Mayeul Dalleau
email: mayeuldalleau@kelonia.org

Simon Benhamou

Stéphane Ciccione

Gilles Lajoie

Jean-Yves Georges

Jérôme Bourjea

Individual Based Modeling of Green Turtle Spatial Dynamics in the Southwest Indian Ocean

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Objectives

The main objective of the model is to study the influence of SWIO topology to the reproductive potential of green turtle rookeries. As a matter of fact, spatial arrangement of nesting and feeding areas affects population viability. Proximity and accessibility of feeding sites is likely to favor reproduction by reducing the cost of the migration. On the other side overpopulated feeding sites might delay reproduction event through food competition. Thus, population reproductive potential and therefore viability may be a resulting balance between the accessibility of the sites and the number of individuals that share this site.

Material and Methods

Individual-based modeling

Individual-based or agent-based models (IBM) generically refer to models that infer the global dynamic of a system by modeling and simulating the constituents of that system. In ecology, IBMs are models that derivate the properties of ecological systems from the properties of the individuals constituting these systems. This "bottom-up" approach allows the understanding of unexplained properties at population scale from a good knowledge of what is happening at individual scale. We used the publicly available platform NetLogo to implement the model.

Individual cycle

We decided to consider only adult female migration cycle. Every turtle individual owns an energetic level. Similarly feeding sites is associated with a feeding quality. Turtle can intake energy only at feeding site simultaneously affecting the feeding quality of this site. While nesting or migrating every individual expends a relative amount of energy. Three different types of migration are implemented. pre-and post-nesting migrations are go and return movements between a feeding and a nesting site, foraging migration is a movement between two feeding sites. When the energetic level of an individual reaches over a minimum threshold, it starts a pre-nesting migration. When the quality of a feeding site goes under a minimum threshold, the individual starts foraging migration.

Feeding and nesting sites

We choose to study strategically located nesting sites. Those are the French Eparses islands and Mayotte. It should be noted that any nesting sites could easily be added to the model. Feeding sites have been located according to expert knowledge of the region.

Reproductive potential

The population reproductive potential was assess as the product of the number of nesting individuals by their energetic levels.

reproductive potential = number of nesting individuals x mean of energetic level at nesting

Results

With a few number of simulated scenarios, the model already exhibits interesting patterns regarding reproductive potential of green turtle rookeries in the SWIO. Under moderate site depletion, an island (e.g. Tromelin) located far from the feeding sites areas generally displayed a lower reproductive potential while islands (e.g. Mayotte, Glorieuses) closely surrounded by numerous feeding areas displayed a higher reproductive potential. This can easily be explained, the migration cost being higher when turtles have to swim long distances. Indeed, lowering the energetic expenditure due to migration tends to homogenize rookeries reproductive potentials. Remarkably increasing the impact of individuals on feeding site quality also tends to reduce these differences. This is mainly because turtles from remote nesting islands make use of less frequented feeding sites. So our model suggests that longer reproductive migration can be compensated by a lower frequentation of the feeding areas.

Further simulations should confirm these results and guide us in our understanding of the influence of the SWIO spatial topology on green turtles population.

Discussions

Also this model showed very intuitive results, further investigation is needed at this stage to gain confidence in the results. Exploiting this model to its best requires a sensitivity analysis in particular to energetic parameters. Moreover, more individual variability could easily be implemented. We also plan to run more simulations under different case scenarios (e.g. altering local feeding sites quality). Oceanographic and environmental variables also play for sure an important role in the spatial dynamic of the green turtle in the SWIO and could be integrated.

Fig. 1 :

 1 Fig. 1 : Map of the Soutwest Indian Ocean.

Fig. 2 :

 2 Fig. 2 : Individual energy budget cycle.

Fig. 3 :Fig. 5 :

 35 Fig. 3 : Spatial cycle of adult female Fig. 5 : Model outputs

Acknowledgments

The project is funded by Direction de l'Environnement, de l'Aménagement et du Logement -La Réunion Terres Australes et Antarctiques Françaises Kélonia, l'observatoire des tortues marines Institut français de recherche pour l'exploitation de la mer -La Réunion

The PhD candidate is funded by Région Réunion

The presentation is funded by Kélonia, l'observatoire des tortues marines Western Indian Ocean Marine Science Association International Sea Turtle Society Western Pacific Regional Fishery Management Council U.S. Fish and Wildlife Service U.S. National Marine Fisheries Service Disney Worldwide Conservation Fund


Magali Brunet

Benoit Malard

Nicolas Ratel-Ramond

Christophe Deshayes

Sébastien Joulié

Bénédicte Warot-Fonrose

Philippe Sciau

Joël Douin

Frédéric De Geuser

Alexis Deschamps

Precipitation in original Duralumin A-U4G versus modern 2017A alloy

Keywords:

Precipitation in Duralumin, a historic quaternary alloy of the type: Al-Cu-Mg-Si, was never fully studied nor observed by current electron microscopy techniques. This article presents the full characterization and comparison of two alloys: a Duralumin (A-U4G) from the 1950s collected on a vintage aircraft and its modern equivalent: a 2017A alloy. The as-received and peak-aging states were analysed with DSC, SAXS and TEM advanced techniques. It is shown that old Duralumin and modern 2017A present a similar nanoprecipitation in the as-received state and behave similarly upon artificial aging. As opposed to what has been reported in the past, three types of precipitates participating in hardening were found upon aging: 𝜃'-Al 2 Cu, Q'(Q)-AlCuMgSi and Ω -Al 2 Cu.

Introduction

Duralumin, with a composition close to 4 Cu-0.7 Mg-0.6 Mn-0.6 Si wt%, is the oldest age-hardening aluminium alloy. Discovered in 1906 by Alfred Wilm, a German metallurgist [START_REF] Duparc | Alfred Wilm et les débuts de Duralumin[END_REF] , it triggered the development of aeronautics at the beginning of the XXth century. At the scientific and technical research centre in Neubabelsberg, Wilm experimented many treatments on Al-Cu-Mn alloys with small amounts of magnesium (0.5 wt%). He found out that by quenching from temperatures below its melting point (about 450 °C) and by letting it age naturally for a few days, the new alloy exhibited enhanced mechanical properties (strength and hardness). Thanks to its low density and strength, Duralumin soon became the prime choice for airplanes construction, well-illustrated by the airplane Breguet 14 whose production reached 12,000 during World War I [START_REF] Victor | Comment a été construit le premier avion en Duralumin[END_REF] . From then on, aircraft production, previously a craftsmen business, transformed into a real industry. Following on this discovery and for reaching even higher performance, metallurgists in the 1920s tuned aluminium alloys via the addition of microalloying elements. This was done empirically, as no explanation could yet be provided for the age-hardening phenomenon. Merica et al. [START_REF] Merica | Heat treatment of Duralumin[END_REF] were the first team in 1929 to explain that the hardening during aging of duralumin was due to fine and highly dispersed Al 2 Cu precipitates. But the concept of dislocation, essential to a complete understanding of the structural hardening mechanism, was introduced by Orowan in 1934 [START_REF] Orowan | Zur kristallplastizität. III[END_REF] . The discovery of coherent copper-rich zones in the Al-Cu 4% alloy was achieved independently by Guinier [START_REF] Guinier | Structure of age-hardened aluminium-copper alloys[END_REF] and Preston [START_REF] Preston | The diffraction of X-rays by an age-hardening alloy of aluminium and copper. The structure of an intermediate phase[END_REF] with X-ray experiments in 1938. It took even longer to be able to observe directly the nanoprecipitates responsible for the hardening: first observations were made on oxide replicas in France in 1952 by Saulnier and Syre [START_REF] Saulnier | Maturation, revenu et réversion des alliages Al-Cu 4%, Al-Cu-Mg et Al-Cu-Mg-Si de haute pureté[END_REF] .

Original Duralumin was thus improved experimentally by increasing the magnesium content (up to 1.5 wt%), leading to a new alloy, called Duralumin FR in France in the early 1930s [START_REF] Duralumin | notice générale[END_REF] . Duralumin and Duralumin FR became A-U4G and A-U4G1 respectively in 1943, to follow a new designation defined by AFNOR 1 [START_REF] Lebouteux | La symbolisation de l'aluminium et de ses alliages en France et dans quelques pays étrangers[END_REF] . From the 1930s to the 1950s, because of its lack of strength and ductility, A-U4G was replaced gradually in aeronautics construction by A-U4G1. In 1954, international standardization of aluminium alloys took place [START_REF]The-Aluminum-Association, International alloy designations and chemical composition limits for wrought aluminum and wrought aluminum alloys[END_REF] : the numerical designation with 4 numbers was adopted. A-U4G was designated as 2017A and A-U4G1 as the well-known 2024.

Original Duralumin has not received a major focus in the scientific literature related to the precipitation at nanoscale. In fact, published works on the precipitation occurring in Duralumin and even on the modern 2017A alloys are scarce. Original studies on the precipitation of industrial Duralumin were published in the 1950s. In 1950, Lambot studied the precipitation of an industrial Duralumin by X-ray abnormal scattering [START_REF] Lambot | Etude cristallographique de la précipitation structurale dans les duralumins[END_REF] . In accordance with results found by Guinier and Preston in Al-Cu alloys, after quenching and room temperature aging, he detected scattering due to copper clusters or Guinier Preston Zones (GPZ). Then, upon aging at moderate temperatures, a first form of precipitates was evidenced, with a structure of three aluminium-rich planes 1 Created in 1926, AFNOR is the French association for standardization. E-mail address: magali.brunet@cemes.fr (M. Brunet). in between two copper-rich planes. This phase evolved to the welldistinguished 𝜃'-Al 2 Cu phase when aging at higher temperatures. Other scattering patterns, superimposed to the ones originating from Al 2 Cu precipitates, were detected. They were attributed to the Al 4 CuMg 5 Si 4 phase. This phase is expected at equilibrium state, in alloys of Mg/Si ratios lower than 1.73 [START_REF] Mondolfo | Metallography of Aluminum Alloys[END_REF] . Two years later, Saulnier and Syre [START_REF] Saulnier | Maturation, revenu et réversion des alliages Al-Cu 4%, Al-Cu-Mg et Al-Cu-Mg-Si de haute pureté[END_REF] identified in a similar industrial Duralumin, thanks to electron microscopy (using oxide replicas), the presence of the so-called phase X (renamed 𝜃'' later) after aging at 260 °C evolving into the 𝜃'-Al 2 Cu phase at 290 °C, confirming the findings of Lambot. Another phase was evidenced with thermal dilatometry. However, it was attributed to Mg 2 Si needles. In this work, the Q-Al 4 CuMg 5 Si 4 phase could not be evidenced. Similarly, Yano and Koda in 1968 [START_REF] Yano | Electron microscope observation of duralumin[END_REF] , observed by transmission electron microscopy (TEM), the presence of the 𝜃'-Al 2 Cu and Mg 2 Si phases in Duralumin. More recently, Härtel et al. [START_REF] Härtel | Effects of particle reinforcement and ecap on the precipitation kinetics of an Al-Cu alloy[END_REF] studied the 2017A alloy and observed the 𝜃' phase and another phase, Ω -AlCu. However, they did not mention the presence of Q-AlCuMgSi, neither Mg 2 Si. It is to be noted though that the silicon content was, in their case, very low (0.08 wt%).

As shown here, literature does not provide a clear and comprehensive report on precipitation occurring in Duralumin. This article focuses on the observation and identification of the metastable phases and precipitation sequences occurring in a Duralumin (A-U4G) of 1958 collected on an old aircraft and aged at ambient temperature for more than 60 years. It will be compared to a modern equivalent, the 2017A alloy. When looking at the international standards [START_REF]The-Aluminum-Association, International alloy designations and chemical composition limits for wrought aluminum and wrought aluminum alloys[END_REF] , it is interesting to note that, as opposed to the A-U4G1 which composition is significantly different from the modern equivalent 2024, the composition of A-U4G/2017A has barely evolved over time. As the collected Duralumin is more than 60 years old, by comparing its behaviour and microstructure to the modern equivalent, it is also the occasion to study long-term aging on this type of alloy. This constitutes a first motivation for studying this alloy. The second motivation is related to Cultural Heritage. Duralumin constitutes most of the airplanes' fuselage and structure before and during World War II. Precious specimens conserved in Museum should be preserved. By understanding and documenting the intrinsic constitution of the alloy and its behaviour over time, it will be easier to anticipate problems such as lack of mechanical resistance and/or corrosion.

The microstructure of the A-U4G and 2017A in different states, asreceived and after artificial aging, will be reported hereafter by combining different techniques: conventional bright and dark field TEM, selected area diffraction patterns (SADP), scanning transmission electron microscopy coupled with energy dispersive X-ray spectroscopy (STEM-EDS), with high angle annular dark field (HAADF) imaging or with electron energy loss spectroscopy (EELS), differential scanning calorimetry (DSC) and small angle X-ray scattering (SAXS) experiments. It is shown that, compared to what could be expected from the literature on quaternary Al-Cu-Mg-Si alloys [START_REF] Eskin | Decomposition of supersaturated solid solutions in Al-Cu-Mg-Si alloys[END_REF][START_REF] Nie | 20 -Physical metallurgy of light alloys[END_REF] with a range of 2.5-4.5 wt% of copper, and a ratio of Mg:Si close to 1 or what was observed in the 1950s and 1960s on Duralumin [START_REF] Saulnier | Maturation, revenu et réversion des alliages Al-Cu 4%, Al-Cu-Mg et Al-Cu-Mg-Si de haute pureté[END_REF][START_REF] Yano | Electron microscope observation of duralumin[END_REF] , a complex and specific precipitation sequence occurs in these alloys.

Materials and characterization techniques

Materials

The studied alloy is a Duralumin, identified as A-U4G in the shape of plate (about 1 mm thick) collected on an aircraft from 1958, a Breguet Sahara 765: the sample was extracted from a flap of the landing gear, a part not subjected to hot temperatures related to engine proximity (see Fig. S1 of supplementary information). The equivalent modern alloy, EN AW-2017A (labelled hereafter 2017A) was bought from Bikar Metalle GmbH. It was received in T4 state. In general, for 2017A and 2024 alloys, the states T3 and T4 are common for parts of low thickness ( < 12 mm). In these states, the alloy exhibits a good corrosion resistance, a high toughness and a good fatigue resistance [START_REF] Develay | Propriétés de l'aluminium et des alliages d'aluminium corroyés[END_REF] . The plate collected on the plane received also a T4 treatment upon fabrication: it was clearly mentioned in the Breguet standards [START_REF] Breguet | Standard forms, Vélizy factory[END_REF] . However, for this alloy, the asreceived state corresponds in fact, to a T4 state plus 10 years of service and 50 years in outdoors conditions.

For each alloy, the inductively coupled plasma-optical emission spectrometry (ICP-OES) measurements, performed by Evans Analytical Group SAS, is reported in Table 1 . It is confirmed that the compositions are similar. Currently, 2017A contains some zinc whereas old version of this alloy does not. The ratios Mg:Si are 1.1 and 1.16 for 2017A and A-U4G, respectively, which correspond to reported values for Duralumin [START_REF] Saulnier | Maturation, revenu et réversion des alliages Al-Cu 4%, Al-Cu-Mg et Al-Cu-Mg-Si de haute pureté[END_REF] .

For the study of hardening precipitation, two states were compared: as-received and peak-aging conditions. To find peak-aging condition, received samples were heat-treated at 180 °C during different times. The samples were then cut along three perpendicular directions relative to the rolling direction and measurements of micro-hardness (Vickers) were carried out in the corresponding planes (RP: rolling plane; NP: normal plane; TP: transverse plane).

Characterization techniques

The alloys were analysed by different techniques, which required adapted preparations. For hardness measurements as well as for the observation of the microstructure by optical microscope (OM) and scanning electron microscope (SEM), each specimen (in the 3 planes) was embedded in an epoxy resin and mechanically polished on waterlubricated abrasive papers (silicon carbide): P600 then P1200 and eventually on polishing cloths with diamond paste (from 3 μm down to 1 μm).

For TEM observations, the preparation was the following: to bring the sample thickness down to 25 μm, a mechanical polishing with paper grade up to 2400 SiC, was performed. The specimens were then electrochemically thinned using a Tenupol-5 Struers apparatus operating at 60 V in a solution of methanol and nitric acid (3:1) at -15 °C. Observations of the nanostructure were carried out on a JEOL 2010 operating at 200 kV. Bright field images were taken with an orientation slightly off the [100] Al zone axis. Chemical elements were identified with a CM20 FEG TEM/STEM microscope, operating at 200 kV and equipped with an energy dispersive X-ray spectroscopy (Quantax EDS system with silicon drift detector from Bruker). For EDS mapping, an area of 235 × 88 pixels was scanned with a probe size of about 5 nm and a high statistic (20 000 cps).

HAADF scanning TEM micrographs of as-received and heat-treated samples were acquired using a JEOL cold-FEG JEM-ARM200F instrument operated at 200 kV (energy resolution, 0.3 eV) and equipped with a probe Cs corrector with a spatial resolution of 0.078 nm. To identify the phases and chemistry, energy electron loss spectroscopy (EELS) spectra were acquired using a GIF Quantum ER imaging filter, between 850 and 1850 eV in order to include the Cu-L 2,3 (931-951 eV), the Mg-K (1305 eV), the Si-K (1839 eV) and the Al-K (1560 eV) edges. EELS was performed on line scans across precipitates as well as on some fixed positions. DSC experiments were also carried out. For this, samples with a thickness of 1 mm and a weight of about 20 mg were prepared. They were subjected to ramp heating between -10 °C and 550 °C at 20 °C/min in a Mettler-Toledo DSC3 + . Results were corrected for baseline and normalized for the sample weight.

SAXS measurements were performed on an Xeuss equipment proposed by Xenocs. The X-ray source is equipped with a Cu anode and the SAXS setup is equipped with anti-scattering slits and 6 M Pilatus detector. The beam size is 1 mm 2 . For SAXS experiments, samples were thinned down to 100 μm. In order to investigate the large q-range available, 3 sample-to-detector distances were considered (40 cm, 1.2 m and 2.5 m). In addition, a fixed detector collected the intensity scattered at wide angles, and allowed to obtain the first diffraction peaks of the material. Transmission and background corrections were made using standard protocols. The quantitative analysis was carried out according to the approach described in a previous article [START_REF] Deschamps | Quantitative characterization of precipitate microstructures in metallic alloys using small-angle scattering[END_REF] .

Results

As received state

DSC results for A-U4G and 2017A in as-received state are presented in Fig. 1 . Six different regions can be identified. Peaks are directly related to precipitates nucleation and growth in the alloy. The first region (I), from room temperature up to 150 °C is flat. It is generally assumed that in this temperature range, Guinier-Preston zones should appear. Here, in the as-received alloys, GPZ are supposedly already formed. A second region (II) contains an endothermic peak between 150 °C and 180 °C and a small exothermic peak around 200 °C, ascribed to the dissolution of GPZ and the formation of 𝜃'' phase, respectively. The third and fourth regions (III and IV) exhibit two large exothermic peaks: one between 220 °C and 280 °C (III) and one between 280 °C and 320 °C (IV). In literature on Al-Cu-Mg-Si alloys, these peaks were attributed to different phases. It was first observed by Saulnier and Syre [START_REF] Saulnier | Maturation, revenu et réversion des alliages Al-Cu 4%, Al-Cu-Mg et Al-Cu-Mg-Si de haute pureté[END_REF] that an exothermic peak at 290 °C coincided with the presence of the 𝜃'-Al 2 Cu phase whereas the peak at 350 °C was marked by the apparition of a needle shape precipitation, supposedly Mg 2 Si. More recent works [START_REF] Dutta | Role of Al 2 O 3 particulate reinforcements on precipitation in 2014 Al-matrix composites[END_REF][START_REF] Biswas | Compositional evolution of Q-phase precipitates in an aluminum alloy[END_REF][START_REF] Bassani | Thermal analysis of Al-Cu-Mg-Si alloy with Ag/Zr additions[END_REF] report a peak around 250 °C as the growth of the quaternary (Q' or 𝜆')-AlCuMgSi phase and a peak around 280 °C as the apparition of the 𝜃'-Al 2 Cu phase. The fifth region (V) is a small exothermic peak between 350 °C and 400 °C, associated by some authors to the incoherent 𝜃-Al 2 Cu phase precipitation [START_REF] Bassani | Thermal analysis of Al-Cu-Mg-Si alloy with Ag/Zr additions[END_REF] . However, instead of a precipitation peak, this shoulder could correspond to the start of the dissolution of precipitates present in the alloy, which occurs here between 450 °C and 520 °C (region VI) and then proceeds to full solid solution at the solutionizing temperature.

The main observation from Fig. 1 is the similarity in between the A-U4G and 2017A behaviour. In the measurements, the observed peaks are close to what is reported for the Duralumin in 1952 by Saulnier and Syre [START_REF] Saulnier | Maturation, revenu et réversion des alliages Al-Cu 4%, Al-Cu-Mg et Al-Cu-Mg-Si de haute pureté[END_REF] and for the 2017A alloy by Härtel et al. [START_REF] Härtel | Effects of particle reinforcement and ecap on the precipitation kinetics of an Al-Cu alloy[END_REF] . Similar precipitation should thus be expected. Peak V is however slightly shifted to higher temperatures (around 400 °C) for 2017A compared with A-U4G (around 350 °C). This could be the sign of an earlier start for precipitate dissolution in the A-U4G alloy.

Representative bright-field TEM images in conventional mode for the as-received samples oriented in [001] Al zone axis are shown in Fig. 2 (a) and (b). In the as-received samples, only dispersoids (AlMnSi) can be observed in bright field mode: they are larger in size (125 nm ± 51 nm in average) in 2017A than in the A-U4G (57 nm ± 25 nm in average), both presenting a large standard deviation. Diffraction patterns (not shown in Fig. 2 ) do not present any visible spot that can be attributed to coherent precipitation.

Atomic resolved STEM-HAADF experiments were performed on the as-received alloys, see Fig. 3 for the images of A-U4G in [001] Al zone axis. Precipitation in the matrix was hardly detectable. In HAADF-mode, see Fig. 3 (b) and(d), the intensity is proportional to Z 1.5 -1.8 , Z being the atomic number [START_REF] Pennycook | Z-contrast stem for materials science[END_REF] . Bright dots/areas correspond to the heavier element and are thus representative of copper atoms. The EDS analysis confirmed the presence of copper in higher concentration in these zones compared with the matrix. Thus, Cu-rich zones, not very well defined and having a size of a few nanometres, were detected here and there in the matrix (see Fig. 3 (a) and (b)). Some clusters were also located at the matrix/dispersoid interfaces (see Fig. 3 (c) and(d)).

SAXS measurements, performed on A-U4G and 2017A alloys in the as-received state, are shown in Fig. 4 . The SAXS data observed on both specimens show a very similar behaviour that can be described by two main contributions. In the low q -range (0.01 < q < 0.1 Å -1 ), a clear q -4 contribution is observed, which can be attributed to large precipitates such as the dispersoids. In the range 0.1 < q < 1Å -1 , a plateau is observed, followed by an intensity decrease as q increases. We have modelled this contribution by a distribution of spherical precipitates of uniform electron density (i.e., composition) with a polydispersity given by the Schultz distribution (see [START_REF] Kotlarchyk | Analysis of small angle neutron scattering spectra from polydisperse interacting colloids[END_REF] for the detailed expression of this contribution). A least-square fit of the data gives mean diameters of about 8 Å, which is consistent with the existence of GPZ or small clusters in the alloy matrix. The exact nature of the clusters is at this stage not determined.

Artificial aging treatment at 180 °C and identification of precipitates

Fig. 5 presents the Vickers hardness as a function of aging time for the two studied alloys: A-U4G and 2017A. Only RP hardness was plotted as measurements on other planes gave similar results. The A-U4G is harder than the modern alloy 2017A by about 10 HV throughout the aging curve except for long over-aging times where the A-U4G hardness drops faster. In both cases, the conditions corresponding to the maximum peak hardness were reached after 8 h at 180 °C. These peak ageing conditions were selected for the investigation of the microstructure.

Fig. 6 shows the SAXS measurements after aging for 8 h at 180 °C. Once again, both alloys behave similarly. From these measurements, it is seen that the signal, in the range of 0.1 < q < 1 Å -1 , disappeared in favour of a signal in the range 0.04 < q < 0.2 Å -1 , which translates into the disappearance of small clusters and the formation of larger particles. This contribution to the scattered intensity can be reproduced by using a flat cylinder model. Dimensions of precipitates are: 4 nm thick and 70 nm long for A-U4G and 3 nm thick and 40 nm long for 2017A.

The microstructures corresponding to this peak-aged condition in both alloys, observed by TEM in conventional mode are shown in After 8 h at 180 °C, a very dense precipitation is observed inside the matrix ( Fig. 7 ) and at grain boundaries ( Fig. 8 ). In both alloys ( Fig. 7 (a) and (c)), precipitates lying in {100} Al planes are observed: some as platelets and others as rods, with precipitation on dislocations. As the observed precipitation is similar in both A-U4G and 2017A alloys, advanced analyses results are then only shown for A-U4G alloy.

In Fig. 8 , precipitates at grain boundaries were identified by EDS: they are (Mg, Si)-precipitates and copper precipitates. On these EDS maps, dispersoids of AlMnSi nature are present. A precipitation in the matrix is also visible but hardly identifiable.

As a matter of fact, different precipitates populations co-exist inside the grains and their identification was made possible by combining different electron microscopy techniques. From SADP taken along [001] Al zone axis, shown in Fig. 7 (b) and (d), the presence of 𝜃'-Al 2 Cu phase was evidenced thanks to the strikes along < 001 > Al directions and faint spots (110) 𝜃' at 1 2 of (220) Al [START_REF] Wang | Precipitates and intermetallic phases in precipitation hardening Al-Cu-Mg-(Li) based alloys[END_REF] . They correspond to three vari- SADP, this phase was identified thanks to STEM-EELS. Rods appeared to be the Q-AlCuMgSi phase or one of its precursors (Q'). The composition found for this phase by STEM-EELS was in average 73 ± 10 Al-10 ± 4 Cu-12 ± 6Mg-5 ± 5 Si at%. Aluminium content for this phase is overestimated as part of the matrix is also probed. Moreover, as observed on the STEM-HAADF image ( Fig. 10 (b)) and confirmed by STEM-EELS, the precipitates have a core-shell structure with a core rich in silicon and magnesium and a copper-rich shell. This core-shell structure observed here coincides with results reported by Matsuda et al. in 2007 [29] and Biswas et al. in 2014 [21] .

In Fig. 9 (b), one of the quaternary precipitate is seen edge-on, coprecipitated with a 𝜃'-Al 2 Cu precipitate. In this orientation, by STEM-HAADF, the Q'(Q)-AlCuMgsi phase can be recognized by the Cu-rich columns (high intensity due to high atomic number) surrounded by lower intensity columns (Si, Al and Mg) similar to the cross-section structure viewed along < 100 > Al zone axis shown in Ding et al. [START_REF] Ding | Study of the Q ′ (Q)-phase precipitation in Al-Mg-Si-Cu alloys by quantification of atomic-resolution transmission electron microscopy images and atom probe tomography[END_REF] .

Eventually, a third phase is present, detected thanks to the spots at 1/3 and 2/3 of the [022] Al orientation in SADP ( Fig. 7 (b) and(d)). STEM-HAADF in two orientations [110] Al and [112] Al (see Fig. 11 ) was necessary to identify this phase as the Ω -Al 2 Cu. This phase is known to have an orthorhombic structure, space group Fmmm with a = 0.496 nm; b = 0.859 nm and c = 0.848 nm and to precipitate in the {111} Al planes [START_REF] Knowles | The structure of {111} age-hardening precipitates in Al-Cu-Mg-Ag alloys[END_REF] . Measurements of Cu-Cu distances on STEM-HAADF images along Fig. 6. SAXS curves for 2017A and A-U4G at peak-aging state: after 8 h at 180 °C. The model was superimposed to the experimental curves.

[001] Ω , [100] Ω and [010] Ω correlate with the expected lattice parameters. These precipitates grow preferentially on dispersoids (AlMnSi).

All the observations provided a clear evidence on the phases present upon artificial aging: at grain boundaries, copper and Mg-Si precipitates were formed while in the matrix, 𝜃'-Al 2 Cu, Q'(Q)-AlCuMgSi and Ω -Al 2 Cu were clearly identified.

Discussion

Hardness evolution

The higher hardness of A-U4G compared to 2017A in the as-received state (about 10 HV difference) could be explained by different factors. When considering precipitation at nanoscale, SAXS experiments provided evidence of the presence of clusters in as-received state whether in A-U4G or in 2017A. DSC confirmed the presence of this population in the as-received state, as a dissolution event occurred around 180 °C. These clusters, with a diameter in the range of 8 Å, are the principal cause of hardness in Duralumin. However, it was shown that the nanoprecipitation is identical in both alloys, which rules out this factor as the primary cause in the hardness difference. Secondly, dispersoids contribute partly to hardness by controlling the grain size. It was shown that they are smaller and more numerous in A-U4G than in 2017A. This could be an explanation for the difference in hardness. However, since the grain size is close in both alloys (see Fig. S2 in Supplementary information), the contribution of dispersoids should not be significant. The third explanation comes from the plates' conditions. The old plate was bent when collected. This action generated strain hardening, with a higher density of dislocations (see Fig. S3 in Supplementary information). The presence of these dislocations is believed to be the main factor in the higher hardness of the A-U4G alloy in the as-received state.

For over-ageing conditions, the hardness of the A-U4G alloy drops faster until reaching the hardness of the modern alloy. It can be hypothesized that after artificial ageing, the historical alloy loses part of its additional dislocation density through faster recovery. The final part of the hardness curve could also be explained be the fact that alloys age faster when additional dislocations are introduced, because the dislocation network transports solute atoms through pipe diffusion [START_REF] Legros | Observation of giant diffusivity along dislocation cores[END_REF] and thus accelerates growth and coarsening of 𝜃' precipitates, known to nucleate on dislocations [START_REF] Liu | In-situ stem imaging of growth and phase change of individual CuAlX precipitates in Al alloy[END_REF] . This growth and coarsening will induce a decrease in hardness.

Nano-structure in the as-received state

About the nano-clusters present in the as-received states, several points can be made. The copper-rich clusters observed by atomicallyresolved TEM images in HAADF mode were not sufficiently numerous to correlate with the measured SAXS signal. Electron microscopy is thus not fully succeeding in observing and analysing nanoclusters. On another hand, at this stage, our experiments cannot differentiate between GPZ (flat clusters growing in the planes parallel to the < 100 > Al axes) and disordered clusters. GPZ are usually found in the model Al-4%Cu alloy aged at room-temperature, as it was firstly proved by Guinier and Preston by XRD in 1938. Lambot [START_REF] Lambot | Etude cristallographique de la précipitation structurale dans les duralumins[END_REF] , who studied both Al-4%Cu and Duralumin by X-ray scattering, proved that the scattering signal was perfectly identical in both alloys, concluding thus that GP zones are present in Duralumin aged at room temperature. It is to be noted that their studies were carried out on single crystals. In the present case, SAXS signal is an average of the measurements made on several grains: the information related to orientation and/or anisotropy is then cancelled out. Further investigation is required to confirm the structural nature of these clusters.

Upon artificial aging -identification of precipitation

Upon artificial aging (at 180 °C for 8 h), it was demonstrated that three populations of precipitates coexist in the matrix: 𝜃'-Al 2 Cu, Q'(Q)-AlCuMgSi and Ω -Al 2 Cu. Linking back the TEM observations with DSC results and thanks to results reported in the literature for Al-Cu-Mg-Si alloys, it can be hypothesized that region III (between 220 °C and 280 °C) is related to the growth of Q'(Q)-AlCuMgSi phase and region IV (between 280 °C and 320 °C) to the growth of the 𝜃'-Al 2 Cu phase. As for the Ω -Al 2 Cu phase, having the same stoichiometry as the 𝜃'-Al 2 Cu phase and if its growth is controlled by diffusion, there could be superimposition of peaks in the DSC curve.

The 𝜃'-Al 2 Cu phase, found in large quantity, is not surprising. It constitutes the main hardening phase in the 2xxx family alloys and was reported in all studies on Duralumin or 2017A alloy [START_REF] Saulnier | Maturation, revenu et réversion des alliages Al-Cu 4%, Al-Cu-Mg et Al-Cu-Mg-Si de haute pureté[END_REF][START_REF] Lambot | Etude cristallographique de la précipitation structurale dans les duralumins[END_REF][START_REF] Yano | Electron microscope observation of duralumin[END_REF] .

The presence of the Q-AlCuMgSi phase or one of his precursors (Q') was evidenced, in both old and modern alloys. Although not systematically revealed in previous studies of Duralumin, these findings confirm the equilibrium diagrams data reported by Mondolfo in 1943 [START_REF] Mondolfo | Metallography of Aluminum Alloys[END_REF] which predicts that with Mg:Si ratio < 1.73, hardening constituents should lead to the Al 2 Cu and Al 4 CuMg 5 Si 4 equilibrium phases. The silicon content is indeed higher than that required for the formation of Mg 2 Si (Mg:Si = 1.08): the excess of silicon will combine with copper and magnesium to form the quaternary phase. The global stoichiometric composition for the Q phase has been, however, a controversial matter. Mondolfo predicts two possible quaternary phases: Al 4 CuMg 5 Si 4 or Al 4 CuMg 4 Si 4 . In more recent literature, mainly for alloys of the 6xxx family (Al-Mg-Si with additions of copper) [START_REF] Eskin | Decomposition of supersaturated solid solutions in Al-Cu-Mg-Si alloys[END_REF] , but also for 2014 alloys [START_REF] Dutta | Role of Al 2 O 3 particulate reinforcements on precipitation in 2014 Al-matrix composites[END_REF] , the stoichiometry reported in bulk alloys were Al 5 Cu 2 Mg 8 Si 6 , Al 4 Cu 2 Mg 8 Si 7 , Al 3 Cu 2 Mg 9 Si 7 etc. However, the nanometre scale of these precipitates makes the quantitative analyses challenging. Biswas et al. [START_REF] Biswas | Compositional evolution of Q-phase precipitates in an aluminum alloy[END_REF] used atom probe analysis and proved that the Q precipitates evolved, upon aging, from a copper-rich composition (44Al-22Cu-16Mg-16,5Si at%) to a magnesium and silicon-rich composition (28Al-9Cu-37Mg-26Si at%.) after 4 h at 260 °C. In a very recent paper, Ding et al. [START_REF] Ding | Study of the Q ′ (Q)-phase precipitation in Al-Mg-Si-Cu alloys by quantification of atomic-resolution transmission electron microscopy images and atom probe tomography[END_REF] were able to provide, by combining atom probe tomography and atomic column intensity quantification in STEM-HAADF images, the reason why varying composition were reported in past work: it was indeed proved to be due to the occupancy change of the atomic columns. The stoichiometry of the Q phase and its precursor, the Q ' phase were found to be: Al 5 Cu 3 Si 6 Mg 9 and Al 12 Cu 1 Si 4 Mg 5 , respectively. Even if matrix effects cannot be excluded with EELS, the elemental composition found here by EELS in Duralumin (73 ± 10 Al-10 ± 4 Cu-12 ± 6 Mg-5 ± 5 Si at%) is in all cases much leaner in magnesium and silicon than what is usually reported. This results should be corroborated with other techniques to confirm the tendency. The heterogeneous nucleation of 𝜃'-Al 2 Cu phase and the Q'(Q)-AlCuMgSi phase, observed in Fig. 9 (b), was also reported by other authors [START_REF] Biswas | Compositional evolution of Q-phase precipitates in an aluminum alloy[END_REF][START_REF] Wenner | A hybrid aluminium alloy and its zoo of interacting nano-precipitates[END_REF] . It could be caused by the strain field of Q-phase precipitates on which the 𝜃'-Al 2 Cu nucleate.

Another phase, Ω -Al 2 Cu, is here observed. The Ω phase is usually found in high density in Al-Cu-Mg-Ag quaternary alloys with a high Cu:Mg ratio [START_REF] Wang | Precipitates and intermetallic phases in precipitation hardening Al-Cu-Mg-(Li) based alloys[END_REF][START_REF] Knowles | The structure of {111} age-hardening precipitates in Al-Cu-Mg-Ag alloys[END_REF] . It presents one of the highest strengthening potential in aluminium alloys. In particular, Al-Cu-Mg-Ag alloys have been developed for elevated temperature aerospace industry applications [START_REF] Polmear | After concorde: evaluation of creep resistant Al-Cu-Mg-Ag alloys[END_REF] due to the enhanced thermal stability of the Ω phase. Some authors proved that silver was not required for the formation of Ω precipitates although its presence increases their density [START_REF] Garg | Precipitation of the Ω phase in an Al-4.0Cu-0.5Mg alloy[END_REF] . The Ω phase was indeed observed in 2024 and 2124 alloys by Wang et al. [START_REF] Wang | Precipitation of the Ω Phase in 2024 and 2124 Aluminum Alloys[END_REF] and was actually found in 2017A alloy as well [START_REF] Härtel | Effects of particle reinforcement and ecap on the precipitation kinetics of an Al-Cu alloy[END_REF] . In 2024 and 2014 alloys, the Ω phase was found to form preferentially on manganese-based dispersoids. This heterogeneous growth is clearly evidenced in our reported results. To understand why such a phase appeared in Duralumin, it is interesting to discuss its nucleation process. In Al-Cu-Mg-Ag alloys, the process has been widely investigated: upon aging, magnesium and silver co-cluster, then copper atoms aggregate to the {111} Al planes. During coarsening, when copper concentration becomes close to Al 2 Cu, silver and magnesium start to migrate from the interior to the interface of the precipitate. The interfacial structure was proved to be a doublelayer composed of Ag in hexagonal structure and Mg and Cu below the centre of the hexagon [START_REF] Kang | Ab initio study of growth mechanism of omega precipitates in Al-Cu-Mg-Ag alloy and similar systems[END_REF] . Magnesium is considered necessary for the nucleation of the Ω phase. As for silver, its role is to accelerate the aggregation of magnesium atoms by retaining vacancies and to assist in this way the magnesium diffusion [START_REF] Winkelman | Stimulation of the Ω phase in an Al-1.1 at% Cu-0.5 at% Mg alloy by a duplex ageing treatment involving initial natural ageing[END_REF] . Silicon is also an element that influences the presence of Ω phase. From Polmear et al. [START_REF] Polmear | After concorde: evaluation of creep resistant Al-Cu-Mg-Ag alloys[END_REF] , silicon (and iron) should remain low (less than 0.10 wt%) to minimise the formation of low melting point eutectics at grain boundaries and to maximise Ω precipitation. In fact, high levels of silicon interfere with precipitation of Ω because silicon interacts with magnesium and thus reduces the formation Mg-Ag co-clusters, which generally facilitate nucleation of Ω . This was also observed in the work by Ünlu et al. [START_REF] Ünlü | The effect of cold work on the precipitation of Ω and 𝜃′ in a ternary Al-Cu-Mg alloy[END_REF] . However, the authors showed that it is rather the Mg:Si ratio that had to be considered: some Ω precipitation was found in alloys with silicon levels as high as 0.5 and 0.65 wt% with corresponding Mg:Si ratios of 3.0 and 2.23. In the present alloys, the observation that Ω forms principally on dispersoids in the Duralumin and 2017A can be explained by the presence of copper clusters at the matrix/dispersoid interface in the as-received state. Moreover, it does not form as a phase dispersed in the matrix because of the absence or the very low density of magnesium clusters in the matrix. The magnesium clusters are in this case impeded by silicon, which rather facilitates the formation of the Q-AlCuMgSi phase.

Conclusion

An old Duralumin (A-U4G) and its modern equivalent (2017A) were studied with different characterization techniques to clarify the precipitation processes. The main findings are:

-The precipitation sequence in Duralumin and modern 2017A upon artificial aging is similar. It consists in 𝜃'-Al 2 Cu, Q'(Q)-AlCuMgSi and Ω -Al 2 Cu in the matrix. At grain boundaries, various precipitates are found: copper-based and (Mg,Si) precipitates with unknown stoichiometry. -The Q-AlCuMgSi phase or one of its precursors (Q') is observed in the matrix but the 𝛽-Mg 2 Si phase, previously reported in Duralumin, is not present. This phase facilitates the nucleation of the 𝜃'-Al 2 Cu phase through strain field. -The Ω phase is observed and it grows preferentially on dispersoids.

It presumably nucleates due to the presence of copper clusters at the matrix/dispersoid interface. This phase is a minor phase due to the absence or rare occurrence of magnesium clusters in the matrix. The nucleation mechanisms for the Ω phase on dispersoids in these Al-Cu-Mg-Si alloys remains to be evidenced. -In the as-received state (naturally aged), the presence of clusters is evidenced, with copper as the main element but the structural nature of these clusters is yet to be determined. -Despite some differences in the microstructure, such as the dispersoids' size and density, Duralumin exhibits a similar nanoprecipitation as its modern equivalent in the naturally aged conditions, and it behaves also similarly upon artificial aging. These observations lead to the conclusion that the conditions experienced by the collected plate (10 years of service plus 50 years at ambient temperature) had no significant impact on the nanostructure.

Overall, these results constitute a base for assessing the very longterm aging of aluminium alloys in aeronautics.

Fig. 1 .

 1 Fig. 1. DSC curves of A-U4G and 2017A, as-received state. The heating rate is 20 °C/min. 2017A curve is shifted upward for a better visualisation.

Fig. 2 .

 2 Fig. 2. TEM bright field images of as-received alloys (a) 2017A ; (b) A-U4G showing mainly AlMnSidispersoids in the matrix.

Fig. 3 .

 3 Fig. 3. STEM images of as-received A-U4G alloy in [001] Al zone axis; (a) and (c) Annular bright field; (b) and (d) high angle annular dark field. Images (a) and (b) on one side and (c) and (d) on the other side correspond to the exact same area. A cluster is indicated with the white arrow at the interface dispersoid/matrix.

Fig. 7 :

 7 Fig.7: bright-field images and corresponding selected-area electron diffraction patterns are presented. Fig.8shows the STEM-EDS mapping for the A-U4G alloy.After 8 h at 180 °C, a very dense precipitation is observed inside the matrix ( Fig.7) and at grain boundaries ( Fig.8). In both alloys ( Fig.7 (a) and (c)), precipitates lying in {100} Al planes are observed: some as platelets and others as rods, with precipitation on dislocations. As the observed precipitation is similar in both A-U4G and 2017A alloys, advanced analyses results are then only shown for A-U4G alloy.In Fig.8, precipitates at grain boundaries were identified by EDS: they are (Mg, Si)-precipitates and copper precipitates. On these EDS maps, dispersoids of AlMnSi nature are present. A precipitation in the matrix is also visible but hardly identifiable.As a matter of fact, different precipitates populations co-exist inside the grains and their identification was made possible by combining different electron microscopy techniques. From SADP taken along [001] Al zone axis, shown in Fig.7 (b) and (d), the presence of 𝜃'-Al 2 Cu phase was evidenced thanks to the strikes along < 001 > Al directions and faint spots (110) 𝜃' at1 2 of (220) Al[START_REF] Wang | Precipitates and intermetallic phases in precipitation hardening Al-Cu-Mg-(Li) based alloys[END_REF] . They correspond to three vari-

  ants of plate-shape semi-coherent precipitates growing in the {100} Al planes: one face-on and two edge-on. The 𝜃'-Al 2 Cu precipitate nature was confirmed thanks to STEM-HAADF as shown in Fig.9 (b). Similarly to HAADF images taken on as-received states, bright dots correspond to copper-rich atomic columns. Seen edge-on, the crystal structure of this platelet is body-centred tetragonal as previously shown by Bourgeois et al.[START_REF] Bourgeois | Structure and energetics of the coherent interface between the 𝜃′ precipitate phase and aluminium in Al-Cu[END_REF] and Shen et al.[START_REF] Shen | Atomic-scale mechanism of the 𝜃″ →𝜃′ phase transformation in Al-Cu alloys[END_REF] . The presence of pre-𝜃' -1 (Al 2 Cu) precipitates coherent on the {100} Al planes, was also revealed with STEM-HAADF, as shown in Fig.9( c)): these types of precipitates were recently identified by Liu et al.[START_REF] Liu | Multiple precipitation reactions and formation of 𝜃'-phase in a pre-deformed Al-Cu alloy[END_REF] . They consist of three parallel copper planes, each separated by two aluminium planes. On SADP, diffraction patterns of the pre-𝜃' -1 coincide with diffraction patterns of 𝜃'-Al 2 Cu. Both types are thus, difficult to differentiate. At peak-aged conditions, this phase, which corresponds to the first stage of 𝜃'-Al 2 Cu formation, is not predominant: most of the population has evolved to the 𝜃'-Al 2 Cu phase.Other precipitates in the shape of rods were observed in the A-U4G alloy after 8 h at 180 °C, as shown in the bright field images of Fig.7 (a) and in the HAADF image of Fig. 9 (a). The rods, seen end-on, are oriented along < 001 > Al directions. No clear identification being possible with

Fig. 4 .

 4 Fig. 4. SAXS curves for 2017A and A-U4G in the as-received state. The model was superimposed to the experimental curves.

Fig. 5 .

 5 Fig. 5. Hardness measurement on rolling plane for old (A-U4G) and modern (2017A) alloys heat treated at 180 °C.

Fig. 7 .

 7 Fig. 7. TEM bright-field images and corresponding SADP in [001] Al zone axis after treatment at 180 °C-8 h for 2017A: (a) and (b) and for A-U4G: (c) and (d). (110) 𝜃' spots and strikes along [001] Al directions for the 𝜃' phase are indicated with arrows. Spots at 1/3 and 2/3 of (022) Al corresponding to Ω phase are surrounded.

Fig. 8 .

 8 Fig. 8. STEM-EDS mapping after treatment at 180 °C-8 h in A-U4G showing precipitation at grain boundaries: MgSi and Cu as well as in the matrix.

Fig. 9 .

 9 Fig. 9. (a) STEM Bright-field image of A-U4G alloy in [001] Al zone axis showing 𝜃'-Al 2 Cu (black arrows) and Q-AlCuMgSi rods (white arrows), (b) HAADF images of 𝜃'-Al 2 Cu platelets seen edge-on with co-precipitation of Q'(Q)-AlCuMgSi and (c) pre-𝜃' -1 Al 2 Cu disks seen edge-on.

Fig. 10 .

 10 Fig. 10. (a) STEM Bright-field image of A-U4G alloy in [001] Al zone axis (b) STEM-HAADF image of Q-AlCuMgSi rods seen top-on.

Fig. 11 .

 11 Fig. 11. STEM Bright-field images and HAADF images of A-U4G alloy in (a) and (b): [110] Al zone axis; (c) and (d): [ 1 1 2 ] Al zone axis showing Ω -Al 2 Cu platelets (white arrows) growing on dispersoids.

Table 1

 1 Elemental composition of alloys (in wt%) obtained by ICP-OES.

		Al	Cu	Mg	Mn	Si	Fe	Ti	Zn	Cr
	2017A	Base	4.32 ± 0.08	0.68 ± 0.01	0.611 ± 0.002	0.618 ± 0.006	0.34 ± 0.01	0.043 ± 0.001	0.20 ± 0.04	0.029 ± 0.003
	A-U4G	Base	4.18 ± 0.04	0.710 ± 0.005	0.67 ± 0.01	0.61 ± 0.01	0.285 ± 0.008	0.010 ± 0.001	0.082 ± 0.008	0.007 ± 0.0001

Acknowledgments

The authors which to thank Pierre Roblin from FR FERMAT, Université de Toulouse, CNRS, Toulouse, France, for SAXS measurements, Teresa Hungria-Hernandez, from Raimond Castaing Microanalysis Centre, Toulouse for STEM (HAADF, EDS and EELS) experiments and the volunteers of Les Ailes Anciennes Toulouse for providing us with the plate collected on the Breguet Sahara 765 aircraft.

This study has been partially supported through the grant NanoX n°A NR-17-EURE-0009 in the framework of the "Programme des Investissements d'Avenir ".

Declaration of Competing Interest

None

Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.mtla.2019.100429 .


Ivan Redor

Eric Barthélemy

Hervé Michallet

Miguel Onorato

Nicolas Mordant

Experimental evidence of a hydrodynamic soliton gas

We report on an experimental realization of a bi-directional soliton gas in a 34 m-long wave flume in shallow water regime. We take advantage of the fission of a sinusoidal wave to inject continuously solitons that propagate along the tank, back and forth. Despite the unavoidable damping, solitons retain adiabatically their profile, while decaying. The outcome is the formation of a stationary state characterized by a dense soliton gas whose statistical properties are well described by a pure integrable dynamics. The basic ingredient in the gas, i.e. the two-soliton interaction, is studied in details and compared favourably with the analytical solutions of the Kaup-Boussinesq integrable equation. High resolution space-time measurements of the surface elevation in the wave flume provide a unique tool for studying experimentally the whole spectrum of excitations.

In 1965 Zabusky and Kruskal coined the word "soliton" to characterize two pulses that "shortly after the interaction, they reappear virtually unaffected in size or shape" [START_REF] Zabusky | [END_REF]. This property, that makes solitons fascinating objects, is a common feature of solutions of integrable equations, such as for example the celebrated Kortewegde Vries (KdV) equation that describes long waves in dispersive media, or the Nonlinear Shrödinger equation, suitable for describing cubic nonlinear, narrow-band processes. Those equations find applications in many fields of physics such as nonlinear optics, water waves, plasma waves, condensed matter, etc [START_REF] Dauxois | Physics of Solitons[END_REF]. In analogy to a gas of interacting particles described mesoscopically by the classical Boltzmann equation, in the presence of a large number of interacting solitons, Zakharov in 1971 derived a kinetic equation for the velocity distribution function of solitons [START_REF] Zakharov | [END_REF], see also [4][5][6]. Some of the theoretical predictions have been confirmed via numerical simulations of the KdV equation in [7]. The wave-counterpart of the particle-like interpretation of solitons is known as "integrable wave turbulence"; such a concept was introduced more recently by Zakharov [8]. The major question in this field is again the understanding of the statistical properties of an interacting ensemble of nonlinear waves, described by integrable equations, in the presence or not of randomness; the latter may arise from initial conditions which evolve under the coaction of linear and nonlinear effects, [9][10][11][12][13][14][15][16][17]. In contrast to many nonintegrable closed wave systems that reach a thermalized state characterized by the equipartition of energy among the degrees of freedom (Fourier modes) [START_REF] Onorato | Proc. Nat. Acad. Sc[END_REF][START_REF] Pistone | [END_REF], integrable equations are characterized by an infinite number of conserved quantities and their dynamics is confined on special surfaces in the phase space. This prevents the phenomenon of classical thermalization and it opens up the fundamental quest on what is the large time state of integrable systems for a given class of initial conditions. So far the question has no answer and, apart from recent theoretical approaches [20,21], most of the results on the KdV problem relies on numerical simulations, see e.g. [7,16,22,[START_REF] Pelinovsky | Challenges in Complexity: Dynamics, Patterns, Cognition[END_REF]. Soliton gas have been extensively observed in optics [e.g. [START_REF] Mussot | [END_REF]25] while experimental evidence in a hydrodynamic context is scarce. In [26] it has been claimed that the low frequency component of sea surface elevations measured in the Currituck Sound (NC, USA) behave as a dense soliton gas, displaying a power law energy spectrum with exponent equal to -1; another approach is described in [27] where the soliton content in laboratory shallow water wind waves is estimated.

In this Letter we describe a unique experiment that is designed to build and monitor a hydrodynamic soliton gas in a laboratory. We focus on shallow water gravity waves where the dynamics is described to the leading order in nonlinearity and dispersion by the KdV equation. In planning the experiment, many issues had to be faced. The main one is that dissipation is present in any experimental set-up; thus, the integrable equations cannot describe experiments over long time scales. In order to reach a stationary regime, energy must be injected by a forcing device. These two features, dissipation and forcing, are clearly at odds with the integrable turbulence framework. Therefore, is it possible to produce in the laboratory a soliton gas described in statistical terms by integrable turbulence? Answering positively this question in laboratory experiments would strongly support the application of the integrable turbulence framework to in-situ data, as reported in [26,[START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF].

Experiments are performed in a 34 m-long wave flume, 55 cm-wide with a water depth at rest equal to h = 12 cm. Waves are generated by a piston-type wavemaker (see a more detailed description in [29,30]). The free surface vertical displacement is measured along the flume by imaging through the lateral glass walls. We use seven synchronized monochrome cameras with resolution 1920 × 1080 pixels; each camera records the image of the contact line of water over 2 m. Thus, the field of view consists of the 14 m-long central part of the flume. The spatial resolution is close to 1 mm/pix. A grid is used to calibrate the images and correct their geometrical distortions. The contact line is extracted by detecting the strongest grayscale gradient in a vertical line of pixels. Subpixel resolution is obtained by polynomial interpolation in the vicinity of the pixel of steepest gradient. We estimate that the resulting error on the surface elevation is close to 0.1 mm. The cameras are operated at 20 frames/s.

The second important issue to deal with during the design of a soliton gas experiment is that wave flumes are usually not long enough to observe many soliton collisions in the KdV regime which account only for wave propagating in one direction. To cope with such limitation, the end of the flume, opposite to the piston, consists of a vertical wall that reflects the waves. Thus the waves propagate in both directions and are also reflected on the piston. We take advantage of the reflections to observe the propagation over time scales larger than the duration restricted to a one-way trip along the flume. An integrable system of equations that deals with bidirectional wave propagation in shallow water is the Kaup-Boussinesq (KB) system:

∂ t η + ∂ x [(h + η)u] = - h 3 3 ∂ xxx u ∂ t u + u∂ x u + g∂ x η = 0, (1)
where η = η(x, t) is the free surface elevation, u = u(x, t) is the fluid velocity, g is the acceleration of gravity and h is the water depth. Single soliton, as well as multisolitons solutions, are known for this system and reported in [31]. The multi-solitons solutions of KB include the collision of two solitons: either overtaking collisions (two solitons of distinct amplitude propagating in the same direction) or head-on collisions. Such collisions are elastic, i.e. the amplitudes of the solitons are not altered after the interaction. The effect of the collision is to induce a phase shift. For head-on collisions such a phase shift is much smaller than for overtaking solitons For the latter, the larger solitons "jump" forward, while the smaller solitons experiences a negative delay. For co-propagating solitons, the collision is very close to the KdV one [31].

The amplitude of a single soliton traveling in the flume decays exponentially with an e-fold time scale of 90 s (see supplemental material), i.e. a clear indication of the presence of viscous dissipation (in the bulk and along the walls and bottom) and of the non integrable long term dynamics. Nevertheless, the shape of the soliton is fitted correctly by the single soliton solution, suggesting that the shape of the soliton changes adiabatically and highlighting the short time integrable dynamics [32].

Before discussing the realization of the soliton gas, we analyze the two-soliton solution in the laboratory. A space-time representation of a head-on collision of large amplitude solitons (a/h 0.4, with a the amplitude of each soliton) is shown in Fig. 1(a). The left-going soliton is generated first and reflects at the right end wall. The collision appears quasi-elastic and the resolution of the picture does not allow us to visualize here the phase shift FIG. 1. Two-soliton interaction from experiment: (a) head-on interaction of large amplitude solitons in a space-time representation. The wavemaker is located at x = 0 m. Small dispersive waves radiated from the collision can be seen in the background, a sign of small departure from integrability. (b) Snapshots of head-on interaction of small amplitude solitons (a/h = 0.1) traveling in opposite directions. Red: 2 s before the collision, blue: at the collision, green: 5 s after. Black dashed lines: solution of the KB equations (see text for details). (c) Collision of solitons traveling in the same direction (overtaking interaction), same colour code as in (b).

induced by the collision. The value a/h=0.4 is below the critical threshold, (a/h) c = 0.6, for which the headon collision leads to the formation of a residual jet, see [33]. Very small amplitude dispersive waves that move slower than the solitons are emitted after the collision, see [34][35][36]. These waves, that are a sign of nonintegrability, remain of small amplitude and are only visible for collisions of very large solitons. Fig. 1(b) shows the headon collision and Fig. 1(c) the overtaking collision for small amplitude solitons together with the two-soliton solutions of the KB eqs [START_REF] Zabusky | [END_REF]. The only free parameters are the amplitudes of the solitons that are obtained by extracting the amplitude of each soliton when they are widely separated (before and after the collision) and extrapolated to the collision point by compensating the dissipation. The two-soliton theoretical solution matches nicely both experiments. Again, the observed short term dynamics is well reproduced by the integrable model.

In order to obtain a bidirectional soliton gas, a large number of solitons must be injected in the wave flume. We face a technical difficulty when trying to inject individual solitons: the piston must recede slowly so as to not induce undesired waves and then move forward quickly to generate the soliton. The receding phase is so long that only a few solitons can be introduced in the entire flume by this procedure. We circumvent this issue by continuously forcing a sinusoidal wave at the wave maker. As observed experimentally in [37,38] and numerically in [39,40], a sine wave in shallow water spontaneously steepens and decomposes, after a certain propagation distance, into a train of solitons of various amplitudes. Our generation setup is similar to [41,42] but with a longer flume to ensure soliton fission. These solitons then interact with solitons emitted earlier that survive until dissipated by viscosity. The number of solitons depends proportionally on the Ursell number U = 3Aλ 2 16π 2 h 3 that measures the dispersive effects over the nonlinear ones (A is taken as twice the standard deviation of η and λ is the wavelength), see [START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF]38] for details. In this way, we can inject a large number of solitons in the flume and their amplitude and density can be tuned by changing the amplitude and frequency of the sinusoidal forcing.

An example of wave elevation of the obtained soliton gas can be seen in Fig. 2(a) in a space-time representation. The density of solitons is large enough so that solitons often interact. One can clearly see solitons that propagated in both directions as bright straight ridges, either increasing or decreasing in the (x, t) space. The lines are not parallel due to the distribution of amplitudes of the solitons and because larger solitons propagate faster than smaller ones. The dashed line shows a line of slope equal to the shallow water linear long wave phase velocity c 0 = √ gh R (where h R is the reference level for solitons, slightly below h, see e.g. [27,43] and Fig. 2(c)). The nearby soliton clearly propagates faster than a linear long wave. The head-on collisions are visible by the fact that the amplitude is maximum at each crossing of two counter-propagating solitons (as in Fig. 1(b)). Note that, although the solitons are injected periodically, there is no obvious sign of such periodicity in the plots. The large number of interactions among solitons seems to randomize the gas.

Waves propagating towards positive x can be separated from those propagating towards negative x by computing the time-space Fourier transform η(k, ω) of the measured wave field η(x, t). Waves with k > 0 & ω > 0 (or k < 0 & ω < 0 as the field is real) propagate towards negative x and waves with k > 0 & ω < 0 to positive x. Fig. 2(b) shows the time-space representation of the waves traveling with increasing x. Indeed, only the solitons going to the right are retained. An example of overtaking interaction of a large soliton and a smaller one is highlighted in Fig. 2(c) which strongly resembles the isolated collision shown in Fig. 1(c). The observation of such events supports the fact that our wave field is indeed dominated by solitons.

A way to investigate all possible excitations in the system is to compute the space-time Fourier spectrum, E(k, ω), of the wave elevation. Here the Fourier trans- Green: surface elevation at t = 13.5 s corresponding to the short dashed line in (b). These two curves correspond to preand post-collision states of a collision that occurs at t = 11.5 s and x = 19 m. They illustrate an example of overtaking soliton interaction in the soliton gas.

form is taken over the entire available spatial window (14 m long) and using the Welch method in time (i.e. using successive temporal windows of duration 25 s). The spectrum is shown in Fig. 3(a): several curves with a significant level of energy can be identified. The strongest is a symmetric pair of straight lines (labeled 'bb'). They correspond to solitons whose Fourier components travel all at almost the same phase velocity. The thickness of the lines is due to the fact that solitons of various size coexist and travel at slightly distinct velocities depending on their amplitude. A second set of curves is labeled as 'aa'. They correspond to linear dispersive waves following closely the dispersion relation curve ω = gk tanh(kh). As mentioned above, these waves may be radiated during collisions and are a signature of non integrability; also dissipated solitons may end up as linear dispersive. The lines marked 'cc' correspond to very small standing transverse modes that are hard to avoid in wave flumes. The most energetic lines visible are a clear sign of a soliton gas; to better visualize the spread of energy across the soliton line, we compute the frequency spectrum E(ω) averaged spatially, see Fig. 3(b). The spectrum is shown for the full field (black line), the waves propagating towards positive x (red line) and those reflected (blue line). In the curve corresponding to the waves leaving the wavemaker, a small peak at the forcing frequency is present. Energy is also present at frequencies lower and higher than this forcing. The high frequency content appears to decay exponentially; this result is consistent with numerical simulations of pure integrable equation containing a large content of solitons, see [16]. The spectral band below the forcing frequency is rather flat; this is especially true for the case of left-going waves with no forcing peak visible. Amongst the solitons moving away from the wavemaker (right-going) are "old" solitons reflected on the piston and new ones emerging from the sinusoidal wave fission. The latter are responsible for the forcing peak as they have not experienced much col-lisions and have kept the memory of their initial phase coherence. In contrast the blue curve relates to solitons having undergone at least one reflection and traveled a longer distance, thus having experienced more interactions. These interactions have randomized their phase. Our observation of a flat spectrum for low frequencies is consistent with numerical simulations of the KdV equations (unidirectional soliton gas) described in [16]. Field data measured at Duck Pier, North Carolina, are also consistent with a flat spectrum in a regime which is interpreted as dominated by solitons [START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF]. A flat spectrum is actually reminiscent of the shot noise due to discrete carriers in electronics or photonics. The electrons or photons have a random distribution in time that leads to a flat spectrum [START_REF] Papoulis | Probability, random variables, and stochastic processes[END_REF]. All these results are in contrast with the observations of Costa et al. [26] who showed that a dense soliton gas in a narrow and shallow sound has low frequency power spectra that behave as ∼ ω -1 . A deeper analysis of our data would require to perform the Inverse Scattering Transform (IST) [START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF][START_REF] Smirnov | [END_REF] for estimating the content of the signal in terms of solitons or cnoidal waves. It is expected that such an approach might help to understand the low frequency power spectra trend in our experiments.

Further statistical information can be obtained from the skewness S = (η -η ) 3 / (η -η ) 2 3/2 and kurtosis K = (η -η ) 4 / (η -η ) 2 2 of the surface elevation of waves travelling to positive x; those moments of the probability density function of the surface elevation measure the departure from Gaussian statistics (S = 0, K = 3). For the dataset shown in Fig. 3, U = 0.93, in the case of waves propagating towards positive x, we obtain S = 0.9, K = 3.45; those numbers are very close to the numerical values S = 0.8, K = 3.45 obtained at U = 0.95 in [16]. Although our setup is not integrable, many features appear consistent with numerical simulations of the integrable KdV equation. This is most likely due to the fact that dissipation operates over very long times scales (90 s, see Supplemental Material) compared to collision timescales (about 4 s from Fig. 2(b)). These collisions are the elementary process that leads to the randomization of the relative phases of the solitons. The observed largest soliton is approximately 5 cm high and the progressive damping ensures that all soliton amplitudes, from 5 cm down to zero, are present in the flume. This is characteristic of a warm soliton gas [6] and promotes many strong interactions as the one highlighted in Fig. 2(c). If sufficient scale separation is present between collision and dissipation time scales, then the dynamics is ruled by the integrable wave equation.

An open question is the role of head-on collisions on the statistical properties of the surface elevation. For each head-on collision, the resulting maximum amplitude exceeds that calculated by the superposition of the incident solitary waves [35]. Thus one may expect that the extreme events of the bidirectional gas to be of larger magnitude than that of the unidirectional KdV gas. Such analysis is left for future investigations.

In conclusion, in the present Letter we have given evidence of a hydrodynamic soliton gas produced in the lab. Despite the fact that dissipation unavoidably exists in the flume, we were able to produce a gas in a steady state regime whose properties are consistent with the conservative integrable dynamics. It is remarkable that the energy injected exits the wave system without changing the global picture of integrable turbulence. We hope that our results will stimulate new theoretical and experimental work in other nonlinear dispersive media, such as optics, BEC etc, where soliton dynamics plays an important role.

FIG. 2 .

 2 FIG. 2. (a) Space-time representation of the soliton gas. The horizontal scale is the same for all subfigures. Solitons traveling in both directions are visible. Many head-on collisions can be seen. The dashed line is the long wave phase velocity c0 = √ ghR, with hR the reference level of soliton propagation. (b) Part of the signal propagating away from the wave maker (to the right) obtained by the Hilbert transform. (c) Extraction of the surface elevation profile corresponding to the dashed line in (b) at t = 11.5 s. Magenta: fit of a single soliton solution to an isolated pulse. The horizontal dotted line is the corresponding reference level hR. Red: the surface elevation at t = 10.3 s corresponding to dashed line in (b).Green: surface elevation at t = 13.5 s corresponding to the short dashed line in (b). These two curves correspond to preand post-collision states of a collision that occurs at t = 11.5 s and x = 19 m. They illustrate an example of overtaking soliton interaction in the soliton gas.

4 FIG. 3 .

 43 FIG. 3. (a) E(k, ω) spectrum of the soliton gas. Dashed curves: 'aa' is the dispersion relation of linear waves ω = gk tanh(kh). 'bb' is the relation ω = c0k. 'cc' shows the part of the spectrum related to transverse waves. (b) Frequency spectrum of the wave elevation spatially over locations accessible to the cameras. Black line: full signal. Red and blue lines: signal propagating to the right and to the left, respectively. The vertical dashed line corresponds to the forcing frequency at 0.6 Hz.

  This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 647018-WATU). M. O. has been funded by Progetto di Ricerca d'Ateneo CSTO160004. M.O. was supported by the "Departments of Excellence 2018-2022" Grant awarded by the Italian Ministry of Education, University and Research (MIUR) (L.232/2016). P. Suret, S. Randoux and G. El are acknowledged for discussions during the early stage of the work.


D Wagner

Nicolas Ranc

Claude Bathìas

D W A G N E R

Fatigue crack initiation detection by an infrared thermography method

Keywords: infrared pyrometry, number of cycles at crack initiation, temperature recording, very high cycle fatigue

HAL is

N O M E N C L A T U R E

b = burger vector E = Young's modulus K eff = effective stress intensity factor K = stress intensity factor σ = stress amplitude σ y = yield stress σ y = cyclic yield stress

I N T R O D U C T I O N

In fatigue tests, according to the strain or stress level, three domains exist. For few number of cycles at fracture (N f < 10 4 cycles), it is the low cycle fatigue domain; for intermediate number of cycles at fracture (10 4 < N f < 10 7 cycles), it is the high cycle fatigue domain (megacycle domain) and for high number of cycles at fracture (N f > 10 7 cycles), it is the very high cycle fatigue domain (gigacycle domain). 1 The latter domain is now investigated with the development of devices (piezoelectric fatigue machines) working at high frequency (20 or 30 kHz), allowing to obtain 10 8 or more cycles in reasonable tests time.

Correspondence: D. Wagner. E-mail: daniele.wagner@u-paris10.fr

These tests have shown that fracture can occur at 10 9 or more cycles which is problematic because many components and structures in several industries require design fatigue life often superior to 10 8 cycles. At a macroscopic scale, according to the fatigue domain, different types of crack initiation occur in cylindrical samples with a polished surface depending on whether it is low cycle, mega or giga-cycle fatigue range. For the smallest number of cycles at failure, the initiation sites are multiple and located on the surface. For intermediate number of cycles at failure, there is only one surface initiation site, whereas in gigacycle fatigue domain, the initiation site may be located in an internal zone or at the surface.

At the microscopic level, Mugrahbi et al. [START_REF] Mughrabi | Specific features and mechanisms of fatigue in the ultrahigh-cycle regime[END_REF] show that the initiation of fatigue crack in the gigacycle fatigue regime can be described in terms of microstructurally irreversible portion of the cumulative cycle strain. It seems that there is no basic difference between fatigue mechanisms in low, mega and giga-cycle fatigue except for the strain localization. For the low cycle fatigue, an important plastic deformation of the specimen bulk governs the initiation. In the megacycle fatigue, the plastic deformation is governed by the plane stress effect and the presence of flaws at the surface which is the critical location of fatigue initiation. However, at lower stress, the plastic deformation in plane stress condition is vanishing and the macroscopic behaviour of the metal is fully elastic except around flaws, metallurgical defects or inclusions. As the probability of occurrence of a flaw is greater in a volume than at a surface, it is normal that the initiation site in gigacycle fatigue will be often in subsurface. In this case, the important parameters are the defect size and the position of the defect. When the initiation is located on an internal defect, the crack propagation leads to a fish-eye propagation around the defect. The geometry of the fish-eye initiation is a circle that collapses on reaching the surface of the specimen. In an internal initiation, it is difficult to determine the number of cycles at initiation. To predict the number of cycles to initiate a fatigue crack from an inclusion, several models are used more or less successfully. 1 In the gigacycle fatigue range, the integration of Paris' law [START_REF] Paris | The relationship of effective stress intensity factor, elastic modulus and Burger's-vector on fatigue crack growth as associated with "Fish Eye" gigacycle fatigue phenomena[END_REF][START_REF] Marines-Garcia | Fatigue crack growth from small to large cracks in gigacycle fatigue with fish-eyes failure[END_REF][START_REF] Marines-Garcia | Fatigue crack growth from small to long cracks in VHCF with surface initiations[END_REF] allows one to predict the number of cycles in the fish-eye growth and obtain the number of cycles at crack initiation. As the crack initiation appears, a short crack around the defect propagates followed by a long crack. In all cases, a cyclic plastic zone around the crack exists. During fatigue cycles, the irreversible part of the rate of the plastic work is responsible for the intrinsic dissipation. When the crack initiation occurs, the plastic deformation at the crack tip increases, and the recording of the surface temperature of the sample during the test allows to follow the crack propagation and to determine the number of cycles at the crack initiation. [START_REF] Ranc | Study of thermal effects associated with crack propagation during very high cycle fatigue[END_REF] In the literature, the infrared pyrometry was already used by many authors, [START_REF] Luong | Fatigue limit evaluation of metals using an infrared thermographic technique[END_REF][START_REF] Liaw | Thermographic detection of fatigue damage of pressure vessel steels at 1,000 Hz and 20 Hz[END_REF][START_REF] Yang | Thermographic investigation of the fatigue behavior of reactor pressure vessel steels[END_REF][START_REF] Fargione | Rapid determination of the fatigue curve by the thermographic method[END_REF][START_REF] Curà | A new iteration method for the thermographic determination of fatigue limit in steels[END_REF][START_REF] Boulanger | Calorimetric analysis of dissipative and thermoelastic effects associated with the fatigue behavior of steels[END_REF][START_REF] Meneghetti | Analysis of the fatigue strength of a stainless steel based on the energy dissipation[END_REF][START_REF] Morabito | Analysis of heat sources accompanying the fatigue of 2024 T3 aluminium alloys[END_REF][START_REF] Beghi | Temperature variations around the crack tip during fracture test[END_REF][START_REF] Harvey | Effects of sequential cyclic and monotonic loadings on damage accumulation in nickel 270[END_REF] but it is the dissipation before the crack initiation which was investigated. Often, the purpose was to get a rapid estimation of the fatigue limit by recording the temperature during the beginning of the test on fatigue machines working at low frequency. To further in number of cycles, the duration of the tests was prohibitive.

In this study, we investigate the temperature evolution measured by an advanced infrared imaging system on various materials between 10 5 and 10 8 cycles on a piezoelectric fatigue system at the frequency of 20 000 Hz. In this case, the duration of the test is compatible with the storage of the data in the camera and allows the recording of the temperature during the crack propagation. 1). The cast aluminium alloys showed a characteristic dentritic microstructure with porosities and the low alloyed steels a martensitic microstructure. The experimental mechanical properties of these alloys and the fatigue test conditions (stress amplitude) of the materials are given in Table 1. These alloys have been tested at 20 kHz using a piezoelectric fatigue system with different R ratio (Table 1). The specimens from AISI 4240 were prepared to be transversal (T) with respect to the rolling direction.

Experimental fatigue tests

From an historical point of view, the first ultrasonic fatigue machine was constructed in 1950 by Mason. [START_REF] Mason | Piezoelectronic Crystals and their Application[END_REF] With the development of computer techniques, C. Bathias and co-workers [START_REF] Wu | Automatic in Ultrasonic Fatigue Machine to Study Low Crack Growth at Room and High Temperature[END_REF][START_REF] Bathias | Determination of Fatigue Limit between 10 5 and 10 9 cycles using an Ultrasonic Fatigue Device[END_REF] have built a fully computer-controlled piezoelectric fatigue machine working at 20 ± 0.5 kHz. Figure 1 shows the principal aspect of this machine in which the following three components are present:

-A power generator that transforms 50 or 60 Hz voltage signal into ultrasonic 20 kHz electrical sinusoidal signals. -A piezoelectric (or magnetostrictive) transducer excited by the power generator, which transforms the electrical signal into longitudinal ultrasonic waves and mechanical vibration of the same frequency. -An ultrasonic horn (in titanium alloy, portion EDC in Fig. 1) that amplifies the vibration coming from the transducer in order to obtain the required strain amplitude in the middle section of the specimen. This amplifier maintains automatically the intrinsic frequency of the mechanical system in the range 19.5-20.5 kHz. Depending on the specimen loading, the horn is designed so that the displacement is amplified between E and C usually three to nine times. It means that the geometry of the horn must be determined. The finite element method may be required when the geometrical shape is complex.

The resonant length of the specimen and concentrator is calculated using Finite Element Method (FEM) by Ansys code. The converter, horn and specimen compose a mechanical vibration system where there are four stress nodes (null stress) and three displacement nodes (null displacement for an intrinsic frequency of 20 kHz) (Fig. 1). During the test, these nodes don't move and remain around at the same position and even damage occurs, because the crack remains small compared to the specimen size.

In our machine, in order to keep the stress constant, during the test, via computer control, there is:

-a linear relation between the electric potential and the dynamic displacement amplitude of the ceramic. -a linear relation between dynamic displacement amplitude of the ceramic and displacement amplitude of the horn. -a linear relation between displacement amplitude of the horn and electric potential.

The dynamic displacement amplitude at the horn extremity and at the specimen extremity is measured by an optic fibre sensor, which permits measurements of the displacement from 1 to 199.9 μm, with a resolution of 0.1 μm. During ultrasonic fatigue tests, the maximum strain values can be measured directly using miniature strain gauges, suitably positioned on the sample surface.

The macroscopic pure homogeneous elastic behaviour during the test (except the case of plastic instabilities dues to phase transformations . . . , the change of yield stress during the test . . .) allows the determination of the dynamic Young's modulus of the tested material. To avoid this use of a load sensor, once the knowledge of the dynamic modulus is achieved, the stress in the mid-section of the specimen is computed from the displacement of the piezo-ceramics system after calibration. When a crack initiates, the dynamic modulus decreases and the device cannot maintain the required frequency (19.5-20.5 kHz), and the test is automatically stopped when the frequency falls below 19.5 kHz. The uncertainty is given by the ratio 0.5/20 which corresponds to minus 2.5% of error.

Temperature measurement

In order to determine the temperature field on the surface of the specimen, a nonintrusive measurement technique by infrared pyrometry was chosen. In the test, an advanced, high-speed and high-sensitivity infrared camera from Cedip Infrared Systems (made up of a matrix of 320 × 240 Mercury Cadmium Telluride detectors) was used to record the temperature changes during ultrasonic fatigue tests. The spectral range of the camera lies between 3.7 and 4.9 μm. This method allows the visualization of temperature cartography with very good time resolution. In this study, the spatial resolution is 0.17 mm and the aperture time can vary between 10 μs (two cycles) and 1500 μs (around 30 cycles). The refresh time of the camera is between 0.83 and 100 Hz. It corresponds respectively to a noise of 0.4 K and 20 mK. The pyrometer was calibrated on a blackbody reference. In pyrometry, the error in temperature is often related to uncertainties on the emissivity of the surface. To eliminate this problem, the specimen surface was covered with a strongly emissive black paint. Figure 2 shows our experimental device.

R E S U L T S

Thermal results

Figures 3a and4a show the temperature variation at the centre of the specimen as a function of the number of cycles for different stress levels in the case of a cast aluminium-silicon alloy (AlSi7Mg06) and a low alloyed steel (AISI 4240). The result for the AISI 5120 is given in Ref. [6]. The global behaviour is the same for all studied alloys.

At the first time, we observe an increase of the temperature just at the beginning of the test. In the same material, the temperature increases depending on the maximum stress amplitude for a given number of cycles. It should be noted that this increase in temperature at the beginning of the test is followed by a stabilization corresponding to a balance between the mechanical energy dissipated and the energy lost by convection and radiation at the specimen surface and by conduction inside the specimen. The temperature variation depends on the material and on stress amplitude level ( σ ) for a same material. Table 2 gives the maximum temperature variation ( T) obtained just before fracture, and N f the number of cycles at fracture. The number of cycles at fracture was between 3 × 10 5 and 8 × 10 7 cycles, which is in the mega and gigacycle fatigue domain. An enlargement of the experimental results at the failure is represented in Figs 3b (corresponding to the test with σ = 52.5 MPa) and 4b. From the thermographies of the specimens recorded by the camera during the fatigue tests, the point with the highest temperature is found. It is at this point that the evolution of the temperature variation versus the number of cycles is drawn. The behaviour is the same for all studied materials. At the end of the test, there is a local and abrupt increase of the temperature.

It is shown that this rapid increase in temperature before failure is related to a very local increase of the temperature (thermographies of Figs 3c and4c). The waypoints marked on the graph in Figs 3b and 4b correspond to the captured pictures in Figs 3c and4c.

Fracture surface analysis

Scanning Electron Microscopy (SEM) was performed for fracture surface analysis in order to determine the origin and localization of the initiation (Table 2). In all materials, the initiation of the crack was initiated on inclusions (steels) or porosities (cast aluminium alloys). Figures 5 and6 show the crack initiation zone on the fracture surface in the cast aluminium alloy AlSi5Cu3Mg and the bearing steel AISI 5120 where the initiation was subsurface. In the steel, the initiation was on a non-metallic inclusion and leads to a typical fish-eye propagation whereas in the cast aluminium alloy, it is on a porosity. In this case, the fracture crack growth is circular around the porosity and produces like a fish eye.

D I S C U S S I O N

H. Mughrabi 2 has studied for many years the microstructural evolution of different materials during fatigue tests. He proposed to distinguish two kinds of materials which correspond at two extreme materials: the type I material is a ductile single-phase material without inclusions and the type II material is high-strength steel containing non-metallic inclusions. For the two types, the S-N curve can be described in a multi-stage W öhler-type S-N plot where there are four ranges characterized by: (I) the LCF (Low Cycle Fatigue domain) Coffin Manson range. (II) the PSB (Permanent Slip Bands) threshold related to the HCF (High Cycle Fatigue or mega cycle domain) plastic strain fatigue limit. (III) the transition from HCF limit to the UHCF range (gigacycle fatigue domain) (IV) the irreversibility threshold corresponding to the UHCF limit.

Whatever the fatigue range, glide of dislocations are observed in the matrix. The microstructural evolution is always the same and leads to local plastic deformation and thus to mechanical energy dissipated into heat. When inclusions (or porosities are present) in the case of type II materials, subsurface fatigue crack initiation at inclusions is dominant in the UHCF range III, 1,2 provided that the inclusion density lies below a critical value. [START_REF] Mughrabi | Specific features and mechanisms of fatigue in the ultrahigh-cycle regime[END_REF] It is now accepted that in this case, 1 the total lifespan is mainly consumed by the process of crack initiation. 4.0x10 [START_REF] Ranc | Study of thermal effects associated with crack propagation during very high cycle fatigue[END_REF] 6.0x10 6 8.0x10 6 1.0x10 7 1.2x10 7 1.4x10 [START_REF] Luong | Fatigue limit evaluation of metals using an infrared thermographic technique[END_REF] Cycle number

(4) (3)
(2)

(1) When the crack is initiated, two plastic zones exist at the root of the crack: [START_REF] Bathias | Fatigue des matériaux et des structures[END_REF] the monotonic plastic zone r y = K 2 /6πσ 2 y and the reverse (cyclic) plastic zone r R = K 2 /24πσ 2 y in which the plastic deformation is more in-tense (with σ y , the yield stress, σ y , cyclic yield stress and K, the stress intensity factor). When the crack initiates from a defect, such as inclusions or pores, it is said that probably a relation must exist between the fatigue limit and the crack growth threshold; 1,2,21 and the propagation of the crack in the fish eye can be modelled by the Paris-Hertzberg law [START_REF] Paris | The relationship of effective stress intensity factor, elastic modulus and Burger's-vector on fatigue crack growth as associated with "Fish Eye" gigacycle fatigue phenomena[END_REF] da/dN = b( K eff /Eb 1/2 ) 3 with E Young's modulus, b the Burger vector, K eff effective stress intensity factor.

In the first approach, the integration of this law without transition short crack-long crack from a o to a radius of the fish eye (Fig. 7) conducts [START_REF] Marines-Garcia | Fatigue crack growth from small to large cracks in gigacycle fatigue with fish-eyes failure[END_REF] to N prop = πE 2 /2 σ 2 with N prop number of cycles for the propagation of the fish eye, σ the experimental nominal stress and a o = a int /0.94 (a int , radius of the defect: inclusion, . . .).

In our experiments, the temperature fields on the specimen surface are measured by infrared thermography. Just before the fracture, thermographies show a significant and local increase in the temperature. In order to better understand these thermal effects and to make a connection with the initiation and the propagation of the fatigue crack, a thermo-mechanical model was developed for the bearing steel. [START_REF] Ranc | Study of thermal effects associated with crack propagation during very high cycle fatigue[END_REF] From the Paris-Hertzberg law, the evolution of crack length versus time can be obtained: a(t) = a 0 /(1-t/t c ) 2 with t c = 2a 0 /bf (with f the loading frequency). Besides, the energy dissipated at each cycle per unit of crack length noted ξ is proportional to the surface of the reverse plastic zone r R : ξ = ηr 2 R where η is a coefficient depending only on the material properties. By replacing the expression of r R , K and a(t), the dissipated power per unit length of crack P (P = ξ f ) can be calculated. The fatigue crack is modelled by a circular ring heat source located in the reverse plastic zone at the crack tip whose radius increases with time. From the heat transfer equation and considering adiabatic conditions on the surface of the specimen, and normalizing dimensions in the problem, the evolution of the non-dimensional temperature during the fatigue testing can be obtained. The numerical resolution of the thermal problem allows the determination of the evolution of the temperature field with time in the specimen. The comparison between test and model shows a good correlation (Fig. 8). In particular, the propagation duration of the crack is well estimated by the model.

So, the increase in temperature at the end of the experimental test corresponds to the fracture initiation, and the number of cycles at initiation can be determined accurately. The number of cycles at the crack initiation (Fig. 8) corresponds to a temperature increase of 0.07 • C (noise of camera after filtration of temperature signal for an aperture time of 10 μs).

For our materials, the results of the number of cycles at initiation N i and the ratio of N i over N f (number of cycles at fracture) is given in Table 3. These results confirm that in the gigacycle domain, more than 92% of the total life is devoted to the initiation of the crack. Figure 9 shows the relation between N i /N f versus N f , and the portion of the total life devoted to the initiation increases with the number of cycles to failure. The greater the number of cycles to fracture, the larger the part of total life for initiation of the crack. Consequently, the number of cycles for the propagation of the fish eye is weak and the comparison (Table 3) of the Paris model with our experiments is good.

C O N C L U S I O N

The final goal of this research is to understand the damage mechanism inside the metal, around defects, when the fatigue life reaches 10 9 . Of course, it is experimentally difficult to catch observations at 10 9 cycles. In order to approach this objective, the experimental study has been carried out for a fatigue range between 10 6 and 10 9 . Further to this, an analytical model has been derived from the Paris law.

A temperature measurement with an infrared camera was performed during fatigue tests on a piezoelectric fatigue machine on cast aluminium-silicon alloys and three low alloyed steels. Different stress levels were applied. At the beginning of the test, the temperature rapidly increases, followed by a stabilization. The higher the applied stress, the larger the increase in temperature and the more significant the energy dissipated. At the end of the test, the temperature increases very rapidly until the fracture. The numerical resolution of the thermal problem allows the determination of the evolution of the temperature field with time in the specimen. The comparison between test and model shows a good correlation, and the sudden increase of the temperature allows to determine the cycle number at crack initiation and it proves that more than 92% of the total life is devoted to the initiation of the crack. The portion of the total life devoted to the initiation increases with the number of cycles to failure. When the initiation is subsurface and leads to a fish-eye formation, the simplified Paris model allows a prediction of the number of cycles to fish eye formation. To conclude, it is proven that the key damage in VHCF is initiation and not slow crack propagation.

Fig. 1

 1 Fig. 1 Scheme of the piezoelectric fatigue machine.

Fig. 2

 2 Fig. 2 Thermal measurement device.

FractureFig. 3

 3 Fig. 3 Evolution of the temperature variation for the AlSi7 Mg06 alloy.

Fig. 4

 4 Fig. 4 Evolution of the temperature variation during the test for AISI 4240 steel (T = 610 • C, σ = 345 MPa).

Fig. 5

 5 Fig. 5 (a) Crack initiation zone in the AlSi5Cu3 Mg alloy. (b) Detail of the crack initiation zone in the AlSi5Cu3 Mg alloy.

Fig. 6 Fig. 7

 67 Fig. 6 (a) Fracture surface of the AISI 5120 steel (with fish eye). (b) Non-metallic inclusion in the fish eye in the AISI 5120 steel 6 .

Fig. 8

 8 Fig.8Comparison between model and experiment for crack initiation detection (AISI 5120)[START_REF] Ranc | Study of thermal effects associated with crack propagation during very high cycle fatigue[END_REF] .

Fig. 9 N

 9 Fig. 9 N i /N f evolution versus N f for the studied materials.

Table 1

 1 Mechanical properties and fatigue test conditions

		UTS		Stress
	Alloy	(MPa)	R ratio	amplitude (MPa)
	AlSi7Mg06	344	0.01	52.5
				50
				45
	AlSi5Cu3Mg	222	-1	5 5
	AISI 4240 (T = 430 • C)	1508	-1	460
	AISI 4240 (T = 610 • C)	1035	-1	345
	AISI 5120	2000	-1	335
	E X P E R I M E N T A L C O N D I T I O N S
	Materials
	Two cast aluminium-silicon alloys (an AlSi7Mg06 and an
	AlSi5Cu3Mg) and three low alloyed steels (two 42CrMo4
	or AISI 4240 and a bearing steel AISI 5120) were studied.
	The aluminium alloys differed by the composition (silicon
	amount . . .), and the two AISI 4240 steels by the heat
	treatment. Both the low alloyed AISI 4240 steels were
	austenized at 830 • C during 30 min, oil quenched but one
	was tempered during 1 h at 430 • C and the second one at
	610 • C.
	The differences result in various Yield Stress and Ul-
	timate Tensile Stress (Table

Table 2

 2 Results of fatigue tests

		σ	T		Locali-
	Alloy	(MPa) ( • C) N f	zation
	AlSi7Mg06	52.5	5.2 8.6932 × 10 6 Surface
		50	3.8 8.3276 × 10 6
		45	1.3 1.25132 × 10 7
	AlSi5Cu3Mg	50	14	8.11392 × 10 7 Subsurface
	AISI 4240 (T = 430 • C) 460	28	3.3094 × 10 5 Surface
	AISI 4240 (T = 610 • C) 345	80	1.76197 × 10 7 Surface
	AISI 5120	335	230 8.3700 × 10 7 Subsurface

Table 3

 3 Number of cycles at initiation and comparison with Paris' model for fish-eye propagation

	Alloy	N i	N f	N i /N f	Localization	N f -N i	N prop
	AlSi7Mg06	8.1010 × 10 6	8.6932 × 10 6	0.932	Surface	5.92 × 10 5	-
		8.0826 × 10 6	8.3276 × 10 6	0.971	Surface	2.45 × 10 5	-
		1.15310 × 10 7	1.25132 × 10 7	0.922	Surface	9.82 × 10 5	-
	AlSi5Cu3Mg	7.89184 × 10 7	8.11392 × 10 7	0.973	Fish eye	2.23 × 10 6	2.73 × 10 6
	AISI 4240 (T = 430 • C)	3.05 × 10 5	3.3094 × 10 5	0.922	Surface	2.6 × 10 4	-
	AISI 4240 (T = 610 • C)	1.76155 × 10 7	1.76197 × 10 7	0.998	Surface	4.2 × 10 3	-
	AISI 5120	8.3620 × 10 7	8.3700 × 10 7	0.999	Fish eye	0.8 × 10 5	6.16 × 10 5
	E AlSi5Cu3Mg = 72560 MPa, E AISI5120 = 210000 MPa
			Evolution Ni/Nf en fct de Nf
		1,1
		1
		0,9
	Ni/Nf	0,8
		0,7
		0,6
		0,5
		1,E+05	1,E+06	1,E+07	1,E+08	1,E+09
				Nf