Mariem Ben Ameur
email: mariem.ben_ameur.etu@univ-lemans.fr

Abderrahim El Mahi

Jean-Luc Rebiere

Isabelle Gimenez

Moez Beyaoui

Moez Abdennadher

Mohamed Haddar

Investigation and identification of damage mechanisms of unidirectional carbon/flax hybrid composites using acoustic emission

Keywords: Carbon fibers, flax fibers, hybrid composites, acoustic emission, damage mechanisms

The objective of the present experimental work is to determine the damage mechanisms appearing in unidirectional carbon/flax hybrid composites during tensile tests. The specimens tested consist of unidirectional carbon and flax fiber plies with different stacking sequences. Laminated composites were manufactured using a manual lay-up process. The specimens were tested under uniaxial tensile loading in static and cyclic fatigue tests. The tests carried out were monitored by the acoustic emission (AE) technique in order to identify the damage mechanisms evolutions. This identification was made with an unsupervised clustering technique. The recorded events were classified with the k-means algorithm based on temporal classification parameters. For each kind of specimen, four classes of AE events were obtained. Then, the contribution of each damage mechanism to overall failure was evaluated by the amplitude range, the cumulative number of hits and the acoustic energy activity. The AE classes obtained were correlated with scanning electron microscopy (SEM) observations in order to identify the damage mechanisms observed.

Introduction

Natural fibers such as flax, sisal and hemp have great advantages, in comparison with synthetic fibers, that make them interesting and competitive materials for composite structure applications. The use of polymer with agrobased fibers has been the subject of many research works due to the awareness of researchers to the environmental challenges. For instance, their availability, their ecological aspect and their biodegradability are some factors for reducing the environmental impact [START_REF] Pickering | A review of recent developments in natural fibre composites and their mechanical performance[END_REF][START_REF] Faruk | Biocomposites reinforced with natural fibers: 2000-2010[END_REF]. Also, they are characterized by their low costs. More especially, they present a potential alternative to conventional fibers owing to their specific properties. In fact, agro-based fibers exhibit a stiffness comparable to those of glass fibers. Moreover, they are about 50% lighter than glass ones [START_REF] Al-Oqla | Natural fiber reinforced polymer composites in industrial applications: feasibility of date palm fibers for sustainable automotive industry[END_REF][START_REF] Wambua | Natural fibres: can they replace glass in fibre reinforced plastics?[END_REF]. The use of flax fibers as reinforcement of polymer matrix composites has been of particular interest to many researchers. Flax fibers are used in different forms (short fibers, monofilaments, unidirectional flax tape, woven fabrics, etc…) for reinforcing polymer matrix (thermoplastic or thermoset resin) with different types of manufacturing processes (stacking sequence, vacuum bag molding process, liquid resin infusion, resin transfer molding, etc…) [5 -9]. Monti et al. [START_REF] Monti | Experimental and finite elements analysis of the vibration behaviour of a bio-based composite sandwich beam[END_REF] studied the damping properties of unidirectional flax fiber reinforced composites. Yan et al. [START_REF] Yan | Microstructure, flexural properties and durability of coir fibre reinforced concrete beams externally strengthened with flax FRP composites[END_REF], Monti et al. [START_REF] Monti | Mechanical behaviour and damage mechanisms analysis of a flax-fibre reinforced composite by acoustic emission[END_REF] and Haggui et al. [START_REF] Haggui | Static and fatigue characterization of flax fiber reinforced thermoplastic composites by acoustic emission[END_REF] evaluated the mechanical properties of unidirectional flax polymers. Duc et al. [START_REF] Duc | Damping of thermoset and thermoplastic flax fibre composites[END_REF] carried out an experimental study to evaluate the mechanical and damping properties of polymer reinforced with carbon, glass and flax fibers. The results revealed that flax fiber laminates present good damping properties but they present lower mechanical properties than carbon and glass fiber laminates. In order to enhance the mechanical performance of agro-based composites, hybridization of flax fibers with carbon ones in the same matrix was proposed [START_REF] Dhakal | Development of flax/carbon fibre hybrid composites for enhanced properties[END_REF]. Therefore, hybridization offers to the composite structures' lightness, good damping properties and sufficient mechanical performance. Consequently, they can be suitable for structural use.

In addition, by hybridization with carbon fibers, the property of variability and moisture sensitivity of flax fibers can be reduced [START_REF] De Moura | Mode II fracture characterization of a hybrid cork/carbon-epoxy laminate[END_REF][START_REF] Atiqah | Development of kenaf-glass reinforced unsaturated polyester hybrid composite for structural applications[END_REF]. The hybrid composites have been recently exploited in industrial applications such as manufacturing tennis rackets (Artengo) and bicycles (Museeuw bikes). Ben Ameur et al. [START_REF] Ameur | Damping Analysis of Unidirectional Carbon/Flax Fiber Hybrid Composites[END_REF] studied the damping properties of unidirectional carbon-flax hybrid fiber reinforced composites. Flynn et al. [START_REF] Flynn | Hybridized carbon and flax fiber composites for tailored performance[END_REF] analyzed the mechanical behavior of hybridized carbon-flax fiber composites. These hybrid composite structures may have different damage mechanisms which occur when they are subjected to service loading.

In the field of structural health monitoring (SHM), several studies used the electrical resistance variation to assess and monitor the internal damage [START_REF] Lemaitre | Damage measurements[END_REF][START_REF] Todoroki | Strain and Damage Monitoring of CFRP Laminates by Means of Electrical Resistance Measurement[END_REF][START_REF] Chen | Damage in carbon fiber-reinforced concrete, monitored by both electrical resistance measurement and acoustic emission analysis[END_REF][START_REF] Niccolini | Scaling in damage by electrical resistance measurements: an application to the terracotta statues of the Sacred Mountain of Varallo Renaissance Complex (Italy)[END_REF]. A breakdown of percolation network, formed by the contacting fibers, results in an increase in electrical resistance. So that, the variation in electrical resistance reflects the damage during the loading process. Other several works used a numerical approach based on the lattice simulation and the Acoustic Emission (AE) technique [START_REF] Iturrioz | Acoustic emission detection in concrete specimens: Experimental analysis and lattice model simulations[END_REF][START_REF] Rescalvo | Monitoring of Carbon Fiber-Reinforced Old Timber Beams via Strain and Multiresonant Acoustic Emission Sensors[END_REF]. Therefore, they analyzed and verified the relationships between the numerical simulations and the parameters of the AE monitoring. Mostly, the AE technique has been frequently used by different research work to monitor in real time the spread of damage in composite structures subjected to different loading types [START_REF] Dahmene | Acoustic Emission of Composites Structures: Story, Success, and Challenges[END_REF]. It is important to note that this technique is the most advanced method for the non-destructive evaluation of damage progression. AE associated with microscopic analyses can be an effective method to investigate damage mechanisms in composites. In this context, the clustering effect was studied in order to know which damages (matrix cracking, fiber-matrix interface debonding, delamination, fiber pull-out and fiber breakage defects) occurred in the laminates. The use of such method needs a fair understanding of the relationships between the damage process and the recorded signal. Different AE signal analysis methods were utilized in research, such as frequency band, AE counts, amplitude ranges, AE energy or a combination of several AE parameters. Different damage mechanisms have been identified for flax fiber reinforced composites in several works [START_REF] Monti | Mechanical behaviour and damage mechanisms analysis of a flax-fibre reinforced composite by acoustic emission[END_REF][START_REF] Haggui | Static and fatigue characterization of flax fiber reinforced thermoplastic composites by acoustic emission[END_REF][START_REF] Habibi | Combining short flax fiber mats and unidirectional flax yarns for composite applications: Effect of short flax fibers on biaxial mechanical properties and damage behavior[END_REF][START_REF] El Mahi | Analyse par émission acoustique de l'endommagement des matériaux éco-composites. 10ème[END_REF][START_REF] Romhány | Tensile fracture and failure behavior of technical flax fibers: Technical Flax Fibers[END_REF][START_REF] Bravo | Characterization of Tensile Damage for a Short Birch Fiberreinforced Polyethylene Composite with Acoustic Emission[END_REF]. According to the work of Bravo et al. [START_REF] Bravo | Characterization of Tensile Damage for a Short Birch Fiberreinforced Polyethylene Composite with Acoustic Emission[END_REF], the AE events can be classified by their amplitudes.

Different amplitude ranges of AE events are attributed to matrix cracking (from 35-42 dB to 45-60 dB). The amplitudes from 45-60 dB to 60-70 dB are assigned to fiber-matrix debonding. The amplitudes of delamination are between 50 dB and 80 dB. The amplitudes between 60 dB and 100 dB are usually attributed to fiber pull-out and fiber breakage. Depending on the type of matrix and fibers used, the ranges of these intervals can change. Some works focus on the classification of damage mechanisms in laminated carbon/epoxy composites using the obtained frequency results [START_REF] Aggelis | Acoustic Emission Characterization of Damage Sources of Lightweight Hybrid Concrete Beams[END_REF][START_REF] Groot | Real-time frequency determination of acoustic emission for different fracture mechanisms in Carbon/Epoxy composites[END_REF]. Otherwise, the wavelet transform analysis in the frequency domain can provide more information about the level of damage in composite materials [START_REF] Martínez-Jequier | Real-time damage mechanisms assessment in CFRP samples via acoustic emission Lamb wave modal analysis[END_REF][START_REF] Stankevych | Investigation and identification of fracture types of structural materials by means of acoustic emission analysis[END_REF][START_REF] Loutas | Damage evolution in center-holed glasspolyester composites under quasi-static loading using time-frequency analysis of acoustic emission monitored waveforms[END_REF][START_REF] Qi | Wavelet-based AE characterization of composite materials[END_REF]. In several works [START_REF] Barile | Acoustic emission waveform analysis in CFRP under Mode I test[END_REF][START_REF] Barile | Innovative mechanical characterization of CFRP by using acoustic emission technique[END_REF][START_REF] Barile | Acousto-ultrasonic evaluation of interlaminar strength on CFRP laminates[END_REF],

the Continuous Wavelet Transform (CWT) was used in frequency domain analysis of the acoustic signals during mechanical testing for detecting and identifying damage mechanisms in carbon fiber reinforced polymers CFRP.

The Discrete Wavelet Transform (DWT) has been used by Gutkin et al. in [40] to identify AE signals based on the frequency range and to interpret the types of damage of CFRP laminates. From AE signals recorded during tensile tests on specimens with various layups and orientations, five failure modes are clearly identified; matrix cracking, fiber/matrix debonding, delamination, fiber pull-out and fiber fracture. Barile et al [START_REF] Barile | Experimental wavelet analysis of acoustic emission signal propagation in CFRP[END_REF] have reported in their work on analyzing the effect of thickness of the material and its geometry in the amplitude and the acoustic energy of the AE signals. Bussiba et al. [42] classified the damage mechanisms of C/C composites during cyclic tests with the acoustic emission technique. They tracked the damage evolution through the AE count rates and cumulative AE counts. They found three active mechanisms of failure: matrix cracking, debonding and fiber fracture. Bourchak et al. [START_REF] Bourchak | Acoustic emission energy as a fatigue damage parameter for CFRP composites[END_REF] tested carbon fiber reinforced plastic composites under static and fatigue loading with monitoring of damages by AE. They found that AE energy provides a valid and useful damage parameter and effective failure criteria for fatigue life prediction.

For hybrid laminates, Fotouhi et al. [START_REF] Fotouhi | Detection of fibre fracture and ply fragmentation in thin-ply UD carbon/glass hybrid laminates using acoustic emission[END_REF] investigated the damage modes occurring in thin-ply UD carbon-glass hybrid laminates using acoustic emission. They used a criterion based on amplitude and energy of the AE event values. Saidane et al. [START_REF] Saidane | Damage mechanisms assessment of hybrid flax-glass fibre composites using acoustic emission[END_REF] tested flax-glass fiber hybrid composites in order to assess the damage mechanisms occurring during tensile tests. Their results showed that all studied laminates present three classes of AE signal (matrix cracking, debonding and fiber breakage). Several research works found that the conventional monoparametric AE signal analysis may be insufficient and limited in terms of damage identification. This is why, multiparameter statistical analyses have already been used in order to better discriminate and identify the damage modes.

This paper deals with relatively new materials: carbon-flax hybrid composites made of an epoxy matrix reinforced by unidirectional flax and carbon fibers with different content and different lay-up configurations. The damage mechanisms of this type of materials have not been deeply studied yet. The main objective of this study is to identify the failure mechanisms occurring under service loading for the flax fiber laminates, carbon fiber laminates and carbon-flax hybrid laminates. AE damage analysis presented is based on the results of static and fatigue tensile testing. This identification is made using an unsupervised classification with a multivariable statistical technique, based on a pattern recognition method. In fact, the recorded events are classified with the kmeans algorithm. By using AE amplitude, vector of hits and contribution of AE energy, the damage mechanisms are classified. Post-test microscopy is performed on failed specimens. The observed damage mechanisms are correlated with the AE classification. Finally, it is found that the used AE technique is an effective method for accurate detection of failure mechanisms, whether in non-hybrid flax and carbon laminates or in carbon-flax hybrid laminates.

Materials and manufacturing

The composite materials studied in this work are reinforced with unidirectional flax tape and unidirectional carbon fabric, and a liquid thermoset epoxy resin. The flax tape, provided by the LINEO Company, has an areal density of 200 g/m 2 [START_REF] Khalfallah | Innovative flax tapes reinforced Acrodur biocomposites: A new alternative for automotive applications[END_REF]. The carbon fabric, provided by the SICOMIN Company, has an areal density of 300 g/m 2 . The thermoset epoxy resin used for the composite materials is the SR 1500 epoxy resin mixed with SD 2505 hardener provided by SICOMIN.

First, the fibers were cut from the roller. Flax fibers were dried in an oven for 1 hour at 110°C to remove water and to enhance the mechanical properties [START_REF] Baley | Influence of drying on the mechanical behaviour of flax fibres and their unidirectional composites[END_REF] but carbon fibers were used as received. After that, carbon and flax layers were impregnated in the desired stacking sequence using a manual lay-up method with the resin at room temperature (20°C). Then, non-hybrid and hybrid composites were elaborated using a vacuum bag molding process. They were cured at 50 kPa depression and room temperature for 7 hours. There are six stacking sequences with unidirectional laminated plates (non-hybrid: [F3]s and [C3]s, and hybrid: [C2/F]s, [C/F2]s, [F/C2]s and [F2/C]s), as presented in Fig. 1, consisting of 6 layers with fibers in 0° orientation plies. Finally, specimens were cut from the laminated plates with nominal dimensions of 200 mm x 15 mm x h where h is the thickness.

The thickness h, fiber volume fraction Vf and density of different stacking sequence are given Table 1.

[C2/F]s h = 2.4 mm [C/F 2 ] s h = 2.8 mm [C3]s h = 2 mm [F/C 2 ] s h = 2.4 mm [F 2 /C] s h = 2.8 mm [F 3 ] s h = 3.1 mm
Carbon fiber Flax fiber

Experimental setup

Static and fatigue tensile tests were performed on non-hybrid and hybrid laminates until failure. They were carried out on a servo-hydraulic machine INSTRON-8516 equipped with a 100 kN load cell (Fig. 2). The tensile test machine was interfaced with an acquisition system for monitoring and data acquisition. The specimens were tested according to the standard test method ASTM D3039/D3039 M [START_REF]Standard test method for tensile properties of polymer matrix composite[END_REF]. Three to four specimens were tested for each configuration sample in order to check the reproducibility. For the static tests, specimens were loaded with a displacement rate of 1 mm/min at room temperature.

For the fatigue tests, specimens were tested under load amplitude control. The tests were performed using a sinusoidal type of waveform with a frequency of 10Hz. The applied load ratio used was equal to RF=Fmin/Fmax=0.1 and the applied load level used was equal to rF=Fmax/Fu=0.75, where Fmin, Fmax and Fu are respectively, the minimum applied load, the maximum applied load and the ultimate failure load determined from the static tests.

During the static and fatigue tensile tests, damage and fracture progression in the tested specimens were analyzed by the AE method. Two resonant piezoelectric sensors (PCA MICRO-80) with a frequency bandwidth of 100 kHz-1 MHz and a resonance peak of 300 kHz, provided by Euro Physical Acoustics (EPA) corporation, were used. They were clamped to the specimens on the same side with a plier using a coupling agent (silicon grease)

between the specimen and the sensors. For best results, we do not take too much of the coupling agent.

Therefore, a thin layer of grease was used on the sensor against the specimen surface. These AE sensors were used to record the transient waves propagated through the material under loading. These acoustic waves were generated as a consequence of elastic deformations and damage mechanisms. In fact, events created in the gauge length and under grips were recorded. The gauge length was 100 mm. In the interest of reducing the attenuation of the waves originating from the source, the distance between the two AE sensors was set to 50 mm. The recorded AE signals were amplified and filtered using two pre-amplifiers with a gain of 40 dB, provided by Mistras Group Company. The acquisition of the acoustic emission signals was processed using the AEwin software, provided by EPA corporation, with a sampling frequency rate of 5 MHz. This system was equipped with a PCI acquisition card connected to the micro-computer (Fig. 3). The AE data were dependent on temporal acquisition parameters, namely: PDT (Peak Definition Time), HDT (Hit Definition Time) and HLT (Hit Lockout Time) [START_REF] Bravo | Characterization of Tensile Damage for a Short Birch Fiberreinforced Polyethylene Composite with Acoustic Emission[END_REF]. The values of these temporal parameters were set to: PDT = 50 µs, HDT = 100 µs and HLT = 200 µs. The amplitude of the acquisition threshold was determined using the Pencil Lead Breaking (PLB) procedure [START_REF] Masmoudi | Fatigue behaviour and structural health monitoring by acoustic emission of E-glass/epoxy laminates with piezoelectric implant[END_REF][START_REF] Nielsen | Acoustic emission source based on pencil lead breaking[END_REF] and was equal to 40 dB in order to filter the AE signals coming from external sources (noise). For each stacking sequence fractured in the tensile test, macroscopic and microscopic observations and analyses of the failure mode were carried out. Samples extracted from the specimens tested under tensile loading were prepared.

For the macroscopic analysis, visual inspection was done to identify damage mechanisms in the broken specimens such as delamination or fiber pull-outs. For the microscopic analysis, scanning electron microscopy (SEM) was used to detect smaller damage mechanisms of failure profiles.

Results and discussion

AE data classification

The AE method of classification was used to identify the damage mechanisms for the detected AE signals. The AE data sets were processed using NOESIS software [START_REF] Kattis | Noesis -Advanced Data Analysis[END_REF]. Five temporal parameters were selected for the -Threshold: is a level set to distinguish signal from noise.

-Amplitude: is the peak value reached by the signal in an AE event which covers the range from 40 dB to 100 dB -Rise time: is the time required for the signal to reach the amplitude after crossing the threshold.

-Duration: is the time difference between the first and last threshold crossings with the event signal.

-Number of counts to peak: is the number of times the signal crosses the threshold between the peak amplitude and the threshold.

-Energy: is the area under the curve of amplitude vs time for an event.

The amplitude of the signal was measured by the data acquisition system in real time. The other parameters were calculated by the waveforms recorded using the acquisition system. After several initial trials, these parameters settled, allowing a good repeatability for data classification.

The AE signals were classified by the K-means algorithm [START_REF] Likas | The global k-means clustering algorithm[END_REF]. This algorithm is unsupervised pattern recognition analysis. It aims to split a set of n events into an optimal number of k clusters in which the sum of squared distances between all the vectors of a class and its center is minimized. The normalization of the descriptors of the vector representative of an acoustic emission signal is indispensable. The difference in scale of the different parameters can lead to one parameter hiding the others because of its important values. The data sets were normalized by the following transformation: , : .

p p i x p i x x m i p F z σ - ∀ ∈Ω ∈ = (1)
where i x is the event i of a set of data p Ω related to the acoustic feature p ,

Acoustic emission under static tensile tests 4.2.1. Investigation of the damage evolution by monitoring the AE activity

The acoustic activity is presented by the evolution of the cumulative number of vector of hits. Fig. 5 presents the correlation between the evolution of the acoustic activity and loading data versus strain within the specimen. An exponential type of curve is obtained for the AE response. It can be noticed from Fig. 5 that the acoustic activity and the evolution of the loading take place in three phases [START_REF] Roundi | Acoustic emission monitoring of damage progression in Glass/Epoxy composites during static and fatigue tensile tests[END_REF]:

-Phase I: First linear elastic phase characterized by very few detections of AE activity. It seems to be a silent period. In this phase, no damage is initiated. In fact, this loading not having any effect on the specimen's integrity.

-Phase II: Here the stress curve begins to lose its linearity slightly. Its slope decreases, reflecting the onset of viscoelastic behavior. This phase is characterized by a steady increase in acoustic activity. At first, some events are detected. These events reflect the beginning of micro-cracks in the resin. Then, the small increase may reflect the onset of fiber-matrix debonding and the micro-cracks evolve and join together to form the delamination.

-Phase III: Finally, the third phase which is just before the specimen's final failure. The stress curve is nonlinear, reflecting the beginning of plasticity. AE activity becomes greater. Its curve evolution changes and exhibits a linear form until the specimen's final breakage. An assumption can be made that a new damage mechanism has occurred which is fiber breakage.

In addition, it can be noted from the comparison between all these non-hybrid and hybrid specimens that when the carbon fiber volume fraction increases the cumulative number of counts increases. For instance, the acoustic activity reaches a maximum value of 80000 counts for [C3]s specimen, followed by 34000 and 30000 counts

Amplitude Distribution versus time

In order to identify the damage mechanisms occurring in the stacking sequences studied and to pursue their temporal evolution until failure, the clustering methodology based on the five temporal parameters was applied. In addition, the attribution of each damage mechanism to the event's class was done using the waveforms of AE signals. Typical forms of the AE waveforms of the four damage mechanisms are given in Table 2. For all specimens studied, the number of hits of every class has a significant increase just before the failure of the specimen. Matrix cracking (class A) is the first damage mechanism appearing in the chronology, and propagates until the specimen's final failure. As shown in Fig. 10, this damage mechanism is the most dominant damage from all types of the specimen and it presents more than 40% of cumulative damage at failure. It is greater for flax fiber laminates than other laminates. This result can be explained by the low fiber volume fraction of flax fibers in the composite structure (fiber volume fraction is 32%). Then, fiber-matrix debonding (class B) appears in the second time after matrix cracking. After that, delamination (class C) and fiber pull-out and fiber breakage (class D) mechanisms occur after around 40% of the breaking load and increase slightly. Fig. 10 also shows that the percentage of fiber breakage, delamination and fiber-matrix debonding for the carbon and hybrid fiber composites is greater than that of flax fiber composites. This is mainly due to the difference in the fiber volume fraction of carbon and flax fiber laminates (0.55 for the carbon fiber laminates vs. 0.32 for the flax fiber laminates). On the other hand, it can be explained by the nature of the adherence between an epoxy resin and carbon fibers compared to the adherence to flax fibers. The radar chart in Fig. 11 presents a comparison between the normalized global vector of hits for the six kinds of the laminates studied for each class. It is clearly seen that when the carbon fiber volume fraction increases the number of vectors of hits increases, for all the AE classes. This fact can be explained by the increasing of rigidity with carbon fibers and then with the increase in applied load. Moreover, the difference between the dimension of carbon fibers and flax fibers may also contribute to these increases (8µm for carbon fiber [START_REF] Ammar | Contrôle de santé vibro-acoustique de l'endommagement des matériaux composites à base polymère pour l'aéronautique[END_REF] vs. 15 à 30 µm for flax fiber [START_REF] Monti | Elaboration et caractérisation mécanique d'une structure composite sandwiche à base de constituants naturels[END_REF]). Class A

[F/C 2 ] s [C 3 ] s [C/F 2 ] s [C 2 /F] s [F 3 ] s [F 2 /C] s 0 20 40 60 80 100 [L3]s [L2C]s [LC2]s [C3]s [C2L]s [CL2]s Class B [F/C 2 ] s [C 3 ] s [C/F 2 ] s [C 2 /F] s [F 3 ] s [F 2 /C] s 0 20 40 60 80 100 [L3]s [L2C]s [LC2]s [C3]s [C2L]s [CL2]s Class C [F/C 2 ] s [C 3 ] s [C/F 2 ] s [C 2 /F] s [F 3 ] s [F 2 /C] s 0 20 40 60 80 100 [L3]s [L2C]s [LC2]s [C3]s [C2L]s [CL2]s Class D [F/C 2 ] s [C 3 ] s [C/F 2 ] s [C 2 /F] s [F 3 ] s [F 2 /C] s

Analysis of the contribution of each damage mechanism by the AE energy

The contribution of each damage mechanism to global failure was evaluated for the different stacking sequences.

For each damage mechanism, the damage contribution (Di) is obtained by the ratio of the cumulative AE energy of the given mechanism (Ei) by the cumulative AE signal energy at failure (ET) [START_REF] Saidane | Damage mechanisms assessment of hybrid flax-glass fibre composites using acoustic emission[END_REF][START_REF] Bravo | Development of novel green and biocomposite materials: Tensile and flexural properties and damage analysis using acoustic emission[END_REF]:

1 . i i i n T j j E E D E E = = = ∑ ( 2
)
where n is the total damage mechanism number, [START_REF] Wambua | Natural fibres: can they replace glass in fibre reinforced plastics?[END_REF]. n = Fig. 12 reports for each damage mechanism the relative contribution Di in terms of energy to the final failure for all composite specimens studied. A clear similarity in the Di distribution is observed for all types of composites.

It is evident that the most significant energy mechanism is the fiber breakage, with more than 72% of the contribution for flax laminates and reaching more than 90% for carbon laminates. The failure of fibers generates a higher absolute energy caused by the great stiffness of fibers. It is evident that the Di of the fiber pull-out and fiber breakage in flax fiber laminates (72%) is lower than that of carbon fiber laminates (90%). This fact can be explained by the stiffness of carbon fibers being greater than the stiffness of flax fibers, as it can be explained by the difference in the fiber volume fraction of carbon and flax fiber laminates. This is followed by the contribution of delamination, which is between 10-20% for all composites. And after that, the contribution of fiber-matrix debonding. However, in contrast with fiber pull-out and fiber breakage, the Di of delamination and fiber-matrix debonding was the greatest in flax fiber laminates (20% for delamination in flax fiber laminates vs 10% for carbon fiber laminates and 7% for fiber-matrix debonding in flax fiber laminates vs 3% for carbon fiber laminates). As a matter of fact, this difference can be explained by the good adherence between carbon fibers and epoxy resin compared to flax fibers. Finally, although the high cumulative number of vectors of hits for matrix cracking, the contribution of this damage mechanism had very low participation in the energy contribution. [52]:

-Phase I: This phase corresponds to the first fatigue cycles. In this phase, we observe the presence of the four damage mechanisms for all the specimens tested. The great damage mechanism corresponds to the initiation and propagation of mico-cracking. The amplitudes of AE events in this phase are between 40 dB and 100 dB.

-Phase II: Followed by the intermediate phase in which AE activity becomes low. This phase involves the propagation of microscopic damage and especially matrix cracking, fiber-matrix debonding and delamination between plies.

-Phase III: Final phase with a high number of AE events followed by specimen failure with the most dominant event being fiber breakage.

Distribution of cumulative number of hits versus the number of cycles

Matrix cracking events (class A) have the greatest activity for all specimens. Fiber-matrix interface debonding (class B) has less important activity. Delamination (class C) events appear from the start of experimentation and are less numerous than those of class A and class B. This is shown in Fig. 14 which illustrates the evolution of the cumulative number of vector hits versus time for the different stacking sequences, showcasing the different evolution, chronology and amount of the activity events observed throughout the fatigue test. Finally, fiber pullout and fiber breakage also occur at the beginning of the cyclic period but they are reduced in the intermediate phase and after they increase before final fracture.

Although the relative load level applied is the same for all kinds of specimens, we find that flax fiber laminate composites have a much higher lifetime compared to carbon fiber laminate composites. However, the number of events is greater for carbon fiber laminates in a short lifetime. The results obtained can be due to the low resistance of carbon fiber laminates in reversible loading compared to flax fiber laminates. As a matter of fact, for the hybrid laminates, we find an important number of events in a short lifetime compared to flax fiber laminates. Also, when the flax fiber layers are internal and their volume fraction is higher than those of carbon fiber layers, it is observed that the number of cycles is much lower than hybrid composites with external flax fiber layers. This result can be explained by the fact that active layers are the internal flax layers but the load level applied is imposed in consideration of the carbon layers.

Macroscopic and microscopic analyses

To confirm the effectiveness of the multivariable statistical analysis, a correlation with macroscopic and microscopic analyses is carried out. Fig. 15 presents the fractured surface of the six kinds of specimens studied under static tests. At a macroscopic scale, a lot of information can be obtained from these failure profiles. For all these unidirectional specimens, initial cracks a few millimeters long can be detected, followed by propagation of cracks through the tensile axis. Several fibers pull-outs can also be noticed. These failed specimens also exhibit delamination between plies over several millimeters, especially when the carbon fiber volume fraction is higher (Fig. 15c, Fig. 15e and Fig. 15f). In addition, according to the visual inspection of the fractured surface of the materials studied, it is clearly shown that laminates made with carbon fibers present more delamination and fiber breakage in comparison to flax fiber laminates.

Several microscopic analyses were performed by scanning electron microscopy (SEM) of failure profiles and are shown in Fig. 16. The micrographic section of the fractured surface of the flax fiber laminates, for the carbon fiber laminates and for the hybrid fiber laminates are shown in Fig. 16a-c was the greater. The results obtained reveal that the use of five temporal parameters seems to be sufficient to separate the main groups of events, which have the most noticeable consequences on the mechanical behavior of the composite. However, for flax-carbon hybrid composites, the method used did not allow the identification of AE events related to flax or carbon fibers separately. Therefore, in future work, researchers may use other methods such as the supervised classification method to classify all the AE signals and correlate them with the different damage mechanisms already existing. Finally, in addition to the mechanical properties in the choice of a composite material for a particular application, we should consider the various damage mechanisms and their relative importance, which may be relatively different for different materials and may be crucial for long-term applications.

Fig. 1 .

 1 Fig. 1. Different stacking sequences of composite laminates with six plies.

Fig. 2 .

 2 Fig. 2. Experimental setup of tensile test.

Fig. 3 .

 3 Fig. 3. Experimental setup of acoustic emission method.

  of the acoustic emission signals: amplitude, rise time, duration, number of counts to peak and energy. They are explained below and illustrated in Fig.4.

σFig. 4 .

 4 Fig. 4. Typical acoustic emission waveform and its parameters.

(

  [C2/F]s and [F/C2]s specimens, respectively) and 12000 and 21000 counts ([C/F2]s and [F2/C]s specimens, respectively). The minimum acoustic activity is observed near 2500 counts in [F3]s specimen.

Fig. 5 .

 5 Fig. 5. Correlation between the evolution of the acoustic activity and the behavior law for all specimens: a) [F3]s, b) [F2/C]s, c) [F/C2]s, d) [C/F2]s, e) [C2/F]s and f) [C3]s.

Fig. 6

 6 Fig.6shows the distribution of the amplitude of the AE signals versus time during experimental tests, whereas the evolution of the applied load was superposed in the same plot.

  in amplitude shows small areas of intersection between classes for all unidirectional specimens which confirms the application of multi-parameters approach. Four classes were obtained for the nonhybrid ([F3]s and [C3]s) and hybrid laminates ([F2/C]s, [F/C2]s, [C/F2]s, and [C2/F]s). Only mathematical considerations were used in this classification to constitute the best separation of the AE data, without taking physics into account. According to the literature [12, 13, 53], each class corresponds to a damage mechanism: matrix cracking (class A), fiber-matrix debonding (class B), delamination (class C) and fiber pull-out and fiber breakage (class D). The matrix cracking (class A) is characterized by an amplitude in the range of [40-48 dB]. The second class of signals (class B) induced by fiber/matrix debonding occurs at an amplitude of [44-57 dB]. The class C (delamination mechanism) is characterized by an amplitude between 48 and 65 dB, however, for carbon laminates the amplitude can reach 79 dB. Finally, fiber pull-out and fiber breakage (class D) have an amplitude range of [55-100 dB].

Fig. 7

 7 Fig. 7 presents box and whiskers plots for the five classification features with respect to the four AE classes for all laminates. The first quartile q1 and the third quartile q3 correspond to the borders of box plots. The second quartile q2 represents the median corresponding to the line cutting through the box plots. The lower and upper values of the amplitude range of each class correspond to the whiskers in the plot. It is clear that the four classes have quite similar statistical characteristics for the six kinds of specimen. A comparison of the average values of the chosen classification features in the six specimen configurations is shown in Fig. 8. The radar chart clearly shows that the acoustic signatures can be assigned to each class. The average properties for all the specimens are very similar. In fact, according to the average values of descriptors in Fig. 8 and to the typical waveforms in table 2, we can confirm that each damage is characterized by the properties of descriptors. The AE signals induced by the matrix cracking (class A) are characterized by a waveform with low amplitude, slow rise time, relatively long duration and low energy. The second class of signals induced by fiber/matrix debonding (class B) is characterized by low amplitude, short rise time, short duration and low energy. The waveform of the delamination mechanism (class C) is characterized by high amplitude, slow rise time, relatively long duration and low energy. And finally, fiber pull-out and fiber breakage (class D) have a high amplitude, short rise time, very short duration and very great energy.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Distribution of amplitude versus times of AE signals clustering under static tests for all specimens: a) [F3]s, b) [F2/C]s, c) [F/C2]s, d) [C/F2]s, e) [C2/F]s and f) [C3]s.

Fig. 8 . 2 Fiber 3 .

 823 Fig. 8. Average properties of the five classification features with respect to the six kind of specimen for the four damage mechanisms.Table 2 Typical form of AE signals obtained for each damage mechanisms Damage mechanisms Form of signal

Fig. 9 .

 9 Fig. 9. Cumulative number of vector of hits under static tests for all specimens: a) [F3]s, b) [F2/C]s, c) [F/C2]s, d) [C/F2]s, e) [C2/F]s and f) [C3]s.

Fig. 10 .

 10 Fig. 10. Percentage of cumulative event classes for the studied materials at failure.

Fig. 11 .

 11 Fig. 11. Comparison between normalized global vector of hits for the studied laminates for each class.

Fig. 12 . 4 . 3 . Acoustic emission under fatigue tests 4 . 3 . 1 .

 1243431 Fig. 12. Damage mechanisms contribution for the six studied materials.

Fig. 13 .

 13 Fig. 13. Distribution of amplitude versus times of AE signals clustering under fatigue tests for all specimens: a) [F3]s, b) [F2/C]s, c) [F/C2]s, d) [C/F2]s, e) [C2/F]s and f) [C3]s.

Fig. 14 .

 14 Fig. 14. Cumulative number of vector of hits for all specimens under fatigue tests: a) [F3]s, b) [F2/C]s, c) [F/C2]s, d) [C/F2]s, e) [C2/F]s and f) [C3]s.

  Fig. 16f label 3. A massive fiber pull-out was observed for every category of specimen (Fig. 16a label 1, Fig. 16g label 1 and Fig. 16i label 2). We can also observe several pulled-out fibers in Fig. 16b label 2 for the flax composite structure and on Fig. 16e and Fig. 16f label 2 for the carbon composite structure. Finally, delamination was also observed in Fig. 16d and Fig. 16h label 1. In addition to the delamination between flaxflax and carbon-carbon fiber layers, a delamination between flax and carbon fiber layers was noticed for the hybrid composites (Fig. 16i label 1).

Fig. 15 .Fig. 16 .

 1516 Fig. 15. Observation macroscopic of rupture facets in the gauge section for all specimens: a) [F3]s, b) [F2/C]s, c) [F/C2]s, d) [C/F2]s, e) [C2/F]s and f) [C3]s.

  Monotonic and cyclic fatigue tensile tests were conducted to investigate the evolution of damage mechanisms using the AE technique of hybrid flax-carbon fiber composites. The mechanical results showed that carbon fibers bring about remarkable improvement in the mechanical properties of composites with natural fibers. The multivariable analyses with five classification features (amplitude, absolute energy, duration, counts to peak and risetime) were used to classify the AE events. The damage mechanisms were interpreted by using the AE amplitude, vector of hits and the contribution of the AE energy. During static and fatigue tensile testing, the composites studied exhibited four classes of damage mechanisms. Microscopic analyses of the fracture surfaces were carried out in order to correlate the classification obtained from the acoustic events and the damage mechanisms. AEbased damage-mechanism identification revealed that, although the event numbers associated with the fiber breakage were the lowest, their contribution in the cumulative of AE energy to the global failure of composites

Table 1

 1 Thickness, fiber volume fraction and density of non-hybrid and hybrid laminates.

	Laminates Nominal thickness h (mm) Fibre volume fraction Vf (flax/carbon)	Density (Kg/m 3 )
	[F3]s	3.10	0.32/0.00	1140
	[F2/C]s	2.80	0.22/0.19	1181
	[F/C2]s	2.40	0.12/0.37	1266
	[C/F2]s	2.80	0.22/0.19	1175
	[C2/F]s	2.40	0.12/0.37	1234
	[C3]s	2.00	0.00/0.55	1340


Rudy Chocat
email: rudy.chocat.pro@gmail.com

Paul Beaucaire

Lo C Debeugny

Jean-Pierre Lefebvre

Caroline Sainvitu

Piotr Breitkopf

Eric Wyart

Damage tolerance reliability analysis combining Kriging regression and support vector machine classication

Keywords: Damage Tolerance, Fracture Mechanics, Reliability, Kriging, Support Vector Machine, Subset Simulation Nomenclature d

Damage tolerance analysis associates a Fracture Mechanical model with the Failure Assessment Diagram to dene the state of a space engine component. The reliability analysis treats the variability of numerical models assessing the probability of failure within Linear Elastic Fracture Mechanics (LEFM) hypotheses. However, these models, while providing quantitative information in the safe domain, give only qualitative information for failed components. This work proposes an original methodology to combine Kriging regression and the Support Vector Machine classication along with transition criteria between both approaches. To accurately describe the limit state, we dene a specic enrichment strategy. The eciency of the proposed methodology is illustrated on reference test cases.

Introduction

In the aerospace sector, designing a component under damage tolerance hypotheses involves considering the structure as inherently awed. It means that in a conservative way, each defect is considered as a crack and it is veried that the structure can withstand the loads throughout its lifetime. In space engine components context, a primary value of interest is the Failure Assessment Diagram (FAD) margin dened by the R6-rule [START_REF] Milne | Assessment of the integrity of structures containing defects[END_REF]. If the FAD margin is positive, the component is considered as safe. Otherwise, it fails. The FAD is used in the post-processing phase of the crack analysis, performed by quickly evaluated analytical models or forms [START_REF] Nasgro | Fracture mechanics and fatigue crack growth analysis software version 7[END_REF], but also by numerical approaches such as the extended nite element method (XFEM) [START_REF] Bordas | Enriched nite elements and level sets for damage tolerance assessment of complex structures[END_REF][START_REF] Bordas | Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment[END_REF][START_REF] Wyart | Substructuring fe{xfe approaches applied to three-dimensional crack propagation[END_REF] developed for complex structures.

However, as shown experimentally by Virkler [START_REF] Virkler | The statistical nature of fatigue crack propagation[END_REF], the crack propagation is subjected to uncertainties about geometry, material properties, loads [START_REF] Mattrand | Random load sequences and stochastic crack growth based on measured load data[END_REF] or considered defects [START_REF] Grooteman | A stochastic approach to determine lifetimes and inspection schemes for aircraft components[END_REF]. The approach to set the properties to the worth case [START_REF] Greenwood | A new tolerance analysis method for designers and manufacturers[END_REF], even if it ensures the strength of the component, may generate over-sizing. Uncertainties may also be considered through probabilistic approaches [START_REF] Sankararaman | Uncertainty quantication and model validation of fatigue crack growth prediction[END_REF]. The structural reliability provides, by setting stochastic models as inputs, the probability of failure which is required to be particularly low in the space application context [START_REF] Echard | Reliability assessment of an aerospace component subjected to fatigue loadings: Appro project, 4th Fatigue Design[END_REF].

In the low probability of failure assessment (< 10 6 ) scope, the zone of interest is localized in the extreme tail of the distribution. Using Monte Carlo Simulation (MCS), the chance is meager to generate failed experiments which drive the convergence of the probability estimator. Therefore, in the FAD context [START_REF] Altamura | Reliability assessment of high cycle fatigue under variable amplitude loading: Review and solutions[END_REF], MCS is limited due to a large number of evaluations required to get accurate results. To limit the number of simulations, advanced reliability methods such as Subset Simulation (SS) [START_REF] Au | Estimation of small failure probabilities in high dimensions by subset simulation[END_REF] restrict the sampling to a subsequence of MCS, xing the associated intermediate probability thresholds, until satisfaction of stopping criteria. The probability to generate failed experiments using SS is higher than with MCS reducing the variance of the estimators.

The use of Multi Level Monte Carlo approaches [START_REF] Clie | Multilevel monte carlo methods and applications to elliptic pdes with random coecients[END_REF][START_REF] Hauseux | Accelerating monte carlo estimation with derivatives of high-level nite element models[END_REF], based on local derivative informations, strongly accelerates the Monte Carlo estimation. However, their intrusive character is limiting in the space engine application. Non intrusive multi-delity techniques [START_REF] Gratiet | Recursive co-kriging model for design of cumputer experiments multiple levels of delity[END_REF][START_REF] Benamara | Adaptive inll sampling criterion for multi-delity optimization based on gappy-pod[END_REF], mainly developed for optimization, are promising but they require high and low delity models.

The lack of quantitative information in the failure domain, resulting from the Linear Elastic Fracture Mechanics (LEFM) hypotheses, limits the application of gradient-based optimization methods such as FORM [START_REF] Zhang | Two improved algorithms for reliability analysis[END_REF][START_REF] Der Kiureghian | Multiple design points in rst and secondorder reliability[END_REF], and SORM [START_REF] Fiessler | Quadratic limit states in structural reliability[END_REF]. More sophisticated mechanical approaches such as plastication are omitted due to the use of dedicated model not required in the space engine component scope of this study. However, the same nding could be observed for any application for which post-failure behavior is not included within the working hypotheses.

To treat the issue of computational cost, advanced reliability methods based on surrogates, also named meta-models, are built according to a Design Of Experiments (DOE) to cover the design space such as Latin Hypercube Sampling (LHS) [START_REF] Helton | Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems[END_REF], Centro dal Vorono tessellation or "Latinized" Centro dal Vorono tessellation [START_REF] Romero | Comparison of pure and latinized centroidal voronoi tessellation against various other statistical sampling methods[END_REF]. Even if the polynomial Response Surface Method [START_REF] Das | Cumulative formation of response surface and its use in reliability analysis[END_REF][START_REF] Gayton | Cq2rs: a new statistical approach to the response surface method for reliability analysis[END_REF][START_REF] Zhang | Diuse response surface model based on moving latin hypercube patterns for reliability-based design optimization of ultrahigh strength steel nc milling parameters[END_REF] is one of the most popular approaches, the interest for Kriging grows for structural reliability [START_REF] Kaymaz | Application of kriging method to structural reliability problems[END_REF][START_REF] Balesdent | Kriging-based adaptive importance sampling algorithms for rare event estimation[END_REF] due to the enrichment possibilities based on the underlying Gaussian process, such as ERGA [START_REF] Bichon | Ecient global reliability analysis for nonlinear implicit performance functions[END_REF] and Adaptive Kriging (AK) [START_REF] Echard | Ak-mcs: an active learning reliability method combining kriging and monte carlo simulation[END_REF]. To assess low probabilities, methods such as AK-SS [START_REF] Huang | Assessing small failure probabilities by ak{ ss: an active learning method combining kriging and subset simulation[END_REF] and AK-SSIS [START_REF] Tong | A hybrid algorithm for reliability analysis combining kriging and subset simulation importance sampling[END_REF] are adopted. However, these methods require a quantitative assessment of the FAD margin in both safe and failure domains. When only qualitative information, allowing only to qualify component as safe or unsafe, is available, classication methods based on Support Vector Machine (SVM) are preferable [START_REF] Hurtado | Structural reliability: statistical learning perspectives[END_REF][START_REF] Song | Adaptive virtual support vector machine for reliability analysis of high-dimensional problems[END_REF]. As for Kriging, several enrichment strategies have been proposed in the SVM context. The Adaptive SVM [START_REF] Pan | An ecient reliability method combining adaptive support vector machine and monte carlo simulation[END_REF] is based on the evaluation of a learning function whereas Max-Min [START_REF] Basudhar | An improved adaptive sampling scheme for the construction of explicit boundaries[END_REF] and Generalized Max-Min [START_REF] Lacaze | A generalized max-min sample for surrogate update[END_REF] solve an optimization problem. For low probability assessment, the 2SMART [START_REF] Bourinet | Assessing small failure probabilities by combined subset simulation and support vector machines[END_REF] method based on a succession of SVM separators, is proposed. We can note that the ASVR -SS [START_REF] Bourinet | Rare-event probability estimation with adaptive support vector regression surrogates[END_REF] method uses the SVM for regression to assess low probabilities.

The present work proposes a specic procedure to assess the failure for damage tolerance using the FAD for Fracture Mechanics [START_REF] Milne | Assessment of the integrity of structures containing defects[END_REF]. To our best knowledge, the existing surrogate-based reliability methods choose between regression and classication approaches. As the information is quantitative for safe components and qualitative for failed ones, the present work proposes to conjointly exploit regression and classication combining advantages of both approaches dealing, respectively, with continuous and binary information. Therefore, the key contribution of this paper is the denition of transition criteria between regression and classication phases. To achieve low probability, the proposed method is based on the subset simulation principle moving step by step to identify the limit state between the safe and failure domain. Moreover, in this contribution, an original adaptive strategy is explored to limit the number of model evaluations for the classication phase. The proposed methodology is called Adaptive Regression and Classication based on Subset Simulation (ARC-Subset).

The paper is organized as follows. The rst section presents the damage tolerance analysis and introduces the denition of the probability of failure. The second section details the proposed ARC-Subset methodology starting with the regression phase. The classication phase is detailed with a new enrichment strategy based on the probability of misclassication [START_REF] Basudhar | Constrained ecient global optimization with support vector machines[END_REF]. Then, the transition between both phases is dened. In the last section, the methodology is applied to two test cases, based on the damage tolerance tool NASGRO [START_REF] Nasgro | Fracture mechanics and fatigue crack growth analysis software version 7[END_REF], and compared with reference methods.

Reliability analysis for damage tolerance

This section introduces the concepts of damage tolerance for Fracture Mechanics and the notion of probability of failure.

Damage tolerance

The damage tolerance approach aims at ensuring component safety during a given number of cycles. In engineering practice, Fracture Mechanics models are often limited to the LEFM hypotheses for computational eciency. At each step of the crack propagation, the outputs are processed considering failure scenarios depending on verication of two fracture criteria: the Stress Intensity Factor K attains the toughness K IC value:

K r = K K IC 1;
(1) the surface between the crack front and the closest free surface, called 'remaining ligament' (see Figure 1), completely plasties: (

) 2
where nom is a non-physical stress value resulting from loads applied on the remaining ligament.

ref and ow are reference stress values depending on the material. Both criteria (1) and ( 2) are correlated by the Failure Assessment Limit (FAL) which is the boundary between the `accepted' and `rejected' domains. In the Failure Assessment Diagram (FAD) shown in Figure 2, for K r and L r values reported as point A, the FAD margin M FAD is dened as the distance ratio jOBj=jOAj from the origin. In the cyclic loading, the targeted lifetime N target is set. At each cycle i, M FAD (i) is evaluated. When point A crosses the FAL, the simulation is stopped, M FAD is not available as the LEFM hypothesis is not veried anymore, and the component is rejected. If the M FAD remains positive at the end of the lifetime, the component is accepted. In Figure 3, the dierent steps of two crack propagation cases are illustrated: one for a safe 150 component and one for a rejected one. Figure 4 illustrates the owchart of the damage tolerance procedure. Therefore, a safe component is characterized by N cycle = N target while a failed one by N cycle < N target . For the safe component, M FAD is a positive quantity, while for the failed one, the obtained negative value is not representative beyond the LEFM hypothesis and may be considered only 155 as qualitative.

Probability of failure

In the reliability context, the uncertainties are modeled by d random variables X which are dened using probability laws characterized by their distributions f X . Random variables are combined in a random vector X of length d dened by a joint density function f X . A component is characterized by the performance function G(X), G(X) > 0 in the safe domain and G(X) 0 in the failure one. In the present work, the considered performance function is: in the standard space where all input variables follow an uncorrelated normal distribution law with zero mean and unit standard deviation U $ x(0; I d ). The performance function is mapped to the standard space G(X) 3 H(U) divided into the failure region where H(U) < 0, the safe region with H(U) > 0 and the limit state H(U) = 0. The probability of failure is expressed as:

G(X) = M FAD (X): (3)
p f = P (G(X) 0) = Z G(X) 0 f X dx = Z H(U) 0 U du (4)
and may be integrated using the MCS method on random samples. Nevertheless, for example, a 10% condence level of a targeted probability around 10 9 requires % 10 11 performance function evaluations limiting the application of MCS for damage tolerance analysis. regression-based methods have diculties establishing H(U) = 0. Thus, this is the motivation for the development of a hybrid method detailed in the following section.

Methodology to evaluate the probability of failure

This section presents an adaptive strategy combining regression and classication approaches to assess the probability of failure within the damage tolerance hypothesis. The algorithm, based on the Subset Simulation principle [START_REF] Au | Estimation of small failure probabilities in high dimensions by subset simulation[END_REF], is divided into two phases:

in the exploration phase, a regression-based approach is coupled with active learning [START_REF] Echard | Ak-mcs: an active learning reliability method combining kriging and monte carlo simulation[END_REF]; the trends of the model are accounted to characterize intermediate thresholds;

in the exploitation phase, a classication-based approach is associated with adaptive strategy because of the lack of quantitative information in the failure space; the goal is to accurately determine the limit state in the last iteration.

This hybrid 'Adaptive Regression and Classication' algorithm is based on Subset Simulation (ARC-Subset). The following paragraphs rstly describe the regression steps. Then, the classication is detailed and an active learning for classication based on the multi-objective optimization is proposed. The 5b: Performance function in the U space for a through crack in a beam in traction using NAS-GRO [START_REF] Nasgro | Fracture mechanics and fatigue crack growth analysis software version 7[END_REF]. Two random variables are considered: the length of the crack c U[0:1mm; 1mm] and the load S 0 N(52:5MPa; 10%). Note the dashed circle lines representing the iso-values of standard deviation to give information about the distance to the failure region. crucial point is the transition phase between the regression and classication 195 steps (Section 3.3). The DOE is enriched by Adaptive Kriging (AK) [START_REF] Echard | Ak-mcs: an active learning reliability method combining kriging and monte carlo simulation[END_REF] chosen for its simplicity and eciency. Nevertheless, the AK stopping criterion proposed in [START_REF] Echard | Ak-mcs: an active learning reliability method combining kriging and monte carlo simulation[END_REF] seems to be too conservative for rst subset steps. As the goal is to cross these steps quickly, Tong [START_REF] Tong | A hybrid algorithm for reliability analysis combining kriging and subset simulation importance sampling[END_REF] presents a new stopping criterion for threshold convergence adapted for the Subset Simulation context.

Regression phase

Classication phase

Due to LEFM hypothesis, the model does not provide quantitative information for failed experiments. An alternative way to identify the limit state is to use SVM classication, based solely on the sign of the performance function. At this step:

the DOE contains at least one failed experiment, the subset population is the last population generated by the regression step.

Description of the algorithms

The owchart of the classication part of the ARC-Subset methodology is detailed in Figure 7 [START_REF] Mattrand | Random load sequences and stochastic crack growth based on measured load data[END_REF] If the quality criterion of the SVM separator (d) is not achieved, the DOE is enriched (d') by the adaptive method improved for classication, presented in the next paragraph. Otherwise, the ARC-Subset stops and the reliability infor-mation is returned.

As the last classication step determines the quality of the reliability assessment, this paper proposes an enrichment strategy based on a compromise between exploration and exploitation.

Enrichment strategy

At this stage, the DOE contains experiments of both classes. The SVM separator denes the limit state, but it may suer from a lack of accuracy around the zone of interest. Enrichment strategies for SVM are developed, principally based on the distance from the DOE. A multi-objective approach proposed in this section couples two criteria: a distance-based criterion and the probability of misclassication.

Distance-based criterion. To improve the accuracy of a boundary, Basudhar [START_REF] Basudhar | An improved adaptive sampling scheme for the construction of explicit boundaries[END_REF] proposes the Max-Min criterion based on the distance from the DOE coupled with a constraint on the distance from the SVM separator. The new experiment is selected by solving the following constrained optimization problem u MM = arg max u min

This exploration approach is ecient enough to globally describe the limit state, but it does not account for the proximity to the center of the standard space. The generalized Max-Min is introduced by Lacaze

Both optimizations may be solved by a local optimizer using the Chebychev norm.

Probability of misclassication. The SVM classier aims at building a binary decomposition of the standard space. Platt [START_REF] Platt | Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods[END_REF] introduces the notion of the Probabilistic SVM (PSVM) which gives the probability of a point to belong to a given class. It proposes the sigmoid formulation mainly based on the distance to the separator P (+1ju) = 1 1 + exp(A HSVM (u) + B) ; [START_REF] Sankararaman | Uncertainty quantication and model validation of fatigue crack growth prediction[END_REF] where A and B are deterministic PSVM parameters obtained by maximum likelihood. Basudhar [START_REF] Basudhar | Constrained ecient global optimization with support vector machines[END_REF] improves this model introducing the Distance PSVM (DPSVM)

P (+1ju) = 1 1 + exp(A HSVM (u) + B( d d++PSVM d+ d +PSVM )) ; ( 11
)
where PSVM is a conditioning parameter, and d + and d are respectively the distance to the closest positive and negative experiments.

Consequently, it is possible to dene the probability of misclassication P mc (u). Basudhar [START_REF] Basudhar | Constrained ecient global optimization with support vector machines[END_REF] includes this notion to select a new experiment which has a high probability of being misclassied.

This paper proposes to combine both Max-Min criteria interpreted either as an exploration for the classical Max-Min or exploitation for the generalized one. The idea is to solve both optimization problems simultaneously and then select the new experiment which has the highest probability of misclassication. The next evaluated experiment is u new = arg max (P mc (u)) ; u P fu MM ; u GMM g: [START_REF] Altamura | Reliability assessment of high cycle fatigue under variable amplitude loading: Review and solutions[END_REF] In this approach, a compromise between exploration and exploitation is based on the probability of misclassication.

Transition between the regression and classication phases

One of the main points of ARC-Subset is to determine when the transition between regression and classication happens. When the limit state is achieved, the Subset Simulation criterion [START_REF] Clie | Multilevel monte carlo methods and applications to elliptic pdes with random coecients[END_REF] where I H(U) 0 is the indicator function. In the following, k DOE is arbitrarily set to twice, the dimension of the reliability problem. The pseudo-code of the transition phase is given in Algorithm 3.

Application to damage tolerance analysis

ARC-Subset methodology is applied applied to test cases based on NASGRO [START_REF] Nasgro | Fracture mechanics and fatigue crack growth analysis software version 7[END_REF], a damage tolerance tool allowing to assess the FAD margin of a component after crack propagation when the targeted lifetime is reached or when the FAL is crossed (Section 2.1). The goal is to limit the number of damage tolerance evaluations required to assess low probabilities comparing with reference methods such as Subset Simulation [START_REF] Au | Estimation of small failure probabilities in high dimensions by subset simulation[END_REF] and 2SMART [START_REF] Bourinet | Assessing small failure probabilities by combined subset simulation and support vector machines[END_REF]. The failure scenarios (1) and (2) are correlated using the R6 rule and the FAL is dened within same limits for the following test cases. The rst case concerns a through crack and the second one refers to a surface crack in a beam. The rst case considers a through-crack in a beam in traction (type TC11 in NASGRO, Figure 5a). Two of the most signicant variables are set as random: the size of the defect c and the load magnitude S 0 . Table 1 describes the distribution of parameters. The ARC-Subset is compared with the classical Subset Simulation [START_REF] Au | Estimation of small failure probabilities in high dimensions by subset simulation[END_REF], adapted for low probability estimation. 2SMART [START_REF] Bourinet | Assessing small failure probabilities by combined subset simulation and support vector machines[END_REF] is also 260 applied to complete the comparison. This method combines SVM and Subset Simulation to reduce the number of model evaluations for low probabilities. intermediate separators. Therefore, when the margin is positive, we can assume that regression surrogates are more ecient than classication. This explains why the ARC-Subset is more ecient than 2SMART which needs an important number of experiments to describe intermediate subset limit states.

4.2. Surface Crack in a beam with 9 random variables A surface crack in a beam is shown in Figure 10 (type SC17 in NASGRO). Nine parameters are set as random variables. The three methods provide nearly the same probability of failure at 1:19 ¢ 10 7 (Table 3). As for the rst test case, the ARC-Subset method saves respectively one and three orders of magnitude regarding the number of simulations compared to 2SMART and Subset Simulation. Even if the target probability is higher than in the rst example, the ARC-Subset requires more damage tolerance evaluations, due to the higher The advantage of ARC-Subset is the possibility to assess a low probability of failure with a reduced number of simulations without quantitative information from failed experiments. This method can be extended to prevent code crash considering it as failed experiment dened as a qualitative information. The main limitation of this methodology is the curse of dimensionality which mainly impacts the regression phase. However, the modularity allows using a more suitable regression surrogate. Moreover, the condence about p f is based on the Subset Simulation estimators computed on the surrogates. The SVM separator gives no condence level because the SVM margin, interpreted as a zone of uncertainty, is only based on support vectors and not on the whole DOE as it is the case for the Kriging variance. Therefore, this information is not available to estimate the condence bounds of p f .

Conclusion

The ARC-Subset methodology provides promising performance for damage tolerance applications by reducing the number of model evaluations while keeping the same level of accuracy as existing approaches. Nevertheless, we can identify some limitations. The test cases, even if issued from industrial practice, are of a relatively low complexity from the computational point of view. The scalability of the proposed approach is limited to relatively low dimensional applications due to the use of Kriging. Future work concerns the application of ARC-Subset on more complex test cases based on the extended nite element method (XFEM). The enrichment strategy may be further improved using a multi-objective optimization of both Max-Min criteria. Moreover, the extension of the ARC-Subset requires surrogate models better adapted to large dimensions.

  M FAD Failure Assessment Diagram margin n DOE Size of the DOE n SS Size of the subset population N cycle Number of cycles N target Targeted number of cycles density function nom Non-physical stress value of the remaining ligament ow Second reference stress value depending on the material ref First reference stress value depending on the material AK Adaptive Kriging ARCSS Adaptive Regression and Classication based on Subset Simulation DOE Design Of Experiments FAD Failure Assessment Diagram FAL Failure Assessment Line LEFM Linear Elastic Fracture Mechanics MCS Monte Carlo Simulation SS Subset Simulation SVM Support Vector Machine XFEM eXtended Finite Element Method

Figure 1 :

 1 Figure 1: Illustration of the principle of the remaining ligament (dashed surface).

Figure 2 :

 2 Figure 2: Failure Assessment Diagram (FAD) for damage tolerance in Fracture Mechanics.The red line is the Failure Assessment Limit (FAL). O is the origin of the FAD, point A is dened by (Kr,Lr) and B is the intersection between the FAL and (OA) allowing to compute M FAD .

Figure 3 :

 3 Figure3: FAD: The dashed-dotted blue line with triangle markers shows the cycles for a safe component and allows the computation of the FAD margin at every step. The purple dashed line with square markers presents the steps of a failed component, the FAD margin is not quantiable beyond the FAL with the LEFM hypothesis.

Figure 5b illustrates theFigure 4 :

 4 Figure5billustrates the performance function evolution in the standard space for the damage tolerance reliability analysis of a cracked beam in traction considering two random variables (Figure5a). In the failure region, the gradient is close to zero impacting negatively the performance of the gradientbased optimization algorithm required for FORM. Moreover, the non-positive M FAD values form a plateau where only the sign of H(U) is available. The

Figure 5 :

 5 Figure 5: 5a: Cracked beam in traction.

Figure 6 and

 6 Algorithm 1 respectively provide the owchart and the pseudocode of the regression part of the ARC-Subset. The initial DOE (b) is generated using optimized sampling or expert judgement and evaluated (c) to identify the global trends of the model based on the rst subset population (a). A Kriging regression surrogate HKrig (u) is trained on the DOE (d) and is used to evaluate the subset population u (kSS) SS (e). The intermediate thresholds q (kSS) are intermediate subset probabilities. To improve the quality of the surrogate, enrichment strategies are employed (g') until quality stopping criteria are satised (g). If the transition criteria are not satised (h), a new subset 200 population is generated (h'). The DOE is enriched selecting 2 ¢ d experiments by k-means clustering [40] of the new subset population (h").

Figure 6 :

 6 Figure 6: Regression steps of the ARC-Subset algorithm. The grey, blue and red boxes illustrate respectively Subset Simulation, surrogate and model requirements.

Figure 7 :

 7 Figure 7: Classication step of the ARC-Subset algorithm. Color meaning is given in Figure 6. enriched by k-means clustering of the last subset population (a) in order to train a SVM separator (b) to assess the nal subset probability p m (e) by surrogate evaluation (c) p m =

  i=1:::nDOE ku u DOE k s.t.HSVM (u) = 0:

  [START_REF] Lacaze | A generalized max-min sample for surrogate update[END_REF] by multiplying the objective (8) by the joint density function U u GMM = arg max u min i=1:::nDOE ku u DOE k

  by the regression model if it is trained on a DOE containing failure experiments. A second criterion prevents a worse evaluation of the threshold due to failed experiments. The classication phase starts when k failure experiments

Figure 8 :

 8 Figure 8: Damage tolerance evaluations required by ARC-Subset in the standard space. The blue triangle markers are the safe evaluations, and the red circles are failed ones. The green dashed line represents the actual limit state, and the solid black line is the one obtained by SVM.

Figure 8 Figure 9 :

 89 Figure 8 shows in the standard space the 115 damage tolerance evaluation sites required by ARC-Subset to compute the failure probability 1:55 ¢ 10 9

Figure 10 :

 10 Figure 10: Surface Crack beam in traction [2].Table 3: Results of the surface crack beam in traction considering nine random variables. Method Evaluations p f Subset Simulation (10000=step) 81978

Table 1 :

 1 Properties of input random variables for the Through Crack in a beam

		Variable Distribution Type Mean	Standard deviation
		c	Uniform	0:55 mm 0:26 mm
		S 0	Normal	52:5 MPa 5:25 MPa
	255	4.1. Through Crack in a beam

Table 2 :

 2 Results of the through crack in a tension beam considering two random variables.

	Method	Evaluations p f
	Subset Simulation (10000=step) 104889 2SMART [37] 3089	1:53 ¢ 10 9 (8:64%) 1:54 ¢ 10 9

ARC-Subset (¢10)

103:8 (8%) 1:55 ¢ 10 9 (8:58%)

Acknowledgments

The authors acknowledge the support of CNES (French National Center for Space Studies) and of ANRT (French National Association for Research and Technology) by the CIFRE convention 2015/0209. This work has been also partially supported by the ORFI project co-funded by the Walloon Region (Convention 7567).

Algorithm 1 ARC-Subset pseudo-code algorithm of the regression phase initialization p f = 1, k SS = 1, p target = 0:1, n DOE = 5 ¢ d, n SS = 10e4

. Preconised


MD Giorgia Battipaglia

Annalisa Ruggeri

MD Radwan Massoud

MD Jean El Cheikh

Matthieu Jestin

Ahmad Antar

Syed Osman

Walid Rasheed

Marwan Shaheen

Ramdane Belhocine

MD Syed Osman Ahmed

PhD Ollivier Legrand

Riad Youniss

MD Mahmoud Aljurf

PhD Mohamad Mohty

PhD Ali Bazarbachi

Efficacy and feasibility of sorafenib as a maintenance agent after allogeneic hematopoietic stem cell transplantation for Fms-like tyrosine kinase 3-mutated acute myeloid leukemia

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

Giorgia Battipaglia, MD

Hôpital Saint Antoine 184 Rue du Faubourg Saint Antoine, 75012 Paris France Email: giorgia.battipaglia@aphp.fr

Authors have no conflicts of interest to disclose

Dear Editor, Patients diagnosed with acute myeloid leukemia (AML) with Fms-like tyrosine kinase 3 (FLT3) mutations have a very poor prognosis [START_REF] Schlenk | Prognostic significance of activating FLT3 mutations in younger adults ( 16 to 60 years ) with acute myeloid leukemia and normal cytogenetics : a study of the AML Study Group Ulm[END_REF]. Use of allogeneic hematopoietic stem cell transplantation is now a standard practice in this setting of patients, but disease relapse may occur, with difficulties to obtain a complete remission despite salvage treatments [START_REF] Brunet | Impact of FLT3 Internal Tandem Duplication on the Outcome of Related and Unrelated Hematopoietic Transplantation for Adult Acute Myeloid Leukemia in First Remission: A Retrospective Analysis[END_REF]. More recently, the incorporation of FLT3inhibitors in the treatment of these patients has shown promising results [3][4].

We previously reported the safety and efficacy of sorafenib, a FLT3-inhibitor, as a maintenance agent after allogeneic hematopoietic stem cell transplantation (allo-HSCT) in patients diagnosed with acute myeloid leukemia (AML) with FLT3 mutations [START_REF] Battipaglia | Efficacy and Feasibility of Sorafenib as a Maintenance Agent After Allogeneic Hematopoietic Stem Cell Transplantation for Fms-Like Tyrosine Kinase 3-Mutated Acute Myeloid Leukemia[END_REF]. Since our report, other retrospective series confirmed our results showing significant overall (OS) and progression-free survival (PFS) gain in patients receiving sorafenib before and/or after allo-HSCT [START_REF] Xuan | Effect of Sorafenib on the Outcomes of Patients With FLT3-ITD Acute Myeloid Leukemia Undergoing Allogeneic Hematopoietic Stem Cell Transplantation[END_REF]. More recently, preliminary conclusions of a prospective trial randomizing use of maintenance treatment with sorafenib versus placebo introduced during the first 60-100 days after allo-HSCT, further supported the use of this drug in this high-risk setting [START_REF] Burchert | Sorafenib as maintenance therapy post allogeneic stem cell transplantation for FLT3-ITD positive AML: results from the randomized, double-blind, placebo-controlled multicentre Sormain Trial[END_REF].

Therefore, we would like to update the follow-up of the 27 patients with FLT3 mutated AML who received sorafenib as a single maintenance agent. In our cohort, FLT3-ITD AML was detected in 25 patients, whereas FLT3-TKD was found in the remaining 2 patients. Median age at time of transplant was 46 (range 16-58) years. Disease status at time of transplant was complete remission (CR) in 26 patients (including 3 CR2 and 1 CR3) and active disease in the remaining one. Of note, minimal residual disease remained detectable at time of allo-HSCT in 12 patients. Conditioning was myeloablative in 20 patients, reduced-intesity in 5 patients and sequential in the two remaining patients. Sorafenib was introduced at a median time after transplant of 70 (range 29-337) days with different starting doses, subsequently adjusted to the best tolerated dose according to side effects. Toxicities occurred in 11 out of 27 patients, mainly being grade 1 or 2 and reversible in all cases. These events were mostly hematologic (2 patients), gastrointestinal (6 patients, including 1 patient with hematologic toxicity), cardiac (1 patient), cutaneous (3 patients, including 1 patient with gastrointestinal toxicity), and biochemical (amylase and lipase elevation [1 patient]). Drug withdrawal or dose reductions due to toxicity were made in 5 and 3 patients, respectively. Thirteen patients developed graft-versus host disease (GVHD) with a median time to onset after sorafenib introduction of 14 (range 5-56) days, prompting to either drug withdrawal or dose reduction in 3 and 4 patients, respectively. Among patients starting with a dose of 400 mg BID and needing a dose reduction, in most of them the final dose was 200 mg BID. A possible synergistic effect of sorafenib and alloreactive donor T cells in facilitating long-term disease control has also been proposed in murine models in which sorafenib apparently exacerbated GVHD [8][9].

After an initial median follow-up of 18 months, we previously reported a 1-year-PFS rate of 92%±5% and 1-year OS of 92±6%, with 2 patients dying of GVHD or infection. At that time point, 18 patients out of 27 were still on sorafenib with a median treatment duration of 14 (range 3-46) months.

We are now at a median follow-up from allo-HSCT of 40 (range 15-68) months and 12 patients are still receiving sorafenib out of 18 patients in the initial report. Interestingly, the main reason for treatment withdrawal in the six patients was physician choice in 3 while in one patient, after intermittent use of sorafenib, the drug was definitively stopped when chronic GVHD occurred two years after allo-HSCT.

In the two remaining patients, withdrawal was related to disease recurrence. One of those patients achieved a second CR with salvage chemotherapy followed by a second allo-HSCT. In the second patient salvage chemotherapy also allowed second CR. Of note, in the previous report we also described a case of sorafenib withdrawal followed by the use of prophylactic and then curative 5azacitidine in relation to molecular relapse, allowing CR at 20 months after allo-HSCT. A second relapse occurred in this patient at 27 months from allo-HSCT and, despite 2 further salvage regimens, death occurred due to disease progression. Interestingly, at time of relapse this patient presented different size of FLT3-ITD mutation as compared to those found at time of diagnosis.

Finally, at the last follow up, 22 patients are still alive. The 2-years PFS is 73%±9% while the 2-year OS is 80%±8% (Figure 1). These results are similar to those previously reported by Brunner et al. who reported 2-year PFS and OS of 82% and 81%, respectively, in FLT3-mutated AML receiving sorafenib after allo-HSCT [START_REF] Brunner | Haematopoietic cell transplantation with and without sorafenib maintenance for patients with FLT3 -ITD acute myeloid leukaemia in first complete remission[END_REF]. Importantly, in the current study, stratification of patients according to their FLT3 mutant-to-wild-type allelic ratio at the time of diagnosis was not possible because it was not systematically performed in the 3 centers.

In conclusion, our results with longer follow-up further support the use of sorafenib as a maintenance agent after allo-HSCT, allowing persistence of impressive long-term disease control.

Figure 1 .

 1 Figure 1. Two-year progression-free (a) and overall (b) survival in patients receiving sorafenib after