Bernard Le Gall
email: blegall@univ-brest.fr

Mohamed Ahmed Daoud

Joël Rolet

Nima Moussa

Joel Rolet

Nima Moussa Egueh

Large-scale flexuring and antithetic extensional faulting along a nascent plate boundary in the SE Afar rift

New structural data from the SE Afar margin in Djibouti leads us to discriminate two recent (<1 Ma) extensional fault networks between the Afar accreted crust, and the South Danakil Block to the east. Emphasis is placed on the N-S-oriented Makarassou fault system which dissects the prominent westward-(riftward-)facing topographic flexure between the two structural domains. The most striking characteristic of the Makarassou fault belt is a closely-spaced network of continentward-dipping extensional faults, bounding domino blocks of 3-1 Ma tilted lavas of the Stratoid series. The Makarassou fault-flexure system is interpreted as an extensional faulted monocline that developed in response to differential subsidence and crustal downwarping of the Afar magmatically-accreted transitional crust. Large-scale flexuring and associated structures along the eastern flank of the Asal-Manda Inakir incipient plate boundary are regarded as indicative of a nascent volcanic rifted margin in SE Afar.

Introduction

Since the recognition of volcanic rifted margins (VRMs) in the 1980's (see [START_REF] Coffin | Large igneous provinces : Crustal structure, dimensions and external consequences[END_REF], for a review), the continent-ocean transition at VRMs is known to be characterized by a typical style of crustal deformation and magmatic construction. Diagnostic structures at VRMs comprise oceanward-thickening wedges of flood basalts displaying a large-scale flexure, facing towards the ocean, and further dissected by a network of continentward-dipping extensional faults [START_REF] Mutter | Seaward dipping reflectors and the continent-ocean boundary at passive continental margins[END_REF]. The resulting seaward-dipping reflector sequences (SDRS) pass laterally into oceanic crust [START_REF] White | Magmatism at rift zones : The generation of volcanic continental margins and flood basalts[END_REF].

Generally, the continent-ocean transition at VRMs is either partly under water, as along modern ocean margins, or severely eroded within older exposed continental margins [START_REF] Klausen | The Lebombo monocline and associated feeder dyke swarm : Diagnostic of a successful and highly volcanic rifted margin ?[END_REF]. Therefore, complete cross-strike transects of VRMs are rarely observed, hence leading to uncertainties about, for example, the isostatic versus tectonic origin of the overall fault-flexure pattern [START_REF] Palmason | A continuum model of crustal generation in Iceland : Kinematics aspects[END_REF][START_REF] Nielsen | The East-Greenland rifted continental margin : an examination of the coastal flexure[END_REF], or the timing and duration of brittle strain with respect to the rifting and breakup events. These structural issues are discussed here in the context of the SE Afar margin, where a magmatic fault-flexure system is exceptionally exposed along the Asal active rift (Fig. 1) and interpreted as a possible analog to a nascent VRM.

SE Afar rift context

The network of still-active tectono-magmatic axes that propagated since 3 Ma throughout the Afar depression, ahead of the Red Sea and Aden oceanic ridges, form onshore incipient plate boundaries between Arabia, Somalia and Nubia (Fig. 1B) [START_REF] Mohr | Nature of the crust under Afar : new igneous, not thinned continental[END_REF]. In SE Afar, the inland prolongation of the Aden ridge is outlined by the <1 Ma northerlypropagating Ghoubbet-Asal and Manda-Inakir en echelon subrifts [START_REF] Manighetti | Propagation of rifting along the Arabia-Somalia plate boundary : Into Afar[END_REF].

This modern rift axis cuts through the 3-1 Ma-old Stratoid flood basalts (Barberi and Varet, 1977) that cover most of the thin (18-22 km) and transitional crust of the Afar depression [START_REF] Dugda | New constraints on crustal structure in eastern Afar from the analysis of receiver functions and surface waves dispersion in Djibouti[END_REF]. The SE Afar magmatically-accreted domain is flanked to the east by the South Danakil uplifted block via a prominent riftward-facing topographic scarp representing the SE Afar margin (Fig. 1). The latter is further dissected by the N-S-trending Makarassou fault system (MFS), which is generally regarded as a transfer zone linking the Ghoubbet-Asal and Manda-Inakir subrifts [START_REF] Tapponnier | La zone de Mak'arrassou en Afar, un équivalent émergé des failles transformantes océaniques[END_REF]. Mechanisms of rift propagation in SE Afar have been extensively discussed by [START_REF] Manighetti | Propagation of rifting along the Arabia-Somalia plate boundary : Into Afar[END_REF], but never addressed in terms of uplift process. The role of vertical tectonics during the recent development of the SE Afar margin is emphasized in the present work from the 3Dmorphostructural analysis of the Asal-South Danakil area.

The South Danakil-Asal rift system

Our structural study of the SE Afar margin is primarily based on 3D-morphostructural data supplied by both Landsat ETM+ (resolution 15 m) and ASTER (60 x 60 km) satellite images. ASTER digital elevation models (resolution 15 m) allow estimates of (1) fault and lava dips, and (2) minimum vertical fault displacement from surface offsets along individual fault scarps. Fault lengths, deduced from map traces, are underestimated because of the loss of resolution below 15 m of vertical offset. Extensive field investigations did not permit the observation of fault surfaces with slip criteria, because the lower parts of fault scarps are systematically concealed by slope deposits, whereas their exposed upper sections usually follow basaltic columnar jointing that rarely preserves the fault slip plane.

Field measurements were mostly devoted to calibrating fault and lava dips, estimated from ASTER cross-profiles. The timing and magnitude of syn-to post-Stratoid regional uplift are deduced from the spatial distribution, thickness variations and elevation of the Stratoid basalts along the SE Afar margin.

The mapped extent of the Stratoid flood basalts on the western flank of the South Danakil block suggests a two-stage uplift history which started in pre-Stratoid times with initiation of the proto-Danakil block as a NE-plunging asymmetrical dome (Fig. 1A). Pronounced uplift is assumed to have continued in syn-to post-Stratoid times, in combination with subsidence of the Afar accreted crust to the west. The resulting crustal-scale downward bending occurred synchronously with the emplacement of the Stratoid basalts as a volcanic wedge thickening westwards from a few 10's m on the Asa Gayla plateau, up to 100 m in the Makarassou fault belt, and > 1200 m in the Asal rift (Fig. 1C) [START_REF] Zan | Geothermal exploration in the Republic of Djibouti: thermal and geological data of the Hanlé and Asal areas[END_REF].

Ongoing uplift of the proto-South Danakil block in post-Stratoid times caused the ~800 m elevation of the initially horizontal Stratoid basalts towards the Day Mountains (Fig. 1A). The onset of uplift might be as young as 125 kyr, by correlation with the Middle Pleistocene conglomeratic fans that currently fringe the foot of the South Danakil range in the Tadjoura coastal plain (Fig. 1B) [START_REF] Gasse | Sédiments plio-quaternaires et tectonique en bordure du Golfe de Tadjoura, République de Djibouti[END_REF].

During the last 3 Ma, the SE Afar margin recorded several episodes of extensional faulting in various tectonic settings. The westernmost edge of the proto-South Danakil block (presentday Asa Gayla plateau) developed as a >10 km-wide half-graben-like structure, filled with a probably thin sequence of Stratoid basalts (S 2 ), and bounded to the east by an inferred submeridian master fault (Fa on Figs. 1B and1C). Uplift and erosion events in the footwall block to the east are currently expressed by discrete inliers of deeply incised Lower Stratoid basalts (S 1 ), resting horizontally on top of Dalha substratum volcanics. Two younger extensional events finally led to the present-day fault arrangement along the SE Afar margin.

The <1 Ma Ghoubbet-Asal-Manda-Inakir en echelon subrift pattern along the subdued part of the margin is not detailed here (see [START_REF] Manighetti | Propagation of rifting along the Arabia-Somalia plate boundary : Into Afar[END_REF]. More interestingly for the present issue is the structural development of the Makarassou fault system in the upper part of the SE Afar topographic scarp.

Structure of the Makarassou fault/flexure system

The NS-trending structures of the Makarassou fault system extend through the western flank of a prominent asymmetrical topographic range that culminates at 1800 m to the south, in the Day Mountains, and slopes down, both northwards at 300 m in the Dorra plain, and more abruptly westwards at <100 m in the Asal-Manda-Inakir rift floor (Fig. 1A). On the Landsat ETM+ satellite image in Fig. 2A structures that are sharply cut, and thus post-dated, by the submeridian Makarassou bounding faults (Fig. 2C). The northern termination of the Makarassou fault belt in the Dorra plain (Fig. 1B) is locally sealed by undeformed Manda Inakir basalts, >0.87 Ma in age (in [START_REF] Vellutini | Carte géologique de la République de Djibouti au 1/100 000. Feuille de Dorra[END_REF].

The ~40 km-long transect in Fig. 1C cuts at high angle the widest exposed section of the Makarassou fault system. Topography is dominated by a westerly-facing monoclinal flexure that lowers the Stratoid surface to about 900 m, from the Asa Gayla plateau down to the Asal rift floor. The hinge line of the Makarassou flexure approximately coincides with the NStrending trace of fault Fb (Fig. 2A). The two fault subdomains correspond to a marked shallowing of topography westwards, with a flexure gradient decreasing from ~1°/km to <0.5°/km. However, the most specific attribute of the Makarassou faults is the easterly (outward) dip of 21 extensional structures (out of a total of 29). With the exception of marginal horst-graben structures in the crestal zone to the east, the antithetic fault network occurs strictly through the downflexed part of the cross-section (Fig. 1C). The resulting inward-dipping Stratoid blocks are arranged in a domino configuration, with a mean spacing of ~500 m, and rotated lava dips as high as 30°, in the upper and steepest part of the flexure (Fig. 2B). Mean fault block width increases, up to 1 km, in the western subdomain, concomitant with decreasing lava dips (10° in average). Average fault dip is about 70°E, and remains nearly constant across-strike. Topographic fault scarps extracted from the ASTER 3D models indicate vertical fault throws in the range 20-700 m, with a mean value of 105 m.

Kinematic model and discussion

From the new dataset above, the SE Afar margin is inferred to comprise two distinct extensional fault networks that differ in terms of geometry, timing and kinematics. The Asal-Manda-Inakir fault pattern bounds NW-SE-oriented en echelon horst/grabens, orthogonal to the N30°E regional extension (e.g. [START_REF] Jestin | The Somalia plate and the East African Rift System : present kinematics[END_REF], in the Afar depressed axis. In contrast, the Makarassou extensional fault network is typically characterized by : (1) its spatial association with a riftward-facing monoclinal flexure, (2) its outward-dipping attitude, away from the rift axis, (3) its NS orientation, parallel to the flexure axis, and oblique to the regional extension, and (4) its relatively young age with respect to the Asal structures.

Most of these structural features display striking similarities with those of coastal flexures involving SDRS-type magmatic prisms along VRMs [START_REF] Mutter | Seaward dipping reflectors and the continent-ocean boundary at passive continental margins[END_REF]. However, one specific aspect of the Makarasou fault-flexure system is the lack of exposed mafic dyke swarms, but these latter probably exist at depth, beneath the Stratoid fissural-type basalts.

These analogies lead us to evaluate whether the models applied to nascent VRMs could be valid for the SE Afar volcano-tectonic pattern. Given the location of the Makarassou flexure along the western edge of the Asa Gayla plateau, i.e. close to the hinge line marking the limit of pronounced lava thickness towards the magmatically-accreted rift domain, we infer that magma-assisted processes acted as driving forces for riftward-facing flexuring along the western flank of the South Danakil block. By analogy with the isostatic model of [START_REF] Palmason | A continuum model of crustal generation in Iceland : Kinematics aspects[END_REF], it is suggested that crustal downwarping along the SE Afar margin was the flexural response to differential loading of the magmatically-accreted crust during the Stratoid (3-1 Ma), and even Dalha (8-4 Ma), flood basalt activity (Fig. 3C). Additional parameters are thought to have interacted during flexuring, and to be responsible for the finite geometry of the Makarassou fault-flexure system. Its NS orientation, oblique to the regional extension, is believed to have followed preexisting structural fabrics. Indeed, the extensional rejuvenation of inherited submeridian faults, such as those cutting the Mablas volcanics to the east, likely accounts for the inferred syn-Stratoid bounding fault Fa (Figs. 1C and3A). A steep and similarly-trending weakness zone (Fb) is suspected to occur at depth, beneath the hinge line of the flexure, further west. This resulted in an extensional monocline, analogous to those developed above the upward-propagating tip of a blind normal fault [START_REF] Willsey | Early evolution of an extensional monocline by a propagating normal fault : 3D analysis from combined field study and numerical modeling[END_REF]. Dominantly dip-slip displacement, down to the west, and probably >500 m, likely occurred along the steep discontinuity Fb, in order to accommodate part of the downflexed bending of the Stratoid sequences at the surface. Assuming that the Makarassou flexure nucleated above a steep weakness zone satisfactorily accounts for (1) its relative tightness (~10 km) with respect to its total length, and (2) its concave profile, with higher flexure gradients in its upper part.

The discontinuity Fb is believed to have once breached the monocline along a surface breaking zone, currently outlined by the crestal horst-graben pattern. The network of antithetic normal faults, splaying upwards into the immediate hangingwall of fault Fb, resemble those directly observed, or obtained experimentally, in extensional monoclines [START_REF] Withjack | Experimental models of extensional forced folds[END_REF][START_REF] Kaven | Growth of surface-breaching normal faults as a threedimensional fracturing process[END_REF]. But, their nucleation and development in the downflexed Stratoid lava pile were probably controlled by volcanic structures, as follows.

Provided that the wedge-shaped pile of initially horizontal (Dalha-Stratoid) lava flows rotated westwards, towards the axis of the active magmatic zone, during early flexuring, the cooling joint pattern, known to generally act as tension fractures under tensile stress [START_REF] Forslund | Structure of Tertiary and Pleistocene normal faults in Iceland[END_REF][START_REF] Gall | A morphotectonic study of an extensional fault zone in a magma-rich rift type : the Baringo Trachyte Fault System, Central Kenya Rift[END_REF], likely evolved from an initially vertical attitude to an easterly-inclined position with time (Fig. 3B). The decreasing dip attitude of the joint pattern at depth resulted in a listric-shaped joint trajectory which might have later triggered nucleation of normal faults, dipping away from the subsiding rift axis to the west, once submitted to the N30°E-directed extension.

The isostatic, instead of tectonic, model proposed here for the Makarassou flexure is reinforced by the fact that structural evidence are currently absent for a continentward-facing detachment underlying the Stratoid wedge, as argued in the 'roll-over anticline' model applied to SDRS prisms elsewhere [START_REF] Geoffroy | The coastal flexure of Disko (West Greenland), onshore expression of the 'oblique reflectors[END_REF]. Furthermore, the constant and steeply-dipping attitude of extensional faults throughout the Makarassou flexure is at odds with highly rotated hangingwall structures, typically developed above the frontal part of a detachment [START_REF] Mcklay | Recent advances in analogue modelling : uses in section interpretation and validation[END_REF].

From the age of pre-(3-1 Ma) and post-flexuring (>0.87 Ma) basalts, the maximum duration of the Makarassou flexure is ~2Ma, i.e. as similarly stated for others VRMs [START_REF] Lenoir | High-rate flexure of the East Greenland volcanic margin : constraints from 40 Ar/ 39 Ar dating of basaltic dykes[END_REF][START_REF] Klausen | The Lebombo monocline and associated feeder dyke swarm : Diagnostic of a successful and highly volcanic rifted margin ?[END_REF]. Such a short time span suggests that regional strain rates were relatively high during flexuring along the SE Afar margin, probably in relation with anomalous thermal gradients at depth.

  , the Makarassou fault system is a 70 km-long fault belt that cuts the frontal part of the Stratoid basaltic wedge. It comprises two adjoining fault subdomains, each 5 km wide, with contrasting structural styles. The high-density fault subdomain to the east encloses a dense network of anastomosed and sigmoidal faults, about 5 km long on average. It is separated from the tabular Stratoid basalts of the Asa Gayla plateau via the highly segmented fault Fb in Fig. 1C. Fault arrangement is more regular in the western subdomain where four wider blocks are bounded by linear, and less segmented faults, >15 km in length. There, the westerly-tilted Stratoid surfaces show remnants of NW-SE (Asal-type)

  The magma-driven vertical tectonics and riftward-facing flexuring that operated during the last 3 Ma, along the eastern flank of the Asal-Manda Inakir future plate boundary, are confidently regarded as the stuctural expression of an incipient VRM in SE Afar. The inferred embryonic stage of continental breakup is in agreement with previous assessments about the transitional type of the Stratoid flood basalt series, and the thinned character of the Afar crust.

  Figure captionsFig. 1A. ASTER digital elevation model of the SE Afar rift in Djibouti (vertical resolution 15 m). Topography ranges from -150 m (Asal) up to 1800 m (Day Mountains). See nomenclature in Fig. 1B. The white line shows the limit of Stratoid basalts (onshore). Ticks are inside the volcanic area. Note its obliquity with respect to the South Danakil reliefs. Fig. 1B. Structural sketch of the SE Afar margin in Djibouti. 1. Moussa Ali volcanics ; 2a. 3-1 Ma Stratoid and Gulf basalts; 2b. Lower Stratoid basalts; 3. >3 Ma synrift volcanics, 3a. Dalha basalts, 3b. Mablas Fm. ; 4. Main faults, 5. Makarassou fault system. A., Asal ; AGP., Asa Gayla plateau ; AS., Ali Sabieh ; D., Dorra ; Dj., Djibouti ; DM., Day Mountains ; G., Ghoubbet ; GT., Gulf of Tadjoura ; MA., Moussa Ali volcano ; MFS., Makarassou fault system ; MI., Manda Inakir; T., Tadjoura. Trace of section 1C is drawn. Inset shows accretionary axes in the Afar Triple junction. 1C. Structural cross-section of the central part of the Makarassou fault system. Topography is extracted from the ASTER DEM. 1 and 2. S 1 and S 2 Stratoid basalts, respectively. 3. Dalha basalts. Vertical exaggeration ~4. Location in Fig. 1B.

Fig. 2A .

 2A Fig. 2A. Map trace of the Makarassou fault belt on the eastern flank of the Asal-Manda Inakir rift framework (Landsat ETM+ image, 15 m resolution). The dashed black line is the trace of the flexure hinge line, close to the trace of fault Fb. 2B. Field view of a closelyspaced network (~500 m) of three Stratoid tilted blocks (dip ~15°W), bounded by easterly-facing extensional faults. View looking south. The main fault scarp in the back is about 150 m high. Location in Fig. 1C. 2C. Focussed view of the Landsat ETM+ image (in Fig. 2A) showing Makarassou submeridian normal faults cutting NW-SEtrending Asal horst-graben structures.

Fig. 3A .

 3A Fig. 3A. Sketch structural map of SE Afar emphazing the role of inherited NS discontinuities on (1) the propagation of the Aden and Ghoubbet-Asal-Manda-Inakir incipient plate boundaries, and (2) initiation of the corresponding volcanic rifted margin (Makarassou belt). 1. <1 Ma rifted zones ; 2. Makarassou flexure ; 3. Onshore distribution of 3-1 Ma Stratoid and Gulf basalts; 4. Pre-Stratoid volcanics. A. Asal ; Ar. Arta ; G. Ghoubbet ; MFS. Makarassou fault system ; MI. Manda Inakir ; MT. Maskali transform. 3B. Conceptual model for the origin of outward-dipping extensional faults in wedges of flood basalts. 3C. Development of riftward-flexuring and antithetic extensional faulting along the SE Afar margin, as a consequence of magma-driven isostatic loading, and reactivated heterogeneities. 1. Stratoid basalts (3-1 Ma); 2. Dalha basalts (8-4 Ma); 3. Volcanic substratum (>8 Ma).

Acknowledgments

Funding and logistical support from CERD of Djibouti (Dir. Dr Mohamed Jalludin), and the MAWARI program, are appreciated. Thanks also to the reviewers G. Eagles, A.

Gudmundsson and J. Rowland, and to the Associate Editor A. Nicolas for helpful and constructive comments that improved a previous version of the manuscript. Stefan Lalonde is warmly thanked for polishing the English.


D Idier
email: d.idier@brgm.fr

A Falqués
email: falques@fa.upc.edu

B G Ruessink
email: b.g.ruessink@uu.nl

R Garnier
email: garnierr@unican.es

Shoreline instability under low-angle wave incidence

The growth of megacusps as shoreline instabilities is investigated by examining the coupling between wave transformation in the shoaling zone, longshore transport in the surf zone, cross-shore transport, and morphological evolution. This coupling is known to drive a potential positive feedback in case of very oblique wave incidence, leading to an unstable shoreline and the consequent formation of shoreline sandwaves. Here, using a linear stability model based on the one-line concept, we demonstrate that such instabilities can also develop in case of low-angle or shore-normal incidence, under certain conditions (small enough wave height and/or large enough beach slope).

The wavelength and growth time scales are much smaller than those of highangle wave instabilities and are nearly in the range of those of surf zone rhythmic bars, O(10 2 -10 3 m) and O(1 -10 days), respectively. The feedback mechanism is based on: (1) wave refraction by a shoal (defined as a crossshore extension of the shoreline perturbation) leading to wave convergence shoreward of it, (2) longshore sediment flux convergence between the shoal and the shoreline, resulting in megacusp formation, and (3) cross-shore sed-

Introduction

Rhythmic shorelines featuring planview undulations with a relatively regular spacing or wavelength are quite common on sandy coasts. In the present study, we focus on undulations that are linked to submerged bars or shoals and are generally known as shoreline sandwaves [START_REF] Komar | Beach Processes and Sedimentation[END_REF][START_REF] Bruun | Migrating sand waves or sand humps, with special reference to investigations carried out on the Danish North Sea Coast[END_REF]. These sandwaves can be classified according to their length scale as short and long sandwaves (see, e.g., [START_REF] Stewart | Morphology, formation and migration of longshore sandwaves; long point, lake erie, canada[END_REF]).

The spacing of short sandwaves ranges from several tens to several hundreds of meters and their seaward perturbations are known as megacusps. Observations show that these megacusps can develop shoreward of crescentic bar systems during the typical "Rhythmic Bar and Beach" morphological beach state or can develop from the shore attachment of transverse bars that characterise the "Transverse Bar and Beach" state [Wright and Short, 1984]. These transverse bars can appear where the horns of a previous crescentic bar approach the shoreline [Wright and Short, 1984;[START_REF] Sonu | Three-dimensional beach changes[END_REF][START_REF] Ranasinghe | Morphodynamics of intermediate beaches: A video imaging and numerical modelling study[END_REF][START_REF] Lafon | Morphodynamics of nearshore rhythmic sandbars in a mixed-energy environment (SW France): I. Mapping beach changes using visible satellite imagery[END_REF][START_REF] D R A F T Castelle | Double bar beach dynamics on the high-energy meso-macrotidal french aquitanian coast : a review[END_REF]. On the other hand, transverse bars can also develop freely, independently of any offshore rhythmic system (e.g. the "transverse finger bars" [START_REF] Sonu | Collective movement of sediment in littoral environment[END_REF][START_REF] Sonu | Three-dimensional beach changes[END_REF][START_REF] Ribas | Characteristics and dynamics of surfzone transverse finger bars[END_REF]). The formation of rhythmic surf zone bars and associated megacusps is believed to be due primarily to an instability of the coupling between the evolving bathymetry and the distribution of wave breaking (bedsurf coupling) [START_REF] Falqués | A mechanism for the generation of wave-driven rhythmic patterns in the surf zone[END_REF]. The developing shoals and channels cause changes in wave breaking, which in turn cause gradients in radiation stresses and thereby horizontal circulation with rip currents. If the sediment fluxes carried by this circulation converge D R A F T September 27, 2011, 5:50pm D R A F T over the shoals and diverge over the channels, a positive feedback arises and the coupled system self-organizes to produce certain patterns, both morphological and hydrodynamic (see, e.g., [START_REF] Reniers | Morphodynamic modeling of an embayed beach under wave group forcing[END_REF]; [START_REF] Garnier | Modelling the formation and the long-term behavior of rip channel systems from the deformation of a longshore bar[END_REF]). In the case of oblique wave incidence, a meandering of the longshore current is also essential to the instability process [START_REF] Garnier | Generation and nonlinear evolution of shore-oblique/transverse sand bars[END_REF]. Two important characteristics of all available models of the selforganized formation of rhythmic surf zone bars are that they (1) are essentially based on sediment transport driven by the longshore current and rip currents only, i.e. ignore cross-shore transport induced by undertow and wave non-linearity and (2) do not consider morphological changes beyond the offshore reach of the rip-current circulation.

Rhythmic shorelines can also develop as a result of an instability not related to bedsurf coupling. [START_REF] Ashton | Formation of coastline features by large-scale instabilities induced by high-angle waves[END_REF] and Ashton and Murray [2006a, b] have shown that sandy shorelines are unstable for wave angles (angle between wave fronts and the local shoreline orientation) larger than about 42 • in deep water, leading to the formation of shoreline sandwaves, cuspate features and spits. [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] have found that the initial characteristic wavelength of the emerging sandwaves is in the range of 3 to 15 km, i.e., much larger than that of surf zone rhythmic bars. This instability caused by high-angle waves will henceforth be referred to as HAWI (High-Angle Wave Instability).

The physical mechanism can be explained as follows. For oblique wave incidence, there are essentially two counteracting effects. On one hand, the angle relative to the local shoreline is larger on the lee of a cuspate feature than on the updrift side. This tends to cause larger alongshore sediment flux at the lee and thereby divergence of sediment flux along the bump, which therefore tends to erode. On the other hand, since the refractive wave ray turning is stronger at the lee than at the updrift flank, there is more wave energy D R A F T September 27, 2011, 5:50pm D R A F T spreading due to crest stretching at the lee causing smaller waves and smaller alongshore sediment flux. This produces convergence of sediment flux at the bump, which therefore tends to grow. For high angle waves the latter effect dominates and, if the bathymetric perturbation associated with the shoreline feature extends far enough offshore, it leads to the development of the cuspate feature. In contrast to the bed-surf instability for rhythmic surf zone bars, this instability mechanism depends essentially on the coupling between the surf and shoaling zones. Indeed, the gradients in alongshore sediment flux that induce bathymetric changes in the surf zone occur because of wave field perturbations induced by bathymetric features in the shoaling zone. Thus, in order to achieve a positive feedback, the shoals (or the bed depressions) in the surf zone must extend into the shoaling zone. This is achieved by the cross-shore sediment fluxes induced, for instance, by wave nonlinearity, gravity and undertow, which force the cross-shore shoreface profile to reach an equilibrium profile. HAWI may provide an explanation for the self-organized formation of some long shoreline sand waves which are reported in the literature [START_REF] Verhagen | Sand waves along the dutch coast[END_REF][START_REF] Inman | Nile delta profiles and migrating sand blankets[END_REF][START_REF] Thevenot | Longshore sandwaves at Southampton Beach, New York: observations and numerical simulation of their movement[END_REF][START_REF] Ruessink | Dunefoot dynamics along the dutch coast[END_REF][START_REF] Davidson-Arnott | Migration and sedimentology of longshore sandwaves[END_REF].

On the other hand, some observations for low incidence angles show that longshore currents can converge on megacusps because of wave refraction [START_REF] Komar | Beach Processes and Sedimentation[END_REF]. Such current convergence may lead to longshore sediment flux convergence and hence to megacusp growth. If the submerged part of the megacusp grows and extends far enough into the shoaling zone (due to the cross-shore transport leading to an equilibrium profile), wave refraction would be enhanced and a positive feedback would arise. This might provide a mechanism for shoreline instability formation under low-angle wave incidence that bears D R A F T September 27, 2011, 5:50pm D R A F T some similitude with HAWI in the sense that the coupling between the surf and shoaling zones is essential, in contrast to the bed-surf instability. We will refer to this potential mechanism as Low-Angle Wave Instability (LAWI). The aim of the present contribution is to investigate this new morphodynamic instability mechanism and discuss whether the resulting shoreline instability may be found in nature.

The lay-out of this paper is as follows. First (Section 2), we introduce the 1D-morfo model [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] that we used to investigate LAWI. Numerical experiments of idealized cases are presented and analyzed in Section 3. We find that shoreline sandwaves can indeed develop because of LAWI, with a length scale comparable to those of megacusps and rhythmic surf zone bars. In Section 4, we analyze the physics of the instability mechanism. We conclude our paper with a discussion and a summary of the main results.

The shoreline stability model: 1D-morfo

Owing to the similitude with HAWI [START_REF] Ashton | Formation of coastline features by large-scale instabilities induced by high-angle waves[END_REF][START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF], the LAWI mechanism is assumed to be un-related to surf zone processes like rip or longshore meandering currents. Thus, the engineering simplification of one-line modeling [START_REF] Dean | Coastal Processes[END_REF] is used, where the shoreline dynamics are based on alongshore gradients in the total alongshore transport rate, Q (i.e., the total volume of sand carried by the wave-driven longshore current that crosses a cross-section of the surf zone area for unit of time (m 3 /s)). Using such a simple model, which neglects numerous aspects of surf zone dynamics, including rip current circulation, longshore current meandering, and thus the bed-surf coupling phenomena, allows to investigate properly whether the LAWI mechanism is supported by the governing equations. Furthermore, the consistency

D R A F T September 27, 2011, 5:50pm D R A F T
of the sediment transport patterns from the one-line modelling has been confirmed by using a 2DH model (delft3D) [START_REF] List | A circulation modeling approach for evaluating the conditions for shoreline instabilities[END_REF], at least in case of HAWI. These reasons, in addition to the fact that HAWI has been studied and reproduced with a linear stability model called 1D-morfo based on the one-line concept, lead us to use the same model to investigate LAWI and its differences with HAWI. A very brief description of the 1D-morfo model is given here. The reader is referred to [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] for further details.

The model describes the dynamics of small amplitude perturbations of an otherwise rectilinear coastline. Following the one-line concept, the dynamics are governed by gradients in the total alongshore wave-driven transport rate Q:

D ∂x s ∂t = - ∂Q ∂y (1)
A Cartesian coordinate system is used, with x increasing seaward in the unperturbed cross-shore direction and y running alongshore (Figure 1). The position of the shoreline Figure 1 is given by x = x s (y, t), where t is time and D is the active water depth (as defined by [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF]), which is of the order of the depth of closure. This active water depth is directly related to the one-line model concept, for more details, see Falqués

and [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF]. The transport rate Q is computed according to the longshore sediment transport equation of [START_REF] Ozasa | Mathematical modelling of beaches backed by seawalls[END_REF]. In this formulation, Q is the sum of two terms: the first one (Q 1 ) is driven by waves approaching the shore at an angle and is equivalent to the CERC formula [USACE , 1984], and the second one (Q 2 ) takes into account the influence of wave set-up induced currents related to the alongshore gradient in the wave height.

D R A F T September 27, 2011, 5:50pm D R A F T

The equation of the sediment transport rate Q can be written as follows:

Q = Q 1 + Q 2 Q = μH 5/2 b sin(2(θ b -φ)) -r 2 β cos(θ b -φ) ∂H b ∂y (2)
where H b is the (rms) wave height at breaking (index b), θ b is the angle between wave fronts and the unperturbed coastline at breaking, and φ = tan -1 (∂x s /∂y) is the local orientation of the perturbed shoreline. β is the beach slope at the instantaneous shoreline (i.e. the waterline). The constant μ is proportional to the empirical parameter K 1 of the original CERC formula and is ∼ 0.1 -0.2 m 1/2 s -1 . For the reference case of this paper, a value, μ = 0.15 m 1/2 s -1 , was chosen, which corresponds to K 1 = 0.525. The nondimensional parameter r is equal to K 2 /K 1 , where K 2 is the empirical parameter of [START_REF] Ozasa | Mathematical modelling of beaches backed by seawalls[END_REF]. According to [START_REF] Horikawa | Nearshore Dynamics and Coastal Processes[END_REF], r ranges between 0.5 and 1.5, whereas [START_REF] Ozasa | Mathematical modelling of beaches backed by seawalls[END_REF] suggest a value of 1.62. The value r = 1 is used for the present reference case, which is equivalent to K 2 = K 1 and has been used in several earlier studies on shoreline instabilities [START_REF] Bender | Potential shoreline changes induced by three-dimentional bathymetric anomalies with gradual transitions in depth[END_REF][START_REF] List | Understanding differences between delft3d and emperical predictions of alongshore sediment transport gradients[END_REF][START_REF] Van Den Berg | Long-term evolution of nourished beaches under high angle wave conditions[END_REF]. However, the term Q 2 is not always taken into account, and its validity and application range are uncertain. In section 3.2, we will study the sensitivity of our results to the value of r.

Some discussion exists about the capacity of the CERC formula (Q 1 ) to predict correctly gradients in alongshore sediment transport in the presence of bathymetric perturbations [START_REF] D R A F T List | Predicting longshore gradients in alongshore transport: comparing the cerc formula to delft3d[END_REF][START_REF] List | Understanding differences between delft3d and emperical predictions of alongshore sediment transport gradients[END_REF][START_REF] Van Den Berg | Long-term evolution of nourished beaches under high angle wave conditions[END_REF]. The results of [START_REF] List | A circulation modeling approach for evaluating the conditions for shoreline instabilities[END_REF] suggest that the CERC formula predicts qualitatively correct transport gradients for large scale shoreline undulations (alongshore lengths of 1-8 km). The term Q 2 was introduced to describe the sediment transport resulting from alongshore variations in the breaker wave height induced by diffraction near coastal structures. These breaker-height variations in-D R A F T September 27, 2011, 5:50pm D R A F T duce alongshore gradients in set-up, which drive longshore currents and hence sediment transport. In our work, alongshore variability in breaker heights is related to wave refraction rather than to wave diffraction, but the subsequent mechanism for alongshore sediment transport remains the same.

To compute the sediment transport rate according to eq. 2, H b (y, t) and θ b (y, t) are needed. The procedure to determine them is as follows. It is assumed that the wave height and wave angle are alongshore uniform in deep water. Then, wave transformation, including refraction and shoaling, is performed from deep water up to the breaking point so that H b (y, t) and θ b (y, t) are determined and Q can be computed. To do the wave transformation, a perturbed nearshore bathymetry coupled to the shoreline changes is assumed:

D(x, y, t) = D 0 (x) -β f(x) x s (y, t) ( 3 )
where D(x, y, t) and D 0 (x) are the perturbed and unperturbed water depth, respectively, and f (x) is a shape function. Figure 2a shows some examples of possible perturbation Figure 2 profiles: constant bed perturbation in the surf zone and decreasing exponentially in the offshore direction (P1), bed perturbation decreasing exponentially in the offshore direction from the coast (P2), and bed perturbation similar to a shoal (P3 and P4).

The offshore extension of the bathymetric perturbation is controlled by its "characteristic" length, xl, which is a free parameter in the model. It was shown by Falqués

and [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] that the coupling between the surf and shoaling zones is crucial for HAWI. This is accomplished only if xl is at least a couple of times larger than the surf zone width. The parameter xl can be seen as a way to parameterise cross-shore sediment transport, especially between the surf and shoaling zones. This makes HAWI essentially

D R A F T September 27, 2011, 5:50pm D R A F T
different from the surf zone morphodynamic instabilities that lead to rhythmic bars and rip channels. The changes in the shoreline cause changes in the bathymetry (both in the surf and shoaling zones), which in turn cause changes in the wave field. The changes in the wave field affect the sediment transport that drives shoreline evolution. Therefore, the shoreline, the bathymetry and the wave field are fully coupled.

Following the linear stability concept, the perturbation of the shoreline is assumed to be

x s (y, t) = ae σt+iKy + c.c. ( 4
)
where a is a small amplitude. For each given (real) wavenumber, K, this expression is inserted into the governing equation (eq. 1), and into the perturbed bathymetry equation (eq. 3). By computing the perturbed wave field and inserting H b and θ b in eq. 1, the complex growth rate, σ(K) = σ r + iσ i , is determined. All of the equations are linearized with respect to the amplitude, a. Then, for those K such that σ r (K) > 0 a sandwave with wavelength λ = 2π/K tends to emerge from a positive feedback between the morphology and the wave field. The pattern that has the maximum growth rate is called the Linearly Most Amplified mode (LMA mode).

Numerical experiments on idealised cases

To investigate the possible mechanism causing shoreline instabilities under low wave incidence angles, numerical experiments on idealised cases are performed. First, numerical experiments and results are given. Then, a sensitivity study is done to assess better the results.

D R A F T September 27, 2011, 5:50pm D R A F T

Instabilities versus beach slope and wave incidence angle

Configuration

A Dean profile (Figure 2b) is chosen as the equilibrium profile, using various beach slopes (Figures 2b andc). The adopted profile is given by:

D 0 (x) = b (x + x 0 ) 2/3 -x 2/3 0 (

Results

Figure 3a shows the growth rate of the LMA mode versus the wave angle and the beach Figure 3 slope. The wave angle is given for a water depth of 25 m. The beach slope is defined as the beach slope at the shoreline. For small beach slopes (< 0.04) the coast behaves as expected: it is unstable only if the wave incidence angle is large enough. In this case, the shoreline instabilities clearly correspond to HAWI. For instance, for a beach slope of 0.02, D R A F T September 27, 2011, 5:50pm D R A F T a wave incidence of 70 • leads to the largest growth rate (3.13 × 10 -9 s -1 ) , corresponding to a wavelength of 7000 m (Figure 3b).

However, for larger beach slopes, instabilities occur for all wave directions, and especially for low wave incidence angles. These instabilities correspond to what will be called LAWI in the present paper. Furthermore, among all the wave incidence angles, the most amplified mode is for shore-normal wave incidence. For a beach slope of 0.1, an angle of 0 • leads to the largest growth rate (1.82 × 10 -6 s -1 ), corresponding to a wavelength of 571 m (Figure 3b).

Sensitivity analysis

The sensitivity analysis is performed using a planar beach, as the aim is to focus on the mechanisms. However, simulations with other (e.g. barred) profiles also lead to LAWI

(not shown).

Wave height and beach slope

Keeping the same reference configuration as above and focusing on shore-normal wave incidence, we investigate the sensitivity of the instability to the wave height for a range of beach slopes. For normally incident waves, instabilities develop only for a beach slope that exceeds 0.04 (Figure 4). In this case, there is an optimum in the wave height for Figure 4 which the growth rate is largest. This wave height increases with the beach slope. For instance, for a beach slope of 0.1 and 0.18), the optimal wave height H rms is 1.75 m and 2.5 m, respectively. A wave height increase also leads to an increase in the LMA mode wavelength, which is due to the corresponding increase in surf zone width.

Bathymetric perturbation length

D R A F T September 27, 2011, 5:50pm D R A F T
The influence of the parameter xl was previously investigated by [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF], who showed that xl must exceed a threshold to initiate HAWI (the perturbation must extend across both the surf and shoaling zones). Similar behaviour is found here by exploring the range between xl = 10 and xl = 10 4 m. Figure 5 shows that there is a The main conclusion is that instabilities occur only if the perturbation extends far enough into the shoaling zone. When the perturbation length is about the width of the surf zone, it influences the wavelength and growth rate of the LMA mode strongly, whereas for larger perturbation length values, this influence is negligible.

Initial perturbation shape

To evaluate the influence of the bathymetric perturbation shape function on the results

we have presented so far, several shapes were investigated using the same wave-boundary conditions and a perturbation length xl of 2000 m. The shape functions we consider are (Figure 2a):

• Perturbation P1: bed perturbation constant in the surf zone and decreasing exponentially in the offshore direction D R A F T September 27, 2011, 5:50pm D R A F T

• Perturbation P2: bed perturbation decreasing exponentially from the coast in the offshore direction

• Perturbation P3: P2 perturbation with a shoal located only in the shoaling zone, from 400 to 800 m, with a maximum height at x 1 = 600 m.

• Perturbation P4: P2 perturbation with a shoal located in both the surf and shoaling zones, from 0 to 1400 m, with a maximum height at x 1 = 600 m.

The reference configuration is still the same (H rms =1.5 m, T = 8 s, θ = 0 • ). The four shape functions result in LMA mode wavelengths of 571, 571, 608 and 608 m, respectively, and growth rates of 1.7, 1.5, 1.7, 1.7 ×10 -6 s -1 , respectively. Thus, the results are slightly sensitive (mean wavelength of 598 m and a standard deviation of 18 m) to the bed perturbation type, but all perturbation types cause LAWI with similar growth rates.

Sediment transport equation

To investigate the sensitivity of the results to the sediment transport equation, computations were carried out with r = 0, which reduces eq. 2 to the CERC equation. This sensitivity study is done in beach slope -wave angle space. The LMA characteristics are quite similar with r = 1 (Figure 3) and r = 0 (Figure 6), except for the case of normal Figure 6 wave incidence. In this case (r = 0), for small beach slopes, all of the perturbations are damped, as for r = 1. For larger beach slopes, the growth rate increases with decreasing perturbation wavelength without reaching a local maximum (Figure 7). Thus there is no Figure 7 LMA mode for shore-normal wave incidence (X symbol on Figure 6). This specific case for shore-normal wave incidence will be discussed in section 4. To summarize, the previous results are not highly sensitive to the second term of the sediment transport equation, except for the case of shore-normal wave incidence.

D R A F T September 27, 2011, 5:50pm D R A F T

The physical mechanism

Here we investigate the physics behind the model prediction of shoreline instabilities caused by low wave incidence angles. The physical processes are analysed based on the study of the growth rate components, and the hydrodynamic and sediment transport patterns, before identifying the main mechanisms.

Growth rate analysis

In the perturbed situation where the shoreline position is given by eq. 4, the wave height and wave angle at breaking are given by:

H b (y, t) = H 0 b + ( Ĥ br + i Ĥ bi )e σt+iKy + c.c. θ b (y, t) = θ 0 b + ( θ br + i θ bi )e σt+iKy + c.c. ( 6
)
where H 0 b , θ 0 b are the wave height and angle for the unperturbed situation. Then, according

to [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF], the growth rate (the real part of the complex growth rate) is:

σ r = 2 μ D H 0 b K 2 cos 2θ 0 b e 0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ -1 e 1 + θ bi aK e 2 + 5 Ĥ bi 4aKH 0 b tan 2θ 0 b e 3 -r Ĥ br aβ cos (θ 0 b ) cos (2θ 0 b ) e 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (7)
A clue to the physical mechanism is provided by a careful analysis of the meaning and behaviour of each term:

• e 0 : common to all terms. It does not contribute to the stability/instability since it is positive. This is because we can assume that θ 0 b < 45 • due to wave refraction. It is the magnitude of the growth rate.

• e 1 : always negative. It represents the contribution due to the changes in shoreline orientation when there is no perturbation in the wave field. This is the only term arising D R A F T September 27, 2011, 5:50pm D R A F T in case of the classical analytical one-line modeling (Pelnard-Considère equation) [START_REF] Dean | Coastal Processes[END_REF]. It is a damping term describing the shoreline diffusivity in that approach.

• e 2 : its sign depends on θ bi . Numerical computations [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] demonstrates that it is always positive. This results from the fact that refracted wave rays tend to rotate in the same direction as the shoreline. Thus e 2 is a growing term.

• e 3 : its sign depends on Ĥ bi . Numerical computations [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] show that Ĥ bi > 0 for long sandwaves and < 0 for short sandwaves. This term is related to energy spreading due to wave crest stretching as waves refract. Thus e3 is a growing or damping term, depending on the sandwave wavelength, 2π/K. Moreover, its magnitude increases with an increasing incident wave angle. These two properties explain that e 3 is an essential growing term for HAWI formation [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF], whereas, for LAWI, its magnitude is smaller and it is, most of the time, negative.

• e 4 : this term stems from the alongshore gradients in H b in the sediment transport equation (eq. 2). Its sign is the opposite to that of Ĥ br , which is numerically found to be always positive. This is related to the fact that the maximum in wave energy is always located close to the sandwave crest (wave focusing). Thus e 4 is a damping term.

The corresponding growth rate contributions, σ 1 = e 0 e 1 , σ 2 = e 0 e 2 , ... are plotted in Figure 8. It can be seen that σ 2 is always positive leading to the development of shoreline Figure 8 sandwaves, whereas σ 1 and σ 4 are always negative, leading to the damping of the sandwaves. The term σ 3 can be either positive, for small beach slope (eg smaller than 0.05 to 0.08), or negative, for larger beach slopes. Even if the behaviour of this term is not monotonous, σ 3 generally increases with the wave angle. It is remarkable that σ 2 becomes D R A F T September 27, 2011, 5:50pm D R A F T very large for large beach slopes and for small wave angles. For normal wave incidence (θ 0 b = 0), σ 3 = 0 and σ 2 is the only contribution leading to the instability. We therefore conclude that wave refraction is responsible for LAWI in the case of very steep beach slopes.

Based on the growth rate results, it is also possible to analyse the relative influence of the wave incidence induced Q 1 and wave set-up induced Q 2 sediment transport. The analysis of the growth rate versus the perturbation wavelength for r = 0 or r = 1 and several wave incidence angles θ (Figure 9) shows that the relative influence of Q 2 decreases Figure 9 with increasing the wave incidence angle. In other words, wave height gradients (wave setup induced sediment fluxes) largely influence (damp) the shoreline instability for low wave angle incidence (LAWI), whereas their impact is almost negligible for large wave incidence angles (HAWI). The main driving term of the LAWI is due to the wave incidence induced sediment transport flux Q 1 . This means that the use of the CERC equation alone, taking into account wave refraction in the shoaling zone, can cause LAWI. The Q 2 term influences this instability by changing the growth rate and the favored wavelength. This influence increases with decreasing wave incidence until the case of perfectly shore-normal waves, for which there is a prefered wavelength (LMA mode) only if the Q 2 term is taken into account (Figure 6).

Model results analysis: Hydrodynamic and sediment transport

To understand better how wave refraction causes shoreline instabilities, hydrodynamic and sediment transport model results are analysed next. Here, we focus on the case of shore-normal wave incidence, considering the LMA mode obtained for a beach slope equal to 0.1 (case a). The model results are compared with the same wave incidence, but D R A F T September 27, 2011, 5:50pm D R A F T a smaller beach slope (0.02) for which the shoreline is stable (case b). The perturbation wavelength (571 m) is the same for both cases, corresponding to the LMA mode of case (a). Although the linear stability analysis is strictly valid only in the limit a → 0 we choose a shoreline sandwave amplitude a = 10 m (for visualization and comparison of the different sources of sediment transport). In addition, to make the analysis simpler, we assume shore-normal waves and r = 0. Figures 10a andb show planviews of the

Figures 10

wave angle, as well as a longshore cross-section of several quantities along the breaking line. First, it can be noted that the breaking line is much farther offshore in case (b) because of the shallower bathymetry. This is directly linked with the bathymetry. These planviews illustrate the wave focussing on the cusp, which implies increasing wave height and converging waves at the cusps.

Looking at the alongshore cross-section (Figure 10c), the wave angle amplitude is much larger for case (a) than for case (b), about 14 oscillation. This clearly shows that the growth rate is positive for case (a) and negative for case (b). The alongshore cross-section of the resulting sediment flux Q (Figure 10c)

• versus 3.2 • ,
illustrates the opposite behaviour for the two cases. Our sign convention is that positive Q represents sediment transport directed in the direction of the increasing y coordinate (i.e. to the right on the cross-sections). A positive (negative) longshore gradient indicates a convergence (divergence), assumed to cause shoreline accretion (erosion). Thus Figure 10c shows that Q for case (a) has a spatial phase-lag compared to the shoreline such that the shoreline perturbation should be amplified, whereas Q for case (b) has an opposite phase-lag, leading to the damping of the perturbation. This spatial phase-lag change results from the continuous amplitude changes of the terms e 1 and e 2 : the phase-lag between shoreline and longshore sediment fluxes is either 90 • or -90 • , implying that there is no migration and either amplification or damping of the shoreline perturbation.

Thus, for shore-normal waves and neglecting the damping term related with wave setup induced sediment flux (second term in eq. 2), the instability of the shoreline results from an alongshore oscillation in the angle of wave refraction, which is stronger than the oscillation in the angle of shoreline orientation.

Mechanism

From the above, we can draw the following conclusion: the main growing term is related to the wave refraction toward the cusp, leading to wave incidence induced sediment transport converging at the cusp. This term strongly increases with beach slope. The damping is due to three components: (1) longshore sediment transport due to the shoreline orientation only (and not refraction, term e 1 ), (2) wave energy spreading (term e 3 ), D R A F T September 27, 2011, 5:50pm D R A F T and (3) wave height gradients (set-up) (term e 4 ), the second component having a smaller damping effect than the two others.

Now we can figure out how LAWI works (Figure 11). Let us consider shore-normal Figure 11 wave incidence. In this case, the wave energy spreading has no influence on the instability (e 3 = 0). If a cuspate feature with an associated shoal develops on a coastline, wave refraction bends wave rays towards the tip of the feature. Depending on the orientation of the refracted wave fronts with respect to the local shoreline along the cuspate feature, the alongshore sediment flux can be directed towards the tip, reinforcing it and leading to a positive feedback between flow and morphology. Whether the transport is directed to the tip, depends on the bathymetry and wave conditions. For a given offshore extent, xl, of the associated shoal and a given wave height, the surf zone will become narrower if the beach slope increases. Then, the shoal will extend a longer distance beyond the surf zone, and the waves will be refracted strongly when they reach the breaking point, increasing the wave incidence related sediment flux (e 2 ) convergence whereas the divergence term (e 1 ) is constant. As shown by the model results (Figure 10), the contribution of the refracted wave angle (e 2 ) can exceed the contribution of the shoreline orientation to the sediment flux (e1), such that Q 1 (resulting from e 1 and e 2 ) converges near the cusp. This leads to the development of the cusp. If the beach slope is mild, the surf zone will be wider, and wave refraction over the shoal before breaking will be less intense, leading to smaller wave incidence angle induced sediment fluxes, which are dominated instead by the diverging sediment flux induced by shoreline orientation changes. In this case, as shown in the model results (Figure 10c), the sediment flux is directed away from the tip of the cusp.

D R A F T September 27, 2011, 5:50pm D R A F T

Discussion

Linear stability analysis validity

The 1D-morfo model has been applied to investigate HAWI and to study the sandwaves generation along the Dutch coast [START_REF] Falqués | Wave driven alongshore sediment transport and stability of the Dutch coastline[END_REF], El Puntal beach -Spain [START_REF] Medellín | Coastline sand waves on a low energy beach at 'El Puntal' spit, Spain[END_REF][START_REF] Medellín | Sand waves on a low-energy beach at el 'puntal' spit, Spain: Linear Stability Analysis[END_REF]. The results indicated similarities with the wavelengths observed in nature.

This supports the use of this linear stability model to investigate the mechanisms of shoreline sandwave formation. In the present paper, it is clear that LAWI is a robust output of the 1D-morfo model, and the physical mechanism causing instabilities is wave refraction induced by an offshore shoal associated with a cuspate feature. This wave refraction leads to two counter-acting phenomena: sediment transport induced by converging waves counteracted by diverging wave height gradients. The present paper investigates the linear generation only. The pros and cons of linear stability analysis have been discussed

extensively in [START_REF] Blondeaux | Mechanics of coastal forms[END_REF][START_REF] Dodd | The use of stability methods in understanding the morphodynamical behavior of coastal systems[END_REF][START_REF] Falqués | Rhythmic surf zone bars and morphodynamic self-organization[END_REF][START_REF] Tiessen | A field test of a linear stability model for crescentic bars[END_REF]. In any case, the fundamental assumption of infinitesimal amplitude growth makes comparisons to field data questionable. Nonlinear model studies for other rhythmic features, such as crescentic sandbars and sand ridges [START_REF] Calvete | Morphological stability models: Shoreface-connected sand ridges[END_REF][START_REF] Damgaard | Morphodynamic modelling of rip channel growth[END_REF],

have sometimes shown the finite-amplitude dynamics to be dominated by the LMA mode ; in other cases, modes other than the LMA became dominant. We leave the nonlinear modeling of LAWI, including the study on cessation of the growth, to future work.

Analogy with megacusps: growth rates and circulation patterns

Although the model results are given for a planar beach, LAWI is also found in the presence of a shore-parallel bar (not shown). Thus, for intermediate morphological beach predicts LAWI. To survive in the finite amplitude domain, the LAWI must grow at a rate comparable to that of co-existing instabilities. Our sensitivity studies indicate LAWI growth rates to range from 10 -6 s -1 (for a beach slope of 0.05) to 10 -5 s -1 (for a beach slope of 0.2). The typical generation time scale thus ranges from 1.5 to 11.5 days. These time scales were obtained for shore-normal waves having a moderate wave height of 1.5 m and wave period of 8 s. A typical time scale for the LMA mode of crescentic bars is several days [START_REF] Damgaard | Morphodynamic modelling of rip channel growth[END_REF][START_REF] Garnier | Mechanisms controlling crescentic bar amplitude[END_REF]. Thus, for specific beach slope and wave conditions, the LMA shoreline instabilities have comparable initial growth rates as those of crescentic bar patterns.

Computations for the idealized cases gives LAWI wavelengths of the same order of magnitude as the observed spacing of crescentic bars and associated megacusps. The distinction between these two kinds of instabilities is therefore difficult and the validation of the presence of LAWI in a Rhythmic Bar and Beach morphological environment is not straightforward. More generally, a proper validation of the present results would need dataset of shoreline evolution, together with bathymetric, wave and current data, starting from an initial longshore uniform beach. To our knowledge, such data do not exist.

Although we cannot validate the model results with wavelengths observed in the field, it is possible to discuss whether the type of nearshore circulation linked to LAWI, that is, a longshore sediment flux pointing toward the cuspate feature at both sides, is realistic or not in nature and in the framework of 2DH modeling. According to [START_REF] Komar | Beach Processes and Sedimentation[END_REF] that if the breakwater is beyond the breaker line, the waves drive longshore currents that converge at the lee of the breakwater to build a salient [Ranasinghe et al., 2006]. This converging type of circulation at a megacusp was also observed by [START_REF] Haller | Experimental study of nearshore dynamics on a barred beach with rip channels[END_REF] in laboratory experiments on barred beaches with rip channels. One of their six experimental configurations may be quite close to a LAWI configuration. This configuration had the largest average water depth at the bar crest and the smallest rip velocity at the rip neck, such that, in addition to the rip current circulation, they found a secondary circulation system near the shoreline, likely forced by the breaking of the larger waves that propagated through the channel. As these waves are breaking close to the shoreline, they drove longshore currents away from the rip channels into the shallowest area. This experiment shows that breaking close to the shoreline counteracts the rip-induced circulation, leading to current convergence in the shallowest area.

The studies of [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] and [START_REF] Orzech | Megacusps on rip channel bathymetry: Observations and modeling[END_REF] give other elements to investigate the plausibility of the LAWI mechanism, in rip channels configurations. For the case of a barred-beach, [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] developed a 2DH linear stability model, having a fixed shoreline, that describes the formation of rip channels from an initially straight shore-parallel bar. For shore-normal waves, the circulation linked to rip channel formation is offshore through the channels and onshore over the shoals or horns of the developing crescentic bar as is clearly observed in nature (e.g. [START_REF] Macmahan | Rip current review[END_REF]).

However, they also noticed small secondary circulation cells near the shoreline flowing in the opposite direction, leading to presence of megacusp formed in front of the horns of the crescentic bar; therefore, the shoreline undulations were out of phase (spatial phaselag of 180 • ) with the crescentic bars, meaning that the amplitude of the wave-refracted D R A F T September 27, 2011, 5:50pm D R A F T terms should dominate the amplitude of the wave set-up terms (Eq. 2). The formation of those megacusps was not part of the instability leading to the crescentic bar development but was forced by the hydrodynamics associated with it. More importantly, the small secondary circulation cells were essentially related to wave refraction: if wave refraction from the model was eliminated, they did not develop. Thus, wave refraction by offshore shoals (those of the crescentic bar in this case) can induce a circulation that may move sediment toward a developing cuspate feature. A recent study, based on both observation (video images) and non-linear morphodynamic modeling [START_REF] Orzech | Megacusps on rip channel bathymetry: Observations and modeling[END_REF]] also showed the occurence of two types of megacusp (shoreward of the shoal or shoreward of the rip), and the associated converging sediment fluxes toward the megacusps. This tends to support our mechanism analysis of LAWI formation.

The similarity between LAWI and megacusps in both wavelength and growth time is certainly intriguing given the fact that 1D-morfo is mainly based on the gradients in longshore sediment transport and wave set-up induced sediment transport (damping term), but neglects many surf zone processes like rip current circulation, which is known to be essential to crescentic bar dynamics [START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF][START_REF] Garnier | Modelling the formation and the long-term behavior of rip channel systems from the deformation of a longshore bar[END_REF].

However, the analysis above tends to show that there could be configurations, where the processes taken into account in 1D-morfo are dominant in the system, at the initial stage.

This could explain why similarities are observed with the various numerical experiments done with more sophisticated models. Furthermore, we should keep in mind the fact that the LAWI mechanism is not related to any longshore bar, and thus that short shoreline sandwaves such as megacusps could develop without a bar, whereas it was thought that, for barred beaches, short shoreline sandwaves develop due to surf zone sand bar variability.

D R A F T September 27, 2011, 5:50pm D R A F T

Conclusions

A one-line linear stability model, which was initially created to describe the formation of shoreline sandwaves under high-angle wave incidence, has revealed shoreline instabilities for low to shore-normal wave incidence (LAWI). The most amplified mode has wavelengths of ∼ 500 m and characteristic growth time scales of a few days, which are smaller than those of the high angle wave instabilities. Sensitivity analyses focusing on wave height, wave incidence angle, beach slope, beach profile, model free parameters and the sediment transport equation show that, for low to shore-normal wave incidence, instabilities develop for sufficiently large beach slopes (e.g. 0.06) and for sufficiently small wave heights (smaller than 2 m for a beach slope of 0.06).

The main process causing the instabilities for low to shore-normal wave incidence is wave refraction on a shoal in the shoaling zone, which focuses wave fronts onshore of it, leading to wave incidence induced sediment transport converging at the cusp. This effect strongly increases with beach slope. The damping is due to three longshore transport components:

(1) that caused by shoreline orientation only (and not refraction), (2) that caused by wave energy spreading (minor effect for low-angle wave incidence), (3) that caused by wave height gradients (set-up). Whether LAWI develops or not depends on the balance between these growing and damping terms. If this shoreline sand accumulation can feed the initial shoal through cross-shore sediment transport, a positive feedback arises.

synthesis, Mar. Geol., 56, 93-118, 1984. indicates that, even if the growth rate was positive, there was a singularity at θ = 0 • (growth rate continuously increasing with decreasing wavelength), and therefore no LMA mode (see Figure 7).

  5)which has been modified from the original Dean profile to avoid an infinite slope at the shoreline. The constants b and x 0 are determined from the prescribed slope β at the coastline and the prescribed distance x c from the coastline to the location of the closure depth D c (see[START_REF] Falqués | Large scale dynamics of sandy coastlines. Diffusivity and instability[END_REF] for details). The forcing conditions are waves with H rms = 1.5 m and T p = 8 s at a water depth of 25 m. The wave direction ranges from 0 • to 85 • in increments of 5 • . A closure depth D = 20 m is chosen. To perform the linear stability analysis, the shape function for the bathymetric perturbations is assumed to be constant in the surf zone and to decrease exponentially seaward. Its cross-shore extent is given by the characteristic distance xl corresponding to the closure depth of 20 m, i.e. xl = 1410 m for the present case (P1-perturbation in Figure2a).

Figure 5

 5 threshold, xl > 100 m, above which shoreline instabilities may occur. This value appears physically reasonable since the width of the surf zone in the reference case is about 73 m. For 125 ≤ xl ≤ 250 m, the shoreline instability wavelength decreases significantly (from 1040 to 530 m) with increasing perturbation length, whereas for xl ≥ 250 m, the wavelength increases only slightly until reaching a nearly constant value of 574 m. The growth rate increases with increasing xl for values below 500 m but decreases for xl exceeding 500 m, reaching a nearly constant value for large perturbation length scales.

  indicating a stronger refraction up to the breaking line in case (a). The corresponding amplitude of the oscillation in the shoreline angle is about 6.3 • , and is thus within the range of those two values. This means that the angle of the wave fronts with respect to the local shoreline reverses when passing from case (a) to (b), implying a reversal in the direction of sediment transport. This can be traced back to equation 6. The wave angle amplitude is much larger for case (a). Coming back to equation 6, in the case of shore-normal waves and r = 0, there are only two terms left: e 1 , which is the contribution due to shoreline change only, and e 2 , which represents the wave refraction-induced sediment flux. The analytical computation for the present case leads to: e 1 = -1 for both cases, whereas e 2 = 2.23 for case (a) and e 2 = 0.507 (case b), consistent with the different amplitudes of alongshore wave angle D R A F T September 27, 2011, 5:50pm D R A F T

  state where crescentic bars and associated megacusp systems usually develop, the model D R A F T September 27, 2011, 5:50pm D R A F T

  , both types of longshore current patterns, either converging or diverging at a megacusp, are observed in nature. Another example showing that this type of circulation is realistic is the case of a submerged breakwater. Both observations and numerical modelling indicate D R A F T September 27, 2011, 5:50pm D R A F T

Figure 1 .Figure 2 .

 12 Figure 1. Sketch of the geometry and the variables. The angle between the wave fronts and

Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 3456 Figure 3. LMA (a) growth rate and (b) wavelength as a function of beach slope and angle of

Figure 7 .Figure 8 .Figure 9 .Figure 10 .Figure 11 .

 7891011 Figure 7.LMA growth rate versus wavelength for various beach slopes, using r = 0 and

Acknowledgments. The authors thank the reviewers (including A. Ashton) for their comments and suggestions, D. Calvete for fruitful discussions, J. Thiébot for his comments on this paper. M. Yates-Michelin is also acknowledged for her careful English corrections. Funding from the ANR VMC 2006 -project VULSACO n • ANR-06-VULN-009, the Span-D R A F T September 27, 2011, 5:50pm D R A F T ish Government (CTM2009-11892/IMNOBE project and TM2006-08875/MAR project), the Netherlands Organisation for Scientific Research (NWO) project 818.01.009, the University of Nottingham and the Juan de la Cierva program are also acknowledged.


Z Aboura

N Talbi

R Ayad

S Allaoui
email: samir.allaoui@utc.fr

M Benzeggagh
email: malk.benzeggagh@utc.fr

AN HOMOGENEIZATION PROCEDURE FOR CARDOBORAD AND STITCHED SANDWICHES USING RESPECTIVELY ANALYTICAL AND NUMERICAL SIMULATION

Keywords: Homogenization, cardboard, Stitched Sandwich, Finite element

We present in this paper two analytical homogenizations of corrugated cardboard and stitched sandwiches with emphasis on experimental procedures and numerical modeling.

. Then powerful elements are developed to reproduce the mechanical behaviors of these structures.

Two types of sandwich structures were the subjects of this study. The first is a corrugated cardboard considered as thin (4 mm thickness) and the second is a stitched sandwich with PU core and whose skins are out of woven glass. The tot al thickness is 22 mm. The results of the two analytical methods are compared with the experimental results and finite elements. They are promising and show a rather good correlation.

INTRODUCTION

The sandwich structures are used more and more. The applications go from the corrugated cardboard used in the packaging to the new generations of stitched sandwich structures used in aeronautics. The association of two rigid skins to a light core makes that this type of structure is ideal for bending solicitatio n. However, the use of these materials as structural materials inevitably requires calculations, which can very quickly prove expensive in time and elements. Thus, the mesh of corrugated cardboard sample required 900 elements and 1070 nodes (fig. 1).

Figure 1 -3D meshing of corrugated cardboard sample. 900 elements

The idea that this study proposes is in two steps:

• First make an homogenization of the elastic behavior of this type of structure. The methods used must be not very complex and inexpensive. • Second, after obtaining these homogenized properties; powerful finite element is using in order to simulate the bending behavior of these structures. Thus, in the case of the corrugated cardboard of which the thickness does not exceed 4mm, a shell eleme nt will be used. A modeling 2D will thus be made. Its formulation will have to take into account shearing. As regards the stitched sandwich structures, solid elements will be used in order to modeling this kind of structures.

HOMOGENEISATION OF CORRUGATED CARDBOIARD

2-1 ANALYTICAL METHOD

To analyze the elastic behavior of the corrugated cardboard, the proposed technique is a point-wise lamination approach using the classical laminate theory. This approach is inspired from Ishikawa et al [START_REF] Ishikawa | Stiffness and strength behavior of woven fabric composites[END_REF], Aboura [START_REF] Aboura | Etude du processus de délaminage Mode I, Mode II et Mode mixte (I et II) de matériaux composites à renforts tissés à différentes vitesses de sollicitation[END_REF][START_REF] Aboura | Analytical model of woven composite laminate. Superposition effect of two plies[END_REF] and Scida et al. [START_REF] Scida | Prediction of elastic beha vior of hybrid and non-hybrid woven composite[END_REF] related to the elastic behavior modeling of woven composites materials. The core consists of in-plane sinusoidal cells extending vertically between top and bottom face laminates. A unit cell representative of the corrugated cardboard is defined as in figure 2 and considered the skins and the undulated fluting as an assembling of many infinitesimal elements "dx" of unidirectional lamina oriented at different angles.

The classical laminate theory is then applied to each element. The relationship between the in-plane stress and moment resultants N i and M i , and in-plane strains, curvature ε j and κ j is given by:

            =       j j ij ij ij ij i i k D B B A M N ε (i, j =1,2 and 6) (1)
∫ - = 2 : 2 / 2 ) , , 1 ( ) , , ( h
h ij dz Q z z D B A ( 2
)
Q ij is evaluated for each constituent of the corrugated cardboard unit cell. This means that the superior and inferior skins and the fluting undulation are taken into account. The local stiffness of each infinitesimal element depends on the constituent elastic properties as well as on the fluting orientation defined by the local off-axis angle θ(x). This angle is calculated from the fluting median fiber function H t (x):

) ) ( ( tan ) ( 1 dx x dHt x - = θ ( 3
)
H t (x) is assumed to be of sinusoidal form with a maximum thickness of hc:

) 2 ( sin 2 ) ( P x hc x Ht π = (4)
A detail of this approach is explained on previous work (see [START_REF] Aboura | Elastic behavior of corrugated cardboard: Experimental and modeling[END_REF]).

2-2 FINITE ELEMENTS METHOD

The cell has been modeled using the finite element method (figure 3). Numericals tests are conducted on it for extract the equivalent elastic constants for corrugated cardboard. The unit cell is first subjected to uni-axial and equal bi-axial extension tests for determination of Poisson's ration and the tensile elastic moduli. The equivalent shears moduli G LT is determined by constraining the displacement of the upper and lower faces and imposing a horizontal displacement for the nodes of the upper face.

The sinusoidal and flat laminates of the corrugated cardboard are modeled with rectangular 4-node isoparametric shell elements. The homogenized elastic analytical model result and their good correlations with FE analyses and experimental [START_REF] Aboura | Elastic behavior of corrugated cardboard: Experimental and modeling[END_REF] predictions are presented in table 1.

Figure 3 -Finite elements homogenization of cell unit of corrugated cardboard

2-3 COMPARISON BETWEEN ANALITICAL AND FINITE ELEMENTS METHODS

The two methods have results close to the experimental results (tab.1), with an advantage to the analytical method. This advantage is accentuated if the computing time is taken into account. On the other hand the advantage of finite elements the method is in the possibility of obtaining the properties in the third direction (these results are not presented for this study

HOMOGENEISATION OF STITCHED SANDWICH

Advanced composite structures, such as three-dimensional woven and stitched stiffened composite laminates are used in many engineering applications. Stitching or other through the thickness reinforcement greatly reduces the driving force for propagation of the delamination crack. The stitching creates closing traction acting across the laminate thickness that limits damage extension.

Definition of unit cell

Elementary volume is defined by thickness H (figure 4) that is represented by the foam's dimension and the skin's thickness. The width py is determined by the spacing of the rows of seams, while the length px is variable and depends on the various structural parameters (step and angle of stitch).

Modeling procedure:

The stitched sandwich is considered as a stacking of three layers: higher skin, the reinforced core (foam and stitch) and lower skin. To simulate the behavior of the skins (composed by polyester matrix reinforced by woven glass fibers), MesoTex software [START_REF] Scida | Prediction of elastic beha vior of hybrid and non-hybrid woven composite[END_REF] has been used.

Concerning the core, a homogenization procedure was carried out in order to consider both of foam and stitches. The used analytical model [START_REF] Scida | Prediction of elastic beha vior of hybrid and non-hybrid woven composite[END_REF] consists in determining the global rigidity, starting from the basic rigidities of the components according to relation (1):

∑ = = n k k ij k t Global ij C V V C 1 ' , , . 1
, i and j from 1 to 6 (

C ij: is the rigidity matrix of each constituent V k : volume fraction of each constituent From the characteristics of each of the three layers, the theory of the conventional laminates is used to modelize the elastic behavior of the complete structure (2). ? ? ? ?

                          (6)
The properties in the third direction according to the procedure presented in reference [

FINITE ELEMENTMODELING

The homogenized corrugated structure has been modeled using a new multilayer shell finite element, labeled DMTS ( §4.1). A 3D volumetric finite element procedure using the new SFR Hexahedra solid element model has been used for the stitched structure. Performances of both finite element models are studied across some experimental and analytical problem tests.

3-node shell element DMTS

The shell element DMTS (Discrete Mindlin Triangular for Shell) (figure 5) has three nodes and six degrees of freedom per node. It's obtained by combining the well-known me mbrane element CST (Constant Strain Triangle) and a bending/shear plate element DDMT (Displacement Discrete Mindlin Triangle) [START_REF] Ayad | Un nouvel élément fini de coque triangulaire à 3 noeuds pour les coques composites multicouches, 6ème Colloque National en Calcul des Structures[END_REF]. It has been developed for the analysis of isotropic and multilayered plate and shell structures.

8-node solid element SFR

The new 8-node solid element SFR [START_REF] Ayad | A 3D 8-node solid finite element based upon the Space Fiber Rotation concept[END_REF] (figure 6) is based on a kinematical concept of a space fiber rotation. This concept allows to improve the accuracy of the displacement field {U}, getting a higher element with comparison to the 20-node quadratic hexahedral element.

{ } { } ( ) i z i i i i i x x N u N U i r r r - ∧ + = ∑ ∑ = =
U θ X V θ Y W θ Z X Y Z U θ X V θ Y W θ Z q i X Y Z 5 NUMERICAL VALIDATION

Cardboard sandwich (3 points bending)

The specimens of the 3-points bending tests do not require any particular preparation. Their dimensions are 200 x 60 x 4.01 mm. The span value is 160 mm giving the span-tothickness ratio (L/h) equal to 39 minimizing therefore the shearing effect. The specimens are cond itioned at 23°C and 50% RH for at least 24h. The FE equivalent shell model consists of 1-layer shell elements with material properties [START_REF] Talbi | Validation du modèle DDM multicouches homogénéisé pour le carton ondulé à travers un test de flexion à trois points. 20 th Cana[END_REF] defined by the equivalent orthotropic values. The overall dimensions of the models are the same with respect to the actual geometry models, and the same boundary condition is used.

Stitched sandwich (4 points bending)

The performance of the SFR element is tested on a homogeneous orthotropic beam. The length of the specimen was 550 mm length 50 mm width and 22 mm high. The outer span was 440 mm while the inner load span was 220 mm, see figure 8. It's analyzed with a mesh of 1x2x50. The SFR solution compared with H8 element solution is in good agreement with experimental data (table 5). The error on the bending module is less then 8%

CONCLUSIONS

Our research and development is intended to contribute developing a fast and accurate tool for computing the homogenized behavior for a given periodic sandwich structures. Both analytical model and finite element predictions, for corrugated cardboard and stitched structure correlate reasonably well with experimental results.

The results of the two simulations using the new finite element are in good agreement with previous analytical and experimental and should increase modeling efficiency and diminish the numerical errors.

Figure 2 -

 2 Figure 2 -Unit cell representative of the corrugated cardboard. in which A ij , B ij and D ij are the in-plane stiffness for each infinitesimal element dx. These are defined by

Figure 4 -

 4 Figure 4 -Unit cell representative of stitched sandwich

Figure 5 -

 5 Figure 5 -Multilayered shell finite element model DMTS

Figure 6 -

 6 Figure 6 -Hexahedral 8-node Space Fiber Rotation model SFR

Figure 7 -

 7 Figure 7 -Three points bending test

Figure 8 -

 8 Figure 8 -Four point bending test

Table 1 -

 1 ). Comparison between analytical and finite elements methods

		E x (Mpa)	Ey (Mpa)	ν xy	G xy
	Analytical Method	712	392	0,24	201
	Finite Element Method	796	496	0,26	292
	Experiment	656 ± 40 412 ± 25 0,21 ± 0,018 211 ± 5

Table 2 -

 2 Material properties of the stitched sandwich

	5]

step ? px H

Table 3 -

 3 Homogenized material properties of the corrugated cardboardThe central normalized value P/WC obtained from the present elements, together with those from experimental solution given[START_REF] Talbi | Validation du modèle DDM multicouches homogénéisé pour le carton ondulé à travers un test de flexion à trois points. 20 th Cana[END_REF] are presented in Table4. It is obvious that the result from element DMTS are in good agreement with the experimental.

	H		E 1			E 2	N 12	G 12
	(mm)	(MPa)		(Mpa)	(Mpa)
	4.01	719.44 390.53 0.24 203.11
	L	h	∆	exp	∆	num	=	P	num W	%
	20 22.798		20.118	11.75

Table 4 -

 4 Results of the corrugated cardboard bending rigidity

Table 5 -

 5 Results of the stitched sandwich bending rigidities.

		W(mm) P/w N/mm) *E f (N/mm²)	%
	SFR	6.2484	128.033	4301.901	7.81
	H8	5.2909	151.203	5080.420	8.86
	Experiments		138.89	4666.704