Foued Abroug

Etienne Pessard

Guenaël Germain

Franck Morel

A probabilistic approach to study the effect of machined surface states on HCF behavior of a AA7050 alloy

Keywords:

ABSTRAC T

The aim of this study is to understand the impact of periodic surface micro-geometry patterns (obtained by High Speed Machining process) on the fatigue behavior of the AA7050 aluminium alloy and to define a proper defect acceptability criterion. It must be able to account for a large range of surface defects and of component sizes and geometries (wings, brackets, frames, etc.). A vast experimental campaign under fully reversed plane bending loads containing different surface states has been undertaken to characterize the effect of the surface topography on the fatigue behavior. The results show that the fatigue strength decreases when the surface roughness is significantly degraded. In order to predict the influence of the surface condition on the fatigue behavior, a numerical approach based on the real surface topology has been developed. It is shown that the numerically identified crack initiation sites are in agreement with the experimental results. A probabilistic approach based on the weakest link concept, associated with the definition of a stress based crack initiation threshold has been integrated in a FE model. This approach leads to a probabilistic diagram inspired from the Kitagawa type diagram, which explains the relationship between the surface and the scale effect on the fatigue strength.

Introduction

Machining is a manufacturing process largely used in the aerospace industry. Milling is one type of machining process that consists in using rotary cutters to remove materials from a workpiece. The process induces residual stresses, surface roughness and, to a lesser extent, work hardening. These surface integrity parameters can delay or enhance crack initiation at the part surface. The latter is a critical location for fatigue crack initiation and milling is most of the time seen as detrimental to the fatigue life due to the microscopic mechanical concentrators created. One of the greatest challenges when designing milled parts against fatigue is to find out to what extent a given surface microgeometry is critical and to identify the relevant roughness parameters to be used for the fatigue strength prediction. When dealing with this issue, the effects of the loading mode, the microstructure and the component size must also be taken into account.

The work presented in this paper deals with the machined surface states of structural aeronautical components. The material under investigation is the 7050-T7451 aluminium alloy (Al Zn6CuMgZr). During HSM (High Speed Machining) of aircraft parts, geometrical defects can be created. Mismatch may appear due to the gap left between two consecutive passes of the machining tool. Chatter can also be created due to the tool/part vibrations which may occur during different types of machining. One of the main objectives of the research project is to build a fatigue life predictive model that accounts for the surface integrity parameters induced by milling, and from which a proper defect acceptability criterion can be proposed (defect size, depth, surface roughness, etc.). It must be able to account for a large range of surface defects and of component sizes and geometries. By reducing costly trial and error experimental campaigns, this approach would be an important asset for the quality control of aeronautical parts.

Several studies in the literature pointed out that, by the local thermomechanical effects induced, machining can change the surface integrity of manufactured parts and therefore modify their fatigue strength [START_REF] Brunet | PhD thesis: Influence des contraintes résiduelles induites par usinage sur la tenue en fatigue des matériaux métalliques aéronautiques[END_REF][START_REF] Shahzad | Influence of anodizing process on fatigue life of machined aluminium alloy[END_REF][START_REF] Guillemot | Prise en compte de l'intégrité de surface pour la prévision de la tenue en fatigue des pièces usinées en fraisage[END_REF][START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF][START_REF] Moussaoui | Influence de l'usinage sur la durée de vie en fatigue de pièces aéronautiques en alliage de titane[END_REF]. Changing the cutting conditions can also generate a different surface topology and therefore a new fatigue life [START_REF] Brunet | PhD thesis: Influence des contraintes résiduelles induites par usinage sur la tenue en fatigue des matériaux métalliques aéronautiques[END_REF][START_REF] Limido | Etude de l'effet de l'usinage grande vitesse sur la tenue en fatigue de pièces aéronautiques[END_REF]. For instance, Shahzad [START_REF] Shahzad | Influence of anodizing process on fatigue life of machined aluminium alloy[END_REF] pointed out that the fatigue resistance of machined aluminium alloy shows a significant decrease when surface roughness increases. Different literature studies [START_REF] Tang | The influence of tool flank wear on residual stresses induced by milling aluminum alloy[END_REF][START_REF] Rao | Analysis on high-speed face-milling of 7075-T6 aluminum using carbide and diamond cutters[END_REF][START_REF] Campbell | Microstructural characterization of Al-7075-T651 chips and work pieces produced by high-speed machining[END_REF] addressed the issues of residual stresses and microstructure variation due to the machining process. For the plain milling process, Brunet [START_REF] Brunet | PhD thesis: Influence des contraintes résiduelles induites par usinage sur la tenue en fatigue des matériaux métalliques aéronautiques[END_REF] showed that residual stresses induced by the machining process have an important effect on the fatigue response. On the contrary, for the end milling process of the AA-7XXX alloys, the studies tend to show that residual stresses and the microstructure variations are mainly located in the first https://doi.org/10.1016/j.ijfatigue.2018.06.048 ⁎ Corresponding author. E-mail address: Foued.Abroug@ensam.eu (F. Abroug).

T the surface topology [START_REF] Brunet | PhD thesis: Influence des contraintes résiduelles induites par usinage sur la tenue en fatigue des matériaux métalliques aéronautiques[END_REF][START_REF] Suraratchaï | Modelling the influence of machined surface roughness on the fatigue life of aluminum alloy[END_REF]. Regarding the fatigue life prediction of AA-7XXX alloy and the effect of machining, different approaches proposed in the literature could be good candidates to account for the effect of the surface topology due to turning, shaping, milling or even abrasion. In the first category of approach, both roughness parameters (i.e. the arithmetic linear roughness R a or surface roughness S a ) and geometrical characteristics of groove due to machining are used to define scalars that are supposed to be relevant fatigue damage indicators [START_REF] Inglis | Stresses in a plate due to the presence of cracks and sharp corners[END_REF][START_REF] Arola | Estimating the fatigue stress concentration factor of machined surfaces[END_REF][START_REF] Neuber | Theory of notch stresses[END_REF][START_REF] Souto-Lebel | Rôle de l'intégrité de surface dans la tenue en fatigue d'un acier bainitique après fraisage de finition[END_REF]. FE calculations can also be conducted to assess the stress concentration in the case of complex geometric defects [START_REF] Suraratchaï | Modelling the influence of machined surface roughness on the fatigue life of aluminum alloy[END_REF]. Another way is to directly predict the fatigue strength via the empirical criterion of Murakami [START_REF] Murakami | Metal fatigue: effects of small defects and nonmetallic inclusions[END_REF] in its adapted formulation to the machined surface states.

The second category is composed of non-local approaches, like the critical distance [START_REF] Ås | Fatigue life prediction of an aluminum alloy automotive component using finite element analysis of surface topography[END_REF] originally proposed by Taylor [START_REF] Taylor | Geometrical effects in fatigue: a unifying theoretical model[END_REF][START_REF] Taylor | The theory of critical distances[END_REF] and that considers a mean stress value evaluated over a line, a surface or a given volume. The efficiency of non local approaches has been demonstrated in a number of applications, for instance for notched components [START_REF] Taylor | A mechanistic approach to critical-distance methods in notch fatigue[END_REF], specimens containing internal defects [START_REF] Le | Investigation of the effect of porosity on the high cycle fatigue behaviour of cast Al-Si alloy by X-ray microtomography[END_REF] or surface defects [START_REF] Gerin | A non-local approach to model the combined effects of forging defects and shot-peening on the fatigue strength of a pearlitic steel[END_REF].

The last category is the one of probabilistic approach. In the high cycle fatigue regime, the weakest link concept [22,[START_REF] Weibull | A statistical distribution function of wide applicability[END_REF] is often used to predict the effect of manufacturing processes on the fatigue life of components [START_REF] Hild | A statistical approach to the rupture of brittle materials[END_REF][START_REF] Chantier | A probabilistic approach to predict the very high-cycle fatigue behaviour of spheroidal graphite cast iron structures[END_REF][START_REF] Bomas | Application of a weakest-link concept to the fatigue limit of the bearing steel SAE 52100 in a bainitic condition[END_REF][START_REF] Delahay | Estimation of the fatigue strength distribution in high-cycle multiaxial fatigue taking into account the stress-strain gradient effect[END_REF][START_REF] Pessard | A mechanistic approach to the Kitagawa-Takahashi diagram using a multiaxial probabilistic framework[END_REF]. Initially developed to take into account the effect of defects on cast alloys such as spheroidal graphite (SG) cast iron, this type of approach has been extended to account for the scatter on different kind of hetereogenities population (inner defects population [START_REF] Pessard | Modelling the role of non metallic inclusions on the anisotropic fatigue behaviour of forged steel[END_REF], grains mechanical resistance [START_REF] Morel | Plasticity and damage heterogeneity in fatigue[END_REF] and surface defects population [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF]).

In the case of machining, [START_REF] Guillemot | Prise en compte de l'intégrité de surface pour la prévision de la tenue en fatigue des pièces usinées en fraisage[END_REF][START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF] proposed to characterize the microgeometry by defect size distributions. A two-scale approach taking account of the probabilistic description of the geometrical defects induced by the milling process is used to predict the endurance limit of specimens subjected to four-point bending fatigue tests. The approach developed by [START_REF] Guillemot | Prise en compte de l'intégrité de surface pour la prévision de la tenue en fatigue des pièces usinées en fraisage[END_REF][START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF] takes into account all the surface defect size distribution and allows for a link between the fatigue limit scatter and the studied defect population.

Despite the great number of studies and models dealing with the effect of machining in the literature, the fatigue strength of machined components is still an open question. This is all the most true when, as in the present study, a wide range of defects and surface conditions of machined components presenting different sizes and shapes has to be faced.

The investigation described in this paper is both experimental and numerical. A large experiment campaign is conducted on specimens with different surface topology obtained by end milling process with different milling conditions. The experimental results allow to plot a diagram inspired from the Kitagawa-Takahashi type diagram [START_REF] Kitagawa | Applicability of fracture mechanics to very small cracks[END_REF] showing the effect of the surface roughness on the fatigue life. The observed behavior is modeled using Finite Element simulations together with the weakest link concept. Two different approaches are tested, surface and volume-based approaches developed and presented in a previous study [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF]. The suitability and limitations of these approaches are discussed based on the comparisons with experimental results.

2. The AA7050-T7451 alloy and the experimental procedure

The material

The material under investigation is the AA 7050 Aluminum alloy (Al Zn6CuMgZr) and is provided in a 30 mm thick sheet form.

The chemical composition of the AA7050-T7451 alloy is given in Table 1. The rolling process to obtain the material sheet generates a high recrystallization rate. Areas of consolidation of recrystallized and non-recrystallized grains can reach a few millimeters of length. Grains have dimensions ranging from 5 to 300 μm in rolling and transverse directions, and 5-50 μm in depth (see Fig. 1). Different types of inter- metallic particles (Mg Si Al Cu Fe MgZn AlCuZn , , ,

, etc…) are present at the grain boundaries and in the recrystallized grains (Fig. 1). These particles can be at the origin of crack initiation under fatigue loading due to their brittle nature [START_REF] Shahzad | Influence of anodizing process on fatigue life of machined aluminium alloy[END_REF][START_REF] Gupta | Fatigue crack surface crystallography near crack initiating particle clusters in precipitation hardened legacy and modern Al-Zn-Mg-Cu alloys[END_REF].

Monotonic tensile tests are conducted, in respect to the standard ASTM E8M-04, using two extensometers to measure the strain in the length and width of the specimen (geometry shown in Fig. 2). Two to Three specimens are tested for each direction ( °°0 , 45 and °90 to the rolling direction). The anisotropy is slightly marked and the ductility is limited with a maximum elongation around 13% (Table 2). In addition, microhardness measurements are conducted along the sheet depth and show slight variations around 157 Hv/0.5.

Specimens are machined out of the center of the aluminium sheet. The machining process can change the surface integrity of manufactured parts and therefore change their fatigue strengths. Several studies have shown that in the case of end milled AA7xxx alloys, the geometrical aspect of the surface is the dominant factor on fatigue behavior [START_REF] Brunet | PhD thesis: Influence des contraintes résiduelles induites par usinage sur la tenue en fatigue des matériaux métalliques aéronautiques[END_REF][START_REF] Shahzad | Influence of anodizing process on fatigue life of machined aluminium alloy[END_REF][START_REF] Suraratchaï | Modelling the influence of machined surface roughness on the fatigue life of aluminum alloy[END_REF]. Residual stresses as well as the microstructural changes are, in this context, localized to a small thickness layer and will therefore be neglected in the presented study.

Fatigue experimental conditions

All fatigue tests are carried out under fully reversed plane bending (R = -1). A specific specimen geometry is developed for this study (Fig. 3). Four chanfrens are machined to avoid corner cracks. Under bending loading, maximum stress is located in the upper (or lower) surface of the gauge length specimen. Tests are conducted at a frequency of 75 Hz with a resonant Rumul Cracktronic type machine at room temperature and ambient air. Fatigue tests are stopped when a frequency drop of 0.7 Hz is reached (wich corresponds to a minimum crack length of 5 mm) or when 2.10 6 cycles are reached.

Fatigue tests are carried out according to the StairCase method described by Dixon and Mood [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] and according to the standard ISO12107 [START_REF]Metallic materials -Fatigue testing -Statistical planning and analysis of data[END_REF]. The step for the StairCase method is 10 MPa and the fatigue strength assessed for 2.10 6 cycles. 15 samples per batch are used except for the M5.5 surface state batch, where 10 specimens are tested. Specimens that survived in the staircase procedure were retested at a higher load in order to identify the Wöhler curve shape. These retested specimens simply give an overview of the SN curve in the finite fatigue life regime and do not offer any additional information since they already survived a lower loading level and hence do not represent the original population.

In order to assess the machining surface state effect on the fatigue behavior, three surface state batches are prepared via milling operation with a KX30 type high speed machining center. Some specimens are mirror-polished so as to have a surface state with a perfect surface plane geometry and a reference for the fatigue tests. The notation used for the different machining batches is referred to the arithmetic mean surface height S a (i.e: M1.1: Machined surface state with an arithmetic mean surface height value = S 1.1 μm a

). The various surface states used in this study are shown in Fig. 5 and are as follows:

-M1.1 surface, the surface state of industrially machined components. This batch is machined using a tool with a high nose radius and the usual industrial cutting conditions. -M4.2 surface, voluntarily degraded. This batch is machined using a tool with a low nose radius and degraded cutting conditions. -M5.5 surface, voluntarily highly degraded. The M5.5 surface state is machined under the same cutting conditions as for the M4.2 batch, but with a very slight forward inclination of the tool (offset angle of approximately °0.05 ) (Fig. 4). In this case, the rear teeth is not in contact with the surface specimen, which greatly modifies the surface state (Fig. 6). The difference between the two surface states M4.2 and M5.5 is shown in Figs. 6 and7. -P0 surface, this surface state is manually polished in the laboratory and used as the reference in fatigue.

The machining conditions for the batches M1.1, M4.2 and M5.5 are summarized in Table 3.

After machining, the surface topography of the specimens is scanned with a 3D BRUKER type profilometer (see Fig. 6). This profilometer, with a vertical resolution along the Z axis of 10 nm and a spatial resolution along the X and Y axes of 1.9 μm, provides high quality scans and accurate measurements of the different surface parameters. The average usual surface state parameters (Eqs. ( 1)-( 6)) for the different batches are characterized and presented in Table 4. R a , arithmetic mean roughness:

∫ = R l Z x dx 1 | ( )| a n l 0 n (1)
S a , arithmetic mean height:

∬ = S S Z x y dxdy 1 | ( , )| a S (2)
S q , root mean square height:

∬ = S S Z x y dxdy 1 ( ( , )) q S 2 (3)
S z , maximum height:

= + S Z Z | | z m a x m i n (4)
S ku , Kurtosis:

∬ = S S Z x y dxdy 1 ( ( , )) ku q S 4 4 (5)
S vi , groove's fluid retention index:

= S S V h S 1 ( ) vi q V 0.8 (6)
with V h ( ) V 0.8 : the void volume of the valley section that represents 80% of the material ratio curve [START_REF]Leach Characterisation of areal surface texture[END_REF], and S: the surface of the measured area.

The repeatability of the surface states produced on the specimens of each batch is verified by the 3D scanning. The standard deviation range of the arithmetic mean height S a for each batches is presented in Table 4. In order to simplify the study, specimens of each batch will be considered identical in the following analysis. The values presented in Table 4 are therefore an averaging of all specimens measurements of each batch. The arithmetic mean roughness R a measured along the specimen direction depends on the position of the measured line Fig. 7. In the Table 4 the maximum value of the R a value measured on these different lines is presented.

The batch M1.1, machined under the classic industrial cutting parameters, shows low surface roughness. Resulting machining grooves have a maximum depth of 3 μm and around 40 μm in width (Fig. 7).

For the M4.2 and M5.5 batches, 2D profiles extracted from 3D scans show that 2D profile greatly depends on the position of the line considered (Fig. 7). Due to the high value of the feed rate, areas with high difference of altitude are created by the path of the machining tool. The depth of the machining grooves for the M4.2 and M5.5 batches is around 10 μm, whereas maximum height reaches 29 μm for the M4.2 batch and 18 μm for the M5.5 batch (Figs. 6 and7). Even if the M4.2 batch has a higher R amax than the M5.5 batch, which can be due to the machining conditions (i.e. slight difference in the cutting grooves depth), its S a value is lower (Table 4).

Experimental results

Fatigue test results

Fig. 8 shows the results of the fatigue tests in the form of a Wöhler diagram. The well known plateau in fatigue is observed above 3.10 5 cycles. The following analysis will be focused on the High Cycle Fatigue domain and on the level of each plateau for each batch. As observed on the Wöhler curves (Fig. 8) and on the StairCase results (Table 6), the scatter is not pronounced. The StairCase approach allows to estimate a standard deviation of the fatigue strength lower than 5.3 MPa for each batch.

Summarizing the fatigue strength from the different surface states in Table 5, it is clear that the fatigue strength decreases gradually by increasing the surface arithmetic height parameter S a .

For the M1.1 surface state, the fatigue strength is 15 MPa (approximately 10%) lower than the strength of the polished batch. This means that for the studied aluminium alloy it is necessary to highly degrade the surface roughness to observe a more pronounced effect on fatigue. The M4.2 batch presents a more significant decrease of the fatigue strength with 141 MPa. For the M5.5 batch, with a fatigue strength of 119 MPa, decrease of the fatigue strength reaches 32 %.

Crack initiation localization

After all fatigue tests, specimen surfaces are analyzed using optical microscopy. Crack initiation occurs at a single location and coalescence phenomena is not observed. Crack initiation occurred randomly on the gauge length of specimens. SEM observations of the fracture surfaces show that all crack initiations occurred at the upper surface or at few micrometers below the surface of the specimens (see Figs. 9 and10). For the majority of the specimens and whatever the batch, initiation starts mostly (more than 90 percent of the observed fracture surfaces) from particles (Fig. 10a)) or from MgZn 2 precipitates at a nanoscale detected via Energy Dispersive Xray Spectroscopy (EDX) [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF]. In a few cases, initiation starts from material matrix (Fig. 10b)). Only the M4.2 batch at low number of cycles (below 2.10 5 cycles) shows systematically initiation without any particles or precipitates.

For the M1.1 surface states, due to the overlap between the multiple grooves of small depth and width, it is difficult to localize the crack initiation sites on the machining grooves. For the M4.2 and the M5.5 batches however, combination of SEM observations and 3D surface scans after fatigue test allows to precisely locate the crack initiation site on the 3D surface scan (Fig. 10). Fig. 10 shows that all crack initiation sites are located at the bottom of the machining grooves.

The machining tool used is a two-tooth tool. For the M4.2 batch, the slight difference in height between these teeth is at the origin of the difference in depth of the order of 2-3 μm observed on the 2D profiles (Fig. 10). It can be seen that this small difference in depth is sufficient to locate almost all the crack initiation sites at the bottom of the deepest machining groove (Fig. 10). For the M5.5 batch machining grooves created by the two teeth of the machining tool do not show a significant difference in height. The set of crack initiation sites is therefore randomly located at the bottom of the grooves created by each of the teeth (Fig. 11).

In Fig. 11, the crack initiation sites are positioned on the 3D scan of the surface and compared to the measurement of arithmetic roughness R a performed in the length direction of the specimen and at different positions according to its width. For the two batches M4.2 and M5.5, crack initiation sites coexist in areas of low and high roughness (ie crack initiation sites at 1 mm and 3.5 mm across the width of the specimen). Fig. 11 allows to illustrate that the initiation is not systematically located on the lines of higher arithmetic roughness R a and that a fatigue resistance criteria based on this quantity are therefore questionable.

Modeling the AA7050 high cycle fatigue behavior in the presence of different surface states

The aim of this part is to propose a high cycle fatigue criterion that can estimate the fatigue resistance of machined components whose surface topology may be periodic or composed of local defects. It is hence necessary to identify surface state parameters that are capable of characterizing the impact of the surface topology on the fatigue behavior. The correlation between different assessments of the stress concentration factors and the fatigue results will then be analyzed.

Correlation between fatigue strength and surface condition characteristics

Among the various parameters used to characterize a surface state, the arithmetic roughness R a is a frequently used parameter in the field of machining. Using the 3D profilometer, R a measurements were made for each batch of specimens in the direction of loading (Fig. 11). These figures show that R a depends on the position of the measurement across the width of the specimen.

The maximum arithmetic roughness R amax according to the direction of the specimen was identified for each surface condition by taking different measurements over the width of the specimen. The evolution of the fatigue strength as a function of R amax is presented in Fig. 12a. A decrease in the fatigue strength is well observed with the increase of R amax . However, the surface state M4.2 having a maximum arithmetic roughness of 9.3 μm has a better fatigue strength than the surface state M5.5 having a maximum arithmetic roughness of 7.1 μm. In order to take into account the whole surface topology and not the sole roughness over a line, the arithmetic mean height of surface S a was also measured for each batch. Fig. 12b shows that the fatigue strength decreases gradually as S a increases. S a appears therefore to be a more convenient parameter for characterizing the impact of surface topology on fatigue.

The same approach to search for a possible correlation between surface state parameters and fatigue strength has been applied to all the usual surface condition criteria (Sq, Sz, Skewness Ssk and Sku). None of these criteria seems to show a clear link between the fatigue strength and the surface topography of the different tested batches. For example, Fig. 12c and12d illustrate the evolution of the fatigue strength as a function of the Sq and Sz surface parameters that are used to characterize the impact of the surface condition on the fatigue behavior by some authors [START_REF] Guillemot | Prise en compte de l'intégrité de surface pour la prévision de la tenue en fatigue des pièces usinées en fraisage[END_REF][START_REF] Souto-Lebel | Rôle de l'intégrité de surface dans la tenue en fatigue d'un acier bainitique après fraisage de finition[END_REF][START_REF] Griffiths | Manufacturing surface technology: surface integrity and functional performance[END_REF] and for which the correlation is not direct. In the same way as for the R amax parameter, the S z parameter, which takes into account the lowest positions in the surface grooves, and the S q parameter, which describes the standard deviation of the surface roughness [START_REF]Leach Characterisation of areal surface texture[END_REF], are higher for the batch M4.2 than for the batch M5.5.

In conclusion, for the investigated surface states, the obtained results show that the surface parameter S a is the best suited parameter to capture the effect of the surface topology on the fatigue behavior of the end milled parts.

Literature criteria estimating the impact of surface condition on the fatigue strength

This paragraph presents the literature criteria that correlate the fatigue strength with the surface topography [START_REF] Murakami | Metal fatigue: effects of small defects and nonmetallic inclusions[END_REF] or with the stress concentration generated by the surface topography (mainly located at the valley of the grooves and described by a peak value of the stress, generated as a result of a nominal loading, often applied upon the component in a perpendicular direction to the grooves) [START_REF] Inglis | Stresses in a plate due to the presence of cracks and sharp corners[END_REF][START_REF] Arola | Estimating the fatigue stress concentration factor of machined surfaces[END_REF][START_REF] Neuber | Theory of notch stresses[END_REF][START_REF] Souto-Lebel | Rôle de l'intégrité de surface dans la tenue en fatigue d'un acier bainitique après fraisage de finition[END_REF].

The Murakami criterion [START_REF] Murakami | Metal fatigue: effects of small defects and nonmetallic inclusions[END_REF] is one of the most used criteria for predicting the impact of surface or internal defects on the HCF behavior. Murakami also adapted his criterion to machined surface conditions. The defect size parameter noted area is in this case estimated as a function of the depth and the spacing between successive machining grooves [START_REF] Murakami | Metal fatigue: effects of small defects and nonmetallic inclusions[END_REF]:

= ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ( ) area F a 0.65 a b 2 2 (7)
with a the depth of the groove, b 2 the spacing between 2 successive grooves and ( )

F a b 2
is a geometric correction factor which depends on the parameters a and b 2 (see Fig. 13). Let's recall that the fatigue strength is predicted by the Murakami criterion through the equation:

= + σ H area 1.43( 120) ( ) D v 1/6 (8)
with H v the Vickers hardness of the material. Since the Murakami criterion is based on a roughness profile considered over a line, the calculated predictions will be based on the line where the defect size area (and hence the depth value a) is maximal. The fatigue resistance of the polished batch P0 is not predicted with the Murakami criterion but considered as the experiment fatigue resistance found for this batch (174 MPa). Fig. 15 shows that the Murakami cri- terion does not predict successfully the fatigue strength of the machined surface states. Considering the machining grooves as "ideal" defects having the same projected area according to the loading direction via the area parameter does not make it possible to estimate the fatigue strength of the different tested batches.

In the literature, there is a number of empirical fatigue criteria based on the estimation of the stress concentration generated by the surface topology measured over a line. Table 7 contains the expressions of the 3 analytical stress concentration estimations studied in this section.

For these expressions, a is the depth of the groove, ρ the radius at the bottom of the defect (machining groove), b 2 the distance between two successive grooves, R z the average maximum height of the measured profile, R t the maximum height of the measured profile and n a parameter equal to 2 in case of tensile stress loading. For a real profile, the identification of the depth a and the radius of the groove ρ is not easy. As shown in Fig. 14, the radius of the red circle for example describes the machining groove in its totality, the stress concentration factor generated for this radius via the Inglis expression is 1,2. The stress concentration factor generated by considering the radius at a 7 is therefore very dependent on the observation scale and the measurement of the radius and the depth of the groove. Fig. 14 illustrates, in the case of the M4.2 batch, the important error that can be generated by the choice of the observation scale.

For the Neuber and Arola estimations, the profile along the axis of the specimen maximizing the estimated stress concentration factor is used. The depth of the groove is considered equal to the distance between the top and the bottom of the groove and ρ is measured at the local scale (example of the green radius in Fig. 14). Figs. 14 and15 show that, regardless of the approach considered (Inglis, Neuber or Arola), for the different batches, the fatigue strength does not seem to correlate with local stress concentration estimations.

As the estimation of the stress concentration factor in the loading direction via the measurement of a 2D profile has shown its limits, Souto-Lebel [START_REF] Souto-Lebel | Rôle de l'intégrité de surface dans la tenue en fatigue d'un acier bainitique après fraisage de finition[END_REF] proposed a criterion based on surface parameters. This criterion determines, via measurements made over the entire gauge length surface of the specimens, an effective fatigue stress concentration factor presented in Table 7. In our case, the material parameter a c that minimizes the error predictions is 10 μm. Even though surface parameters are considered, the predictions made with the Souto-Lebel criterion do not seem to correlate with the fatigue limits obtained for the different tested batches (Fig. 15).

In conclusion, applying analytical approaches [START_REF] Inglis | Stresses in a plate due to the presence of cracks and sharp corners[END_REF][START_REF] Arola | Estimating the fatigue stress concentration factor of machined surfaces[END_REF][START_REF] Neuber | Theory of notch stresses[END_REF][START_REF] Souto-Lebel | Rôle de l'intégrité de surface dans la tenue en fatigue d'un acier bainitique après fraisage de finition[END_REF][START_REF] Murakami | Metal fatigue: effects of small defects and nonmetallic inclusions[END_REF] that try to estimate the stress concentration factors from 2D or 3D profiles measurements on machined surface states is not easy due to the complex geometries of the real profiles. This complexity can not be approximated by a geometry with a single notch. Even though this type of approach is applicable to turned parts, its use on machined parts by end milling process is questionable.

In order to estimate the stress field generated by the complex surface topology resulting from the end machining process, a finite element calculation based on the 3D scan of the real surface can be used. Being able to have the stress field generated by a machined surface state is a first step towards the development of a HCF design approach. It is also necessary to use a proper fatigue criterion capable of accounting for the gradient and size effect.

A probabilistic fatigue criterion to account for the size effect

This chapter aims at presenting a probabilistic approach built to account in a more efficient way for the roughness due to end milling.

Previous work of the same authors carried out on specimens containing different sizes and numbers of simplified geometry defects has shown that Aluminum AA7050 is very sensitive to the statistical size effect [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF]. The results obtained are summarized on a Kitagawa-Takahashi diagram presented in Fig. 17, in which it can observed that:

• Fatigue limit decreases gradually when increasing defect size • An increase in the number of hemispherical defects of exactly the same size (from 1 to 22) causes the fatigue strength to decrease significantly.

When turning to the case of grooves due to machining, there is every reason to think that the number and the length of those particular defects can affect the fatigue strength for the same reasons as for the defects of simplified geometry.

As the statistical size effect seems to be the reason why the fatigue

Table 7

Expressions of the stress concentration factors K t proposed by Inglis [START_REF] Inglis | Stresses in a plate due to the presence of cracks and sharp corners[END_REF],

Neuber [START_REF] Neuber | Theory of notch stresses[END_REF], Arola [START_REF] Arola | Estimating the fatigue stress concentration factor of machined surfaces[END_REF] and of the fatigue stress concentration factor K f pro- posed by Souto-Lebel [START_REF] Souto-Lebel | Rôle de l'intégrité de surface dans la tenue en fatigue d'un acier bainitique après fraisage de finition[END_REF]. strength is affected, it is proposed to apply a probabilistic approach based on the weakest link concept allowing to integrate the entire stress field at the surface and to consider the impact of the size of the highly stressed area on the fatigue strength. Fig. 17. Predictions of fatigue behaviour a surface and a volume approach in the case of polished specimens with different size and number of hemispherical defects [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF].

K t Inglis K t Neuber K t Arola K f Souto-Lebel + 1 2 a ρ + n 1

Probabilistic framework

This section is devoted to the development of a HCF criterion able to predict the effect of surface defects on fatigue. The proposed model is defined, as is the case of most fatigue criteria, by an equivalent mechanical quantity (i.e. stress, strain, energy, etc.), which is compared to a threshold quantity, for a given number of cycles. In the following, for convenience, stress quantities will be employed. Crack initiation is therefore predicted once the equivalent stress is greater than or equal to the threshold stress value. This condition for crack initiation is defined by: ⩾ σ σ eq th [START_REF] Campbell | Microstructural characterization of Al-7075-T651 chips and work pieces produced by high-speed machining[END_REF] Note that in this section, the maximum principal stress amplitude is considered as responsible for crack initiation. Even though the maximum principal stress criterion is not well suited to model the fatigue behavior of ductile material, this criterion is chosen for its simplicity and is used below to explain the proposed methodology. It has been verified that the use of other equivalent stresses leads to very close predictions (see Section 5.2).

Fig. 18. Steps followed for the numerical simulation. In order to take into account the inherent stochastic nature of the fatigue phenomenon it is proposed, as part of the modelling framework, to use a two parameter Weibull distribution [START_REF] Weibull | A statistical theory of the strength of materials[END_REF][START_REF] Weibull | A statistical distribution function of wide applicability[END_REF] to describe the threshold stress σ th .

The choice to use a two parameter Weibull distribution then makes it possible to define the probability of crack initiation. Eq. ( 10) shows the probability density function of the threshold stress.

⎜ ⎟ ⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ ⎧ ⎨ ⎩ -⎛ ⎝ ⎞ ⎠ ⎫ ⎬ ⎭ - f σ m σ σ σ σ σ ( ) exp th th m th m 0 0 0 1 0 (10)
where σ 0 is the scale parameter and m is the shape parameter (or the Weibull exponent) used to reflect the scatter associated with the threshold stress and hence the fatigue strength. The probability of microcrack initiation corresponds to the probability of finding a threshold stress σ th that is less than the applied equivalent stress σ eq (e.g: maximum principal stress). This can be ex- pressed as:

∫ = < = P σ σ f σ dσ ( ) ( ) F t h e q σ th th 0 0 eq 0 (11)
Using the equation of Weibull's distribution function, the integration allows to express the failure probability as follows:

⎜ ⎟ = - ⎡ ⎣ ⎢ -⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ P exp σ σ 1 F eq m 0 0 (12)
To obtain the probability of crack initiation for the complete structure P F , the weakest link theory is used [22]. The probability of survival of the structure -P

(1 ) F is defined as the product of the survival probabilities of each elementary volume or surface. In this work it will be firstly assumed that crack initiation is essentially a surface phenomenon, where the quantity S 0 corresponds to a reference surface area of 1 mm 2 and S Ω is the total surface area of the domain Ω corresponding to the specimen (see Fig. 16).

∏ -= -

P P 1 ( 1 )
F S F Ω 0 (13)
Hence the probability of survival of the complete structure is given by:

∫ ⎜ ⎟ = - ⎡ ⎣ ⎢ - ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ P exp S σ σ dS 1 1 F S eq m 0 0 Ω (14)
Introducing the surface stress heterogeneity factor on the component surface H ms [39]:

∫ ⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ H S σ σ dS 1 ms S eq max m Ω Ω (15)
with σ max the maximum equivalent stress on the component surface:

= σ max σ ( ) max S eq Ω (16)
The expression for the failure probability of the total structure becomes:

⎜ ⎟ = - ⎡ ⎣ ⎢ - ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ P exp S S H σ σ 1 F m s max m Ω 0 0 (17)
In order to simplify the expression and to make a link with the high cycle fatigue strength, the macroscopic fatigue strength σ a can be ex- pressed via the following expression:

= ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ - ( ) σ σ K S S ln H a t P ms m 0 0 Ω 1 1 1/ max F (18)
where K tmax is the maximal equivalent stress concentration factor which corresponds to the maximal equivalent stress divided by the nominal applied stress.

= K σ σ t max applied max (19)
In the case of polished specimen and a simple uniaxial loading, the surface stress heterogeneity factor H ms could be easily obtained by an analytical calculation [START_REF] Flaceliere | Probabilistic approach in high-cycle multiaxial fatigue: volume and surface effects[END_REF]. In the case of the machined surface states, the surface stress field is more complex and numerical finite element simulations are necessary to assess this quantity.

The proposed criterion is a 2 parameters model with m and σ 0 that have been identified in a previous work [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF]. To identify these parameters, the criterion has been applied on polished specimens with different sizes and numbers of hemispherical defects (Table 8) under a surface and a volume formulations.

For the volume approach, the maximum principal stress distribution is considered in a domain consisting of a volume V Ω with a constant depth * d from the free surface S Ω . Hence, the stress heterogeneity factor H mv is expressed as follows:

∫ ⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ H V σ σ dV 1 mv V eq max m Ω Ω (20)
where σ max , the maximum equivalent stress on the component volume, reads:

= σ max σ ( ) max V eq Ω (21)
In the same way as for the surface approach, fatigue strength can hence be predicted as: (22) where V 0 represents the reference volume.

= ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ - ( ) σ σ K V V ln H a t P mv m 0 0 Ω 1 1 1/ max F
For a value of * d equal to 50 μm, the two parameters m and σ 0 are identified in [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF] and are presented in Table 8. Fatigue strength predictions via the volume approach are also presented in Fig. 17 [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF]. Better correlation with the experimental results is found via the volume approach where the fatigue strength in case of 22 defects of Ø60 μm is predicted with less error (9%).

Predictions for the AA7050 Alloy containing end machined surface states

The objective of this part is to test the predictivity of the two (surface and volume) approaches in the case of machined surface conditions. The parameters of the two approaches will be those determined in [START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF] from tests carried out on specimens containing hemispheric defects and presented in Table 8.

For both surface and volume formulations, parameters H m and K tmax need to be identified for each surface state. The periodicity of the profile allows to express the stress heterogeneity factor of the entire specimen as the product of the heterogeneity factor of a cell by the number of cells considered. A cell is seen as a representative area of the investigated surface state (see Fig. 16). [START_REF] Hild | A statistical approach to the rupture of brittle materials[END_REF] Note that due to the periodicity of the studied surface state, the stress heterogeneity factors H msm and H mvm calculated on the representative area or volume are equal respectively to the stress heterogeneity factors of the sample H ms and H mv .

= H S nS H 1 [ ] ms m ms Ω m (23) = H v nV H 1 [ ] mv m mv Ω m
To identify H m and K tmax , finite element simulations are necessary in order to determine the generated stress in each element on the surface or in the considered volume. For that, FE calculations integrating the real profile of the surface state resulting from scans made with the 3D profilometer, are carried out.

The approach followed is shown in Fig. 18 and begins with a profilometric scan of the surface geometry of the specimens. All the scans containing the and height of each scanned point are recorded using a spatial resolution of 1.9 μm. After verifying the repeatability of the surface states for the different specimens of each batch, a representative area of the surface condition is selected.

All the scans are recorded using a spatial resolution of 1.9 μm. A first filtering step conventionally used in profilometry is applied to the raw data to solve measurement problems (missing pixels, aberrant altitude values at certain points, etc.). In addition, a second Gaussian regression filter is applied in order to remove any stress concentration singularities generated by the measurement noise. This technique is usually employed when dealing with geometries inspired from real measurements [START_REF] Suraratchaï | Modelling the influence of machined surface roughness on the fatigue life of aluminum alloy[END_REF][START_REF] Ås | Fatigue life prediction of an aluminum alloy automotive component using finite element analysis of surface topography[END_REF]. On a global scale, the Gaussian filter used does not change the surface condition (Fig. 19). On the other hand, at the more local scale, the filter allows to smooth the obtained profile and eliminate the discontinuity and singularities problems generated by the raw measurement process (see Fig. 19).

The cloud of points collected from the area of interest is then transformed into a volume geometry that is meshed afterwards using the GMSH software. The mesh size chosen on the surface is the one given by the 3D scan and is equal to 1.9 μm.

Elastic calculation is performed using the Finite Element Method. Elements are tetrahedral with reduced integration. Boundary conditions are as follows:

• Displacement Ux = 0 along of the plane = X 0 (surface A).

• Displacement Uy = 0 along of the line = = X Y 0, 0.

• Displacement Uz = 0 at the point = = = X Y Z 0, 0, 0.

• A bending loading is applied along the surface B (Fig. 20) and is representative of the experimental bending loading applied to specimens.

A post-processing step then makes it possible to extract the stress fields at the surface or in depth and to apply the probabilistic model developed over an area of the size of the cell. Note that in order to avoid boundary effects, the FE calculation is performed over an area larger than the cell size. All results presented in Fig. 21 are obtained applying a nominal maximum stress in bending σ a of 100 MPa.

Due to the simplicity of the applied loading, the analysis of the stress field is focused on the first principal stress σ I . In our work, the stress concentration factor is therefore defined on each point by:

= K σ σ t I a (25)

Surface stress field analysis

The numerical calculation is first carried out for the M1.1 surface state. The comparison of the crack initiation sites with the numerical calculation is not easy given the large number of overlapping machining grooves with shallow depth and width (Fig. 21a). The numerical calculation shows a localized stress concentration at the bottom of the machining grooves. The maximum stress concentration factor value K t for this surface state is of the order of 1.5.

In Fig. 21b and21c, the surface stress field obtained by numerical simulation is compared to the location of the crack initiation sites of the M4.2 and M5.5 surface states. Fig. 21 shows that for these two batches the maximal principal stress is higher at the bottom of the machining grooves and more importantly on the areas between the highest peaks (areas of high roughness values). This result does not agree with the random location of the crack initiation sites at the bottom of these grooves. This result justifies the choice of a non-local approach to predict fatigue behavior.

Individual measurements of the stress concentration factor along the machining ridges are made for the two surface states M4.2 and M5.5. For both surface states, the stress concentration factors are close and K t varies between 1.2 and 1.7 at the machining grooves. On the other hand, K t reaches values of the order of 1.8 in local areas between the highest peaks. With these values of stress concentration factor, the maximum local stress reaches 270 MPa, which remains below the yield stress of the material of 428 MPa (see Section 2.1). The hypothesis of elastic behavior is therefore well verified. Despite their close stress concentration factors, the fatigue strengths of the M4.2 and M5.5 batches are very different, respectively 141 and 119 MPa. This result shows the limits of a local approach and the interest of taking into account the entire stress field over the surface of the component.

Comparison between the predictions of the surface approach and the experimental results. 4).

For different failure probabilities, the surface approach predictions show a good agreement with the experimental results (Fig. 22) except for the M5.5 surface state where the model incorrectly predicts a better performance than for M4.2 surface state. At a failure probability of 50% (PF = 0.5), the estimated error between the experimental results and the predictions via the surface approach reaches 30% for the M5.5 surface state against less than 3% for the M1.1 surface state and 10% for the M4.2 surface state. Note that the use of other equivalent stresses than the first principal stress leads to very close predictions of the fatigue behavior (i.e: the scatter between predictions considering the first principal stress, Von-Mises equivalent stress and Tresca equivalent stress is lower than 3 MPa for the different surface states).

The volume approach integrating the stress distribution in a limited volume with a depth d * = 50 μm (Fig. 23) provides better predictions than the surface approach. Volume approach allows to keep the advantages provided by the surface approach in describing the fatigue behavior and to give a better prediction of the low fatigue strength of M5.5 batch (Fig. 24). The predicted fatigue strength for M5.5 surface state is however non conservative with an error of 9%. The difference between the predictions of the two approaches on the batches M4.2 and M5.5 can be explained analyzing the size of the highly stressed area or volume. The highly loaded area where the first principal stress is 20% higher than the first principal nominal stress applied is close for the two batches and the fatigue limits predicted using the surface approach are almost equal for the two batches studied.

The highly loaded volume is larger for the M5.5 batch compared to the M4. and the fatigue limit predicted using the limited volume approach is therefore lower for the batch M5.5.

To conclude, the proposed model in its surface and volume configurations, whose parameters have been identified by tests conducted on specimens with artificial defects, is able to predict the fatigue behavior of the P0, M1.1 and M4.2 surface conditions. For the M5.5 batch the volume approach shows more accuracy than the surface approach in predicting the fatigue strength.

Prediction of fatigue resistance in the case of different structure size

The proposed probabilistic model has the advantage of naturally integrating the scale effect and thus being applicable to different sizes of structures. For example, the fatigue strength of an end-milled component, with "standard" industrial surface conditions (M1.1 surface state), and larger than the tested specimen can be identified by considering the ratio between the components loaded surface and the specimens loaded surface (Fig. 25). Fig. 25 illustrates that an increase in the size of the structure generates a decrease in fatigue strength. For instance, a drop of 25% is expected when the loaded surface increases by a factor of 100.

Conclusion and prospects

The main conclusions of this work can be presented as follows:

-Fully reversed plane bending fatigue tests campaign have been conducted on specimens with different surface roughness levels obtained via end milling process. It is necessary to greatly increase the surface roughness in order to observe a significant decrease in the fatigue strength of the AA7050 alloy. -The surface roughness S a seems to be the best roughness parameter to characterize the effect of the surface topology on the fatigue behavior of the end milled parts. -Fatigue strength depends on the local stress concentration as well as on the dimension of machining grooves (scale effect). -Fatigue strength predictions are in good agreement with the experimental results. The proposed approach based on Weibull's distribution law and the weakest link concept allows to predict with accuracy the fatigue strength for polished specimens as well as specimens with end machined surface states.

As prospects, the proposed criterion will be applied for different conditions:

-The criterion will be applied to other loading cases, in particular multiaxial loading. -That kind of approach can be applied on other types of materials such as TA6V alloy. -The proposed criterion will be applied in the case of real components with different sizes and the same machined surface state (scale effect).

Fig. 1 .

 1 Fig. 1. (a) AA7050-T7451 alloy RD: Rolling Direction, TD: Transverse Direction, ND: Normal Direction, (b) SEM images of the intermetallic particles observed on the AA7050 alloy sheet.

Fig. 2 .

 2 Fig. 2. Geometry of tensile specimen.

Fig. 3 .

 3 Fig. 3. Geometry of plane bending specimen showing four chanfrens.

Fig. 4 .Fig. 5 .

 45 Fig. 4. (a) M4.2 surface state: machining + heeling. (b) M5.5 surface state: machining with tool offset angle of °0.05 .

Fig. 6 .

 6 Fig. 6. 3D surface scan of (a) M1.1, (b) M4.2 and (c) M5.5 batch.

Fig. 7 .

 7 Fig. 7. Machining grooves 2D profiles measured (a) on the middle of the specimen (b) at 1.3 mm from the middle of the specimen.

Fig. 8 .

 8 Fig. 8. Wöhler curves for different surface states.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Failure surface, initiation start from (a) particle (b) material matrix.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Comparison between crack initiation sites location and arithmetic roughness R a (a) for (b) M4.2 and (c) M5.5 surface state.

Fig. 13 .

 13 Fig. 13. (a) Simulation of surface defects with a succession of grooves, (b) KI factor for periodic surface grooves [15].

Fig. 14 .

 14 Fig.14. Effect of the ρ parameter on stress concentration prediction for a machined surface.

Fig. 15 .

 15 Fig. 15. Evolution of the fatigue strength as a function of the predictions according to the (a) Murakami, (b) Souto-Lebel, (c) Neuber and (d) Arola approach.

Fig. 16 .

 16 Fig. 16. M4.2 surface scan showing the different surface area used for fatigue analysis.

Fig. 19 .

 19 Fig. 19. Illustration of the impact of the gaussian filter on the raw profile.

Fig. 20 .

 20 Fig. 20. FE model based on the actual surface topography of machined specimen.

Fig. 21 .

 21 Fig. 21. Comparison between the surface topology and the principal stress distribution obtained via FE simulation for the (a) M1.1 (b) M4.2 and (c) M5.5 surface states.

Fig. 23 .

 23 Fig. 23. Illustration of the * d parameter in the Finite Element Model.

Fig. 24 .

 24 Fig. 24. Comparison between the predictions of the volume approach and the experimental results.

Fig. 25 .

 25 Fig. 25. Evolution of the fatigue strength of a milled component (M1.1 surface state) as a function of the size of the loaded surface. The point corresponds to the experimental fatigue strength on M1.1 test specimens.

Table 1

 1 Chemical composition of the AA7050-T7451 alloy, in weight percentage.

	Element	Al	Zn	Cu	Mg	Zr	Ti	Si	Fe	Mn
	Weigth (%) bal. 6.027 2.221 1.847 0.102 0.039 0.038 0.095 0.01

Table 2

 2 Monotonic properties of the AA7050 alloy.

	Rolling	Yield Stress	Utlimate Tensile	Tensile Elongation
	Direction	(MPa)	Strength (MPa)	(%)
	°0	475	635	12.3
	°45	428	496	13.9
	°90	475	538	11.9

Table 3

 3 Machining conditions for the AA7050 alloy.

	Surface state N (rpm)	Fz	Z (teeth)	Nose	D (mm)	Orientation
			(mm/		Radius		tool/
			tooth/		(mm)		workpiece
			rd)
	M1.1	24 000	0.15	2	4	Ø10	°0
	M4.2	10 000	0.55	2	0.8	Ø20	°0
	M5.5	10 000	0.55	2	0.8	Ø20	°0.05

Table 4

 4 Surface roughness characteristics of the different surface state batches.

	Surface State	R amax (μm)	S a (μm)	S a (μm)	S q (μm)	S z (μm)	S ku (μm)	S vi (μm)
	P0	0.02	0.02	< 0.02	0.05	0.42	3.55	0.12
	M1.1	1.02	1.1	0.24	1.19	7.12	2.49	0.09
	M4.2	9.27	4.2	0.2	6.41	45.5	7.36	0.05
	M5.5	7.1	5.5	0.76	5.74	29.1	1.94	0.04

Table 5

 5 Average fatigue strength for the different surface states batches.

	Batch	Polished	M1.1	M4.2	M5.5
	σ a (MPa)	174	159	141	119

Table 6

 6 Test history for StairCase tested specimens.

	Surface	Nominal Stress								Specimen Number
	State	Amplitude (MPa)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
	P0	180			X		X		X		X				X		X
		170		O		O		O		O		X		O		O
		160	O										O
	M1.1

Table 8

 8 Parameters of the surface and volume probabilistic approaches as defined in[START_REF] Abroug | HCF of AA7050 alloy containing surface defects: Study of the statistical size effect[END_REF].

	Parameter	d * (μm)	m	σ 0 (MPa)
	Surface approach	-	17	240
	Volume approach	50	22	226

X X X 160 X X O O X O X 150 O O O O O M4.2 150 X X X X 140 O X X O O O X O 130 O O O M5.5 130 X X 1 2 0 O X O X X 110 O O O

Acknowledgments

This work was carried out within the FUI QUAUSI project with the support of industrial (Dassault Aviation, Europe Technologie, Figeac Aéro, Mecachrome, Spring Technologie, Precise, GEBE2, CETIM) and academic partners (LSN2-IUT Carquefou).


Cuong Pham-Nguyen

Serge Garlatti
email: serge.garlatti@telecom-bretagne.eu

Pervasive Technology-Enhanced Learning System Integrating Working and Learning Situations

Keywords: Pervasive Learning, Context-Aware, Adaptation, Task/Method Paradigm, Hierarchical Task Model I

In the p-LearNet project, we are interested in TEL systems integrating context-aware corporate learning and working activities for e-retail (shops and hypermarkets). The main issues of the p-LearNet project are: work-integrated learning and customer learning support whatever the place, the time, the organisational and technological contexts of the individual or collective learning and working processes. We propose an adaptive and context-aware model of scenario based on an interdisciplinary approach (education, computer science, social sciences, and business) for a pervasive learning system supporting working and learning situations. This model enables us to choose how to achieve activities according to the current situation. The scenario model is based on a hierarchical task model having the task/method paradigm -methods define how to achieve a task.

INTRODUCTION

Nowadays, technology-enhanced learning (TEL) systems must have the ability to reuse learning resources from large repositories, to take into account the context and to allow dynamic adaptation to different learners based on substantial advances in pedagogical theories and knowledge models [START_REF] Balacheff | 10 issues to think about the future of research on TEL[END_REF]. In computer science, Lyytinen and Youngjin have defined the following terms: mobile, pervasive and ubiquitous [START_REF] Lyytinen | Issues and challenges in ubiquitous computing[END_REF]. Mobile computing is mainly about increasing our capability to physically move computer environment with us, by means of lightweight devices and/or access to computing capacity over the network. In mobile computing, the computing model does not really change while we move. It is an important limitation. In pervasive computing, computers have the capability to obtain information from their environment and use it to dynamically build models of computing. In other words, the computer is able to adapt its model of computing according to its environment. Thus, an important issue is the acquisition and management of a context model which describes the physical environment and the system goals. Ubiquitous computing integrates large-scale mobility with the pervasive computing functionality.

For TEL systems, several terms are used: pervasive, ubiquitous, mobile, ambient and nomadic learning systems. We consider that these expressions are synonymous. They are used for highlighting a specific property in learning systems [START_REF] Hundebol | Pervasive e-Learning -In Situ Leaning in Changing Contexts[END_REF][START_REF] Sharples | Learning As Conversation: Transforming Education in the Mobile Age[END_REF][START_REF] Siobhan | Pervasive Scale: A model of pervasive, ubiquitous, and ambient learning[END_REF]. Generally, such type of learning systems used to have all properties of mobile, pervasive and ubiquitous computing. The main idea is to embed learning activities into everyday life or work by means of lightweight devices and network communications. Nevertheless, mobile, pervasive or ubiquitous learning is not just about learning using lightweight devices, but learning across context and are best viewed as mediating tools in the learning process [START_REF] Sharples | Big Issues in Mobile Learning: Report of a workshop by the Kaleidoscope Network of Excellence Mobile Learning Initiative[END_REF]. Many definitions about these type of learning systems are given in literature [START_REF] Hundebol | Pervasive e-Learning -In Situ Leaning in Changing Contexts[END_REF][START_REF] Siobhan | Pervasive Scale: A model of pervasive, ubiquitous, and ambient learning[END_REF][START_REF] Bomsdorf | Adaptation of Learning Spaces: Supporting Ubiquitous Learning in Higher Distance Education[END_REF][START_REF] Jones | Ubiquitous Learning Environment: an Adaptive Teaching System using Ubiquitous Technology[END_REF]. We can cite the following one:"Pervasive learning environment is a context (or state) for mediating learning in a physical environment enriched with additional site-specific and situation dependent elements -be it plain data, graphics, information -, knowledge -, and learning objects, or, ultimately, audio-visually enhanced virtual layers" [START_REF] Hundebol | Pervasive e-Learning -In Situ Leaning in Changing Contexts[END_REF]. In pervasive learning, some issues are as follows: combination of formal (at school) and informal (outdoor, at home, at workplace etc.) learning, integration of mobile devices in broader educational scenarios, combination of collaborative, mobile learning and inquiring learning, context-as-construct and seamless learning across different contexts [START_REF] Balacheff | 10 issues to think about the future of research on TEL[END_REF][START_REF] Sharples | Big Issues in Mobile Learning: Report of a workshop by the Kaleidoscope Network of Excellence Mobile Learning Initiative[END_REF].

In the p-LearNet project, we are interested in TEL systems integrating context-aware corporate learning and working activities in a particular framework: e-retail (retail through shops and hypermarkets). The main issues of the p-LearNet project are: work-integrated learning and customer learning support, continuous professional training at workplace, professional learning whatever the place, the time, the organisational and technological contexts of the individual or collective learning and working processes context-as-construct and seamless learning. In this project, our corporate partnerships are international companies in the retails through shops and hypermarkets. The design and engineering of pervasive learning systems must be considered as an interdisciplinary problem requiring the integration of different scientific approaches from computer science, education, commerce, social sciences, etc. While learning focuses mainly on how to support individual's and group's learning process through the pedagogical guidance and how to enhance the learner's knowledge. The workplace identifies the problems and requirements about quality and efficiency of information and services to increase market share. Moreover, working activities are about solving tasks and particularly in knowledge-intensive organizations which implies continuous learning. Inexperienced persons learn from their experienced colleagues and they acquire the knowledge necessary to solve their work tasks. However, learning is a process that takes place mainly unconsciously. Solving the particular working tasks is prior -learning is just a means [START_REF] Farmer | AD-HOC -Work-integrated Technology-supported Teaching and Learning[END_REF]. In such a framework, several innovative scenarios have been setting up according to two main learning and/or working situations for the seller and the potential customers as learners: i) Seller or customer, outside the shop counters: seller in the back office or storage areas, client at home or other place; ii) Seller in his department, alone or with a customer having resources from the Smart spaces surrounding them (large LCD screen, printers, RFID, etc.) [START_REF] Derycke | P-Learning and e-retail: a case and a flexible Sotware Architecture[END_REF]. Scenario analysis reveals that current learning and working situations could be modeled by an explicit task model partly because working and learning activities are well structured and stable. Activities, represented by tasks, cannot be achieved in the same way according to different situations. It is necessary to have a context-aware and adaptive mechanism to decide how to achieve an activity according to a given situation.

The main contribution of this paper is an adaptive and context-aware model of scenario based on an interdisciplinary approach (education, computer science, social sciences, and business) for a pervasive learning system supporting working and learning situations. This model enables us to choose how to achieve activities according to the current situation. The scenario model is based on a hierarchical task model having the task/method paradigm -methods define how to achieve a task. We defined a context model to deal with dynamic contextscontext-as-construct. The context-aware and adaptive process can be viewed as the selection of the relevant methods for a given task according to the current working and/or learning situation.

First of all, the context model and its organization are analyzed and explained. Secondly, we present our scenario model and its relationships with the context model. Thirdly, the context management, the detection and creation of new situations are explained. Fourthly, the adaptation process which links the context model, the scenario model and the context management is presented. Finally, the conclusion highlights the main results of this study and some perspectives.

II. CONTEXT MODELING

There have been numerous attempts to define the notion of context in different fields. We only focus on definitions giving the relationships between the physical world and users' activity. We claim that activities embedded in a particular physical world (or environment) are key issues to give us intention and meaning according to different situations and finally to determine the relevant features describing the different situations. These definitions are as follows: « learning context is used to describe the current situation of a person related to a learning activity; in addition to attributes relying on the physical world model » [START_REF] Derntl | Modeling Context-Aware E-learning Scenarios[END_REF]; « information and content in use to support a specific activity (being individual or collaborative) in a particular physical environment » [START_REF] Kurti | Designing and implementing ubuquitous learning activities supported by mobile and positioning technologies[END_REF].

In pervasive learning systems, a context model must consider a crucial context features: dynamic and/or "contextas-construct". According to Dourish, context can be viewed as an interactional problem: i) Contextuality is a relational property that holds between objects or activities; ii) The context is not defined in advance. On the contrary, the scope of contextual features is defined dynamically; iii) Context is an occasioned property, relevant to particular settings, particular instances of activities; iv) Context arises from the activity. Context cannot be separated from activities. It is actively produced, maintained and enacted in the course of activity [START_REF] Dourish | What we talk about when we talk about context[END_REF].

In "Big issues in Mobile Learning", context is viewed as follows: "context-as-construct", i.e. "context should be reconceived as a construct that is continually created by the interaction of learners, teachers, physical settings, and social environments" and "learning not only occurs in the context, it also creates context through continual interaction" [START_REF] Balacheff | 10 issues to think about the future of research on TEL[END_REF].

In retail framework, context is continuously produced during selling process: a seller can communicate with customers while revising his knowledge, checking the inventory or contacting the provider about products, etc. Our purpose is to enable learning scenarios in workplace on different mobile and ubiquitous devices to support learning and working activities. Our definition of context is not focus on static information. Instead, we pay attention to a mechanism that refines and generates dynamically context information following interactions for such learning situations in workplace.

A. Context model

Our context model has to consider the context-as-construct property, the combination of formal (at school for professional) and informal (outdoor, at home, at workplace etc.) learning, the integration of mobile devices in broader educational scenarios, and the seamless learning across different contexts. Our context model is composed of: i) a set of dimensions D c , defining a set of relevant features (represented as triplet) and their structure; ii) a set of Views V c , a view consists of a subset of features which are relevant to a given content category (activity, resource , web service) for adaptation. Thus, an adaptation process does not manage the same features for different categories of content. For a given content category, its features and those of the corresponding view can be matched in the adaptation process; iii) a set of situations S c , organized in historical dependencies. A situation is a partial instantiation of the context model consisting of the relevant features describing the current learning situation and its physical environment. A user activity can be influenced by his previous work and learning activities. As soon as a new situation is occurring, the next activity or the activity continuation can be chosen according to the historical dependencies, for instance to ensure seamless learning.

Definition 1(Context model):

Let CM be a context model, CM = <D c , V c , S c , P c >, D c be a set of dimensions D c = {d 1 , d 2 , … , d n }, where n is the number of dimensions and d i is a context dimension; V c be a set of different context views V c = {v 1 , v 2 , … , v m }, where m is the number of views, v i is a context view. Each v i represents a grouping of properties from different dimensions in term of their usage relating to a domain type, S c be a set of situations (or historical situations) S c =< {s 1 , s 2 ,… , s p },≤ s >, where s p is the current situation. A total order ≤ s on S c , where s i ≤ s s j implies that the situation s i precedes temporally the situation s j , P c be a set of predicates that is used to manage situations including: the change, detection, creation, storage.

Definition 2 (Context dimension):

Let d i be a context dimension, d i ∈ D c , > =< p i m i c i i d d d d , ,
, d C i be a set of characteristics, each of them is defined by a name and a domain, d m i be a subset of context metadata associating to the dimension d i that is used to manage the dimension d i , d P i be a subset of predicates, used for managing the dimension d i . These predicates are using metadata in d m i and dimension characteristics in d C i . The context dimensions are divided into two categories: abstract dimensions and atomic dimensions. An abstract dimension can be recursively decomposed into sub-dimensions which are abstract or atomic. An atomic dimension only consists of a characteristic set. Our context dimensions are the scenario (a hierarchal task model having a task/method paradigm), the user (a user can be a learner, a teacher, a salesman, a customer, etc.) with sub-dimensions: the role, previous knowledge, know-how, preferences, loyalty card, purchase intentions, intention of use), the retail shop, the device, the location, the time, the pedagogical tools, the network, the physical environment and the resource (learning object, services, media resource, system resource, etc.). The management of a context dimension (abstract or atomic dimension) is described by context metadata. It is a set of attributes used by predicates.

Definition 3 (Context view):

A context view v i = > < DT D cv i ,
, where cv i D is a subset of properties formed by a regrouping of some dimensions or dimension characteristics in order to describe a viewpoint about the context usage relating to a particular domain type DT,

c cv i D D ⊆ .
Context views are used to define different viewpoints about the context usage. Consequently, different adaptation categories and seamless learning strategies are specified accordingly. At present, we focus on two views: pedagogical context view and professional context view (cf. Table 1). The pedagogical view represents context for learning activities, the professional view represents context for working activities in workplace.

B. Pervasive learning situations in workplace

A learning situation in workplace is composed of the physical environment, learning and commercial setting of the user's current work situation. Several context dimensions are combined together to describe different pervasive learning situations. In a company, a learner used to work and sometimes learning and working are interweaved in a pervasive environment. A substantial part of learning does not happen during the training but during working activity performance. Learning and professional activity support must be integrated. The learning system must overcome three main obstacles: time pressure, inadequate learning support in the working context and cognitive and structural disconnection between work, knowledge, and learning [START_REF] Farmer | AD-HOC -Work-integrated Technology-supported Teaching and Learning[END_REF].

In retail shops, a seller equipped with a PTA (Personal Training Assistant) close to shelves (without a customer) can revise his knowledge about products or can continue his previous learning activity to improve his/her knowledge or he/she can verify the product's labeling or access to product information. During the selling process or the selection / decision phase, the seller can use his/her mobile device for both revising his/her knowledge about products and selling characteristics or as a coach to help him/her, etc. According to the current and past user activities, the learning system has to select the relevant way to achieve the current activity, the resources and the web services according to context changes. Moreover, the current status of activities, suspended or unfinished in the past, need be restored to ensure the seamless learning and working. The relevant activities and resources can be provided to the user according to the current user's work situation. In a formal way, a definition of the situation is given as follows: defines a complete context state associating to all interactions between the user and the learning system in workplace in a given time interval. It is specified temporally with a starting time t s i and an ending time t s j . e ij is a dimension characteristic in s i . It is acquired as follows: automatically detected by the learning system or gathered from interactions between the user and the system or dynamically inferred or generated by the learning system (based on inferences or adaptation rules defined in the adaptation model).

Our context model is represented by ontology. A situation and its historical dependencies are used to select and/or to continue an activity by means of a scenario model.

III. SCENARIO MODEL

In our framework, the goal of scenarios is to describe the learning, working and tutoring activities to acquire some knowledge domain and know-how to solve a particular problem or to support working activities. A scenario may depend on several dimensions which describe different learning and working situations: the learning domain (course topic), the learner (his know-how and knowledge levels), the tutor/teacher, the learning, working and tutoring activities (their typology, organization and coordination), the resources (documents, communication tools, technical tools, etc.), the activity distribution among learners, teachers and computers, the learning "procedures" according to a particular school or corporate.

The main role of a scenario model is to integrate mobile devices in broader learning and working scenarios, formal and informal learning and to enable us to manage seamless learning across contexts. Scenario analysis reveals that current learning and working situations could be modeled by an explicit task model partly because working and learning activities are well structured and stable. Several research studies in artificial intelligence focus on the hierarchical task model using the task/method paradigm [START_REF] Trichet | DSTM: a Framework to Operationalize and Refine a Problem-Solving Method modeled in terms of Tasks and Methods[END_REF][START_REF] Wielinga | The KADS Knowledge Modelling Approach[END_REF][START_REF] Willamowski | A development shell for cooperative problem-solving environments[END_REF]. In learning environment, hierarchical task models were also used for designing, for instance, authoring tools [START_REF] Ikeda | Task Ontology Makes It Easier To Use Authoring Tools[END_REF], learning systems [START_REF] Betbeder | Structuring collective activities with tasks and plans[END_REF][START_REF] Choquet | Modelling Knowledge-Based Components of a Learning Environment within the Task/Method Paradigm[END_REF][START_REF] Ullrich | Course generation based on HTN planning[END_REF]. The mechanism of hierarchical and recursive decomposition of a problem into sub-problems is one of the basic characteristics of the hierarchical task model [START_REF] Trichet | DSTM: a Framework to Operationalize and Refine a Problem-Solving Method modeled in terms of Tasks and Methods[END_REF][START_REF] Wielinga | The KADS Knowledge Modelling Approach[END_REF][START_REF] Willamowski | A development shell for cooperative problem-solving environments[END_REF].

In pervasive learning systems, activities cannot be achieved in the same way according to the different situations. It is necessary to have a context-aware and adaptive mechanism to decide how to achieve an activity according to a given situation. In the MODALES project, an adaptive and contextaware model of scenarios has been successfully proposed and implemented; it is based on an interdisciplinary approach (didactics, physics and its epistemology, computer science and education), a particular teacher community of practices and a hierarchical task model, having the task/method paradigm [START_REF] Tetchueng | A Context-Aware Learning System based on generic scenarios and the theory in didactic anthropology of knowledge[END_REF]. In such a framework, an activity is represented by a task having several associated methods. A method represents a way to achieve a task in a particular situation. This hierarchical task model has been reused and modified according to our new requirements.

A. The task/method concept

Within the framework of the Task/Method paradigm, tasks represent activities and sub-activities managed by a knowledge-based system. A method describes how a particular task can be achieved. There exist two types of tasks: abstract task and atomic task. An atomic task is not composed of subtasks. It can be achieved by a simple procedure defined into a method. An abstract task represents a high level activity composed of sub-activities. A method defines how an abstract task is composed of sub-tasks which can be abstract or elementary tasks. Methods define the control structure which allows the recursive decomposition of tasks into sub-tasks and the control structure defines sub-task order at runtime. For a given task (abstract or atomic), several methods can be used for achieving it.

Methods are described by a context descriptor and an achievement descriptor. A context descriptor is dedicated to adaptation -selection of relevant methods. It is composed of a property set according to the corresponding context view (i.e. professional or learning activity). An achievement descriptor defines how to achieve the corresponding task. This descriptor consists of one or several structures among the following four structures: i) the control structure defines how an abstract task can be composed of sub-tasks by means of operators. At present, three different operators are used: sequence, alternative and parallel; ii) the content requirement specification identifies how relevant resources are selected for the current task and situation; ii) the service requirement specification describes how to assemble relevant services (or web services) to achieve the current task1 ; iv) a procedure/function specification defines how to execute a simple procedure or function, which is either automatic (executed entirely by the system) or interactive (human computer interaction: interaction between the user and application or between two users via an application).

B. Learning scenario for workplace

Pervasive learning scenarios in workplace are modeled by a hierarchical structure of tasks/methods representing learning and working activities. For a complete presentation and execution of the scenario, another category is needed: mixed activities. Thus, working and learning activities are represented by working, learning and mixed tasks. In Figure 1, the context descriptor and the control structure of some methods are represented for different categories of tasks. A learning task represents a learning activity. It is a learning activity or a learning activity supporting a working situation. The latter aims at supporting the user to reach the working objective and to improve his/her knowledge and howknow about the domain. The selection of the relevant method to achieve the corresponding task takes into account the pedagogical context view. A working task represents a working activity that the user can perform in his professional environment. An abstract working task is maybe achieved by several working sub-tasks. It can be considered as a sub scenario that can be modeled independently. An atomic working task is the smallest unit of work. The selection of the relevant method to achieve the corresponding task takes into account the pedagogical context view. A mixed task is an abstract task which is composed of learning and working subtasks. The goal is to mix tasks of the two categories to support a high level activity in the learning scenario. To select a method in a given task, the corresponding context view and the context descriptor are compared.

IV. CONTEXT MANAGEMENT

To solve complex and dynamic context changes and demands of situated learning, the learning system detects the situation changes. The context management process generates a new situation or updates the current situation for maintaining the activity relevance and continuity. This process is divided into four main steps: i) Stage 1: Context change detection and aggregation: this step determines context changes and check whether these changes leads to the creation of a new situation or an update of the current situation. The changes can come from collaborators (colleagues) or customers or tutors or learner -user interactions, location, network, device, time, scenario, etc.; ii) Stage 2: Gathering of the initial context information for creating the new situation, this is done by a partial or complete copy of the last situation and/or by querying the context ontology; iii) Stage 3 -Selection of the current task: the last task status can be restored in the new situation to ensure seamless learning and working based on the historical dependencies of situations. When the last task is finished, the next one is selected according to the current scenario and the historical dependencies of situations. When the last task is not finished, it is necessary to continue it; iv) Stage 4 -Searching for the relevant couples (Task, Method): According to the current situation and the task, the adaptation process has to select the relevant methods. The main role of the adaptation process is to maintain consistency between the learning system, the physical environment and the current learning or working situation and also to ensure seamless learning and working.

V. ADAPTATION

The fundamental issue in a pervasive learning environment is how to provide learners with the right learning content at the right time in the right way. Thus, adaptation is mandatory to all types of learning activities in pervasive learning environments [START_REF] Bomsdorf | Adaptation of Learning Spaces: Supporting Ubiquitous Learning in Higher Distance Education[END_REF]. At present, we focus on adaptation mechanism dedicated to scenarios: how to achieve an activity according to the current situation, or in other words how to select the relevant methods pour a given task. On other words, the learning system has to select dynamically the relevant way to achieve the different tasks included in a scenario. The main issues are seamless learning and working and the management of dynamic adaptation. Our current scenarios are not sufficiently detailed to enable us to define accurate strategies to deal with seamless learning and working. Thus, we mainly focus on the management of dynamic adaptation. As the context is dynamic, it is not possible to know in advance how the next situation will be structured. In other words, it is not possible to anticipate the set of features composing the different situations. For managing adaptation, it is necessary to trigger rules using the context descriptor of methods which are defined a priori and the set of situation features which can be unexpected -not known in advance. Consequently, it is not possible to define rules for each possible configuration of features in a situation. As in Mobilearn European project [START_REF] Lonsdale | Towards a dynamic process model of context[END_REF], we associate specific metadata to situation properties and/or dimensions for managing adaptation.

Situation features are divided in to two categories: permanent and transitory. Features describing scenarios and users used to be available in all situations. They are permanent features in a situation. Thus, it is possible to manage them as usual because they are known a priori. Other features are transitory. For them, it is necessary to analyse how each property contributes to content adaptation -method selectiondifferently according to its role in the adaptation process. Some of them are used to filter the content while others are used to rank or annotate it -filtering, ranking and annotation are adaptation techniques. For instance, learning and working methods can be filtered according to learning places or used devices while it is annotated according to user's knowledge, know-how or preference). As soon as features are used to filter learning and working methods, it is easy to manage them. In other words, a present transitory feature can filter or not a method; the method is eliminated or not. At present, all transitory features of learning and working methods are used to filter. Nevertheless, we shall have to investigate in depth this issue according to more detailed scenarios in future.

A. Adaptation Model

Our adaptation model is composed of several adaptation categories, a set of adaptation techniques, a set of equivalent classes and their membership rules. An adaptation category is defined by a content type (resource, web service or method) and a context view. For instance, we divide method adaptation into two categories: one is related to learning context view and another is related to professional context view. For a given adaptation category, methods -content -is classified in several equivalence classes -at least two classes, up to five classes {"Very Bad", "Bad", "To Consider", "Good", "Very Good"} -according to all permanent situation features. Equivalence classes are mutually exclusive and totally ordered. An equivalence class contains a set of membership rules which define necessary and sufficient conditions to check method membership. Rules are declarative predicates using the current situation and the method context descriptor. For each transitory feature, methods are classified into two classes {"Good", "Bad"}. For example, the classes {"Good", "Bad"} are associated to the location dimension while the classes {"Very Bad", "Bad", "To Consider", "Good", "Very Good"} are associated to the user's knowledge sub-dimension.

The adaptation technique defines the technique used for adaptive navigation in our framework. It is sorting, hiding, annotation, direct guidance techniques found in Adaptive Hypermedia field [START_REF] Brusilovsky | Adaptive Hypermedia[END_REF]. An adaptation technique can be chosen for an adaptation category at runtime or a priori.

B. Adaptation Principles

Adaptation process in SCARCE is specified for a resource type and an adaptation category. It consists of three stages: i) Classification: content is classified according to the current situation in several equivalence classes: two classes {"Good", "Bad"} for each transitory feature and up to five equivalence classes for all permanent situation features, together. The content belongs to an equivalent class if it satisfies its membership rules; ii) Filtering: all contents belonging to "Bad" classes according to a transitory feature are filtered. In other words, these contents are discarded. For example, with the network dimension, the class "Good" is considered as relevant while the class "Bad" is not. So that the system will eliminate all content that belongs to the class "Bad"; iii) Adaptive navigation: an adaptive technique can be chosen by the system or by the user according to an author decision. All methods belonging to the same equivalence class are treated in the same way. Annotation and sorting are process according to the total order of equivalence classes. For hiding, only methods belonging to the class "Good" are kept.

VI. CONCLUSION

We have proposed an adaptive and context-aware model of scenario based on a hierarchical task model having the task/method paradigm -methods define how to achieve a taskfor a pervasive learning system supporting working and learning situations. This model enables us to choose how to achieve activities according to the current situation. In other words, the relevant methods are selected dynamically according to the current situation. Integration of mobile devices in broader learning and working scenarios, formal and informal learning and management of seamless learning across contexts has been achieved by means of the scenario model and the adaptation process in using the context model. Our context model and adaptation process is able to mange to deal with dynamic contexts -context-as-construct -and seamless learning and working across contexts. These issues are managed in a limited way. Thus, our current scenarios are not sufficiently detailed to enable us to define accurate strategies to deal with seamless learning and working. Our context and scenario models have necessary properties to managed seamless learning (situation and its historical dependencies). We need investigate in depth our scenarios to establish rules able to manage seamless learning. As our scenarios are limited, we also have to study in depth our adaptation policies to manage dynamic context.

Figure 1 .

 1 Figure 1. A small part of the e-retail scenario

TABLE I

 I

			(temperature, noise level); location; time}
		.	A SHORT DESCRIPTION OF CONTEXT VIEWS
	Context		Attributes
	views
	Pedagogical	{user(role, knowledge, how-know); learning collaborator;
	context	scenario (learning activity); pedagogical tool; learning objects;
	view	learning location; time} {device (PTA, RFID tag reader, large
		LCD screen); network (network type, bandwidth); system
		resource; physical environment (temperature, noise level);
		location; time}
	Professional	{user(role, preference, loyalty, purchase intentions, intentions
	context	for use); organisation unit; professional collaborator; scenario
	view	(working activity); resources; working location; time} {device
		(PTA, RFID tag reader, large LCD screen); network (network
		type, bandwidth); system resource; physical environment

Context and service requirements are out of the scope of this paper.