern Senegal, Sierra Leone ) and Hcrcynian belts ( northern Senegal, Mauritania ) . North of the craton, rifting at ;:::: 200 Ma reactivated the Hercynian belt ofMorocco [START_REF] Piqué | The central Atlantic rifting: reactivation of Palaeozoic structures?[END_REF] . South of the North American continent, formation of oceanic lithosphere in the Gulf of Mexico was preceded by the dcvclopment oftriassic extensional ba sins elongated parallel to the Ouachitas mountains ( Worral and Snelson,! 989 ) .

Gondwana Cretaceous break-up be tween South America and Africa is related to the upwelling of the Tristan da Cunha and St. Helena plumes. The initial rift system, however, propagated over more than 3000 km between and beyond these structures, following the Malmesbury-Gariep, Dom Feliciano, Kaoko, Ribeira and West Congolian Neoproterozoic belts (Fig. 3a). The parallelism observed between the South Atlantic rift system and the continental fabric is frequently striking since basins follow the curvatures of these orogenic belts. At the southern tip of Africa, for instance, the initial rift bends parallel to the Gariep-Malmesbury then the Cape fold belts where it became transten sional [START_REF] Burke | Development of graben associa ted with the initial ruptures of the Atlantic ocean[END_REF]. Along the eastern coast of Brazil, [START_REF] Chang | Tectonics and stratigraphy of the East Brazil Rift system: an overview[END_REF] showed that the east Brazilian rift is remarkably parallel to basement struc tures (Fig. 3b). Even where it propa gated in the Sào Francisco craton, the south Atlantic rift followed the steep tectonic fabric and large-scale shear zones of the 2Gyr-old Itabuna belt and Salvador-Esplanada belt of un known age. [START_REF] Burke | Development of graben associa ted with the initial ruptures of the Atlantic ocean[END_REF] emphasized that grabens oblique to the tectonic fabric of the continents rapidly aborted. The central Atlantic is more complex since the ocean curves from N-S to almost E-W and crosscuts various geo tectonic demains. [START_REF] Bonatti | Anomalous opening of the Equatorial Atlantic due to an equatorial mantle thermal minimum[END_REF] sug gests that a cold equatorial belt impeded the propagation of the southem Atlan tic rift northward. An aborted branch of the rift propagated north of this cold belt along the N-S-trending Neopro terozoic Dahomeyide belt, but the main continental breakup occurred through wrench faulting between the St. Helena plume and the North Atlantic spread ing ridge. This Cretaceous wrench tee tonies [START_REF] Fairhead | Differential opening of the Central and South Atlantic oceans and the opening of the West African rift system[END_REF] reactivated Neoproterozoic transconti nental E-W-shear zones.

Rift propagation and rheological heterogenelty of the continental lithosphere

Various interpretations have been pro posed to explain the relationship be tween continental lithospheric rifts and the tectonic fabric of continents. Rheo logical calculations (e.g. [START_REF] Dunbar | How preexisting weaknesses control the style of continental breakup[END_REF] show that an increase in crustal thickness can halve the total strength of the continental lithosphere and thus favour localization of rifting. In active orogens, crustal thickness is expected to vary and to culmina te close to suture zones that represent good candidates for the localization of sub sequent rifting. For rifting beginning shortly after the end of an orogeny (e.g. North Atlantic in the Appalachians/ Mauritanides orogen), variation in crustal thickness across the belt is likely and may result in lateral variation of the total strength of the lithosphere and favour the localization of the rupture process.

Local lithospheric weakness asso ciated with crustal roots does not satis factorily explain rift propagation within an old and stable shield (e.g. South Atlantic or East African rifts). The time lag between the last orogeny and rift initiation is of several hundreds of Myr, a duration long enough for large-scale excess crustal thickness to be removed by surface erosion, isostatic compensa tion or gravitational collapse. The grav ity map ofBrazil, for instance, does not show any anomaly suggesting a residual crustal root beneath late Proterozoic orogens (Ussami et al., 1993), and it may reasonably be assumed that this was already the case during the Cretaceous when the South Atlantic rifting initiated.

Rheological heterogeneity of the continental lithosphere may also arise from old cratonic nuclei incorporated into continents (e.g. [START_REF] Tommasi | Initiation and propagation of shear zones in a heterogeneous continental lithosphere[END_REF]. Even several hundreds of Myr after continent assembly, old cratonic nuclei display a 'colder' geotherm and a thicker lithosphere than the surround ing terranes. Their stiffer lithosphere tends to impede propagation of rifts into cratonic nuclei, and their bound ary with the surrounding lithosphere may represent a preferential site for extensional deformation to localize. This mode) is certainly valid for rifts formed along old cratonic nuclei as the Baïkal and the Kenya rifts, although it remains difficult to evaluate the respec tive contribution of rheological hetero geneity and structural inheritance since usually orogenic belts wrap around the cra tons and rifts are parallel to both the craton boundary and the surrounding belts. Moreover, rheological heteroge neities fail to explain ail cases of rifts developed in an ancient continental lithosphcrc, since in many places rifts formed far from any old cratonic nu cleus (e.g. the southern East African Rift or the South Atlantic rift).

lnherited lithospheric mantle fabric: dues from seismic and eledrical anisotropies

Upper mantle xenoliths and peridotite massifs usually display a well-defined crystallographic fabric due to past de formation of the lithosphere (e.g. Mer cier and [START_REF] Mercier | Textures and fabrics of upper mantle peridotites as illustrated by xenoliths from basalts[END_REF]. Since upper mantle rock-forming minerais (espe cially olivine) are seismically anisotro pie, their crystallographic preferred orientation results in a significant seis mic anisotropy of mantle rocks (e.g. [START_REF] Nicolas | Formation of anisotropy in upper mantle peridotites -A review[END_REF], and is regarded as the main source of teleseis mic shear wave splitting (Silver, 1996). This supports the notion that shear wave splitting is mostly related to the tectonic fabric of the upper mantle, either duc to active (asthenospheric) or 'frozen' (lithospheric) mantle flow. Short-scale spatial variations of split ting parameters on continents and a good correlation with surface geology (e.g. [START_REF] Barruol | Seismic Anisotropy In The Eastern US: Deep Structure Of a Complex Continental Plate[END_REF] point to a dominant contribution of continentàl lithospheric mantle fabric to the split ting of shear waves.

Most commonly, the fast split shear wave is polarized subparallel to the tectonic grain of the orogens (e. g. Sil ver, 1996). The difference in arrivai time between the fast and slow waves is usually around 1 s and may excep tionally exceed 2 s. Considering the intrinsic anisotropy of mantle xeno liths, such time lags require that the waves propagated through at least a 100 krn-thick layer of anisotropie man-tle characterized by a preferred orienta tion of the [0 1 0] and [100] axes of olivine. respectively, perpendicular and close to the strike of the belt (Mainprice and Silvcr, 1993). Thcsc mcasurements hint at the existence of a well-developed crystallographic fabric over the entire thickness of the lithospheric mantle.

Coupled shear wave splitting and magnctotclluric cxperiments tend to support a common lithospheric origin of seismic and electrical anisotropies. Electrical conductivity anisotropy is oftcn interpreted as resulting from the existence of conducting graphite fi lms along grain boundaries, with the high est conductivity parallcl to the foliation trend (e.g. [START_REF] Sénéchal | Seismic and electrical anisotropies m the lithosphere across the Grenville Front[END_REF]. The depth at which electrical anisotropy occurs is related to the freq uency range in which it is observed, and is therefore approximately determinable. [START_REF] Sénéchal | Seismic and electrical anisotropies m the lithosphere across the Grenville Front[END_REF], for instance, carried out shear wave splitting measurements and magnetotelluric soundings along a trans cct in the Canadian shicld: consistent directions of anisotropy and measured electrical anisotropy originating be tween 50 and 150 km, suggested that seismic anisotropy originated from a well-defined fabric in the lithospheric upper mantle. Preliminary results from a magnetotelluric experiments in the eastern US [START_REF] Ogawa | Deep electrical conductivity structures of the Appalachian Orogen in the southeastern US[END_REF] also point to a subcrustal electrical anisotropy with the highest conductivity along the strike of the Appalachians, a result con sistent with shear wave splitting data.

Shear wave splitting measurements in the Appalachians (Barruol et 11!., 1997) and the Pyrenees (Barruol and Souriau, 1995) show (Fig. 4) that most stations located in the external tectonic domains, or even beyond the front of the belts, are characterized by fast split waves polar ized parallel to the belts. This observa tion cannot be explained by a pervasive deformation of the lithosphere during the A ppalachian or the Pyrenean orogenies only. since seismic reflection profiles across the Appalachians [START_REF] Cook | Thin-skinned tectonics in the crystalline southern Appalachians; COC:ORP seismic-reflection profiling of the Blue Ridge and Piedmont[END_REF] and the Pyrenees (Choukroune, 1992) strongly suggest that deforma tion only affected the shallow crust of the cxternal domains ofthese belts. The most Iikely explanation is that the fa b ric of the lithospheric mantle responsi ble for shear wave splitting was not formed during the Appalachian or the Pyrenean orogenies, but was inherited from an earlier orogenic episode (Vau chez and [START_REF] Vauchc7 | Shear waves splitting in the Appalachians and the Pyrenees: Importance of the inherited tectonic fa bric of the lithosphere[END_REF]. These two belts lie on the edge of continents that were mostly built during the Grenvil lian orogeny for the eastern US and the Hercynian orogeny for the western Europe. Considering that, during these episodes of continent assembly, strips of continental Iithosphere > 1000 km wide were deeply deformed. it may be expected that a pervasive fabric was developed over the cntire lithosphere.

Inherited lithospheric fabric had a clear influence on the evolution of east ern North America. The NE-trending Appalachian belt and EW-trending Ouachitas Mountains are consistently parallel with the Grenville belt. In its Canadian branch. the main thrust faults and limits oftectono-metamorphic zones of the Grenville belt trend northeast ward (Hoff man. 1989). Southward, ex posures are scarce, but magnetic and gravimetric anomalies characteristic of the belt domains allow identification of buried terranes and oftheir limits. Post Grenvillian structures systematically formed parallcl to the Grenville bclt. In the Upper Proterozoic-Lower Palaeozoic (i.e. about 400 Myr after the Grenville orogeny), the lithosphere was rifted par allel to the belt, then a passive margin formcd. After a pcriod of occan opening, the eastem American margin turned con vergent, underwent oblique accretion of terranes and finally collision with Africa, resulting in the building of the Appala chian-Ouachitas system that fits almost exactly the Grenville belt. Then a similar cvolution recurrcd [START_REF]Did the Atlantic close and then re-open•)[END_REF] with the opening of the North Atlantic ocean along the Appalachian internai domain.

Reconstitution of the Variscan belt also highlights parallelism betwecn the Pyrenees and pre-existing Hercynian structures. The tectonic fabric of the Pyrenees axial zone is mostly Hercy nian in age, and parallels the belt (e.g. [START_REF] Zwart | The Variscan geology of the Pyrenees[END_REF]). This led us to suggest that the roughly E-W fast polarization di rections found in the southern Pyrenees (Fig. 4) may be generated by an inher ited Hercynian lithospheric fa bric (Bar ruol and Souriau, 1995;Vauche? and Barruol, 1996). A strong mechanical ef fect ofthis inherited fabric is suggested from the subsequent geodynamic evo lution of the pyrenean domain. lberia displacement relative to Eurasia along the North Pyrenean Fault (NPF) started when the Atlantic rift. following the curvature of the Hercynian belt, wrapped around Iberia and produced the open ing of the bay of Biscay. This triggcrcd the break-up between Europe and lber ia, with the initial rupture occurring parallel to the Hcrcynian belt in the future site of the Pyrenees. Subsequently. pull-apart basins elongated parallel to the plate boundary formed � 100 Ma north of the NPF. Finally, N-S con vergence of Iberia and Europa. mainly during the Eocene, led to the formation of the Pyrenees which also parallels the older hercynian grain.

Mechanical anisotropy of the lithospheric mantle

The existence of a penetra live fabric within the lithospheric mantlc suggests an alternative explanation for continental break-up parallel to orogenic belts. If the strength of individual crystals in mantle rocks is ani sotropie. a strong. lattice-preferred orientation of rock forming minerais may entai! a mechan ical anisotropy of upper mantle rocks and therefore of the li thosphere.

Metals, ceramics or ice often display a mechanical anisotropy, i.e. their strength ciepends on the orientation of the ap plied force. Rocks may also display such a mcchanical anisotropy (e.g. Sirieys. 1982): undcr tension. thcir strcngth is generally lower for extension normal than parallel to the foliation plane. This bchaviour is especially clear during elastic and hrittle deformation. hut it also occurs during ductile deformation.

Microscopie studies of deformed peridotitcs and expcrimcntal axial com pression of olivine crystals in different crystallographic orientations provide evidence lhat: (i) few slip systems are availablc to accommodate plastic dc formation. and (ii) these systems dis play a significantly different resistance to flow. The relative strength of the different slip systems in olivine depends strongly on temperature. At high-tem peratures, the (0 10)( 1 00] system is at least two limes weaker than any other system and should accommoda te most of the strain [START_REF] Durham | Plastic tlow of oriented single crystals of olivine. 1. Mechanical data[END_REF]Goctzc, 1977: Bai Pl 111., 1991;Fig. 5). This observa tion agrees with microstructural obser vations of naturally deformed peridotites (e.g. [START_REF] Washington | Crystalline Plasticity and Solid State Flo w in Metamorphic Rocks[END_REF]), which clearly indicate that olivine deforms predominantly by glide on the (0 10)( 1 00] slip system under upper mantle condi tions. Under experimental conditions. a transition from this high temperature [ 100] glide to low-temperature [00 1] glide occurs at c. 1000 CC. Howcvcr. at nat- (After Durham and Goetze, 1997.) ural strain rates this transition occurs at lower temperatures. Thus we might ex pect that upper mantle rocks displaying a crystallographic preferred orientation of olivine will be mechanically aniso tropie. Recent experimental axial com pression of dunite under high temperature and pressure (Chopra, pers. comm. 1995;Wendt and Mainprice in prep.) confïrm this hypothesis: the strength of pretex tured dunites depends on the orienta tion of the foliation with the lowest strength and the most ductile behaviour for compression at 45 ° to the foliation.

The mechanical behaviour of an ag gregate displaying an initial lattice-preferred orientation may also be ap proached through a self-consistent nu merical mode! [START_REF] Lebensohn | A self-consistent anisotropie approach for the simulation of plastic deformation and texture development of polycrystals: Application to zirconium alloys[END_REF]; for a dunite, strength ratios larger than 2 are obtained depending on the orientation of the stresses relative to the crystallographic fabric. Thus if, as sug gested from shear wave splitting mea surements, the continental lithosphere is characterized by a steep foliation and a low-angle lineation, preferential or ientation of the weak (010)[100] slip system should induce reactivation of the initial fabric involving a strike-slip component. This theoretical considera tion is consistent with the large propor tion of earthquakes with a significant component of strike-slip faulting that occur within continental rift zones [START_REF] Doser | Strike slip faulting in continental rifts: examples from Sabukia, East Africa (1928), and other regions[END_REF].

Conclusion

Evidence from xenoliths and peridotite massifs, shear wave splitting and mag netotelluric soundings strongly support that the lithospheric mantle beneath continents retains a penetrative fabric developed during past orogenies. In the Pyrenees or the eastern US, seismic aniso tropy studies together with the tectonic history of these continental domains sug gest that break-up occurred preferen tially along a mantle fabric formed during the Grenvillian or the Hercynian orogeny.

Considering that olivine crystals dis play a significant mechanical anisotropy, we suggest that after a main episode of continent assembly, the lithosphere dis plays a mechanical anisotropy due to a pervasive crystallographic/tectonic fabric. Anisotropie strength of the lithosphere may favour break-up of continents par allel to existing fa bric even if the exten sion direction is not normal to the structural trend of the belt. In this latter case, rifting will be characterized by a transtensional deformation associating extension and strike-slip faulting.

Rifts tend to propagate where the energy required for lithosphere break up is minimum. Recause cratonic roots are stiffer than surrounding terranes, rifts will generally bypass these do mains and localize in mobile belts around them. When rifting occurs ra pidly after an orogenic crisis, the me chanical anisotropy of the mantle may be enhanced by thermal heterogeneities and lithosphere strength variation due to local crustal thickening. Ali these processes have convergent effects. Me chanical anisotropy, however, should be preserved as long as temperature conditions do not allow a significant modification (by annealing for instance) of the fabric ofmantle rocks to occur; it may therefore satisfactorily account for the observed break-up of continents along their main structural trend several hun dreds of Myr after the last orogeny.

Fig. 1

 1 Fig.1Schematic map (modified from[START_REF] Ring | The influence of preex1sting structure on the evolution of the Cenozoic Malawi rift (East African rift system)[END_REF] illustrating the parallelism between normal faults in the Fast African Rift and basement structural fa bric.

  Fig. 2 Schematic map of circum-North Atlantic Caledonian-Hercynian belts he forc North Atlantic rifting.

Fig. 3

 3 Fig. 3 Schematic map of circum-South Atlantic Neo-Proterozoic belts (insert) and of eastern Brazil showing the paralle lism between the Neoproterozoic belts and the Cenozoic rifts formed along the continent margin. Normal faults from Chang et al. (1992). SFC, Sào Francisco cratoo; RPC, Rio de la Plata cratoo; R, Ribeira belt.

Fig. 4

 4 Fig. 4 Teleseismic shear wave splitting in (a) the Appalachians[START_REF] Barruol | Seismic Anisotropy In The Eastern US: Deep Structure Of a Complex Continental Plate[END_REF] and (b) thePyrenees. Anisotropydatain the Central Pyrenees from Barruol and Souriau {1995); in the Eastern Pyrenees the map shows preliminary results from a portable experiment by the Université de Montpellier II and the Observatoire Midi Pyrénées of Toulouse. Black lines show directions of fast split wave polarization; size of grey circles (a) or squares (b) are proportional to the time lag between fast and slow waves. Abbreviations are for station locations. Grey areas in (b) represent exposed Hercynian massifs.

  -strain rate plol for dry olivine single crystals compressed in three different directions ([J JO]c, [JOl]c, [01 l)c) relative to the crystallographic structure.

Acknowledgements

This paper is a contribution to International Commission of the Lithosphere program III-5. We are indebted to A. Souriau who is co-owner of seismic data in the Eastern Pyrenees. We also thank D. Mainprice and R. Meissner for valuable discussions.


Nicolas Moës

Mathieu Cloirec

Patrice Cartraud

Jean-François Remacle

A COMPUTATIONAL APPROACH TO DEAL WITH COMPLEX MICROSTRUCTURE GEOMETRIES

Keywords: Homogenization, surface of discontinuity, X-FEM, periodicity

In multiscale analysis of components, there is usually a need to solve microstructures with complex geometries. In this paper, we use the extended finite element method (X-FEM) to solve scales involving complex geometries. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing a new enrichment strategy. Numerical experiments on the periodic homogenization of two-phase complex cells demonstrate the accuracy and simplicity of the X-FEM.

INTRODUCTION

The eXtended Finite Element Method simplifies greatly the analysis of structures with complex geometry since the mesh is not required to match this geometry. This property is particularly useful when homogenizing the behavior of micro-structures, complex by nature. We focus here our attention to the application of X-FEM to micro-scale problems involved in the homogenization theory. The homogenization theory is a powerful mean for analyzing and modeling heterogeneous structures. Moreover, in the case of periodic linear media, this theory can be rigorously justified, through convergence results, see e.g. ( 3), [START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF]. Thus, as the ratio of the period length to the structure length goes to zero, it can be shown that the actual solution of the heterogeneous structure tends to the solution of a problem written for a structure with an homogenized behavior. Furthermore, it turns out that the effective properties are obtained from the solution of a boundary value problem to be solved on a period of the structure, which will be called the basic cell problem in the sequel.

Since the basic cell problem is a boundary value problem, classical numerical methods can be used for computing their solution, see [START_REF] Bőhm | A short introduction to basic aspects of continuum micromechanics[END_REF] for a state of the art in this matter.

The majority of studies have been based on the Finite Element Method (FEM), see [START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF], [START_REF] Michel | Effective properties of composite materials with periodic microstruture : a computational approach[END_REF] and references herein, and only few papers present a Boundary Element Method implementation, see e.g. ( 9), [START_REF] Okada | Homogenization method for heterogeneous material based on boundary element method[END_REF]. Let us also mention the Fast Fourier Transforms based numerical method [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF]. This method is well suited for periodic homogenization, and is very efficient compared to FEM, especially in the linear range [START_REF] Michel | Effective properties of composite materials with periodic microstruture : a computational approach[END_REF], provided that no voids or rigid inclusions are present in the basic cell.

The advantages of the FEM in micromechanical analysis are indeed the same as in standard engineering problems : its flexibility, and its applicability to non-linear problems, anisotropic materials, and arbitrary geometries. Following (4), one may distinguish, in FEM based periodic homogenization studies, four main groups.

In most of the studies, the mesh models the heterogeneities boundaries in the unit cell, i.e. only one phase is present within each finite element. The main drawback of such an approach is the explicit microstructure modeling, which leads to problems for generating the mesh for complex geometries, and requires the use of sophisticated tools, see e.g. [START_REF] Wentorf | Automated modeling for complex woven microstructures[END_REF].

One way to overcome these difficulties is to use a digital image based FEM technique, as initially proposed in [START_REF] Hollister | Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue[END_REF]. It consists in using a uniform mesh that has the same resolution as the digital image, and then to identify each pixel or voxel as a finite element. Such an approach, however, leads to models which are computationally expensive.

A third approach also uses regular meshes, but the interfaces between constituents may be represented independently of element boundaries. For an element cut by an interface, the heterogeneity is treated at the integration point level [START_REF] Zohdi | A description of macroscopic damage through microstructural relaxation[END_REF]. As we shall see in the numerical experiments, this strategy yields a reasonable rate of convergence for the homogenized parameters of the basic cell but suffers a slow rate of convergence for the quality of the overall stress distribution over the basic cell (energy norm in the stresses).

Another technique is the Voronoi cell finite element model [START_REF] Ghosh | Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method[END_REF], in which an element incorporates the effect of an embedded inclusion in the matrix. Both stress and displacement fields need to be discretized with this technique.

The eXtended Finite Element Method provides yet another way to solve the basic cell problem. The advantage of this approach is to retain all the advantages of the finite element approach (applicability to nonlinear and anisotropic constitutive laws, wide range of codes already written, robustness, ...) while considerably easing the meshing step. Indeed, the physical surfaces of the problem do not need to be meshed. They are taken into account by enriching the finite element approximation space through the partition of unity technique [START_REF] Melenk | The Partition of Unity Finite Element Method: Basic theory and applications[END_REF]. The adequate choice of the enrichment function for cracks was discussed in (1; 14), for intersecting cracks and voids in [START_REF] Daux | Arbitrary branched and intersecting cracks with the eXtended Finite Element Method[END_REF]. Concerning material interfaces, an enrichment was introduced in (19). This latter paper was also the first one to use a level set representation of surfaces in the X-FEM context. This representation not only reduces the surface storage to a usual finite element field but also provides a natural way to express the enrichment. In this paper, we improve the enrichment functions for material interfaces and obtain a convergence rate very close to the one obtained with regular finite elements (i.e. conforming meshes).

The remainder of this paper is organized as follows. The basic cell problem is described in Section 2. The X-FEM approach to solve the basic cell problem is detailed in Section 3 and a new enrichment function for material interface is introduced and compared to previous approaches. The fourth section uses the X-FEM to homogenize two different basic cells : a cell with woven fiber and a cell with randomly placed spherical inclusions. The paper ends with conclusions.

THE BASIC CELL PROBLEM

The formulation of the basic cell problem for composites with periodic microstructure can be derived in a systematic way using the two-scale asymptotic expansion method (3), [START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF], or following a process which is also valid for random media [START_REF] Suquet | Elements of homogenization for inelastic solid mechanics[END_REF], [START_REF] Michel | Effective properties of composite materials with periodic microstruture : a computational approach[END_REF]. The latter approach is used in this paper.

The microstructure description is achieved through the definition of a representative volume element, containing all the geometrical and material data. In the case of a periodic medium, the representative volume element is simply a basic cell Ω, which forms the composite by spatial repetition.

At the macroscopic scale, the stress and strain fields are defined by Σ and E, which are the average of the corresponding microscopic field, σ and e, on the basic cell (the macroscopic and microscopic scales are denoted by x and y, respectively):

Σ(x) =< σ(x, y) >= 1 |Ω| Ω σ(x, y)dΩ E(x) =< e(x, y) > (1)
The homogenized behavior relating Σ to E is built from a localization process, which consists in submitting the composite structure to a given macroscopic stress or strain state.

In the case of a periodic microstructure, such a problem can be studied on a single basic cell, thanks to periodic boundary conditions. Moreover, the local periodicity of the strain e(x, y) with respect to the microscopic scale y, and the relation E(x) =< e(u(x, y)) > leads to the following decomposition of the displacement field:

u(x, y) = E(x).y + u m (y), u m periodic on ∂Ω (2)
Therefore, with the data E, the basic cell (or localization) problem is defined by: Find σ, e, u m such that:

         divσ(y) = 0 σ(y) = a(y) : e(y) = a(y) : E + e(u m (y)) u m periodic σ.n anti-periodic (3)
where "periodic" ("anti-periodic") means that values on opposite sides of the boundary ∂Ω are equal (opposite). The problem (3) admits a unique solution, up to a rigid body translation. Once problem (3) is solved, the strain localization tensor D is obtained

e(y) = D(y) : E (4)
and the effective stiffness a hom may be computed as :

a hom =< a(y) : D(y) > (5)
For a linear elastic medium, the complete determination of a hom requires the solution of problem (3) for 6 independent data E. One way to proceed is to consider successively the elementary macroscopic strain states corresponding to one component of E equal to unity, the others being zero. Thus, introducing the second order tensor I ij defined by:

I ij kh = 1 2 (δ ik δ jh + δ ih δ jk ) (6)
and denoting u m ij the solution of problem (3) with the data I ij , one has :

D khij = e(u m ij (y)) kh (7)
The problem (3) corresponds to the strain approach, the data being the macroscopic strain E. One can also use a stress approach, with the data Σ, thus leading to effective compliance A hom , which is equal to the inverse of a hom .

It is important to mention that, in the framework of periodic homogenization, non periodic materials (i.e. with some randomness in the geometrical shape of the heterogeneities and their spatial distribution) may be studied. In that case, however, one has to use a sufficiently large basic cell, in order to obtain a statistically representative response and accurate effective properties. In this paper, such examples of basic cells will be treated.

The variational formulation of the basic cell problems (3), for a data E = I ij is given by:

Find u m ij periodic such that a(u m ij , v) = -a(I ij , v) ∀v periodic ( 8
)
where we use the bilinear form:

a(u, v) = 1 |Ω| Ω e(u) : a : e(v)dΩ (9)
The effective properties are determined from:

a hom ijkl = a(I kl + u m kl , I ij + u m ij ) = a(I kl , I ij ) + a(I ij , u m kl ) (10)
Since in this paper a displacement approach is used, the right-hand side of ( 8) is known. Therefore, the only specific feature of problem ( 8) is the periodicity condition [START_REF] Michel | Effective properties of composite materials with periodic microstruture : a computational approach[END_REF].

X-FEM DISCRETIZATION

In this paper, we consider the homogenization of two-phase basic cells. The material interface separating the two phases is described by a level set function. The level set representation is then used to express a jump in the strain field within the elements crossed by the material interface.

Level set representation

Consider a two-phase basic cell. The two phases are separeted by an interface which we may locate either explicitely (set of CAD entities) or implicitely by assigning to each node I of the mesh, the distance φ I to the interface (with a positive sign if node I is located in one phase and a negative sign if node I is located in the other phase). Next, we may interpolate these nodal informations using the finite element shape functions, N I (x), yielding the level set expression :

φ(x) = I φ I N I (x) (11)
where the sum over I indicates a sum over the nodes of the mesh (more precisely only the nodes belonging to the element containing point x). The iso-zero of the level set function φ approximates the true location of the interface. A strategy for constructing the level set function in the case where the interface is known only by a set of points is described in (2).

Enrichment strategy

Once the level set function is defined, we can express the X-FEM approximation. The classical finite approximation over the basic cell is given by

u FEM = I u I N I (x) (12)
If the finite element mesh is conforming to the material interface, the (linear) approximation above yields an O(h) convergence rate in the energy norm (provided the solution is smooth) [START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF]. On the contrary, as will be shown in the numerical experiments, if the finite element does not conform to the interface, a poor rate of convergence is obtained. For instance, a two-phase 1D problem with a mesh non-conforming to the interfaces gives an asymptotic rate of convergence as poor as O( √ h) [START_REF] Moës | Investigation of the interactions between the numerical and the modeling errors in the Homogenized Dirichlet Projection Method[END_REF]. In order to avoid poor rates of convergence, the X-FEM adds an enrichment to the classical finite element approximation through a partition of unity technique [START_REF] Melenk | The Partition of Unity Finite Element Method: Basic theory and applications[END_REF]:

u X-FEM = I u I N I (x) + u enr , u enr = J a J N J (x)F (x) (13)
The additional degrees of freedom a J are added at the nodes for which the support is split by the interface. The function F is called the enrichment function. It is dicontinuous in its derivative across the interface. In [START_REF] Sukumar | Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method[END_REF], the enrichment is defined as the absolute value of the level set function which indeed has a discontinuous first derivative on the interface:

F 1 (x) = | I φ I N I (x)| (14)
A similar choice(2), is to define u enr as

u enr = J a J N J (x)(F 1 (x) -F 1 (x J )) (15)
A closer look at the two enrichment above indicates that they will yield the same approximation space. However, the conditioning number of the two resulting matrices will be different in general. It was shown in ( 19) and ( 2) that a smooting of F 1 away from the element layer containing the interface somewhat improved the convergence. The effect of this smoothing in 1D is illustrated Figure 1. For 2D problems, the smoothing strategy is detailed in [START_REF] Sukumar | Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method[END_REF]. we suggest another choice for the enrichment function ( 13):

F 2 (x) = I |φ I |N I (x) -| I φ I N I (x)| ( 16
)
This enrichment is shown Figure 1 in the 1D case. For two-and three-dimensional problems, the enrichment function F 2 is a ridge centered on the interface and has zero value on the elements which are not crossed by the interface.

Remark : If the mesh conforms to the material interface, no nodes are enriched since no element is cut by the material interface. The X-FEM then does behave as the FEM.

Convergence tests

In order to compare the convergence rate of the classical and extended finite element method, numerical experiments were carried out in [START_REF] Moës | A computational approach to handle complex microstructure geometries[END_REF]. we consider a 2D circular and a 3D spherical inclusion under tension in an infinite domain. Concerning the material properties, the Poisson ratio of the inclusion is set to 0.25 and to 0.3 for the matrix. The Young modulus ratio between the inclusion and matrix is set to 10. Figures 2 and3 show the decrease in the energy error with respect to the exact solution in four different cases :

• A finite element mesh conforming to the material interface (denoted FEM);

• A finite element computation with a non conforming mesh (denoted FEM-nonconforming). On the elements containing the material interface, the integration of the bilinear form is performed using the appropriate consitutive law at each integration point.

• The extended finite element method using the enrichment F 1 improved by a smoothing (denoted X-FEM-1 + smoothing);

• The extended finite element method using the new enrichment, F 2 , given equation ( 16) (denoted X-FEM-2).

It can be seen that the new enrichment X-FEM-2 improves somewhat the previous enrichment strategy X-FEM-1. The X-FEM convergence rate is now very close to the "optimal" finite element convergence. We also note, as expected, the poor convergence rate of the non-conforming FEM approach.

Periodicity conditions

To take into account the periodicity condition, we use meshes which are periodic. The displacement constraints on the faces of the unit cell are condensed out of the stiffness matrix at the assembly stage. We mentioned earlier that a node is enriched if its support is crossed by the interface. The support must take into account the periodicity of the mesh. For instance, Figure 4, the support of node K is based on the four shaded elements. Thus node K will be enriched since its support is crossed by the material interface. Nodes K and L have the same number of degrees of freedom with the same values.

NUMERICAL EXPERIMENTS

We consider two different cell geometries : a woven composite cell and a randomly filled cell with spherical inclusions. In each case, we compare the X-FEM results to existing simulation based on the classical finite element method.

Woven cell

This example is based on the numerical experiment carried out by [START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF] and was already considered in [START_REF] Moës | A computational approach to handle complex microstructure geometries[END_REF]. The fibers geometry is defined by a disc following a sinus path. The geometry of the unit cell is shown Figure 6 for "two plies". The finite element mesh designed in [START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF] is shown Figure 7 (for one ply). The X-FEM mesh used is a 32x32x32 uniform tetrahedral mesh shown (Figure 5). The boron fibers and aluminum matrix have the following properties : E f = 400GP a, ν f = 0.3, E m = 72GP a, ν m = 1/3. The homogenized behavior obtained by the X-FEM and FEM computations are given in [START_REF] Okada | Homogenization method for heterogeneous material based on boundary element method[END_REF].

Random cell

As a last example, we consider a random media filled with spherical inclusions [START_REF] Moës | A computational approach to handle complex microstructure geometries[END_REF]. This example was solved by (12) using a Fast Fourier Transform approach. In order to

        (17)
take into account the randomness of the spatial distribution of the inclusions, many basic cells have to be considered for the computation of the homogenized properties. Therefore, the X-FEM approach is particularly effective since the same mesh can be used for all the simulations. Moreover, it is also applicable to voids instead of inclusions (in which case the enrichment is not used and the voids are taken into account using the strategy designed in (5; 2)).

The Figures 8 and9 show draws of 8 and 32 particles, respectively. The draws were forced to produce a periodic cell structure. The location of the sphere centers are randomly generated. If a sphere cannot be introduced in the cell because it intersects other spheres already present, a new center location is generated. If a new sphere intersects the cell boundary, its periodic "images" are also introduced.

The particules and matrix properties are E p = 70GP a, ν p = 0.2 and E m = 3GP a, ν m = 0.35, respectively and the particule volume ratio is set to 26.78%. The X-FEM mesh used is still the uniform mesh shown Figure 5. The homogenized stiffness obtained along the x axis (a hom 1111 ) is 7.611 for 8 particles and 7.711 for 32 particles. These results are very close to the ones reported in [START_REF] Michel | Homogénéisation en mécanique des matériaux 1[END_REF]. The latter where computed with many more draws (and thus more reliable) : a hom 1111 = 7.675 for 8 particles, 7.664 for 64 particles and 7.667 for 420 particles.

The paper has demonstrated the applicability of the X-FEM to the homogenization of periodic basic cells. Since the mesh does not need to conform to the material interfaces, complex and random cell geometries can easily be computed with very limited meshing effort. The results obtained are very close to direct FEM and Fast Fourier Transform approaches. Moreover the X-FEM is also applicable to cells containing voids and phases with nonlinear constitutive laws. Finally, a new enrichment to introduce discontinuous strains inside an element has been developped. The initial results are promising since this enrichment provides a very close convergence rate to the "optimal" FEM convergence rate.

Figure 1 :

 1 Figure 1: Several choices for the enrichment function.

Figure 2 :

 2 Figure 2: Convergence rates for a 2D circular inclusion problem. The alpha value indicates the rate of convergence.

Figure 3 :

 3 Figure 3: Convergence rates for a 3D spherical inclusion problem. The alpha value indicates the rate of convergence.

Figure 4 :

 4 Figure 4: The support of a node located on the side of a 2D cell.

Figure 5 :

 5 Figure 5: The uniform mesh used for the X-FEM computations.

Figure 6 :

 6 Figure 6: Woven cell geometry.

Figure 7 :

 7 Figure 7: Mesh designed in the paper by Kikuchi et al. for the FEM analysis of the woven unit cell.