D Barbier

N Gey

S Allain

N Bozzolo

M Humbert

Analysis of the tensile behavior of a TWIP steel based on the texture and microstructure evolutions

Keywords:

The texture and microstructure evolutions of a fine-grained TWIP steel subjected to tensile tests at room temperature were investigated in relation to the mechanical behavior. This steel combines both high ductility and strength owing to the TWIP effect. Also the steel exhibits a high strain hardening rate that evolves according to five stages, which are related to the microstructure and texture evolutions and characteristics. The formation of nano-twins in the initial stage of deformation leads to an increase in strain hardening rate. The development of the pronounced <1 1 1> fiber in the tensile direction sustains mechanical twinning and maintains the strain hardening rate on a high level. The resulting microstructure exhibits several types of twin configurations and sub-boundaries with high misorientations due to intense activities of dislocation glide. The twin volume fraction was estimated to be 9% at the final stage of tensile deformation. The new orientations generated by mechanical twinning do not change considerably the final texture.

Introduction

Since few years, the ongoing research of the steel industry mainly focuses on high strength steels with excellent formability. This optimal combination is particularly attractive for automotive applications such as structural reinforcements and energy absorption parts. The increase in strength enables to reduce the mass of the car bodies whereas the increase in ductility admits complex component design. Among the wide variety of recently developed steels, high manganese austenitic steels with low stacking fault energy (SFE) are particularly promising, especially when mechanical twinning occurs [1,[START_REF] Scott | Proceeding of the MS&T'05 Conference[END_REF]. Outstanding mechanical properties combining a high ductility and high strength are then obtained. This compromise often observed in low SFE fcc materials results from a high strain hardening rate. Asgari et al. [START_REF] Asgari | [END_REF] reported the evolutions of the strain hardening rate according to different stages independent of the grain size in low SFE materials and analyzed each stage in relation to the microstructure evolution.

In Fe-Mn-C austenitic steels, mechanisms responsible for the high strain hardening rate are still under discussion [4][5][6]. Some authors attribute the high strain hardening rate to dynamic strain aging (DSA) mechanisms, interpreted as the interaction between C-Mn bonds and mobile dislocations [4,7,8]. Bracke et al. [9] suppressed the DSA in a Fe-22Mn-0.6C (composition given in mass percent) steels by substituting carbon by nitrogen, while keeping the SFE constant. The resultant mechanical properties were the same. Mechanical twinning can also be a dominant mechanism to increase the strain hardening rate. As the strain increases, the occurrence of twins reduces the mean free path of dislocations. This is called TWinning Induced Plasticity (TWIP) effect that can be interpreted as a dynamical Hall-Petch effect [10][11][12][START_REF] Allain | Proceeding of the International Conference on TRIP-Aided High Strength Ferrous Alloys[END_REF][START_REF] Allain | [END_REF]. Allain [START_REF] Allain | Proceeding of the International Conference on TRIP-Aided High Strength Ferrous Alloys[END_REF][START_REF] Allain | [END_REF] suggested that the high strain hardening rate of Fe-Mn-C TWIP steels is mainly due to mechanical twinning. More recently, Bouaziz et al. [START_REF] Bouaziz | [END_REF] have shown that the strain hardening rate in the Fe-22Mn-0.6C steel is mainly of kinematical origin which is linked to the TWIP effect.

Only few studies are related to the microstructure evolution during the deformation of Fe-Mn-C TWIP steels. The microstructure has been characterized either by optical microscopy (low spatial resolution) or by transmission electron microscopy investigations giving accurate local data but which are insufficient for statistical deductions [START_REF] Allain | Proceeding of the International Conference on TRIP-Aided High Strength Ferrous Alloys[END_REF]16]. These studies have shown that the austenite grains contain bundles of nano-twins with different configurations [10][11][12][START_REF] Allain | Proceeding of the International Conference on TRIP-Aided High Strength Ferrous Alloys[END_REF]. Up to now, few attention has been addressed to the corresponding texture evolution and its interaction with twinning [16,17]. These parameters are the key to better understand the high strain hardening rate of TWIP steels.

The aim of this research is to provide a detailed description of the deformation mechanisms of a Fe-Mn-C TWIP steel deformed in tension. In particular the dependence of twinning on the grain size and grain orientation is analyzed. For this purpose, the microstructure and texture evolutions have been characterized in great detail by performing X-ray diffraction, EBSD with a FEG-SEM and TEM. Interactions between twinning and texture evolution are analyzed. Moreover we estimated the twin volume fraction from its effect on the texture formation. Finally, the transitions of subsequent strain hardening stages were analyzed in relation to microstructure and texture features.

Experimental procedures

As-received material

A fine-grained Fe-Mn-C TWIP steel was supplied by Arcelor-Mittal Research SA Maizières as a 1.3 mm thick sheet. The alloy designation of the steel is 22Mn-0.6C (composition in mass percent, balance iron). The high manganese content stabilizes the austenitic phase at room temperature and decreases the SFE to about 20 mJ m -2 [START_REF] Scott | Proceeding of the MS&T'05 Conference[END_REF]18]. Consequently, mechanical twinning occurs during deformation at room temperature [START_REF] Scott | Proceeding of the MS&T'05 Conference[END_REF]18]. At lower temperature, as the SFE is decreased, ε martensite is formed.

The initial microstructure shows equiaxed ␥ grains with an average size of 2.6 m. However, the grain size varies within a large range from about 0.5-15 m. Grains larger than 5 m are less numerous (about 10%) but they represent 40% of the material surface. The initial texture is rather weak with a texture index (TI) value of 1.23, as shown in Fig. 2a.

Tensile tests

Tensile tests were conducted at room temperature by applying a strain rate of 10 -3 s -1 (quasi-static loading conditions). The tensile samples were cut out of the sheet parallel to the transverse direction (TD). The gauge length was 80 mm and the width 20 mm, respectively. One tensile test was conducted up to the rupture. Six others tensile samples were uniformly strained with 0.02, 0.05, 0.10, 0.20, 0.30 and 0.40 true strain. The true stress and inelastic strain were calculated and the normalized strain hardening rate, defined as (d/dε)/G 0 , as well. G 0 is the shear modulus at room temperature which was determined to be 62 GPa according to the elasticity relation G 0 = E/2(1 + ). The Young modulus E and the Poisson coefficient were determined experimentally. The strain hardening rate was calculated from the experimental true stress true strain curve, fitted with a 7th order polynomial function.

Microstructure and texture characterization

The EBSD technique was used to analyze the local texture in relation to the microstructure. For this purpose, high resolution orientation maps were acquired in a 6500F JEOL field emission gun-scanning electron microscope (FEG-SEM) equipped with the Channel 5 EBSD system (HKL Technology). Orientation maps with a beam step between 20 and 70 nm were taken from a large number of grains of the as strained samples. They were at first prepared by standard grinding and polishing. An additional surface treatment by using ion milling facility (BalTec RES 010) improved the quality of the Kikuchi patterns of the deformed samples with more than 0.3 true strain. Despite this careful preparation, the Kikuchi pattern indexing rate decreased from 95% for the undeformed to 65% for the samples deformed at 0.55 true strain.

The results of the EBSD scans are displayed as Kikuchi pattern quality (KPQ) maps and additional orientation maps. These reveal the local texture, the intra-granular misorientation gradients induced by the deformation and interesting details of the microstructure such as grain and twin boundaries [19], respectively. The orientation coloring code used in this paper is given by the standard triangle of the inverse pole figure (IPF) with TD projected in the cubic reference frame: grains misoriented by less than 20 • with respect to the <1 0 0>, <1 1 0>, <1 1 1>//TD fiber orientations are red, green and blue, respectively.

To analyze the intra-granular misorientations produced by the deformation, the grain orientation spread was quantified as follows: (1) pixels belonging to the same grain are detected with a misorientation criterion of 3 • applied from one pixel to the next, (2) the mean grain orientation is calculated and finally (3) the average misorientation between each pixel orientation and the mean grain orientation are deduced. This average misorientation characterizes the long range orientation spread and is called internal orientation spread (IOS). This parameter is grain size dependant.

The microstructure was also investigated in a FEI CM 200 TEM to take advantage of a higher spatial resolution. The samples were mechanically polished to 80 m in thickness and the TEM foils were electrolytically thinned to electron transparency by using TENUPOL thinning facility and a 5 vol.% perchloric acid/95 vol.% acetic acid solution at 30 V, DC. In order to check that the thin foil preparation did not induce the formation of artifact dislocations and mechanical twins, the as-received material was also prepared for TEM examination.

The deformation textures were determined from the {2 0 0}, {2 2 0} and {1 1 1} pole figures. These were measured by X-ray diffraction (XRD) using a Siemens D5000 goniometer (Cr K ␣ radiation). The experimental data were processed to obtain the orientation distribution function f(g) (ODF) [START_REF] Bunge | Texture Analysis in Material Science[END_REF]. The orthorhombic symmetry, clearly displayed by the pole figures, was imposed for the calculation of the ODF. The TI defined as TI = f (g) 2 dg describes quantitatively the texture sharpness. In this paper, the texture is represented by the {1 0 0} and {1 1 1} pole figures and the ϕ 2 = 45 • ODF section in the Euler space.

The main texture components are described according to the {khl}<uvw> notation ({khl} being the plane parallel to the rolling plane and <uvw> being the direction parallel to the rolling direction at 90 • from the tensile direction. The volume fraction of each texture component was calculated by taking a tolerance of 10 • around the strict orientation into account. The largest Schmid factors (SF) for slip and for twinning were calculated for each texture component as presented in Table 1. The {1 1 1}<1 1 0> slip systems and the {1 1 1}<1 1 2> twinning systems were considered.

Results

Mechanical behavior

The Fe-Mn-C steel exhibits outstanding mechanical properties combining high ductility and extraordinary strength. The ultimate tensile strength amounts to 1000 MPa and more and the elongation to rupture is over 70%. The true stress and the normalized strain hardening rate are plotted versus the true strain in Fig. 1a andb, respectively. Markers indicating each interrupted tensile tests are attached on both curves. Serrations on the stress-strain curve were observed. These are increasing at large deformation and consequently the calculation of the strain hardening rate coefficient is limited at 0.40 true strain. The occurrence of serrations on true stress-strain curves of high manganese austenitic steels is well known. They have been interpreted as dynamic strain aging mechanism or as 'Portevin-Le Chatelier' effect [4,5]. However, the exact nature of the DSA in austenitic steels, i.e. interaction between mobile dislocations and carbon atoms, is not well defined in the literature and is still under discussion [4][5][6].

A remarkably high strain hardening rate was observed and is displayed in Fig. 1b. The amount of (d /dε)/G 0 is greater than 0.04 up to 0.3 true strain. During the deformation, the strain hardening rate evolves according to different stages. After a pronounced decrease up to (d /dε)/G 0 = 0.045, from 0 to 0.02 true strain (stage A), the strain hardening rate increases to reach the first constant level ((d /dε)/G 0 = 0.048) at 0.1 true strain (stage B). Then, it slightly decreases (stage C) and reaches the second constant stage at 0.2 true

Table 1

Main texture components: {hkl}<uvw> notation, Euler angles in the ϕ2 = 45 • section (Bunge convention [START_REF] Bunge | Texture Analysis in Material Science[END_REF]), corresponding fiber to which they belong, color in the transverse direction inverse pole figure TD-IPF triangle, corresponding material state, maximal value of the Schmid factor calculated for the {1 1 0}<1 1 1> slip systems and {1 strain ((d /dε)/G 0 = 0.043) up to 0.3 true strain (stage D). Finally, over 0.3 true strain, the strain hardening rate decreases again (stage E) up to the specimen rupture. This sustained high level of strain hardening rate, observed up to large strains, confers high strength and ductility to this fine-grained Fe-Mn-C TWIP steel.

Texture evolution

Tensile deformation along the prior sheet transverse direction TD leads to the formation of a sharp texture characterized by four distinct texture components: the {1 1 0}<1 1 2> brass, the {1 1 2}<1 1 0> rotated copper, the {1 1 0}<1 1 0> rotated Goss and the {0 0 1}<1 0 0> cube. Fig. 2a-d shows the {1 0 0} and {1 1 1} pole figures and the ϕ 2 = 45 • ODF sections for the as-received material -as cold rolled and recrystallized -and for the samples deformed at 0.2, 0.3 and 0.55 true strain. The horizontal direction of the pole figures corresponds to the tensile direction.

At 0.55 true strain, the highest orientation densities are reached for the brass component (maximal value of 15) and for the rotated copper component (maximal value of 7.6). The rotated Goss and the cube components are weaker with respectively maximum ODF values of 1.8 and 2.3. On the pole figures, a pronounced density of <1 1 1> poles and a lower density of <1 0 0> poles are observed. In fact, the major brass and rotated copper components belong to the pronounced <1 1 1>//TD fiber orientation and the minor rotated Goss and cube components to the weak <1 0 0>//TD fiber orientation. The main texture components presented in Fig. 2 are listed in Table 1. This texture sharpens stepwise as underlined by the evolution of the texture index TI (Fig. 3) and the volume fraction of each texture component (Fig. 4).

Up to 0.2 true strain, the texture remains very weak. The TI is about 1.5. However, the texture components change significantly during this deformation step. The ODF maximum shifts from the Goss position (90 • , 90 • , 45 • ) to the brass position (55 • , 90 • , 45 • ). The deformation components progressively increase at the expense of the initial Goss and copper components of the as-received material. The volume fraction of grains with brass orientation (8% in the asreceived material) enlarges with increasing the deformation.

From 0.2 to 0.3 true strain, the texture significantly sharpens. The TI increases from 1.5 to about 3. In this range of deformation, the development of the texture components induced by the deformation is very strong. The increase of the <1 1 1>//TD fiber is very important in comparison to the one of the <1 0 0>//TD fiber. At 0.3 true strain, 45 vol.% of the grains are oriented close to <1 1 1>//TD fiber and only 10 vol.% are close to <1 0 0>//TD fiber (Fig. 4). Copper and Goss components progressively disappear (corresponding volume fraction in green in Fig. 4).

At 0.3 true strain, the deformation texture is almost fully developed. Beyond this deformation, the TI increases slowly. However, the brass and rotated copper components continue to strengthen, as it can be seen on Fig. 2d. Moreover, new orientations are generated by mechanical twinning, which may influence the texture evolution. This point will be discussed in Section 4.

Microstructure analysis

Low strain regions (ε ≤ 0.1)

In the first steps of the deformation, the grains show twins in addition to numerous dislocations. The twinning activity depends more on the grain size than on the grain orientation.

Primary twins were observed in the samples strained at 0.02 true strain as illustrated in TEM micrographs of Fig. 5a. Thus the onset of twinning coincides with the increase of strain hardening rate at 0.02 true strain. Some grains have already activated two twin systems, see Fig. 5b. The corresponding selected area electron diffraction (SAED) and indexed patterns are shown in Fig. 5c andd. In order to improve the bright field contrast and better see * direction allows the identification of the diffraction spots associated with the twins on the ( 11 1) plane. On the SAED pattern of Fig. 5c, the spots corresponding to each twin systems are labeled TW1 and TW2, respectively. The twin thickness is about 10-40 nm. For these low deformations levels, the grains also present numerous dislocation densities (see Fig. 5e,f). Fig. 6 shows the results of the EBSD analysis performed on the sample strained at 0.1 true strain. The twin thickness is smaller than the resolution of the EBSD map performed with a step of 60 nm. Their orientations cannot be measured by automated EBSD analysis. However, the Kikuchi pattern quality (KPQ) map reveals the presence of straight dark lines within some grains in Fig. 6a. We checked that these low KPQ values are due to two patterns overlapping (twin and matrix). Thus, these lines reveal the deformation twins. Half of the grains present one visible twinning system.

The TD inverse pole figure map of Fig. 6b confirms the weak texture previously observed in Fig. 2. The activated strain mechanisms and the related grain rotations depend on the grain size and its initial orientation. For example, the larger grains close to the Goss and copper orientations, in green in Fig. 6b, preferentially activate twins. These grains, which were already observed in the as-received material, are indeed well oriented for twinning according to the Schmid factor as depicted in Table 1. With increasing strain, these grains rotate towards the deformation components and finally the Goss and copper components disappear at 0.3 true strain as displayed in Fig. 4. The orientations corresponding to the <1 1 1>//TD and <1 0 0>//TD fibers (blue and red colored grains on Fig. 6b, respectively), appear during deformation by dislocation glides. In addition, twins are also observed in these larger grains.

The intra-granular misorientations resulting from the deformation were analyzed. At 0.1 true strain, about 10% of the grains have an internal orientation spread larger than 5 • . The cumulative misorientation profile along a line has also been drawn for different grains. Fig. 6c presents the profile along the thick black line in Fig. 6b. Over a distance of 4 m, the cumulated misorientation reaches 12 • (up to 20 • in some grains). However, the misorientation between adjacent pixels does not exceed 1 • . Such intra-granular misorientations are associated to the formation of dislocations which assure the plastic strain compatibility between neighboring grains and accommodate the deformation gradients [START_REF] Verbeken | [END_REF]22].

Medium strain regions (

0.1 ≤ ε ≤ 0.3)
Between 0.2 and 0.3 true strain significant changes in the twinning activity are observed along with the texture evolution. The main microstructural features are the formation of different twin configurations with the occurrence of secondary twin systems in most of the grains, the strong interaction between grain orientation and twinning activity and the development of high misorientation boundaries within the grains.

TEM analysis shows that twins are organized in bundles. With increasing deformation, the number of twins in bundles increases and the twin thickness remains constant. Fig. 7 presents a typical EBSD map of a sample strained at 0.3 true strain. As the twin density in each bundle has increased, it is possible to measure the twin orientations by automated EBSD analysis. In fact, the twin orientation becomes major in the probed volume. But the twin thickness deduced by EBSD is overestimated. The austenitic matrix between the twins cannot be differentiated. The maximum thickness of the twin bundles is about 250 nm.

In this range of deformation, the number of grains with a secondary twinning system significantly increases, as shown in the EBSD KPQ maps. Fig. 8 illustrates the three main configurations observed in a sample deformed at 0.2 true strain. In grain A, the twinning systems were simultaneously activated. In grain B, two systems were activated in different regions of the grain. In grain C, they were sequentially activated. Similar twin configurations were found in a sheet having an average grain size of 20 m [START_REF] Allain | Proceeding of the International Conference on TRIP-Aided High Strength Ferrous Alloys[END_REF].

The texture evolution and its strengthening around the main texture components, described in Section 3.2, is clearly revealed by the TD-IPF map in Fig. 7b. The majority of grains are blue indicating that they are misoriented by less than 20 • from the <1 1 1>//TD fiber including the brass and rotated copper components. A strong interaction between grain orientation and twinning activity is observed. Grains from the <1 1 1>//TD fiber exhibit a higher fraction of deformation twins than grains belonging to the <1 0 0>//TD fiber. A small volume fraction of grains exhibits the copper and Goss components (<1 1 0>//TD fiber) and has a well developed twin microstructure.

Another feature is the development of medium to high angle boundaries within some deformed grains. This is illustrated by the green/blue colored grain in Fig. 9 (after 0.3 true strain, marked by a white star in Fig. 7). The KPQ map of Fig. 9a reveals the development of sub-boundaries within the grain. The misorientation across these sub-boundaries is about 10 • . The sub-boundaries seem to delimit different twin bundles as shown in Fig. 9b. This leads to the grain fragmentation. The sub-grains gradually rotate towards the deformation components and could be considered as new grains after a certain amount of deformation. The decrease in the indexing rate to 75-80% is caused by the strong lattice distortions and the increase of the boundaries density and broadening. The intra-granular misorientations are also increasing. About 50% of the grains have an IOS greater than 5 • (most of these grains have an IOS between 5 • and 10 • ). This IOS is due to the formation of highly misoriented sub-boundaries and to the dislocation densities leading to the elastic lattice distortion.

Large strain regions including rupture (0.3 ≤ ε ≤ 0.55)

In the final deformation step, the volume fraction of twins still increases. However, the overall twin volume fraction remains low and the twin orientations represent a small contribution to the macroscopic texture. Large regions with numerous dislocation accumulations develop within the grains. These features are illustrated in the EBSD maps of Fig. 10 of the strained sample at 0.4 true strain. Beyond a strain of 0.3 true strain, new deformation twins are formed within the grains. The KPQ maps of the Figs. 7a and10a are showing evidence of an increase in the twinning activity for strains from 0.3 to 0.4. The number of twins in the bundles increases leading to thicker bundles and therefore to larger volume fractions of twins. These bundles are about 500 nm large in some cases. On the EBSD map, these appear as large twinned domains.

At this deformation level, most of the grains are oriented close to the texture components as shown in Fig. 10b. The grains with orientations close to the initial Goss and copper components have disappeared. The ODF shows an orientation density increase around the brass and rotated copper components. The strong relation between the grain orientation and the twinning activity is still observed. Grains of the <1 1 1>//TD fiber exhibit a larger volume fraction of twins than those of the <1 0 0>//TD fiber.

On the sample strained to 0.4, the twin bundles are large enough to measure the twin orientations. Twins formed in the <1 1 1>//TD oriented grains reinforce the <1 0 0>//TD fiber whereas twins within the <1 0 0>//TD oriented grains create orientations in the <1 1 1>//TD fiber. The strict twin orientations generated by each main texture component have been calculated. Fig. 11(a-d) shows in the ODF section at ϕ 2 = 45 • the twin orientations that are related to the largest Schmid factor. These twins are correspondingly observed on the EBSD map. The twin and grain orientations do not overlap. The same calculation has been performed considering a set of grain orientations dispersed around the ideal texture component as presented in Fig. 11e andf. In this case, the deformed grain and twin orientations overlap. It can be deduced that no new texture component appears by twinning. The EBSD map indicates that the volume fraction of grains oriented close to <1 0 0>//TD fiber decreases slightly from 0.3 to 0.4 true strain, see Figs. 7 and10. XRD data indicate the volume fractions of these orientations increase as shown in Fig. 4. This evolution results from the increase of the twin volume fraction in grains oriented close to <1 1 1>//TD fiber. This aspect is detailed in Section 4 where the volume fraction of twins will be evaluated.

The EBSD map Fig. 10b reveals large intra-and inter-granular areas where no diffraction patterns were indexed. These correspond to sets of pixels with very low KPQ index appearing in dark grey color in Fig. 10a. The decrease of the indexing rate to 70% is correlated with the development of high deformation regions which could not be indexed because of the strong lattice distortion induced by accumulation of dislocations densities. As interpreted by Allain [START_REF] Allain | [END_REF], this indicates a pronounce localization of deformation such as local necking which causes the failure of the tensile sample.

Discussion

Analysis of the texture evolution: interaction with twinning

Tensile testing along the TD generates a sharp austenitic texture in the fine-grained Fe-Mn-C TWIP steel. This deformation texture is characterized by four components: the {1 1 0}<1 1 2> brass, the {1 1 2}<1 1 0> rotated copper, the {1 1 0}<1 1 0> rotated Goss and the {0 0 1}<1 0 0> cube. These components belong either to the pronounced <1 1 1>//TD fiber (brass and rotated copper) or to the weak <1 0 0>//TD fiber (rotated Goss and cube) as shown in Figs. 2 and4. Similar densities of <1 1 1> and <1 0 0> poles in the tensile direction have been observed on TWIP steels [17] or more generally on austenitic steels with low SFE [23,24]. The densities of these components increase by specific grain rotations related to the slip activity. In TWIP steels the occurrence of twinning influences the slip activity, which may modify the volume fraction of the different texture components. This was shown by simulation of texture evolutions with or without taking twinning into account [START_REF] Barbier | Proceeding of Materiaux 2006[END_REF].

In the present work, the texture of the as-received TWIP steel was rather weak with grain orientations spread around the Goss and copper texture components. Reinforcement of these orientation densities is not favored by deformation and consequently large grain rotations are required to develop the final texture components. At 0.2 true strain, the main characteristics of the initial texture components disappear without any significant texture strengthening. At 0.3 true strain, the major <1 1 1>//TD fiber is reinforced, whereas the other texture components show a moderate increase. Further deformation up to 0.55 causes an intensification of the texture components.

The analysis of EBSD data shows that this texture evolution strongly influences the twinning activity. This influence obviously occurs at two steps of deformation. Twins appear in the first deformation stage, see Figs. 5 and6. Initial twinning activity in this fine-grained steel is related to the large grain size distribution and to the Goss and copper orientations which belong to the weak initial texture components. These orientations are in fact favorable for twinning, see the Schmid factor values in Table 1. With increasing deformation the development of the major <1 1 1>//TD fiber still promotes twinning as illustrated in Fig. 10. In fact, the main <1 1 1>//TD fiber is well suited for twinning (Schmid factor twin = 0.31 and Schmid factor slip = 0.27, see Table 1) while the weak <1 0 0>//TD fiber is adequate to slip (Schmid factor slip = 0.41 and Schmid factor twin = 0.23). The volume fraction of grains of the <1 1 1>//TD fiber steadily increases during the deformation, as shown in Fig. 4. At 0.3 true strain, the majority of grains belongs to the <1 1 1>//TD fiber. The intensity increase of the main texture components promotes the twinning activity for strains between 0.3 and 0.4 as illustrated in Figs. 7 and 10 in conformity with ref. [17].

However, twinning generates new orientations and may modify the texture and its evolution. After a strain of 0.3, the twin density in bundles is large enough so that the twin orientations can be measured by using EBSD. The results are showed in Figs. 7 and10. Due to the spread of the experimental orientations around the texture components, grain and twin orientations often overlap. Twins formed within the <1 1 0>//TD oriented grains (at low deformation levels) or within the <1 1 1>//TD oriented grains (at intermediate and high deformation levels) reinforced the <1 0 0>//TD fiber. The limited number of twins formed within the grain of the <1 0 0>//TD fiber orientation reinforces the <1 1 1>//TD fiber. As a consequence, no new texture components are generated by twinning.

As a matter of fact, the analysis of XRD and EBSD data indicates that the contribution of twin orientations to the texture evolution is limited. XRD data show that the volume fraction of grains and twins oriented in <1 0 0>//TD fiber slightly increases for strains in the range from 0.3 to 0.4, as shown in Fig. 4. Besides that the volume fraction estimated from EBSD of the individual grains decreases. The difference between both approaches suggests that the overall increase of this texture component is mainly related to twinning. Several authors report a small influence of twinning on the texture evolution [START_REF] Leffers | [END_REF][START_REF] Leffers | Proceeding of ICOTOM[END_REF][START_REF] Duggan | [END_REF]. In these studies the volume fraction of twins generated during deformation in tension parallel to TD is not sufficient to significantly influence the texture.

The volume fraction of twins is an important parameter to understand the mechanical behavior -plasticity and strain hardening rate -of TWIP steels. However, it is difficult to estimate. The study of the literature shows that several authors have tried to quantify the twin volume fraction in low SFE fcc materials by optical microscopy [START_REF] Asgari | [END_REF]11,[START_REF] Allain | Proceeding of the International Conference on TRIP-Aided High Strength Ferrous Alloys[END_REF]] by using morphological criteria [29]. Such an evaluation only gives an estimation of the twin volume fraction.

In the present work the twin volume fraction is estimated by discerning the contributions of the grains and twins in the ODF thanks to the EBSD maps.

The EBSD analysis showed that most of the twins are oriented close to the <1 0 0>//TD fiber with certain spread, see Figs. 10 and11. However, the determination of the overall twin volume fraction is not directly possible. The volume fraction of the deformed grains of this fiber can be determined from EBSD measurements whereas the total volume fraction of twins and grains belonging to this fiber are obtained from the ODF via XRD measurements.

Thus, the difference ( ) between these two volume fractions corresponds to the volume fraction of twins.

increases from 6% to 9% between 0.4 and 0.55 true strain, respectively. These values are in agreement with the twin volume fraction estimated by Allain [START_REF] Allain | Proceeding of the International Conference on TRIP-Aided High Strength Ferrous Alloys[END_REF][START_REF] Allain | [END_REF] for a coarse-grained TWIP steel of the same chemical composition.

Microstructure evolution-influence on the strain hardening rate

TEM investigations show that the transition from stage A to stage B corresponds to the onset of twinning within grains deformed by multi-slip which is considered as necessary to form twins [START_REF] Nabarro | Dislocations in Solids[END_REF]. For this fine-grained steel, the required macroscopic stress for the onset of twinning is about 550 MPa and is reached at 0.02 true strain, see Fig. 1. Specific texture components of the weak initial texture are well suited for mechanical twinning. The onset of twinning causes an increase in strain hardening rate that reaches a plateau at 0.05 up to 0.1 true strain. The twins even in low volume fraction may reduce the effective grain size and act as strong obstacles for dislocation motions. In other low SFE materials, like MP35N (nickel-cobaltbased alloy) or ␣-brass [START_REF] Asgari | [END_REF][START_REF] El-Danaf | [END_REF][32][33], stage B is characterized by a uniform constant strain hardening rate. However, Rémy [11,12] and Mahajan and Chin [34] consider that twin-slip interactions increase the strain hardening rate. A similar increase corresponding to the onset of twinning was also observed in high purity alpha-titanium [35].

From 0.1 to 0.2 true strain, a weak decrease of the strain hardening rate occurs corresponding to stage C. In this range of deformation, twinning is less active as before. The texture remains very smooth. Grains exhibiting the Goss and copper orientations vanish between 0.1 and 0.2 true strain as illustrated in Fig. 4. In those grains the dislocations start to be accumulated and form sub-boundaries. In some grains, these sub-boundaries delimit twin bundles. This rearrangement of dislocations and the slowing down of twinning activity may be at the origin of the weak decrease of strain hardening rate. Asgari et al. [START_REF] Asgari | [END_REF] also suggest that the decrease of the strain hardening rate in low SFE fcc materials is related to the inhibition of twin formation. The twins generated in the initial stage of deformation will decrease the grain size and are acting as obstacles for twinning [36,37].

A constant strain hardening rate occur between 0.2 and 0.3 true strain (stage D). This is concomitant with two microstructural features. At 0.2 true strain the texture strengthens around the main brass and rotated copper components (<1 1 1>//TD fiber) and the weak rotated Goss and cube components (<1 0 0>//TD fiber). The number of grains with the <1 1 1>//TD fiber orientation is continuously increasing as shown in Fig. 4. Those grains are more adequate for twinning than those belonging to the <1 0 0>//TD fiber orientation, see Table 1. As the first twins have reduced the grain size a higher stress is necessary to generate new twins. For higher strains, a large number of grains exhibit a second twinning system. The formation of secondary twins reduces even more the mean free path of dislocations. These mechanisms maintain the strain hardening rate on a high level.

Finally, from 0.3 true strain to rupture, even if the twin volume fraction increases significantly, the strain hardening rate continuously decreases (stage E). The twin bundles become denser and thicker. Large intra-and inter-granular regions with numerous dislocation accumulations are developed, indicating a pronounced localization of the deformation which consequently tends to the sample failure.

Conclusion

The tensile behavior of a fine-grained Fe-22Mn-0.6C TWIP steel with an average grain size of 2.6 m has been analyzed with respect to texture and microstructure evolutions. The strong interaction between crystallographic slip, mechanical twinning and texture components is described and discussed for subsequent deformation steps.

Tensile testing along the TD produces a sharp texture with a pronounced <1 1 1>//TD fiber and a weak <1 0 0>//TD fiber, respectively. The deformation is accommodated by crystallographic slip and mechanical twinning. This provokes variations in the microstructure and consequently in the texture.

• Primary twins occur due to the presence of Goss and copper orientations of the weak initial texture. Twins are activated in the first stage of deformation at about 0.02 true strain and 550 MPa.

It is shown that with increasing deformation the development of the main <1 1 1>//TD fiber which is well suited for twinning reinforces the twinning activity. • Moreover, the deformation twins developed in grains having orientations close to the two main texture components -<1 1 1>//TD and <1 0 0>//TD fibers -generate orientations close to those orientations. Thus, mechanical twinning does not create additional orientations but increase slightly the density of the existing orientations.

A method to determine the volume fraction of the twins has been developed. This combines the global ODFs, determined by XRD, and microstructural information provided by EBSD. A true strain of 0.55 produced a volume fraction of twins estimated to be 9%.

Fig. 1 .

 1 Fig. 1. True stress true strain curve (a) and normalized strain hardening rate curve as a function of the true strain (b) of the Fe-22Mn-0.6C fine-grained TWIP steel.

Fig. 2 .

 2 Fig. 2. {1 0 0} and {1 1 1} pole figures and ϕ2 = 45 • ODF section at different true strain levels: as-received (a), 0.2 (b), 0.3 (c) and 0.55 (d).

Fig. 3 .

 3 Fig. 3. Evolution of the texture index TI = f(g) 2 dg as a function of the true strain.

Fig. 4 .

 4 Fig. 4. Evolution of the volume fraction of the main texture components (considering 10 • of tolerance) as a function of the true strain. The data are calculated from the ODF.

Fig. 5 .

 5 Fig. 5. TEM bright field images taken from samples strained to 0.02 (a) and 0.05 (b, e, f) true strain. First mechanical twins within the grains (a); two non coplanar twin systems (b); corresponding experimental (c) and simulated (d) SAED patterns of the grain in (b) (the spots corresponding to each twin systems are labeled TW1 and TW2, respectively); dislocations within the deformed grains (e and f). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

  the twins, the sample has been tilted a few degrees off the [0 1 1] zone axis. This is the reason why the diffracted intensities are not homogenously distributed. This SAED pattern gives evidence of the existence of two different twin systems. The first one, formed on the (11 1) plane, induces the diffraction spots represented in blue in Fig. 5d. The blue network is obtained by a 180 • rotation of the matrix reciprocal lattice around the [11 1] * reciprocal direction. A similar construction with [ 11 1]

Fig. 6 .

 6 Fig. 6. EBSD maps of sample strained to 0.1 true strain. KPQ map (a). TD-IPF map (b). Misorientation profile along the black line of figure (a) (c).

Fig. 7 .

 7 Fig. 7. EBSD maps of a strained sample to 0.3 true strain. KPQ map (a). TD-IPF map (b).

Fig. 8 .

 8 Fig. 8. KPQ maps of specific grains highlighting different twin configurations-sample strained to 0.2 true strain.

Fig. 9 .

 9 Fig. 9. EBSD representations related to the grain marked by the white star in Fig. 7; KQP map with the misorientation profile along the yellow line (a); KPQ map with specific misorientation lines and corresponding schematic drawings: twin boundaries in red-grain boundaries misoriented by less than 3 • in white and less than 10 • in green (b); TD-IPF map with the corresponding {1 0 0}, {1 1 0} and {1 1 1} pole figures (c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Fig. 10 .

 10 Fig. 10. EBSD maps of a strained sample to 0.4 true strain. KPQ map (a). TD-IPF map (b).

Fig. 11 .

 11 Fig. 11. Calculated twin and parent orientations represented in the ODF section at ϕ2 = 45• . For each parent orientation only one or two twin orientations associated to the highest Schmid factor are considered in agreement with EBSD observations. Twin orientations (red crosses) calculated from the ideal parent orientation (blue squares) brass (a), rotated copper (b), cube (c) and rotated Goss (d). Twin orientation densities calculated from parent orientations dispersed around brass and rotated copper orientations (e) and around cube and rotated Goss orientations (f). The colored iso-densities lines correspond to the parent orientation densities whereas the black ones correspond to the twin orientations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

  1 1}<1 1 2> twinning systems.

	Symbol	Texture components	{hkl}<uvw>	Euler angles	Fiber	Color in the TD-IPF triangle	Material state	Schmid's factor
								Slip	Twinning
		Goss	{110}<001>	90/90/45	<110>//DT	Green	As-received	0.41	0.47
		Copper	{112}<111>	90/35.26/45	<110>//DT	Green	As-received	0.41	0.47
		Brass	{110}<112>	55/90/45	<111>//DT	Blue	Deformed	0.27	0.31
		Rotated copper	{112}<110>	0/35.26/45	<111>//DT	Blue	Deformed	0.27	0.31
		Rotated Goss	{110}<110>	0/90/45	<100>//DT	Red	Deformed	0.41	0.23
		Cube	{001}<100>	45/0/45	<100>//DT	Red	Deformed	0.41	0.23

Acknowledgments

The authors acknowledge gratefully ArcelorMittal Research SA for financial and technical support and for providing the material.

The authors would also acknowledge Dr. B. Bolle (LETAM UMR CNRS 7078 University Paul Verlaine of Metz) for the XRD measurements, Region Lorraine and CNRS for financial support and the research team 'Material Forming' leading by Dr. V. Favier for fruitful discussions.


Paul Donato

Patrick Iglesias-Zemmour

EVERY SYMPLECTIC MANIFOLD IS A (LINEAR) COADJOINT ORBIT

Keywords: Mathematics Subject Classification. 53D05, 58B99. Key words and phrases. Di feology, Symplectic Geometry, Quantization. 1

A

. We prove that every symplectic manifold is a coadjoint orbit of the group of automorphisms of the integration bundle of the symplectic manifold, acting linearly on its space of momenta. And that, for any group of periods of the symplectic form. This result generalizes the Kirilov-Kostant-Souriau theorem when the symplectic manifold is homogeneous under the action of a Lie group, and the symplectic form is integral.

I

It is well known since Kostant, Souriau and Kirillov [START_REF] Kostant | Orbits and quantization theory[END_REF] [Sou70] [START_REF] Kirillov | Elements de la théorie des représentations[END_REF], that a symplectic manifold (X, ω), homogeneous under the action of a Lie group, is isomorphic -up to a covering -to a coadjoint orbit, possibly a ne.

It is less known that any symplectic manifold 1 is isomorphic to a coadjoint orbit of its group of symplectomorphisms (or Hamiltonian di feomorphisms), possibly a ne [START_REF]Every symplectic manifold is a coadjoint orbit[END_REF]. This has been established, in particular, in the rigorous framework of di feology and uses essentially the notion of Moment Map for that category [PIZ10]. But this theorem still seems to lack something. Although this is not a fundamental aw, we would like to get rid of the a ne action, de ned by a twisted cocycle of the automorphisms, and prefer to identify the symplectic manifold with an ordinary coadjoint orbit, that is an orbit of the usual linear coadjoint action 2 . That could be done by integrating the a ne cocycle in some extension of the group of automorphisms rst, and then identify the a ne coadjoint orbit with an ordinary coadjoint orbit of this extension. This is quite a standard procedure in ordinary di ferential geometry, when it is possible.

But we recall that we are no more in the classical framework but in di feology, and we shall see that the di culty to integrate this cocycle in an extension of the automorphisms is absorbed in this category by the ability to treat irrational toruses. The fundamental element is the integration bundle existing for any symplectic manifold, as it has been established in the paper "La Trilogie du Moment" [START_REF] Iglesias-Zemmour | La Trilogie du Moment[END_REF]. This is a principal ber bundle over the manifold, with group the torus of periods of the symplectic form. That is, the quotient of the real line by the group of periods, i.e. the integrals of the 2-form on every 2-cycles. This principal bundle comes equipped with a connexion form, with curvature the symplectic form. Of course the torus of periods is almost never a manifold, but still a regular di feological group, non-trivial as a few examples have shown [START_REF] Donato | Exemple de groupes di érentiels : flots irrationnels sur le tore[END_REF] [IZL90]. And this is here where di feology is inevitable. We establish rst the following T 1. -Let (X, ω) be a symplectic manifold. Let P ω be its group of periods and T ω = R/P ω be its torus of periods. Let π : Y → X be an integration bundle equipped with a connexion form λ with curvature ω. Let Aut(Y, λ) be the (neutral component of the) group of automorphisms of the integration structure. Then, the kernel of the projection pr: Aut(Y, λ) → Diff(X, ω) is reduced to the action of the torus T ω , and its image is the group Ham(X, ω) of Hamiltonian di eomorphisms. In other words, we get an exact sequence of homomorphisms, which is a central extension:

1 T ω Aut(Y, λ) Ham(X, ω) 1.
The group of Hamiltonian di feomorphisms is de ned precisely in terms of di feology in [PIZ13, §9.15]. Then, denoting by * the space of momenta of the group Aut(Y, λ), we prove the following theorem which reveals the universal model of symplectic manifolds.

T 2. -The Moment Map µ Y : Y → * of the action of Aut(Y, λ) on (Y, d λ), is equivariant: µ Y (φ(y)) = Ad * (φ)(µ Y (y)
) for all φ ∈ Aut(Y, λ), and invariant by T ω . Its projection µ X : X → * is injective and caries out an identification of X with the orbit λ = µ Y (Y) = µ X (X). Therefore, every symplectic manifold is a coadjoint orbit of a linear action of a di eological group, at least Aut(Y, λ).

N

1. -The idea that every symplectic manifold is a coadjoint orbit of its group of symplectomorphisms (or Hamiltonian di feomorphisms), is not new. It appeared already at an early age of symplectic mechanics, a few decades ago. It is mentionned for example, in a functional analysis context, by Marsden & Weinstein in their paper on Vlasov equation [MW82, Note 3, p. 398], Taken up later by Omohundro, Weinstein's student, in his book on geometric perturbation theory in physics [START_REF] Malvern | Geometric Perturbation Theory in Physics[END_REF]p. 364]. What was already original in the rst paper [START_REF]Every symplectic manifold is a coadjoint orbit[END_REF] was the rigorous di feology framework in which the result was proved, the role of the universal moment map, the identi cation of Souriau's cocycle of the action of the automorphisms and the a ne action rather than linear if the cocycle is not trivial. What is original in this paper is that the a ne coadjoint orbit is made linear anyway, by integrating the cocycle into a central extension of the group of Hamiltonian di feomorphisms thanks to the integration bundle.

N

2. -We follow the vocabulary introduced in a previous work. We call parasymplectic form any closed 2-form on a di feological space; and a parasymplectic space any di feological space equipped with a parasymplectic form, which we denote in general by (X, ω). We refer to the textbook [START_REF] Di Eology | Mathematical surveys and monographs[END_REF] for all generic constructions in di feology.

R D C

This construction of a universal model for symplectic manifolds builds on three major constructions already established:

(1) The construction of the Moment Map for any parasymplectic form, on any di feological space and any preserving smooth action of any di feological group, which can be found in "Moment Map in Di feology" [PIZ10].

(2) The general construction of the group of Hamiltonian di feomorphisms that follows this construction op. cit. §9.2. (3) The integration bundle of a parasymplectic form on a manifold, in [START_REF] Iglesias-Zemmour | La Trilogie du Moment[END_REF].

To which is added the previous work on symplectic manifolds as affine coadjoint orbits of the group of symplectomorphisms [START_REF]Every symplectic manifold is a coadjoint orbit[END_REF]. In this section we recall the basis of these main constructions.

1. T M M P S -First of all, let G be a diffeological group. We denote by * its space of its momenta, that is, the space of the left-invariant di ferential 1-forms on G,

* = {ε ∈ Ω 1 (G) | L(g ) * (ε) = ε, for all g ∈ G}.
Now, let (X, ω) be a parasymplectic space with a parasymplectic action of G on X. That is, a smooth morphism ρ : G → Diff(X, ω), denoted by g → g X . Where Diff(X, ω) ⊂ Diff(X) is the group of automorphisms of ω, equipped with the functionnal di feology.

Hence, g * X (ω) = ω for all g ∈ G.

To understand the essential nature of the moment map, which is a map from X to * , it is good to consider the simplest case, and use it then as a guide to extend this simple construction to the general case. The Simplest Case. Consider the case where X is a manifold, and G is a Lie group. Let us assume that ω is exact ω = dα, and that α is also invariant by G. Regarding ω, the moment map3 of the action of G on X is the map µ : X → * de ned by µ(x) = x * (α), where x : G → X is the orbit map x(g ) = g X (x).

As we can see, there is no obstacle, in this simple situation, to generalize, mutatis mutandis, the Moment Map to a di feological group acting by symmetries on a di feological parasymplectic space. But, as we know, not all closed 2-forms are exact, and even if they are exact, they do not necessarily have an invariant primitive. We shall see now, how we can generally come to a situation, so close to the simple case above, that modulo some minor subtleties we can build a good Moment Map in all cases.

The General Case. We consider a connected parasymplectic di feological space (X, ω), and a di feological group G acting on X and preserving ω. Let be the Chain-Homotopy Operator, de ned in [START_REF] Di Eology | Mathematical surveys and monographs[END_REF]§6.83]. We recall that

: Ω k (X) → Ω k-1 (Paths(X))
is a linear operator which satis es the property

d • + • d = 1 * -0 * , (♥)
where t (γ) = γ(t ), with t ∈ R and γ ∈ Paths(X). Then, the di ferential 1-form ω, de ned on Paths(X), is related to ω by d and γ : G → Paths(X) is the orbit map γ(g ) = g X • γ of a path γ. The moment of paths is additive with respect to the concatenation,

[ ω] = ( 1 * -0 * )(ω),
Ψ(γ ∨ γ ) = Ψ(γ) + Ψ(γ ).
This paths Moment Map Ψ is equivariant by G, acting by composition on Paths(X), and by coadjoint action on * . Next, de ning the Holonomy of the action of G on X by

Γ = {Ψ( ) | ∈ Loops(X)} ⊂ * ,
the Two-Points Moment Map is de ned by pushing Ψ forward on X × X,

ψ(x, x ) = class(Ψ(γ)) ∈ * /Γ,
where γ is a path connecting x to x , and where class denotes the projection from * onto its quotient * /Γ. The holonomy Γ is the obstruction for the action of G to be Hamiltonian. The additivity of Ψ becomes the Chasles' cocycle condition

ψ(x, x ) + ψ(x , x ) = ψ(x, x ).
Let Ad : G → Diff(G) be the adjoint action, Ad(g ): k → g k g -1 . That induces on * a linear coadjoint action

Ad * : G → L( * ) with Ad * (g ) : ε → Ad(g ) * (ε) = Ad(g -1 ) * (ε).
Next, the group Γ is made of closed forms, invariant by the linear coadjoint action. Thus, the coadjoint action passes to the quotient * /Γ, and we denote the quotient action the same way:

Ad * (g ) : class(ε) → class(Ad * (g )(ε)).
The 2-points Moment Map ψ is equivariant for the quotient coadjoint action. Note that the quotient * /Γ is a legit di feological Abelian group4

Now, because X is connected, there always exists a map

µ : X → * /Γ such that ψ(x, x ) = µ(x ) -µ(x).
The solutions of this equation are given by

µ(x) = ψ(x 0 , x) + c,
where x 0 ∈ X is an arbitrary point and c ∈ * /Γ is any constant. But this map is a priori no longer equivariant with respect to Ad * on * /Γ. Its variance introduces a 1-cocycle θ of G with values in * /Γ such that

µ( g (x)) = Ad * (g )(µ(x)) + θ( g ), with θ( g ) = ψ(x 0 , g (x 0 )) -∆c( g ), and ∆(c) : g → Ad * (g )(c) -c
is the coboundary due to the constant c in the choice of µ. The cocycle θ de nes then a new action of G on * /Γ, that is, a quotient affine action :

Ad θ * (g ) : τ → Ad * (g )(τ) + θ( g ) for all τ ∈ * /Γ.
The Moment Map µ is then equivariant with respect to this a ne action:

µ( g (x)) = Ad θ * (g )(µ(x)).
Note that, in particular, if G is transitive on X, then the image of the Moment Map µ is an a ne coadjoint orbit in * /Γ.

This construction extends to the category {Di feology}, the Moment Map for {Manifolds} introduced by Souriau in the sixties [START_REF] Souriau | Structure des systèmes dynamiques[END_REF].

The group of all automorphisms of a parasymplectic space is denoted by Diff(X, ω) or by G ω , it is a legitimate di feological group. The constructions above give the space of momenta * ω , the universal path moment map Ψ ω , the universal holonomy Γ ω , the universal two-points moment map ψ ω , the universal moment maps µ ω , and the universal Souriau's cocycles θ ω .

T C

S M -Let (X, ω) be a connected parasymplectic manifold. The value of the paths Moment Map Ψ ω at the point p ∈ Paths(X) = C ∞ (R, X), evaluated on the n-plot F : U → G ω is explicitely given by

Ψ ω ( p)(F) r (δr ) = 1 0 ω p(t ) (ṗ(t ), δ p(t )) d t
where r ∈ U and δr ∈ R n , δ p denotes the lifting in the tangent space TX of the path p, de ned by

δ p(t ) = [D(F(r ))( p(t ))] -1 ∂ F(r )( p(t ))
∂ r (δr ) for all t ∈ R.

In that case we have the foolowing theorem, see [START_REF]Every symplectic manifold is a coadjoint orbit[END_REF] for example: T (P.I-Z). Let X be a connected Hausdor manifold. A parasymplectic form ω on X is symplectic if and only if the following two conditions are satisfied:

(1) The manifold X is homogeneous under the action of G ω .

(2) The universal Moment Map µ ω : X → * ω /Γ ω is injective. Hence, the Moment Map identifies the manifold X with a, a priori affine, [START_REF] Donato | Revêtement et groupe fondamental des espaces di érentiels homogènes[END_REF], ω is equipped with the pushforward di feology of G ω by π , and µ ω is then a di feomorphism.

(Γ ω , θ ω )- coadjoint orbit ω of G ω , µ ω (X) = ω ⊂ * ω /Γ ω . The situation is summarized by the diagram G ω X ω π π X µ ω (♣) where π X (φ) = φ(x 0 ), π (φ) = Ad θ * (φ)(µ ω (x 0 )), for all φ ∈ G ω and x 0 ∈ X is any base point. The projections π X is a subduction [Boo69,

H

D -In the construction of the Moment Map of a parasymplectic action of a di feological group G on (X, ω), the holonomy group Γ is the obstruction of the action of G to be Hamiltonian.

D

. A parasymplectic action of a di eological group G on (X, ω) is said to be Hamiltonian if Γ = {0}.

Hence, the moment maps have their values in * . We get then the following theorem [PIZ10, §9.2] T (P.I-Z). Let (X, ω) be a connected parasymplectic di eological space. There exists a largest connected subgroup Ham(X, ω) ⊂ Diff(X, ω) whose action is Hamiltonian, that is, whose holonomy is trivial. The elements of Ham(X, ω) are called Hamiltonian di feomorphisms. An action ρ of a di eological group G on X is Hamiltonian if and only if, restricted to the identity component of G, ρ takes its values in Ham(X, ω).

The group Ham(X, ω) is precisely built as follows. Let G

• ω be the neutral component of G ω = Diff(X, ω). Let π : G • ω → G • ω be the universal covering. Since the universal holonomy Γ ω is made of closed momenta [PIZ10, §4.7], every γ ∈ Γ ω de nes a unique homomorphism k(γ) from G • ω to R such that π * (γ) = d [k(γ)] [PIZ10, §3.11]. Let H ω = γ∈Γ ω ker(k(γ)), then Ham(X, ω) = π( H • ω ),
where H • ω ⊂ H ω is its neutral component. The space of momenta and the universal moment maps objects associated to H ω = Ham(X, ω) are denoted by: * ω , Ψω , ψω , μω , and θω .

T I

B P F -Let (X, ω) be a parasymplectic manifold, connected, Hausdor f and second countable. Let P ω its group of periods, that is,

P ω = σ ω σ ∈ H 2 (X, Z) ⊂ R.
Since X is second countable P ω is di feogicaly discrete, that is, the di feology induced by the standard di feology of R is the discrete di feology. The plots are locally constant. Let T ω = R/P ω be its torus of periods. Except in the case where the group of periods has only one generator, the torus of periods is not a manifold but a legitimate non trivial di feological group. See for example the paper on the irrational torus T α = R/Z + αZ [START_REF] Donato | Exemple de groupes di érentiels : flots irrationnels sur le tore[END_REF], where α / ∈ Q,

for an example of such irrational torus.

Then, we get the following theorem in [PIZ95, Theorem 1.5].

T (P.I-Z)

. Let (X, ω) be a second countable Hausdor parasymplectic manifold. There exists always a T ω -principal fiber bundle π : Y → X equipped with a connexion 1-form λ with curvature ω. That is, π * (ω) = d λ. The various such integration bundles are classified by the extension group Ext(H 1 (X, Z), P ω ).

We may need to precise that a connexion 1-form λ on Y is a T ω -invariant 1-form, such that, for all y ∈ Y, ŷ * (λ) = θ, where ŷ : T ω → Y is the orbit map ŷ(τ) = τ Y (y); and θ is the canonical 1-form on T ω de ned by class * (θ) = d t , with class: R → T ω the projection.

A E S

For a symplectic manifold, the transition from an a ne orbit to a linear orbit5 needs to absorb the Souriau's cocycle somewhere. We do it by building an extension of the group of Hamiltonian di feomorphisms, associated with the the integration bundle of the symplectic form ( §4).

T C E H D

-Let (X, ω) be a symplectic manifold. Let P ω be its group of periods and T ω = R/P ω be its torus of periods. Let π : Y → X be its T ω -principal integration ber bundle, and λ be its connection form. Let Aut(Y, λ) be the group of automorphisms of the integration structure, that is:

Aut(Y, λ) = {φ ∈ Diff(Y) | φ * (λ) = λ and ∃ f ∈ Diff(X), π • φ = f • π}
Actually we reduce Aut(Y, λ) to its the neutral component. Then, the di feomorphism f belongs naturally to the group of Hamiltonian di feomorphisms Ham(X, ω). The mapping η : φ → f is then a surjective homomorphism. Its kernel is the torus of periods T ω , and η is a central extension. Which is summarized by the exact sequence:

1 T ω Aut(Y, λ) Ham(X, ω) 1 η . N .
-The integration bundle of a parasymplectic form is not necessarily unique. They are all classi ed by Ext(H 1 (X, Z), P ω ) [START_REF] Iglesias-Zemmour | La Trilogie du Moment[END_REF]. The theorem above applies to any of them indi ferently, which had been noticed by Bertram Kostant and Jean-Marie Souriau in the integral case P ω = aZ. And by the way, this extension is called the Kostant-Souriau Extension in that case.

It is notable too, that all this construction is purely di feologial, involves only di ferential forms and do not need tangent vectors or integration of vector elds. That aspect of di feology had been already underlined in the construction of the Moment Map in particular in [PIZ10].

Proof. Let us begin by xing our notation. The action of an element τ ∈ T ω on y ∈ Y will be noted indi ferently by τ • y or by τ Y (y).

Now, let φ ∈ Aut(Y, λ) and f = η(φ). Since f • π = π • φ, φ * (λ) = λ and π * (ω) = d λ, π being a subduction, we get f ∈ Diff(X, ω).
(A) Let us to prove that ker(η) = T ω , acting on Y by τ : y → τ •y. Let φ ∈ ker(η), that is, π • φ = π. Then, for all y ∈ Y, there exists a unique τ(y) ∈ T ω such that φ(y) = τ(y) • y.

(a) Let us rst check that τ : Y → T ω is smooth. Let r → y r by a plot in Y, composed by φ, we get the plot r → τ(y r ) • y r . We need to prove that r → τ(y r ) itself is smooth. The pullback of π : Y → X by the plot r → x r = π(y r ) is localy trivial, then we can restrict these plots to a ball B over wich the pullback

[r → x r ] * (Y) = {(r, y) ∈ B × Y | π(y) = x r }
is trivial. Any T ω -principal bundle isomorphism F from this pullback to the product B × T ω writes F(r, y) = (r, t (r )(y)), and the smooth map t with values in T ω satis es the equivariance t (r )(τ • y) = τ • t (r )(y). Thus, r → t (r )(y r ) is smooth as well as r → t (r )(τ(y r )(y r )) = τ(y r ) • t (r )(y r ). Hence, r → τ(y r ) is smooth. Therefore, the function τ is smooth.

(b) Let us prove now that the function τ is constant. The invariance φ * (λ) = λ implies λ(r → τ(y r ) • y r ) = λ(r → y r ), for all plots r → y r . That is, thanks to the partial derivatives formula [PIZ13, §8.37 ♣]

λ(r → y r ) = λ(r → τ(y r ) • y r ) = λ(r → y r ) + τ * (θ)(r → y r ),
where θ is the canonical 1-form on T ω . Thus, τ * (θ) = 0. Then τ is constant. Hence, ker η = T ω .

(B) Let us prove that η takes its value in Ham(X, ω). That is, that the holonomy group of Aut(Y, λ) vanishes when acting on (X, ω) through the action

φ X (x) = f (x) with f = η(φ).
We shall denote by * the space of momenta of Aut(Y, λ). The Moment Map Ψ X of the action of Aut(Y, λ) on (X, ω) is given, according to previous notations by:

Ψ X : Paths(X) → * with Ψ(γ) = γ * ( X (ω)),
where γ : φ → f • γ is the orbit map. Let us prove now that Ψ X ( ) = ˆ * ( X (ω)) = 0, for all ∈ Loops(X).

Let us recall rst of all that the principal ber bundle π : Y → X induces, in particular, a subduction of loops spaces:

π * : Loops(Y) → Loops(X) by pushforward π * ( ) = π • , see [PIZ13, §8
.32] and [PIZ19]. That is, every plot r → r in Loops(X) has a local smooth lifting r → r , everywhere, in Loops(Y). Note that we shall underline the paths in Y, to distinguish them from paths in X. Now, let and such that π • = . We have

ˆ (φ) = f • = f • π • = π • φ • = π • ˆ (φ), that is, ˆ = π * • ˆ . Thus, ˆ * ( X (ω)) = (π • ˆ ) * ( X (ω)) = ˆ * π * * X (ω)
Then, let us recall the variance of the chain-homotopy operators X and Y , relatives to X and Y [PIZ13, §6.84], summarized by the commutative diagram:

Ω k (Y) Ω k-1 (Paths(Y)) Ω k (X) Ω k-1 (Paths(X)) Y X π * (π * ) *
We have then:

π * * X (ω) = Y π * (ω) = Y (d λ).
Hence:

ˆ * π * * X (ω) = ˆ * Y (d λ) .
Thus:

Ψ X ( ) = ˆ * X (ω) = ˆ * Y (d λ) = Ψ Y ( ),
where Ψ Y is the Moment Map for the action of Aut(Y, d λ) acting on (Y, d λ). Note that, that could have been deduced directly from [PIZ13, §9.13]. Now, according to the fundamental property of the Chain-Homotopy Operator, we have:

ˆ * Y (d λ) + ˆ * d Y (λ) = ˆ * ( 1 * (λ) -0 * (λ)) = ( 1 • ˆ ) * (λ) -( 0 • ˆ ) * (λ) = 0,
because is a loop. Therefore,

ˆ * Y (d λ) = -d ˆ * Y (λ) .
But, for every plot r → φ r in Aut(Y, λ), for all r in its domain :

ˆ * Y (λ) (φ r ) = Y (λ)(φ r • ) = φ r * ( ) λ = φ * r (λ) = λ.
Hence ˆ * Y (λ) is constant, its derivative vanishes and therefore

Ψ X ( ) = ˆ * X (ω) = ˆ * Y (d λ) = 0. (♦)
And that achieves to prove that η : Aut(Y, λ) → Diff(X, ω) takes its values in Ham(X, ω).

(C) Let us show now that

T ω ⊂ Aut(Y, λ) is central, that is, η : Aut(Y, λ) → Ham(X, ω) is a central extension. Let φ ∈ Aut(Y, λ).
We have seen that T ω = ker(η). Thus, for all τ ∈ T ω there exists τ ∈ T ω such that τ = φ • τ • φ -1 . Obviously, h φ : τ → τ de nes a group isomorphism of T ω : h φ (τ 1 τ 2 ) = h φ (τ 1 )h φ (τ 2 ), and h φ (τ

) -1 = φ -1 • τ -1 • φ.
But φ is connected to the identity map 1 Y via a smooth path s → φ s ∈ Aut(Y, λ), de ned on an open interval open interval containing [0, 1], with φ 0 = 1 Y and φ 1 = φ. That de nes a smooth path of isomorphisms

h φ s = φ s • τ • φ -1 s .
Let us denote h s for h φ s . The map (s, t ) → h s (class(t )) is a plot de ned on × R, in T ω . By the monodromy theorem [PIZ13, §8.25], it has a global lifting (s, t ) → H s (t ), de ned on × R, which is a smooth plot in R. And the lift is unique with H 0 (0

) = 0. × R (s, x) H s (x) ∈ R × T ω (s , class(x)) h s (class(x)) = class(H s (x)) ∈ T ω H 1 × class class h
For every parameter s, the restriction H s : R → R is a smooth lifting of the isomorphism h s : T ω → T ω . Thus, up to a constant b s ∈ R, H s is a smooth morphism from R to R. Hence, H s (x) = a s x + b s . Where s → a s and s → b s are smooth, and a s = 0 since H s lifts an isomorphism. Now, for all s ∈ , all x, x ∈ R, h s (class

(x + x )) = h s (class(x)) + h s (class(x )), that is, class(H s (x + x )) = class(H s (x)) + class(H s (x )),
i.e. a s (x + x ) + b s = a s x + b s + a s x + b s + p, and then b s ∈ P ω for all s ∈ P ω .

Since s → b s is smooth and P ω is (di feologically) discrete in R, b s is constant and equal to b 0 which is 0. Thus, H s (x) = a s x. Next, h s (class(x)) = class(H s (x)) implies that, for all p ∈ P ω there exists p ∈ P ω such that H s (x + p) = H s (x) + p . That is, a s (x + p) = a s x + p , and then a s p ∈ P ω for all p ∈ P ω . Again, since P ω is discrete in R, s → a s is constant and the lifting H writes H s (x) = a x, for some number a = 0. Finally, since H 0 lifts the identity h 0 = 1 T ω by a morphism, H 0 (x) = x and a = 1. Therefore H s (x) = x for all s and h s

(τ) = τ, that is, φ • τ Y = τ Y • φ, and the extension η : Aut(Y, λ) → Ham(X, ω) is central. (D) Let us show nally that η : Aut(Y, λ) → Ham(X, ω) is surjective. Let f ∈ Ham(X, ω).
There exists a smooth path t → f t ∈ Ham(X, ω), such that f 0 = 1 X and f 1 = f . We de ne, for all x ∈ X, the path γ X in X by γ x (t ) = f t (x). It satis es γ x (0) = x and γ x (1) = f (x). The map x → γ x is smooth.

Given any y ∈ Y and x = π(y) ∈ X, we will denote by γ y the unique horizontal lifting of γ x with origin y. Moreover, the map y → γ y is smooth and equivariant under the action of T ω [PIZ13, §8.32]. We de ne then:

φ(y) = γ y (1) and φ ∈ C ∞ (Y) (op. cit.)
The map φ is a smooth lifting of f , that is,

π • φ = f • π. Moreover, the equivariance of γ y by T ω also implies that τ Y • φ = φ • τ Y for all τ ∈ T ω . But if φ is equivariant, it
has no reason to preserve the contact form λ. We shall show then that there exists a map τ ∈ C ∞ (Y, T ω ) such that

Φ : y → τ(y) • φ(y),
which is still a smooth lifting of f , preserves the contact form λ, that is, Φ ∈ Aut(Y, λ).

Thanks to the partial derivatives formula (op. cit.), for any plot r → y r of Y, we get:

Φ * (λ)(r → y r ) = λ(r → τ(y r ) • φ(y r )) = θ(r → τ(y r )) + λ(r → φ(y r )) = τ * (θ)(r → y r ) + φ * (λ)(r → y r ). That is, Φ * (λ) = τ * (θ) + φ * (λ). Consider now Φ * (λ) -λ = τ * (θ) + β with β = φ * (λ) -λ. L 1. The 1-form β is the pullback of a closed 1-form ε on X : β = π * (ε).
Let us check that β is closed:

d (φ * (λ) -λ) = φ * (d λ) -d λ = φ * (π * ω) -π * ω = π • φ) * ω -π * ω = ( f • π) * ω -π * ω = π * f * ω -π * ω = 0. Also, β is invariant by T ω : τ * (β) = τ * (φ * (λ)-λ) = (φ•τ) * (λ)-τ * (λ) = (τ•φ) * (λ)-λ = φ * (τ * (λ))-λ = φ * (λ)-λ = β.
0 = Ψ X ( ) = ˆ * X (ω) = ˆ * Y (d λ) = -ˆ * d Y (λ) = -d ˆ * Y (λ) ,
for all ∈ Loops(X) and all ∈ Loops(Y) over , because 1 • * = 0 • * . Now, evaluating the Moment Map on the plot t → φ t connecting 1 Y to φ, using = π • and φ t • = φ t * ( ), we get:

d ˆ * Y (λ) (t → φ t ) = d Y (λ)(t → φ t * ( )) = d t → φ t * ( ) λ = d t → φ * t (λ) = φ * (λ) -λ = φ * (λ) -λ = β = π * ε = ε.
Thus, for all ∈ Loops(X), ε = 0. Therefore, according to [PIZ13, §6.89], there exists ν ∈ C ∞ (X, R) such that ε = d ν.

Now we can achieve to prove that

Φ ∈ Aut(Y, λ). Indeed, let ν = ν • π ∈ C ∞ (Y, R). Let us de ne τ ∈ C ∞ (Y, T ω ) by τ = -class • ν = -class • ν • π,
where class: R → T ω . Hence:

τ * (θ) = -π * (ν * (class * (θ))) = π * (ν * (d t )) = -π * (d ν) = -π * (ε) = -β.
Thus τ * (θ) = -φ * (λ) + λ. Therefore:

Φ * (λ) = τ * (θ) + φ * (λ) = -φ * (λ) + λ + φ * (λ) = λ, and Φ ∈ Aut(Y, λ).
Until now, we have proved that η : Aut(Y, λ) → Ham(X, ω) is surjective, we have to prove then that it is a subduction [START_REF] Di Eology | Mathematical surveys and monographs[END_REF]§1.46]. For this, we need to check that any plot P: r → f r in Ham(X, ω), admits a local lifting P such that P = locally η • P, everywhere.

Thanks to the functional di feology and to both subductions π * : Paths(Y) → Paths(X) [PIZ13, §8.32] and π : Y → X, the map (r, t , x) → f r,t (x) is smooth and then admits a smooth lifting on Y. Thus, for x = π(y), the time t = 1 of this lifting de nes the smooth family φ r (y) of di feomorphisms, the shift by

τ ∈ C ∞ (Y, T ω ) preserves the smoothness of r → Φ r ∈ Aut(Y, λ). M M U E B A
In this section we will show how the sympletic manifold (X, ω) identi es, through the Moment map of the Hamiltonian action of Aut(Y, ω) with an orbit of this group for its linear coadjoint action on its space of momenta. We again emphazise the fact that this result generalizes the Kostant-Kirilov-Souriau theorem when the symplectic manifold is homogeneous under the action of a Lie group, and the symplectic form is integral. In the non-integral case but homogeneous, the optimal result in the category of manifolds states that the symplectic manifold is, up to a covering, an a ne coadjoint orbit of the group. That result had been extended to the group of all Hamiltonian di feomorphism in [START_REF]Every symplectic manifold is a coadjoint orbit[END_REF].

6. S M A (L ) C O -Let (X, ω) be a symplectic manifold, and as it is described in ( §5), let P ω be its group of periods, π : Y → X be an integration bundle with connection λ, and Aut(Y, λ) be the group of automorphisms of the integration structure.

Let * be the space of Momenta of Aut(Y, λ), that is, the space of left-invariant 1forms on Aut(Y, λ). The action of Aut(Y, λ) on Y has a moment map, relatively to the parasymplectic form d λ, given by

µ Y : → * , with µ Y (y) = ŷ * (λ).
Then,

(1) The moment µ Y projects on µ X : X → * , µ Y = µ X • π.

(2) µ Y is equivariant under the coadjoint action of Aut(Y, λ).

(3) µ X is injective. (4) µ X de nes a di feomorphism from X onto the coadjoint orbit

* ⊃ λ = µ Y (Y) = µ X (X).
Therefore the symplectic manifold X inherits the structure of a coadjoint orbit. And this is a universal characterization of symplectic manifolds:

Every Symplectic Manifold is a (Linear) Coadjoint Orbit.
Which completes the statement made in [START_REF]Every symplectic manifold is a coadjoint orbit[END_REF] that Every Symplectic Manifold is a (Affine) Coadjoint Orbit of its group of Symplectomorphisms.

Proof. Let us begin by checking that µ Y is constant on each ber. The action of T ω is central in Aut(Y, λ), so for any τ ∈ T ω , for all y ∈ Y and for all φ ∈ Aut(Y, λ) we have: τ

• y(φ) = φ(τ • y) = τ • φ(y) = τ • (ŷ(φ)), hence τ • y = τ • ŷ. Thus, µ Y (τ • y) = ( τ • y) * (λ) = (τ • ŷ) * (λ) = ŷ * (τ * (λ)) = ŷ * (λ) = µ Y (y).
Now, let us denote, for all φ, ψ in Aut(Y, λ), R(φ)(ψ) = ψ • φ -1 , the right action of the group on its momenta. Then, the equivariance follows from: µ Y (φ(y)) = φ(y) * (λ) = (ŷ • R(φ -1 ) * (λ) = R(φ -1 ) * (ŷ * (λ)) = R(φ -1 ) * (µ Y (y)) = R(φ -1 ) * L(φ -1 ) * (µ Y (y)) = Ad(φ -1 ) * (µ Y (y)) = Ad(φ) * (µ Y (y)).

Finally, pushing forward the moment maps [PIZ13, §9.12 & 9.13] leads to the commutative diagram below, where μX is the Moment Map for the group Ham(X, ω), and * denotes its space of momenta.

Y * X * π µ Y μX µ X η *
The Moment Map μX is known to be injective [START_REF]Every symplectic manifold is a coadjoint orbit[END_REF], as well as η * since η is a subduction ( §5, Proof (C)). Hence, µ X = η * • μX is injective and a subduction on λ = µ Y (Y). Therefore, µ X is a di feomorphism from X to λ = µ Y (Y), equiped with the quotient di feology of Aut(Y, λ) by the stabilizer of any point y ∈ Y.

C

This paper answers the question of the ontological nature of symplectic manifolds, if we can use such a big word. But that question has indeed arised in social networks, for example in mathoverfolw.net [Com17]. That is a good justi cation a posteriori of this work.

As we have seen in this construction, for a symplectic manifold, to pass from an orbit of the a ne action of the Hamiltonian di feomorphisms, to an orbit of a linear action needs the integration of the Souriau's cocycle. This integration is done by considering the integration bundle of the symplectic manifold, which adds a oor to the construction (♣) and is summarized in the following diagram (♠). We have denoted by G λ the group of automorphisms of the integration structure, and π the subduction from Y onto its orbit, by quotient.

G λ Y λ X π π Y µ Y π µ X (♠)
It is important to emphasize this, if the identi cation of the symplectic manifold as an a ne-coadjoint orbit may possibly be adapted in another framework, dealing with in nite dimension spaces and groups, it seems not possible to avoid di feology to transform the symplectic manifold into a linear-orbit of a group of di feomorphisms. Indeed, the key of this construction being the integration bundle, there is no other known smooth theory for which it exists, except in the trivial case of an integral symplectic manifold, where it is a manifold. That is due to the dense nature of the group of periods of the symplectic form in general. Think for example to the simple case of the product S 2 × S 2 , equiped with the symplectic form Surf ⊕ 2 Surf. Its group of periods Z + 2 Z ⊂ R is dense and its torus of periods not Hausdor f, but indeed not di feologically trivial.

  and ω is invariant by G (op. cit. §6.84). Considering ω = ( 1 * -0 * )(ω) and ᾱ = ω, we are in the simple case: ω = d ᾱ with ᾱ invariant. We can apply the construction above and de ne then the Moment Map of Paths by Ψ : Paths(X) → * with Ψ(γ) = γ * ( ω),

  Moreover, β vanishes vertically. Indeed, let us rst remark that τ • φ = φ • τ, for all τ ∈ T ω , implies φ • ŷ = ŷ , for all y ∈ Y and y = φ(y). Then, ŷ *(β) = ŷ * (φ * (λ)λ) = ŷ * (φ * (λ)ŷ * (λ) = (φ • ŷ) * (λ)ŷ * (λ) = (ŷ ) * (λ)ŷ * (λ) = θ-θ = 0. Thus λ = λ+β is a new connection 1-form, the di ference β is then the pullback of a 1-form on X, according to [PIZ13, §8.37, Note]. L 2. The 1-form ε is exact: ε = d ν, ν ∈ C ∞ (X, R).Indeed, considering the fundamental property of the Chain-Homotopy Operator• d + d • = 1 * -0 * (♥), on the one hand, and the vanishing of the holonomy of the action of Aut(Y, λ) on Y (♦) on the other hand, we get,

  address: paul.donato@univ-amu.fr address: piz@math.huji.ac.il URL: http://math.huji.ac.il/~piz

Precisely, one moment map, since they are de ned up to a constant.

For the quotient of the functional di feology of * ⊂ Ω 1 (G) by Γ. In particular, for Lie groups, it is always a product R k × T , k, ∈ N.

We recall that we say "linear orbit" as a shortcut for "orbit of a linear action".