Yolaine Tissier
email: yolaine.tissier@ifsttar.fr

Véronique Bouteiller
email: veronique.bouteiller@ifsttar.fr

Elisabeth Marie Victoire
email: elisabeth.marie-victoire@culture.gouv.fr

Suzanne Joiret
email: suzanne.joiret@upmc.fr

Thierry Chaussadent
email: thierry.chaussadent@ifsttar.fr

Study of an electrochemical chloride extraction treatment on a both carbonated and chlorinated reinforced concrete

Keywords: reinforced micro-concrete, chloride, carbonation, cathodic treatment

Civil engineering structures and historical buildings can suffer from corrosion of the embedded reinforcing steel once the concrete cover is totally carbonated and/or when chloride ions reach the steel/concrete interface. On field, these two types of contamination can be encountered separately or combined requiring implementation of proper repair methods. In this study, in a first step, corrosion of both chlorinated and carbonated reinforced concrete specimens were followed during casting and accelerated contaminations. In a second step, electrochemical chloride extraction was performed as a repair treatment (cathodic polarization, 1 A/m² of steel surface during 8 weeks). The efficiency of the treatment and its impact on the cement matrix and at the concrete-steel interface were studied during the treatment, after rebar depolarization and in the long term (several months) in order to evaluate its durability. In order to achieve these aims, three analysis means were used: electrochemical characterization, scanning electron microscopy and Raman microspectroscopy. In this paper, results obtained on specimens cast with CEM III cement (often encountered on reinforced concrete historical monuments) with chlorides addition into the mix water and further accelerated carbonation are presented. Study of rebar's electrochemical characteristics during the artificial aging highlighted an increase of corrosion rate from negligible value of 0.1 µA/cm² after fabrication to about 10 µA/cm² after carbonation. SEM observations confirmed this phenomenon with the identification of a corrosion layer on most of the concrete/steel interface after carbonation. The ECE efficiency was demonstrated by a chloride extraction of about 97% at the rebar level which yields a decrease of chloride ions concentration below the theoretical threshold value of 0.4 % by mass of cement after two weeks. Simultaneously a realkalisation ring was observed on concrete around the reinforcement bar which reached almost 1 cm after 56 days. After depolarization, results showed that a duration of 28 days of ECE treatment allowed the stabilization of the corrosion state of the rebar. Raman microspectroscopy allowed to study in situ, thanks to specific cells, the corrosion products' reduction during the treatment.

Introduction

Complete cover carbonation and chloride ions penetration until the steel/concrete interface are the two main causes of reinforced concrete degradation induced by corrosion. These two types of contamination can be encountered separately or combined on civil engineering structures as well as on historical buildings requiring implementation of proper repair methods. This study aims to characterize the corrosion processes then the electrochemical chloride extraction (ECE) treatment in a both carbonated and chlorinated reinforced concrete. For this purpose, four series of specimens based on two cements and two aging processes were casted. In this paper the study's complete protocol is presented. First results obtained on one series of specimens during the aging then during the treatment are presented and discussed in a second time. SEM-EDS analyses and Raman microspectroscopy were used to characterize the corrosion product layer at the steel/concrete interface. SEM-EDS gave information about corrosion products distribution, morphology and elements of composition while Raman microspectrometry allowed the identification of crystallized corrosion phases formed. Nondestructive-testing (NDT) electrochemical characterization was used to monitor the evolution of the state of corrosion of the rebar (active/passive).

Methodology

In this section the specimens' characteristics and the experimental program established to follow the evolution of corrosion state during the aging and during the electrochemical treatment are presented.

Specimens formulation and design

Design and composition of specimens used for this study must allow and facilitate chemical reactions within the concrete cover. To achieve this, cylindrical specimens of reinforced concrete of 10 cm in height and 4 cm in diameter (Fig 1 ) were casted using the composition already used in Tong's thesis [START_REF] Yy | Traitement électrochimique de réalcalinisation pour la réparation du béton armé dégradé par carbonatation[END_REF] that allows on a reduced scale to stay close to the granular distribution of a concrete (Tab 1). Two types of cement were selected: -CEM I (52.5 N CP2; Holcim, Lumbres) which is a usual Portland cement which will allow results comparisons. The specimens were named type 1. -CEMIII/A (52.5 L PM ES CP1 NF; Lafarge, Le Havre), which is a cement frequently encountered on concrete historical monuments. Furthermore, the use of this cement which contains blast furnace slag is promoted for the construction of new structures because of its lower impact on environmental issue (less CO2 produced). Therefore, the results will help improve knowledge about the behavior of reinforced structure using this cement. The specimens were named type 3.

Rebar was a smooth round steel type S 235JR of 5 mm in diameter with an active surface of 10 cm². The open porosity measured after curing (28 days for CEM I cement and 90 days for CEM III cement) was about 18 % ± 1 %.

Figure 1 : Specimen design and sampling for the analytical characterizations

Experimental set-up A specific set-up (Fig 2 ) was designed in order to study, in a first step, the corrosion of a both carbonated and chlorinated reinforced concrete and then, in a second step, the efficiency and impact of the electrochemical repair (during the treatment and after its end). and then chloride penetration by wetting/drying cycles (five days in a solution of 35 g/L NaCl and two days in the laboratory at room temperature) (CO2+Cl). The specimens were named type I. Evolutions of carbonation and chloride contamination by wetting/drying cycles were followed thanks to colour indicators, respectively phenolphtalein (0.5 % in ethanol) and silver nitrate solutions (0.1 N).

After the aging step, the electrochemical chloride extraction treatment was then conducted on 37 specimens of each series (G1, I1, G3 and I3). The treatment principle is presented on Fig 3 . As indicated in the NF EN 14038 part 2 standard [START_REF]Electrochemical realkalisation and chloride extraction treatments for reinforced concrete Part2: chloride extraction[END_REF], a current density of 1A/m² of steel was applied between the rebar and a titanium/platinum grid (counter-electrode) during 8 weeks. The electrolyte used was a disodium tetraborate solution (25g/L of Na2B4O7 10H2O) to promote ionic exchanges and facilitate chloride ions entrapment. The treatment monitoring was performed controlling and/or following the room temperature, the applied current, the voltage of each specimen and the voltage of the whole specimens on a one hour frequency base.

Three control specimens were taken from the treatment after 3, 7, 14, 28 and 56 days, 56 days corresponding to the end of the treatment. These times were selected as several authors observed that the extraction of chloride ions is more efficient during the treatment's beginning [START_REF] Herrera | Electro-chemical chloride extraction: Influence of C3A of the cement on treatment efficiency[END_REF][START_REF] Swamy | Effectiveness and structural implications of electrochemical chloride extraction from reinforced concrete beams[END_REF][START_REF] Fajardo | Electrochemical chloride extraction (ECE) from steel-reinforced concrete specimens contaminated by "artificial" sea-water[END_REF][START_REF] Arya | Factors influencing electrochemical removal of chloride from concrete[END_REF]. Electrolyte was renewed at each term (except 3 days) to avoid a loss of treatment efficiency due to the increase of chloride ions concentration into the electrolyte. Regarding the chloride extraction evolution, concrete's free and total chloride concentrations were measured on two of the control specimens by AgNO3 potentiometric titration (on the first centimetre around the rebar and on the 0.75 cm of external cover). Moreover the electrolyte chloride titration was also performed. Phenolphtalein test was performed on the surface of part c (Fig 1 ) of the two same control specimens. The third control specimen was kept to perform an electrochemical characterization after depolarisation.

Simultaneously to the treatment, two specimens (which were not submitted to the current) were used to observe lixiviation phenomenon. They were immerged in the tetraborate based solution and Cl -titration was performed after 14, 28 and 56 days of treatment.

After treatment, the electrochemical characteristics of the reinforced specimens were checked every month to verify rebars are depolarized (storage at 60 %HR and 20°C). Cycles of relative humidity are currently running and electrochemical characterizations are performed to ensure the durability of the treatment.

Figure 3 : Simplified principle of electrochemical chloride extraction

Analysis means

To study the corrosion process of the rebar, three main analyses means were used on different parts of the cylinder (Fig 1 ): -A Jobin-Yvon Horiba (LABRAM) Raman spectrometer was used. Spectra were obtained with a helium-neon red laser (0.1 mW for these measurements). For these study acquisitions a x50 objective was used to focus laser light and collect spectra for duration of 600 seconds. Specific designed cells were prepared from the regular one after the aging step to follow the concrete/steel interface "in-situ" during the corrosion initiation and propagation and then during the treatment by Raman microspectroscopy. After cutting as presented Figure 1, a thin glass slide was pasted on the polished surface to isolate the steel/concrete interface from a direct contact with the external environment. After preparation, these samples were immerged in a 25g/L Na2B4O7 10H2O solution. Regular observations at a magnification of 50 were made until the detection of pitting corrosion. Raman spectra were then acquired in different areas of interest. The same areas were followed during the ECE treatment and after treatment. -A JEOL 5600LV scanning electron microscope (SEM) equipped with a microanalyzer X in dispersal of energy Oxford Link Pentaflet was to observe corrosion at the concrete/steel interface. Two types of samples were examined: polished cross sections to evaluate corrosion layers evolution and fractured samples to observe both corrosion crystallised phases' morphology and eventual modifications of the cement matrix. -An electrochemical characterization was performed with a Bio-Logic VMP2Z potentiostat (5 channels) at each step of the process. The half-cell potential (Ecorr), the linear polarization resistance (LPR) and the electrical concrete resistance (Re) were measured and the corrosion rate iCorr (µA/cm²) was calculated according to the formula (1). iCorr =

With: B =26 mV, Rp (ohm) the linear polarization resistance and S (cm²) the active steel surface (10 cm² in this study)

Corrosion processes

In this section results obtained on G3 specimens casted with a CEM III/A cement, aged first by NaCl addition to the mix and then by carbonation are presented.

Electrochemical characterization

Fig 4 present the half-cell potential (a), the resistance (b) and the corrosion rate (c) of specimens casted with the CEM III/A cement (S), G3 specimens casted with chloride addition (Cl) and G3 specimens after complete cover carbonation (Cl+CO2).

From Fig 4-a, half-cell potentials measured for the three studied contamination states are similar with values in the range of -500 mV to -600 mV. The fact that the half-cell potential is significantly electronegative even without contamination (S) can be explained by the use of the CEM III/A cement [START_REF] Bouteiller | Corrosion initiation of reinforced concretes based on Portland or GGBS cements: Chloride contents and electrochemical characterizations versus time[END_REF][START_REF] Macphee | Theoretical description of impact of blast furnace slag (BFS) on steel passivation in concrete[END_REF][START_REF] Jarrah | Electrochemical behaviour of steel in plain and blended cement concretes in sulphate and/or chloride environments[END_REF][START_REF] Garcia | Potential measurement to determine the chloride threshold concentration that initiates corrosion of reinforcing steel bar in slag concretes[END_REF]. From Fig 4-b, the resistances of non-contaminated specimens (S) and chlorinated specimens (Cl) are of the same order with average values and standard deviations respectively of 746 ohm and 35 ohm for specimens "S" and 971 ohm and 31 ohm for specimens "Cl". In the case of both chlorinated and carbonated specimens (Cl+CO2) the resistance decreased by a 10 factor with average values and standard deviations respectively of 80 ohm and 12 ohm. This could be explained by a better ionic conductivity due to chlorides freeing during carbonation. Finally, the corrosion rate after curing (Fig 4-c) was below 0.1 µA/cm² for both I3-S and G3-Cl specimens which indicates a negligible level of corrosion according to the Rilem [START_REF] Andrade | Rilem TC 154-EMC Test methods for on-site corrosion rate measurement of steel reinforcement in concrete by means of the polarization resistance method[END_REF]. For both chlorinated and carbonated specimens a drastic increase of corrosion rate (8±1 µA/cm²) demonstrated a high level of corrosion [START_REF] Andrade | Rilem TC 154-EMC Test methods for on-site corrosion rate measurement of steel reinforcement in concrete by means of the polarization resistance method[END_REF]. This phenomenon is in good agreement with the hypothesis of chlorides freeing during carbonation.

Figure 5 : (a) BEIC-SEM image (x600) of the steel/concrete interface on cross section of G3 specimen, after casting (Cl) without evidence of corrosion ; (b) SEI-SEM image (x35) of the rebar print of G3 specimen, after casting (Cl) without evidence of corrosion

after carbonation (Cl+CO2) with evidence of a corrosion layer; (b) SEI-SEM image (x35) of the rebar print of G3 specimen, after carbonation (Cl+CO2) with evidence of corrosion products cristallisation Figure 7 : SEI-SEM image (x 1300) of corrosion products critallisation on the rebar print of G3 specimen, after carbonation (Cl+CO2) and EDS associated spectrum

These observations are in good agreement with electrochemical characterization results. Indeed corrosion products were observed only after carbonation of the G3 specimen, chlorinated during the casting, which corresponds to the drastic increase of the corrosion current.

Corrosion products characterization by Raman Spectroscopy

Corrosion in-situ following were made on a specimen G3 both chlorinated and carbonated. Chlorinated green rust (Fig 4) was the predominant phase found (main Raman bands at 434 cm -1 and 506 cm -1 and a chloride presence characteristic band around 320 cm -1 ) [START_REF] Boucherit | Raman studies of corrosion films grown on Fe and Fe-6Mo in pitting conditions[END_REF] which evidenced the presence of chloride into the corrosion processes. Moreover, green rust oxidation into FeIII oxyhydroxyde did not occur which proved the good isolation of the interface from a direct exposure to oxygen from the air.

Chloride concentration and pH evolution

Fig 9 shows the evolution of chloride concentration during the ECE treatment as well as the pH evolution at the rebar vicinity. The chloride threshold for preventing corrosion on new structures from the EN206 standard [START_REF] Jarrah | Electrochemical behaviour of steel in plain and blended cement concretes in sulphate and/or chloride environments[END_REF] has also been added.

Regarding the chloride extraction, the evolution of chloride ions extraction was in good agreement with the increasing chloride ions concentration titrated into the electrolyte. After 3, 7, 14 and 28 days the chloride ions extraction was respectively of about 58, 70, 86 and 93 % in the external cover and 40, 81, 94 and 97 % in the first centimetre around the rebar. The extraction was faster during the seven first days of treatment which is in accordance with literature results [START_REF] Herrera | Electro-chemical chloride extraction: Influence of C3A of the cement on treatment efficiency[END_REF][START_REF] Swamy | Effectiveness and structural implications of electrochemical chloride extraction from reinforced concrete beams[END_REF][START_REF] Fajardo | Electrochemical chloride extraction (ECE) from steel-reinforced concrete specimens contaminated by "artificial" sea-water[END_REF][START_REF] Arya | Factors influencing electrochemical removal of chloride from concrete[END_REF]. Results show that the free chloride concentration became less than the EN206 chloride threshold of 0.4 % mass of cement after 7-14 days of treatment. The extraction of chloride ions decreases with time as the chloride concentration gets slower and therefore the resistance increases. Regarding non polarised specimens, chlorides extraction efficiency was similar to the one obtained under cathodic polarization from 28 days of immersion. Fig 9 highlights that there was initially no difference between total and free chloride ion concentrations which means that after carbonation, as electrochemical characterization results suggested, most chlorides were freed. This can also explain the particularly good extraction results obtained during this study even without polarization.

Regarding the pH evolution (Fig 9), observed pH increase was probably due to the hydroxyl ions formation around the rebar and maybe the cations migration towards the rebar. Based on results obtained and the NF EN 14038 part 1 standard [START_REF]Electrochemical realkalisation and chloride extraction treatments for reinforced concrete Part1: Realkalisation[END_REF], concerning required thickness of concrete with basic pH around the rebar, the minimum treatment duration required is either 56 days to obtain a 10 mm realkalised concrete zone around the rebar or around 28 days considering the experimental rebar diameter (5 mm). show a stabilisation of the rebar corrosion state to a negligible corrosion level after treatment with a half-cell potential of -161±19 mV and a corrosion current of 0.03 ±0.01 µA/cm². This last is however higher than the corrosion current of a non-contaminated specimen (S).

Regarding non polarised specimens, results obtained show that, even with a similar amount of extracted chloride than for polarised specimens, corrosion current is still high. This underline the fact that both chloride extraction and pH increase are needed to stop or at least decrease the corrosion state of the rebar.

Conclusion

In this experimental work, the set-up design adopted to study, first the corrosion process on both chlorinated and carbonated reinforced concrete specimens, and second, the electrochemical cathodic treatment were presented. Results obtained on G3 specimens allowed establishing the following conclusions:

-Non-contaminated specimens as well as chloride contaminated G3 specimens showed a negligible corrosion after curing as indicated by corrosion rates less than 0.1 µA/cm². -After the carbonation process, as expected, corrosion increased rather drastically as evidenced by corrosion rates in the range of 10 µA/cm². SEM results confirmed this behaviour with the observation of corrosion products on most of the interface (5-15 -Efficiency of the electrochemical treatment based on chloride extraction as well as pH increase of the concrete around the µm corrosion layer on 80-90% of the concrete/steel interface).reinforcement was demonstrated. Two weeks of treatment were needed to obtain a Cl -concentration below the threshold value of 0,4 % mass of cement while 56 days of treatment lead to one centimeter of concrete with pH>9.

After the end of the treatment the corrosion current was below 0.1 µA/cm² corresponding to a negligible level of corrosion which demonstrated that the corrosion was halted. -Comparison with results obtained on immersed and non polarised specimens showed that both chloride extraction and pH increase are needed to stabilise the corrosion level of the rebar. Indeed, without polarization, the pH did not increase and although a significant amount of chlorides were lixiviated, the corrosion current remained in the high corrosion level.

The following of treatment durability and Raman spectroscopy results will complete the experimental work on G3 specimens. Comparison with G1 specimens [START_REF] Tissier | Study of a carbonated and chlorinated reinforced concrete: corrosion processes and ECE repair efficiency[END_REF] and with the two other types of specimens will give a larger view on this subject.

Figure 2 :

 2 Figure 2 : Experimental set-up Two different aging procedures were studied: -Chloride addition to the mix (5 % of cement mass of NaCl) (Cl) and then carbonation until complete carbonation of the concrete cover (50 % CO2; 70 % HR) (Cl+CO2). The specimens were named type G. -Carbonation until reaching the concrete/steel interface (50 % CO2; 70 % HR) (CO2)and then chloride penetration by wetting/drying cycles (five days in a solution of 35 g/L NaCl and two days in the laboratory at room temperature) (CO2+Cl). The specimens were named type I. Evolutions of carbonation and chloride contamination by wetting/drying cycles were followed thanks to colour indicators, respectively phenolphtalein (0.5 % in ethanol) and silver nitrate solutions (0.1 N).

Figure 4 .

 4 Figure 4. (a) Half-cell potential, (b) resistance and (c) curent density of non-contaminated specimens (S) and G3 specimens after curing (Cl) and after carbonation (Cl+CO2) Steel / concrete interface characterization by scanning electron microscopy After casting, the steel/concrete interface of both non-contaminated (S) and chlorinated (Cl) specimens was characterized by an absence of corrosion (Fig 5-a) as well as the rebar prints observations (Fig 5-b).

Fig 6 -

 6 Fig 6-b shows a representative area of the interface of a G3 specimen after carbonation (Cl+CO2) with a corrosion layer of 5 to 15 µm. The observation on fracture after carbonation (Fig 6-b) showed numerous areas of corrosion in the rebar print. The crystallisation morphology of corrosion products mainly containing Fe, O and Cl is presented on Fig 7.

Figure 6 :

 6 Figure 6 : (a) BEIC-SEM image (x600) of the steel/concrete interface on cross section of G3 specimen, after carbonation (Cl+CO2) with evidence of a corrosion layer; (b) SEI-SEM image (x35) of the rebar print of G3 specimen, after carbonation (Cl+CO2) with evidence of corrosion products cristallisation

Figure 8 :

 8 Figure 8 : Chlorinated green rust Raman spectrum

Figure 9 :

 9 Figure 9: Evolution of chloride concentration and ring with pH>9 thickness during ECE treatment Electrochemical characterization Fig 10 compares previous electrochemical characterization results to those obtained after the electrochemical cathodic treatment (56 days) after 10 weeks of depolarization. These resultsshow a stabilisation of the rebar corrosion state to a negligible corrosion level after treatment with a half-cell potential of -161±19 mV and a corrosion current of 0.03 ±0.01 µA/cm². This last is however higher than the corrosion current of a non-contaminated specimen (S).Regarding non polarised specimens, results obtained show that, even with a similar amount of extracted chloride than for polarised specimens, corrosion current is still high. This underline the fact that both chloride extraction and pH increase are needed to stop or at least decrease the corrosion state of the rebar.

Figure 10 .

 10 Figure 10. (a) Half-cell potential, (b) resistance and (c) curent density of non-contaminated specimens (S) and G3 specimens after curing (Cl), after carbonation (Cl+CO2), after electrochemical treatment (ECE 56 days) and after 56 days of immersion (NP 56d) Corrosion products characterization by Raman Spectroscopy During ECE treatment, regular Raman acquisitions (successive the first 5 hours and then every day) were performed to follow the phases transformations induced. After two days of polarization, the main phase detected was magnetite (main Raman band at 670 cm -1 ) (Fig 11).This phase remained stable during 20 days of polarization after which the treatment was ended. No remaining chlorinated green rust was identified at the end of the treatment. Magnetite is a more stable corrosion product which explained the stabilization of the rebar corrosion state obtained by electrochemical characterization. A study of the treatment durability should however be performed.

Figure 11 :

 11 Figure 11 : Magnetite Raman spectrum

Table 1 :

 1 "Micro-concrete" composition

	siliceous Palvadeau aggregates	Cement	Water	W/C ratio
		(kg/m 3 )	(kg/m 3 )	(kg/m 3 )
	0/0.315 159	0.315/1 0.5/1 1/4 2/4 295 192 686 229	387	251.55	0.65


Foued Abroug

Etienne Pessard

Guenaël Germain

Franck Morel

HCF of AA7050 alloy containing surface defects: Study of the statistical size effect

Keywords:

This work investigates the effect of artificial surface defects on the fatigue limit of a 7050 Aluminum alloy (Al Zn6CuMgZr). A large fatigue testing campaign under fully reversed plane bending loading is undertaken on specimen with artificial surface hemispherical defects. The defect number was varied from 1 to 44 defects per specimen and the diameter size ranged from 60 μm to 800 μm. The test results allow the characterization of both the defect effect and scale effect on the fatigue behavior of the material. A probabilistic approach based on the weakest link concept together with a fatigue crack initiation criterion are used to account for the stress dis-tribution and the size of the highly stressed volume. This approach leads to a probabilistic Kitagawa-Takahashi type diagram, which in this case explains the relationship between the defect size and the scale effect on the fatigue strength. The predictions show good agreement with the experimental results and illustrate the im-portance of taking the scale effect into account when designing components containing different surface defects types or roughness patterns.

Introduction

The fatigue data transferability from laboratory specimens to real industrial components or structures is often a tough task due to the high number of parameters affecting the fatigue strength. The loading mode, the microstructural heterogeneities, the defects and residual stresses induced by the manufacturing process, the gradient and the size of the loaded volume are known to greatly affect the fatigue response. For instance, for the same size and geometry of a specimen or a component, it is well known that the loading mode affects the fatigue limit: a difference is usually observed between torsion, uniaxial tension, biaxial tension, rotating or plane bending. These differences can be explained by the stress state and by the macroscopic gradient introduced in the different cases. When extra stress concentrators are introduced (notch, welded joint …), the local gradient is affected and if the critical fatigue area is concerned, the resulting macroscopic fatigue strength can be substantially altered.

It has long been recognized that fatigue limit depends on the spatial stress distribution (volumetric stress distribution) and also on the size of the loaded component. However, the so-called "stress gradient effect" and "size effect" are very closely linked and lead sometimes to confusions. A good review of these two effects was proposed by Papadopoulos and Panoskaltsis [START_REF] Papadopoulos | Gradient-dependent multiaxial high-cycle fatigue criterion[END_REF]. In this paper, the authors try to carefully explain the difference between the pure "size effect" and the pure "stress gradient effect". They clearly show that, very often, the two effects can act concurrently. Let us take the example of a smooth cylindrical specimen submitted to a rotating or plane bending fatigue loading. An increase of the specimen radius leads at the same time to a decrease of the stress gradient and to an increase of the loaded volume. In this case, the two effects are simultaneous. On the contrary, when a smooth cylindrical specimen of constant radius is subjected to a uniaxial tension loading and its length is increased, there is no stress gradient and the stressed volume gets bigger. All the investigations in the literature regarding that kind of experimental studies lead to the well-known conclusions:

-"The higher the gradient, the higher the fatigue strength" -"The higher the stressed volume, the lower the fatigue strength"

In short, although they are confused in the literature, the two effects are distinct phenomena that need to be accounted for by different approaches.

The models dedicated to the statistical size effect are often of two types. The first type of approach, usually used for nonmetallic inclusions in steel, is based on the extreme value analysis of the inclusion population. It predicts the maximum defect size that is possible to be observed in a given volume [START_REF] Murakami | Quantitative evaluation of effects of non-metallic inclusions on fatigue strength of high strength steels. II: Fatigue limit evaluation based on statistics for extreme values of inclusion size[END_REF][START_REF] Makkonen | Size effect in fatigue based on the extreme value distribution of defects[END_REF]. The details of the methodology are described in the ASTM standard [START_REF] Astm E | Standard practice for extreme value analysis of nonmetallic inclusions in steel and other microstructural features[END_REF]. The second type of approach ⁎ Corresponding author. E-mail address: Foued.Abroug@ensam.eu (F. Abroug). https://doi.org/10.1016/j.ijfatigue.2018.01.012 T makes use of a two scale probabilistic model to consider the stressed volume size. Applying the weakest link concept [START_REF][END_REF], the failure probability of a component is calculated from the failure probability of several elementary volumes [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF].

The models dealing with the gradient effect in a deterministic way are often built by using a non local approach like the one initially 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]. 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 steelthe[END_REF].

Even though several models considering the size effect and the gradient effect are available, just a few of them [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] Flaceliere | Probabilistic approach in high-cycle multiaxial fatigue: volume and surface effects[END_REF][START_REF] Morel | Plasticity and damage heterrogeneity in fatigue[END_REF] incorporate both effects in a combined manner as discussed in [START_REF] Luu | Formulation of gradient multiaxial fatigue criteria[END_REF]. Moreover, the role of material defects is most of the time investigated by considering the most critical one and by applying a deterministic criterion like in the work of Murakami [START_REF] Murakami | Quantitative evaluation of effects of non-metallic inclusions on fatigue strength of high strength steels. II: Fatigue limit evaluation based on statistics for extreme values of inclusion size[END_REF]. The present study aim is to get a better knowledge of the wrought aluminium components considering both statistical size and gradient effects due to the presence of one or several surface defects of different size.

The investigation is both experimental and numerical. Some plane bending fatigue tests are first carried out on specimens showing the same size and geometry but containing one or several identical surface defects. From the experimental results, a Kitagawa-Takahashi diagram is built showing the difference between the 1-defect and x-defects cases. Then the observed specific size effect is modeled by using Finite Element simulations together with the weakest link concept. Two different approaches are tested, surface and volume-based approaches both reflecting a statistical size effect. The suitability and limitations of these approaches are discussed based on the comparisons with experimental results. The main objectives are:

-Characterization of the statistical size effect -Description of the crack initiation mechanisms with or without the presence of surface defect -Modeling the size effect 2. AA7050-T7451 alloy and experimental procedure

The material

The material under investigation is the Al Zn6CuMgZr Aluminum alloy, also called AA 7050. In this study, specimens are machined out of a 30 mm thick sheet. By the local thermomechanical effects induced, machining 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 behaviour [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][START_REF] Shahzad | Influence of anodizing process on fatigue life of machined aluminium 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.

The chemical composition of the AA7050-T7451 alloy is given in Table 1. For this range of thickness, the material shows a high recrystallization rate. Areas of consolidation of recrystallized and nonrecrystallized grains can reach a few millimeters of length. Grains have dimensions ranging from 5 to 300 μm in rolling and transverse direc- tions, and 5 to 50 μm in depth (see Fig. 1). Different types of inter- metallic particles are present at the grain boundaries and in the recrystallized grains (Mg 2 Si, Al 7 Cu 2 Fe, MgZn 2 , AlCuZn, etc.) (Fig. 2). These particles are brittle and can be at the origin of crack initiation under fatigue loading [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 to determine the mechanical properties of the material at °0 , °45 and °90 to the rolling direction (Table 2). Geometry of tensile specimen is shown in Fig. 3. The anisotropy is slightly marked and the ductility is limited with a maximum elongation around 13%. In addition, microhardness measurements are conducted along the sheet depth and show slight variations around 157 Hv/0.5.

Fatigue experimental conditions

All the fatigue tests are carried out under fully reversed plane bending (R = -1). A specific specimen geometry is developed for this study (Fig. 4). 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 of the 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 (which corresponds to a crack length of 5 mm or more) or when 2 × 10 6 cycles are reached. The endurance limit is determined at 2 × 10 6 cycles using staircase or locati methods. Stress value is decreased and increased by 10 MPa step for staircase and locati method.

Fatigue tests are first 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 materialsfatigue testingstatistical planning and analysis of data[END_REF] on 15 polished specimens. The mean fatigue limit with the associated standard deviation are then identified. Specimens with artificial defects of hemispherical shape are tested via the locati technique. The locati technique allows to characterize the fatigue limit for each specimen individually. For each specimen, fatigue test is conducted at a given stress amplitude, assumed to be lower than the fatigue limit. If failure does not occur within the limit cycle number (2 × 10 6 cycles in our case), the specimen is recycled at a higher stress amplitude [START_REF] Maxwell | A rapid method for generation of a Haigh diagram for high cycle fatigue[END_REF]. This is repeated until the specimen is fractured after less than 2 ×10 6 cycles. For a low number of cycles by step, a damage accumulation calculation must be used to estimate the fatigue limit for a given cycle number [START_REF] Pessard | A new approach to model the fatigue anisotropy due to non-metallic inclusions in forged steels[END_REF]. In our case, due to the high number of cycles applied on each steps, fatigue limit is directly estimated as the mean stress value between the final loading step leading to failure and the step before. For the locati method, the whole surface of each specimen (including around defect) is observed between each step to check that no crack initiates from the previous step.

Specimens are polished and artificial defects of 6 different sizes are introduced by micro-drilling technique (Ø60, 100, 200, 400, 600, and Ø800 μm). After machining, defects are scanned with a 3D BRUKER type profilometer (see Fig. 6). As it can be seen in Fig. 6, defect depths and diameters are carefully adjusted and controlled.

Using SEM observations of fracture surfaces, no microstructure change is detected due to the machining operation (Figs. [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] Flaceliere | Probabilistic approach in high-cycle multiaxial fatigue: volume and surface effects[END_REF][START_REF] Morel | Plasticity and damage heterrogeneity in fatigue[END_REF]. Moreover, SEM observations and 3D profile measures showed that machined surfaces present a low roughness.

In order to characterize the scale effect, 2-3 specimens per defect size are tested with 1 and 22 defects per specimen (see Table 3). In addition, 3 specimens are tested in presence of 44 defects of Ø400 μm (Fig. 5). To avoid interaction effects, the distance between defects is chosen to be more than 2 mm. Fig. 5 shows the distribution of defects on the surface of the gauge length of the specimen. For multiple defect cases, defects are introduced on the specimen along 2 or 3 straight lines with the specimens length direction (i.e. Fig. 5b-d). The position of the defects is shifted so as to have a single defect per section along the gauge length of the specimen. In order to characterize the scatter of the fatigue limit, 3 staircases are conducted on: polished specimens, polished specimens with a defect of Ø400 μm and polished specimens with 22 defects of Ø400 μm. Specimens that survived in the staircase pro- cedure were retested at a higher load in order to identify the Wöhler curve shape (polished specimens) or to determine fatigue limit via locati method (Ø400 μm defect specimens).

Fatigue test results and fatigue damage mechanisms

All the fatigue test results are gathered in a Kitagawa-Takahashi diagram shown in Fig. 7. Fatigue data are also shown in Tables 345.

The errors bars in Fig. 7 show, depending on the method used, the fatigue strength range obtained via the locati or the standard deviation estimated via the staircase.

For the polished specimens, the staircase method allowed to identify a mean fatigue limit of 174 MPa with a low standard deviation of 5.3 MPa (Fig. 8). Fig. 8 shows a plateau over 3 × 10 5 cycles which corresponds to the well known fatigue limit observed in the HCF domain between 3 × 10 5 and 10 7 cycles. This result is in agreement with other results in the literature that show the presence of a plateau for this type of material [START_REF] Shahzad | Surface characterization and influence of anodizing process on fatigue life of Al 7050 alloy[END_REF][START_REF] Chaussumier | A predictive fatigue life model for anodized 7050 aluminium alloy[END_REF]. It assumes that this asymptote on the case of specimen without a defect is also present on the case of specimen with a defect since the used material is the same.

In presence of a single defect, fatigue limit decreases progressively with the increase of the defect size. This behavior is known as the Kitagawa effect [START_REF] Kitagawa | Applicability of fracture mechanics to very small cracks[END_REF] (Fig. 7).

As shown on the Kitagawa-Takahashi diagram (Fig. 7) the number of introduced defects affect the fatigue response. A drop of 20 MPa is observed when the number of defects is multiplied by 22. This decrease can be explained by a scale effect. When increasing the number of defects, the highly stressed volume increases and the probability of finding an heterogeneity (i.e particle) or an unfavorably oriented grain increases. Hence, crack initiation is more likely to occur at a lower stress value than in the case with a single defect.

As shown in Fig. 7, the slope of decrease in fatigue limit is not dependent on the defect number. In case of 1 and 22 defects, the slope is around (-1/10). This is quite different from the (-1/6) slope given by the Murakami criterion [START_REF] Murakami | Metal fatigue: effects of small defects and nonmetallic inclusions[END_REF] (Fig. 7). Let us recall that the Murakami criterion allows to predict the fatigue strength of a specimen based on its hardness and the area parameter (square root of the projected area of a defect). Murakami proposes to use the following equation to predict the limit of fatigue in the case of fully reversed push-pull loads and a surface crack initiation:

= + σ H area 1.43( 120) ( ) D v 1/6 (1)
In our case, the Murakami criterion shows non-conservative predictions of the fatigue limit. In addition, Murakami's criterion gives the same prediction in presence of a one or several defects [START_REF] Murakami | Metal fatigue: effects of small defects and nonmetallic inclusions[END_REF].

Fatigue damage mechanisms

After the fatigue tests, specimen surfaces are analyzed. For all specimens, crack initiation occurs at a single location and coalescence phenomena are not observed. In presence of 22 surface defects, initiation starts from a single artificial defect randomly located on the specimen surface (Fig. 9).

For polished specimens, SEM observations showed the presence of intermetallic particles at the crack initiation site of several specimens (66%) (Fig. 10). These particles are identified via energy dispersive Xray spectroscopy (EDS) analysis as Mg Si . For some specimens, SEM observations showed no defects at the crack initiation site. However, in some cases, EDS analysis showed the presence of Mg elements with the Al and Zn elements at the crack initiation site (Fig. 10). As shown in the literature, the presence of these elements may indicate the presence of MgZn 2 precipitates at a nanoscale [START_REF] Li | Simulation study on function mechanism of some precipitates in localized corrosion of Al alloys[END_REF].

For the case of specimens containing artificial defects below the size of 100 μm, cracks do not necessarily initiate from the artificial defect (see Fig. 11). Above this size, the crack initiation is always located at an artificial defect (see Figs. 121314).

In the case of 1 and 22 defects, around 50% of initiation occurs from particles. These particles can be observed at the initiation site along the artificial defect for example close to the corner defect (Fig. 13) or in the middle of the defect (Fig. 14). In the case of the absence of particles, crack initiates at the upper side of the hemispherical defect but without any preferential side (same case as Fig. 12)). This location corresponds to the maximum stressed area (in terms of principal or Von-Mises equivalent stress) due to the applied bending loading. Stress distribution around the defect is shown further in this paper (see Fig. 17).

The fatigue limit decrease with the increase of defects number cannot be explained only by the presence of intermetallic particles because the percentage of initiation site with particle is the same for specimen with 1 and 22 defects. Other factors can have an effect on the crack initiation localisation. Grain size at the defect's location, orientation as well as their vicinity are possible causes for crack initiation. The effects of grain size and orientation on fatigue behaviour is an actual and an open question that is discussed in several papers [START_REF] Sauzay | Physically-based modeling of the cyclic macroscopic behaviour of metals[END_REF][START_REF] Guerchais | Micromechanical study of the loading path effect in high cycle fatigue[END_REF].

A probabilistic fatigue criterion to account for the size effect

After the previous comprehensive experimental investigation of the size effect due to surface defects, it is now proposed to describe a modelling strategy accounting for both the change in size of the loaded volume and the stress gradient.

Probabilistic framework

This section is devoted to the modelling of the crack initiation damage mechanism in a polycrystalline aggregate containing geometric defects.

As a general rule most fatigue criteria can be defined by an equivalent mechanical quantity (i.e. stress, strain, energy, etc.), which is compared to a threshold quantity, for a given number of cycles. Crack initiation is subsequently predicted if the equivalent quantity is greater than or equal to the threshold quantity. In the following, for convenience, these will be referred to stress quantities. Hence, the condition for crack initiation is defined by:

⩾ σ σ eq th (2)
Note that in this section, the maximum normal stress amplitude is considered as responsible for crack initiation. Even though the maximum normal stress criterion is not well suited to model the fatigue behaviour of ductile material, this criterion is chosen for its simplicity and is used below to explain the proposed methodology.

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 .

This choice then makes it possible to define the probability of crack initiation. Eq. ( 3) shows the probability density function used to define the threshold stress.

⎜ ⎟ ⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ ⎧ ⎨ ⎩ -⎛ ⎝ ⎞ ⎠ ⎫ ⎬ ⎭ - f σ m σ σ σ σ σ ( ) exp th th m th m 0 0 0 1 0 (3)
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 limit. 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 . This can be expressed as:

∫ = < = P σ σ f σ dσ ( ) ( ) F t h e q σ th th 0 0 eq 0 (4)
Using the simple equation of Weibull's distribution function, the integration allows to express the failure probability as follows:

⎜ ⎟ = - ⎡ ⎣ ⎢ -⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ P σ σ 1 exp F eq m 0 0 (5)
To obtain the probability of crack initiation for the complete structure P F , the weakest link theory is used [START_REF][END_REF]. 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 and S Ω is the total surface area of a specimen.

∏ -= -

P P 1 (1 ) F S F Ω 0 (6)
Hence the probability of survival of the complete structure is given by:

∫ ⎜ ⎟ = - ⎡ ⎣ ⎢ - ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ P S σ σ dS 1 exp 1 F S eq m 0 0 Ω (7)
Introducing the surface stress heterogeneity factor on the component surface H ms [13]:

∫ ⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ H S σ σ dS 1 ms S eq max m Ω Ω ( 8
)
with σ max the maximum equivalent stress on the component surface:

= σ max σ ( ) max S eq Ω ( 9
)
The expression for the failure probability of the total structure becomes:

⎜ ⎟ = - ⎡ ⎣ ⎢ - ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ P S S H σ σ 1 exp F m s max m Ω 0 0 (10)
In order to simplify the expression and to make a link with the high

X a X b X X X b X Polished 170 O O b O O b X a O a O 160 O a O a 150 X X a Ø400 140 O X a X O 1 defect 130 X a O O a 120 O a 130 X X X X a Ø400 120 O O a X a O O a 22 defects 110 O a
a Particle at fracture site. b MgZn 2 traces found at fracture site. Fig. 13. Crack initiation site from the corner defect with a particle (specimen D400-22-7).

cycle fatigue limit, the macroscopic fatigue limit σ a can be expressed via the following expression: [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 steelthe[END_REF] where K tmax is the maximal equivalent stress concentration factor which correspond to the maximal equivalent stress divided by the nominal applied stress.

= ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ - ( ) σ σ K S S ln H a t P ms m 0 0 Ω 1 1 1/ max F
= K σ σ t max applied max ( 12
)
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 our study, the surface stress field due to hemispherical defect is more complex and numerical finite element simulation must be conducted to assess this quantity.

Finite element simulations to account for artificial defect

The surface stress heterogeneity factor H ms is used to characterize the heterogeneity of the surface stress field on the entire component. To simplify the H ms calculation and to avoid simulating the entire com- ponent, the total surface is divided in different cells. An example of the choice of the cell size and its position regarding surface defects is shown in Fig. 15. The cell surface containing a defect is noted S d and the po- lished surface cell without a defect is noted S p .

The cell size must be adapted to the size and to the number of defect introduced in the specimen. For defects higher than Ø500 μm, cell size is chosen as 3.1 mm × 3.1 mm. For smaller defects, cell size is chosen as 2.1 mm × 2.1 mm that allows to introduce 44 cells with defects of Ø400 μm in the total specimen surface.

For a specimen containing n defects, the proposed notation allows to obtain a simple expression of H ms based on the surface stress hetero- geneity factor of cells with defects H msd and without defects H msp .

= ⎡ ⎣ ⎢ + - ⎤ ⎦ ⎥ H S nS H S nS K H 1 ms d ms d t m ms Ω Ω d max p ( 13
)
with H msd the surface stress heterogeneity factor of a cell containing a defect: [START_REF] Morel | Plasticity and damage heterrogeneity in fatigue[END_REF] and H msp the surface stress heterogeneity factor of a cell without any defect, equal to one in our case:

= H 1 msp (15)
To calculate the surface stress heterogeneity of defect cells, a finite element simulation is performed on a cube containing a hemispherical defect on its upper surface (see Fig. 16). Calculation is conducted in order to assess the stress in each element on the surface. Elements are tetrahedral with reduced integration and convergence of calculation results is verified with the chosen size of elements. The material is supposed to be homogeneous and an elastic behaviour is used. The potential residual stresses are also neglected. The boundary conditions are as follows:

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

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

• Displacement Uz = 0 at the point

= = = X Y Z 0, 0 , 0.
A bending pressure function representative of experimental bending loading is applied along the surface B (Fig. 16).

The maximal stress concentration factor K tmax is the same for the different defects size and is equal to 2.2. Due to the bending loading, the maximum principal stress is not localized at the bottom of the hemispherical defect but close to the surface (Fig. 17).

Fig. 18 compares the maximal principal stress gradient in depth for different defect size and from the bottom of the defect. Fig. 18 shows that the maximal stress value is the same for the different defect size and that the gradient associated to the bending loading (presented for the polished surface state without a defect) is negligible compared to the gradient generated by the defect.

The identification procedure

The proposed criterion is a 2 parameters model with the shape parameter m and the scale parameter σ 0 to be identified. To get them, results of staircase obtained on specimens with a single defect of Ø400 μm is used as reference. For this surface state, the average fatigue limit is σ a = 137 MPa with a standard deviation σ a of 9.5 MPa.

The shape parameter m associated to the scatter can be calculated from the standard deviation and the average fatigue strength using the following relationship ( 16):

= + - + + ( ) ( ) ( ) σ σ Γ 1 Γ 1 Γ 1 a a m m m 2 2 1 1 (16)
with Γ the Gamma function (or Euler function) given by:

∫ = ∞ -- t x e d x Γ( ) t x 0 1 (17)
Using Eqs. ( 16) and ( 17), m is found to be 17. The scale parameter σ 0 is then identified using Eq. ( 11) and the fi- nite element simulation results H ms and K tmax calculated for a specimen containing a hemispherical defect of Ø400 μm. For a S 0 of 1 mm 2 , σ 0 is equal to 240 MPa.

Predictions using a surface approach

The surface stress heterogeneity factor can be calculated for different defect sizes and numbers. Fig. 19 shows that H ms m 1/ increases with the defect size as well as the defect number due to the fact that the increase of defect size or number leads to the increase of the highly stressed area. Stress gradient in depth also depends on defect size as shown in Figs. 17 and18.

Eqs. ( 11) and ( 13) are used to predict the fatigue limit in the presence of 1 and 22 defects. A comparison of experimental results and predictions is shown in Fig. 20. Predictions are in good agreement with experimental results. The drop of the fatigue limit with the defect size increase is successfully predicted. Also, scale effect is correctly reflected. In addition, fatigue limit for the polished specimen without defect is correctly predicted (see Fig. 20). However, for defects lower than Ø100 μm, predictions are conservative and error reaches 15% (case of Ø60 μm). The physical asymptote presented experimentally at this defect size is not well described by the surface approach.

For the Ø400 μm defect, the predicted fatigue limit for different number of defects is compared to experimental results (Fig. 21). Predictions show a good agreement with experimental results with less than 8% of error for the 22 defects case and less than 5% for the 44 defects case. Furthermore, predictions show that the decrease in fatigue limit slows down with the increase of defect number. The probabilistic model predicts that after a number of defects around 30, fatigue limit decrease is no longer significant. This tendency is usual for statistical size effect [START_REF] Papadopoulos | Gradient-dependent multiaxial high-cycle fatigue criterion[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]. The surface applied stress is 100 MPa.

Predictions using a volume approach

The surface approach allows good predictions of the fatigue limit in presence of defects of different size and number apart from defects smaller than Ø100 μm where predictions are too conservative. This could be due to the fact that the surface approach only considers the stress on the extreme surface of the specimen. Thus, even the smallest size defect has an impact on the predicted fatigue limit which is clearly not the case. In literature, some studies [START_REF] Taylor | Geometrical effects in fatigue: a unifying theoretical model[END_REF][START_REF] Taylor | The theory of critical distances[END_REF][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 steelthe[END_REF] consider the stress not in the highly stressed point but at a given distance or in a non local way for Ø400 μm specimen with different defects number and predictions of fatigue limit using the surface approach.

as a mean value over a given volume. This is a simple way to reflect the asymptotic response of small defects.

For the proposed approach, the stress distribution is considered in a domain consisting of a volume V Ω with a constant depth * d from the free surface S Ω (see Fig. 23). Failure surface observations show that inter- metallic particles are mostly localized in the first 50 μm in depth. Based on these experimental observations, a value of * d equal to 50 μm is used to reflect the possibility to have an intermetallic particle at the initiation site (see Figs. 22 and23).

For the volume formulation, stress heterogeneity factor H mv is ex- pressed as follows:

∫ ⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ H V σ σ dV 1 mv V eq max m Ω Ω (18)
where σ max , the maximum equivalent stress on the component volume:

= σ max σ ( ) max V eq Ω (19)
In the same way as in the previous paragraph, fatigue limit can hence be predicted as: [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] with 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
Like in the surface approach simulation, the specimen volume is divided in different cells containing defect or not. The stress hetereogeneity factor H mv can be expressed as a combination of H mvd and H mvp :

= ⎡ ⎣ ⎢ + - ⎤ ⎦ ⎥ H V nV H V nV K H 1 mv d mv d t m mv Ω Ω d max p ( 21
)
with H mvd the stress heterogeneity factor of a cell containing a defect and H mvp the stress heterogeneity factor of a cell without a defect. The experimental point taken as reference for the identification procedure is the case of 1 defect of Ø400 μm. As shown in Fig. 24, different couples m σ ( , ) 0 allow to predict the fatigue strength of 1 defect of Ø400 μm batch. The one that provides the lower error value between predictions and experimental results is the couple ( = = m σ 22; 226 0 MPa). Fig. 25 shows that the volume approach keeps the advantages provided by the surface approach in describing the fatigue behavior in presence of different size and number of defects. Predictions are in good agreement with experimental results and reflect well the drop of the fatigue limit with the increase of the size and the number of defects. Also, volume approach allows to successfully predict fatigue limit of a component without defect. In addition, it predicts the fatigue limit in presence of small defects in a better way and reduces the error in case of Ø60 μm from 15% for the surface method to 9%. Using a defined volume and not the entire component could be questionnable because this approach does not take into account the probability that initiation could occur in the total volume of material. To discuss this specific question, calculation have been conducted considering the total specimen volume. . The predictions obtained using the previous approach and the total volume approach are compared in Fig. 26. The total volume approach cannot predict the fatigue strength of the polished specimens without defects. A possible explanation is related to the fact that crack initiation in fatigue is mostly a surface phenomenon. At the local scale the crack initiation threshold is therefore probably higher in the bulk of the material compared to the surface. Fig. 26 illustrates that using the same Weibull distribution for the total component, i.e in the bulk and on the surface gives less accurate predictions.

Analysis and conclusion

The main conclusions of this work are:

-An original fatigue tests campaign have been conducted in order to identify the scale effect and the defect size effect on fatigue behaviour of AA7050 alloy. Fully reversed plane bending fatigue tests have been conducted on specimens with differents size and number of hemispherical defects. -Fatigue limit decreases gradually when increasing defect size.

Furthermore, another decrease in fatigue limit is observed when increasing defects number (scale effect). -The proposed approach based on Weibull's distribution law and the weakest link concept allows to predict the fatigue limit for polished specimens as well as specimens with a single or several artificial defects. Fatigue limit predictions are in good agreement with the experimental results. -Volume approach shows better predictions than surface approach and allows to reflect the asymptotic behavior for small defects.

The experimental results and the developed approach illustrate the importance of taking the scale effect into account when designing components containing several surface defects. The proposed criterion could be a useful tool when designing components containing different surface defects types or roughness patterns. As prospects, the proposed criterion will be applied to industrial machined surfaces with different roughness patterns.

Fig. 1 .

 1 Fig. 1. AA7050-T7451 alloy RD: Rolling Direction, TD: Transverse Direction, ND: Normal Direction.

Fig. 2 .

 2 Fig. 2. SEM images of the intermetallic particles observed on the AA7050 alloy sheet.

Fig. 3 .

 3 Fig. 3. Geometry of tensile specimen.

Fig. 4 .

 4 Fig. 4. Geometry of plane bending specimen showing four chanfrens.

Fig. 5 .

 5 Fig. 5. Plane bending specimens with (a) 1 defect of Ø800 μm, (b) 22 defects of Ø400 μm, (c) 22 defects of Ø600 μm and (d) 44 defects of Ø400 μm.

Fig. 6 .Fig. 7 .

 67 Fig. 6. 3D surface scan of a Ø800 μm defect and corresponding 2D profile.

2 and

 2 Al Cu Fe

Fig. 8 .Fig. 9 .

 89 Fig. 8. S-N curve for polished specimens, AA7050 alloy.

Fig. 10 .

 10 Fig. 10. SEM observations on polished specimens: (a) particles at the crack initiation and (b)MgZn 2 traces found at fracture site.

Fig. 11 .

 11 Fig. 11. Specimen with a single defect (D100-1-1): (a) specimens fracture occurs away from the defect, (b) and (c) crack initiation site.

Fig. 12 .

 12 Fig. 12. Crack initiation site from a defect without a particle (specimen D400-44-1).

Fig. 14 .

 14 Fig. 14. Crack initiation site from the middle of the defect with a particle (specimen D400-1-5).

Fig. 15 .

 15 Fig. 15. The two different cells defined Sp and Sd, for fatigue analysis purpose (a) 1 surface defect; (b) 44 surface defects.

Fig. 16 .

 16 Fig. 16. FE simulation of a cube with hemispherical defect under bending loading.

Fig. 17 .Fig. 18 .

 1718 Fig. 17. Stress distribution in case of a Ø200 μm and Ø800 μm defect. The surface applied stress is 100 MPa.

Fig. 19 .Fig. 20 .

 1920 Fig. 19. Evolution of H ms 1/m with the defect size for a specimen containing 1 and 22 defects.

Fig. 21 .

 21 Fig. 21. Comparison between experimental resultsfor Ø400 μm specimen with different defects number and predictions of fatigue limit using the surface approach.

Fig. 22 .

 22 Fig. 22. Illustration of the * d parameter on the specimen D600-1-4 failure surface, = * d 50 μm.

Fig. 23 . 13 ;Fig. 24 .

 231324 Fig. 23. Illustration of the * d parameter in the Finite Element Model.

  The identified model parameters, using the same procedure previously presented are =

  d

Fig. 25 .Fig. 26 .

 2526 Fig. 25. Predictions of fatigue behaviour using a surface and a volume approach.

Table 1

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

	Element	Al	Zn	Cu	Mg	Zr	Ti	Si	Fe	Mn
	Weight (%) 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 direction	Yield stress	Ultimate tensile	Tensile elongation
		(MPa)	stress (MPa)	(%)
	°0	475	635	12.3
	°45	428	496	13.9
	°90	475	538	11.9

Table 3

 3 Test history for locati tested specimens.

	Specimen	Artificial defect	Loading at	Step	N f	Initiation
	name			failure	number		from
				(MPa)			artificial
							defect
		Diameter	Defects	σ a		(× 10 ) 5	Yes/No
		in μm	number	(R = -1)
	D60-22-1	60	22	170	4	7.72	Y
	D60-22-2	60	22	170	3	2.86	Y
	D100-1-1	100	1	170	4	7.59	N
	D100-1-2	100	1	170	3	4.66	N a
	D100-1-3	100	1	150	2	5.65	Y a
	D100-22-1 100	22	150	4	3.7	Y
	D100-22-2 100	22	140	2	14.5	Y
	D200-1-1	200	1	160	3	11.26	Y
	D200-1-2	200	1	150	2	7.81	Y
	D200-1-3	200	1	150	2	11.11	Y a
	D200-22-1 200	22	130	4	4.48	Y
	D200-22-2 200	22	140	4	16.39	Y
	D200-22-3 200	22	120	2	15.93	Y
	D400-1-1 b	400	1	150	3	1.33	Y
	D400-1-5 b	400	1	140	3	5.4	Y a
	D400-1-6 b	400	1	150	3	1.97	Y
	D400-1-8 b	400	1	140	2	5.31	Y a
	D400-1-9 b	400	1	150	2	2.76	Y
	D400-22-3 b 400	22	130	2	3.67	Y
	D400-22-6 b 400	22	120	2	9.86	Y a
	D400-22-7 b 400	22	130	2	7.55	Y a
	D400-22-9 b 400	22	130	2	8.94	Y
	D400-44-1 400	44	120	3	7.88	Y
	D400-44-2 400	44	120	3	4.93	Y
	D400-44-3 400	44	120	2	18.33	Y a
	D600-1-1	600	1	150	6	6.67	Y
	D600-1-2	600	1	130	2	12.28	Y
	D600-1-3	600	1	140	3	10.63	Y
	D600-22-1 600	22	130	2	16.48	Y
	D600-22-2 600	22	120	3	10.88	Y a
	D600-22-3 600	22	120	3	17.69	Y a
	D800-1-1	800	1	140	5	3.11	Y
	D800-1-2	800	1	130	4	13.69	Y
	D800-1-3	800	1	140	4	2.89	Y a
	D800-22-1 800	22	110	3	19.89	Y a
	D800-22-2 800	22	130	6	2.9	Y a
	D800-22-3 800	22	110	2	16.6	Y a

a Intermetallic particle at fracture site. b Retested after staircase.

Table 4

 4 Test history for staircase tested specimens.

	Surface state	Stress [MPa]

Table 5

 5 Average fatigue limit and associated standard deviation for tested surface states.

	Surface state	Average fatigue	Standard deviation	Covariance σ σ / D D
		limit σ D	σ D
	Polished a	174	5.3	0.03
	Ø400 1 defect a	137	9.5	0.069
	Ø400 22 defects a	123	5.3	0.043
	Ø400 44 defects	115	X	X

a Results based on staircase method described by Dixon and Mood

[START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF]

.

Aknowledgments

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).


Didier Grondin

Philippe Breuil

Jean-Paul P Viricelle

Philippe Vernoux

Modeling of the signal of a resistive soot sensor, influence of the soot nature and of the polarization voltage

Keywords: Soot sensor, Resistive sensor, electrical conductivity, Joule effect

come

Modeling of the signal of a resistive soot sensor, influence of the soot nature and of the polarization voltage

Introduction

Impact of vehicles exhaust, especially diesel particle matter, on health and environment has been clearly evidenced, with possible carcinogenic effects [START_REF]Cancer, Diesel and Gasoline Engine Exhausts and Some Nitroarenes[END_REF]. To limit diesel engine emissions, new legislation has emerged. On-board diagnosis sensors able to detect diesel particulate filter failures are required. Therefore, the development of a cheap sensor effective for the measurement of mass concentration is necessary.

Among the available technologies to measure soot concentration, resistive soot sensors are a realistic solution for a massive industrialization due to similarities concerning material and technology with lambda sensor and their robustness to withstand the hard conditions found in exhaust [START_REF] Hagen | Conductometric soot sensor for automotive exhausts: initial studies[END_REF][START_REF] Grob | Conductivity for soot sensing: possibilities and limitations[END_REF][4][START_REF] Ochs | Particulate matter sensor for on board diagnostics (OBD) of diesel particulate filters (DPF)[END_REF][START_REF] Noulette | Sonde de mesure de depot de suie dans l'echappement et son procede de fabrication WO2014044965 A1[END_REF]. This type of sensor allows the measurement of the electrical conductance (or resistance) of the deposited soot, and their periodical thermal regeneration. Indeed, as soot is mainly composed of carbon, particles are conductive. The conductance between platinum electrodes (sensor response) as a function of time increases under a particles flow with soot deposition. At the beginning of the loading, the slope of the curve of the conductance (resistance, or current) over the time is approximately proportional to the soot deposit, itself representative of mass concentration [4,[START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF]. This approach needs a regular self-cleaning of the sensor by burning the deposited soot thanks to a platinum "heater". Such a resistive soot sensor was developed and extensively studied in previous works [START_REF] Noulette | Sonde de mesure de depot de suie dans l'echappement et son procede de fabrication WO2014044965 A1[END_REF][START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Influence of electrodes polarization on the response of resistive soot sensor[END_REF][START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF][START_REF]Système de caractérisation de nanoparticules conductrices[END_REF]. It has been shown that when a significant polarization voltage is applied, which value depends on the sensor geometry, soot bridges are formed between the electrodes contrary to a continuous soot layer deposited with no or low polarization. These "dendrite-like" paths improve the sensitivity of the sensor [START_REF] Ochs | Particulate matter sensor for on board diagnostics (OBD) of diesel particulate filters (DPF)[END_REF][START_REF] Noulette | Sonde de mesure de depot de suie dans l'echappement et son procede de fabrication WO2014044965 A1[END_REF][START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Influence of electrodes polarization on the response of resistive soot sensor[END_REF].

Up to an optimal polarization voltage, the sensitivity increases with this voltage, which is explained by electrophoresis phenomena [START_REF] Fragkiadoulakis | Modeling a resistive soot sensor by particle deposition mechanisms[END_REF].

However, performances are degraded as shown on Figure 1 (sensitivity decrease, noisy signal), beyond the optimal polarization value [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Influence of electrodes polarization on the response of resistive soot sensor[END_REF][START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF]. The conductance variation versus time strongly depends on the polarization voltage. Using various operating point to generate soot, it was also demonstrated that this optimum value is dependent on the soot characteristics, such as their chemical nature and size distribution [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF]. A first objective of this study is the comprehension of this phenomenon (optimal polarization value), its exploitation in order to model the loading of such a sensor. Moreover, from European emission standard "Euro 5", legislation of limits for diesel engine take into consideration the particulate number of emitted soot in addition to their mass. Therefore, a second objective is to improve the measurement method of the sensor resistance in order to obtain additional information on emitted soots.

Experimental

The sensor (Figure 2) as well as the emission and measurement devices are briefly described here; more details can be found in previous publications [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Grondin | Influence of electrodes polarization on the response of resistive soot sensor[END_REF][START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF][START_REF]Système de caractérisation de nanoparticules conductrices[END_REF]. The sensors consist in an alumina substrate with two platinum interdigitated electrodes, the space between electrode is 20 µm. A platinum resistance heater on the other side is used for periodic regeneration. The sensor was tested on a synthetic gas bench. Soot was produced by the combustion of a mixture of propane/air thank to the mini Combustion Aerosol Standard (CAST) burner (Jing Ltd. Switzerland).

Four operating points (OP) were used to produce different soot flux, named OP1, OP1b, OP4 and OP5. Two OPs (OP1 and OP1b) correspond to the same richness but with lower propane and air flows for OP1b, giving the possibility to have two different concentrations of identical particles (chemical nature and size distribution). The tests were performed with gas temperature of 180°C during steady state emission of the selected OP. No heating is applied to the sensor during soot deposition period, so that the device temperature is supposed to be that of the gas,180°C. The particle size and their mass concentration were respectively measured with an EEPS (Exhaust Particle Size Spectrometer, model 3090, TSI Inc) and a PEGASOR particle sensor analyzer [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF][START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF]. The voltage between the sensor electrodes was applied with a Picometer-Source Keithley 6430. High sampling frequency measurement of the electrical conductivity was performed with a resistor bridge and a National Instruments NI-PCI-6034 acquisition card [START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF]. The sensor was exposed to soot flow with its sensing element opposite to the particle stream in order to avoid direct impacts, and without any protection cap. Ex-situ characterizations of soots were also performed. Electrical conductivity was measured at ambient temperature, after their collection at the test bench outlet, their shaping as powder bed and an applied mechanical pressure of 5.6 MPa. Soot electrical conductivity increases with temperature, and a simplified assumption that a linear relationship exists in order to define the conductivity temperature coefficient "a". This coefficient was determined by heating previously loaded sensors, and measuring conductance variation as a function of temperature [START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF]. The temperature of starting oxidation was determined with TPO (Temperature Programmed Oxidation) experiments. Information's on operating points (OP) and corresponding soot properties summarized in Table 1.

Operating Point

Richness

Results and discussion

Soot bridges destruction by Joule effect

A detailed examination of the conductance curves obtained at different polarization voltage (Figure 1) reveals that some "noise" is present for measurements made above the optimum voltage (30V for OP1). In order to quantify this "noise", its standard deviation was calculated, considering only the sensor response after the percolation time. Figure 3 shows that the "noise" standard deviation increases with the polarization voltage. As the "noise" depends both on polarization voltage and operation point, it can be assumed that it does not correspond to a « measurement noise » (signal acquisition), but has a physical origin. These sudden variations in electrical conductivity may be correlated to the creation or destruction of soot bridges between the two electrodes.

To support such an explanation, a first hypothesis is that the destruction of the bridges is due to their combustion, initiated by their heating by Joule effect due to the polarization voltage. The electrical power due to Joule effect is indeed proportional to the square of the polarization voltage. Past TPO measurements [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF] highlighted the sudden soot oxidation under air above a temperature depending on the Operating Point, for example 525°C for OP1 (Table 1). Therefore, we assume that the loss of sensitivity for high polarization voltages is due to the destruction of bridges by Joule-initiated combustion.

Moreover, it must be noted that bridge temperature is not the average temperature of the substrate of the sensor, which is measured thanks to the platinum heater (figure 2) and remains very close to the gas temperature (180°C). Recently, Hagen et al [START_REF] Hagen | Conductometric Soot Sensors: Internally caused thermophoresis as an important undesired side effect[END_REF] showed that thermophoresis can explain loss of sensitivity for polarization voltage high enough to induce overheating by Joule effect. We were able to verify that this effect is indeed present for overheating above 10°C. But such an overheating Joule effect needs a power higher than 100 mW (the static gain of our sensor is about 100°C/W), and can only be achieved with high soot loads of the sensor with OP1 soots (soots with high electrical conductivity). This effect cannot explain the decrease in sensitivity for high polarization voltages, also observed at the beginning of the loading, or for soots with low electrical conductivity (OP4 or OP5).

To validate our hypothesis, the polarization voltage of a preloaded sensor (OP1) set at 180°C (gas temperature), in air, was gradually increased. An irreversible conductance drop was observed beyond 35V under air, or 40V under nitrogen with low residual oxygen concentration (Figure 4), demonstrating that soot bridges are destroyed in the presence of oxygen when applying a high polarization voltage. In previous works [START_REF] Grondin | Influence of electrodes polarization on the response of resistive soot sensor[END_REF][START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF], some optical observations (optical microscopy and avalanche photodiode) revealed light flashes on a pre-loaded sample to which a polarization voltage ramp is applied. Furthermore, strong correlations were observed between these flashes and conductance steps, both being measured with a high sampling frequency.

These results prove the destruction of soot bridges by combustion ignited by Joule effect and also explain the observed optimum value. Indeed, from a certain thermal power supplied by Joule effect, a part of the bridges formed by the particles begins to burn with light emission.

Previous interpretation of soot bridge destruction by Joule effect was also supported by thermal modeling of a bridge. In the case of OP1, we demonstrated [START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF] that 30V polarization which is the optimal voltage is sufficient to reach a temperature higher than 550°C, thus higher than the starting oxidation temperature identified by TPO (table 1).

This calculation, performed by finite differences model, is based on the following assumptions:  Each bridge is assimilated to a cylinder with a diameter of 20 nm (according to transmission electron microscopy pictures [START_REF] Grondin | Influence of key parameters on the response of a resistive soot sensor[END_REF]) and a length of 20 µm (distance between electrodes).  Heating is due to Joule effect, the electrical conductivity at ambient temperature being between 200 S/m (for OP1) and 0.3 S/m (for OP5), and increases linearly with temperature.  Cooling occurs by convection (coefficient 100W/m 2 /°C), radiation of the black body and thermal conductivity towards the bridge connected with platinum electrodes, this last term being largely preponderant thanks to the high thermal conductivity of carbon. However, this parameter that we could not measure is estimated with high uncertainty (between 100 and 1500 Wm -1 K -1 [START_REF] Mahanta | Thermal conductivity of carbon nanofiber mats[END_REF]).  Ambient temperature is 180°C, set by gas flow.

Simulations have been done with large range of parameters, mainly for those having a strong influence like thermal conductivity, electrical conductivity and its temperature coefficient and of course the polarization voltage. Bridges diameter and convection coefficient have a more limited influence. Figure 5 relates a simulation with parameters close to those of OP1, with various soot thermal conductivities. It shows that the temperatures in the middle of the bridge are compatible with a destruction by Joule effect (higher than 550°C). The thermal time constant of a bridge is in the range of a few microseconds.

As a conclusion, both experiments and thermal modeling support the mechanism of soot bridge building and destruction depending on polarization voltage. We therefore attempted to model this transition phase by differential equations corresponding to loading and destruction phenomena. The first step of the model consists in simplified assumptions with a schematic representation of soot bridges summarized in Figure 7: 1. Each bridge is composed of imax elementary particles (depending on particle size and electrode distance). We define Ni as the number of bridges under construction with i particles or more, then 0 ≤ i ≤ imax. Hence, Nimax represents the number of full bridges 2. The maximum number of possible full bridges Nmax, or number of slots (origin of bridge building due to electrode irregular surface [START_REF] Grondin | Influence of electrodes polarization on the response of resistive soot sensor[END_REF]), is constant, and Nmax=N0. 3. These soot bridges can be assimilated to identical resistances (G0) in parallel. Thus, we assume that the measured electrical conductivity (G) increases linearly with the number of bridges (Nimax) joining the two measuring electrodes: 𝐺 = 𝑁 𝑖𝑚𝑎𝑥 . 𝐺 0 , where G0 is the conductance of one bridge.

When all bridges are constructed, Nimax = Nmax , the maximum conductance Gmax is obtained:

𝐺 𝑚𝑎𝑥 = 𝑁 𝑚𝑎𝑥 . 𝐺 0 .

Model without bridge destruction and without electrical field effect

To initiate the model, we will first assume that there is no destruction and that the probability of capture of an additional particle on a bridge under construction is constant (no effect of electrical field depending on the number i of particles already present in the considered bridge). With such hypothesis, the rate of formation of bridges with i particles is proportional to the number of "slots" with i-1 particles, so to (Ni-1 -Ni):

𝑑𝑁 𝑖 𝑑𝑡 = 𝑘(𝑁 𝑖-1 -𝑁 𝑖 ), Equation 1or, with Laplace formalism: 2Then, 𝑁 𝑖 (𝑠) = ) 𝑖𝑚𝑎𝑥 which is proportional to the Laplace transform of the electrical conductivity G(s). In order to calculate Nimax(t), the temporal response to a step of particles concentration, we must compute the inverse Laplace transform of This can be done with the help of Laplace transform table ([14], p 1020-1022):

𝑠𝑁 𝑖 = 𝑘(𝑁 𝑖-1 -𝑁 𝑖 ) Equation
𝑁 𝑖𝑚𝑎𝑥 (𝑡) = 𝑁 𝑚𝑎𝑥 (1 - Γ(𝑖𝑚𝑎𝑥,𝑘𝑡) imax!
) , where Γ(𝑖, 𝑘𝑡) is the "incomplete gamma function".

The calculation of this function allows to represent the ratio "Nimax / Nmax" which is equal to ratio "G / Gmax" according to previous definitions, and represents the normalized sensor signal in regards of the maximum conductance value, which depends on imax ( This first simplified model (only 2 parameters, imax and Nmax) can explain the shape of some load curves with low polarization voltage (Figure 6), including the "percolation time". As imax is related to the particle size in the model (Figure 7), supposing a monodisperse population of particles, Figure 8 also proves that the sensor response versus time is strongly dependent on this parameter. The higher imax is, corresponding to smaller particles in our model, the higher the percolation time is.

Model with bridge destruction and electrical field effect

To improve the previous model, an additional term of bridges destruction was considered, proportional to the number of complete bridges Nimax (because only these bridges can be destroyed by Joule effect) and to a constant h:

Equation 1 becomes:

𝑑𝑁 𝑖 𝑑𝑡 = 𝑘(𝑁 𝑖-1 -𝑁 𝑖 ) -ℎ𝑁 𝑖𝑚𝑎𝑥 Equation 3
Moreover, a second implementation concerns the influence of electrical field due to polarization. It is assumed that the creation rate is proportional to the electric field, then inversely proportional to the free space between an incomplete bridge and the electrode, which is itself proportional to (imax-i):

Equation 3 becomes:

𝑑𝑁 𝑖 𝑑𝑡 = 𝑘 𝑖𝑚𝑎𝑥-𝑖 (𝑁 𝑖-1 -𝑁 𝑖 ) -ℎ𝑁 𝑖𝑚𝑎𝑥 Equation 4
or, with Laplace formalism:

𝑠𝑁 𝑖 = 𝑘 𝑖𝑚𝑎𝑥-𝑖 (𝑁 𝑖-1 -𝑁 𝑖 ) -ℎ𝑁 𝑖𝑚𝑎𝑥 Equation 5
A system of imax equations with imax unknown values (N1 to Nimax) is obtained, and the objective is the determination of the expression of Nimax, proportional to the conductance.

This system cannot be solved in a general case, but only for particular imax values. This resolution can be done with a formal calculation tool, such as MuPAD, a toolbox of MATLAB.

For example, for imax=3, the system can be written with MuPAD as:

solve([s*N1=k*N0-k*N1-h*N3,s*N2=3/2*k*N1-3/2*k*N2-h*N3,s*N3=3*k*N2-3*k*N3-h*N3],[N1,N2,N3])
and the solution is:

𝑁 3 (𝑠) = 9𝑘 3 𝑁 0 9𝑘 3 +18𝑘 2 𝑠+18ℎ𝑘 2 +11𝑘𝑠 2 +11ℎ𝑘𝑠+2ℎ𝑠 2
Equation 6

The temporal step response will be the inverse transform of 1 𝑠 𝑁 𝑖𝑚𝑎𝑥 (𝑠), which can be computed with the MATLAB command "step".

In Figure 9, simulated profiles of conductance versus time are represented for imax between 2 and 6, with k=1 (construction constant) and h=10 (destruction constant) or reverse situation (k=10 and h=1) It has been checked that, for lots of combinations of k and h, the step response to a concentration of particles is the same than the previous model (including the "percolation time"), but sometimes with damped oscillations, which is a consequence of combined imaginary poles in the expression of the equation 6.

These damped oscillations are observed for long loading experiments, with polarization higher than the optimal voltage, when the "Joule effect destruction" becomes important. For example, in Figure 6, a damped oscillation can be seen for OP1 with 60V polarization (optimal polarization 30V), and there is no oscillation for PF4 at 60V (optimal polarization 60V).

It is interesting to note that M. Maricq et al [START_REF] Maricq | The impact of voltage and flow on the electrostatic soot sensor and the implications for its use as a diesel particulate filter monitor[END_REF], with a sensor of a different type (electrostatic), studied the dynamic of creation-destruction of dendrites and obtained the same type of equations and experimental results, including damped oscillations.

Statistical study of "conductance jumps"

When the electrical conductance signal is sampled at a sufficiently high frequency (5 to 10 kHz)during stationary state (about 50 minutes of loading), some "conductance jumps" attributed to bridge creation (positive jump) or destruction (negative jump) can be visualized (Figure 10). Hopefully, a statistical study of these steps may provide interesting information on the quantity and nature of the particles.

Signal treatment -Jump counting

In order to count these jumps correctly, it is necessary to filter out the "useful" noise (steps) from the background noise of the instrumentation. Because of the specific form of this information (sudden jumps), a classic low-pass filtering in the frequency domain is not very efficient. On the other hand, wavelet filtering preserves the shape of the "useful" steps. For this treatment, the Haar wave family has been chosen because of its "stair step" profile, which allows a simpler signal reproduction [START_REF] Maricq | The impact of voltage and flow on the electrostatic soot sensor and the implications for its use as a diesel particulate filter monitor[END_REF][START_REF]Haar 1-D wavelet transform -MATLAB haart -MathWorks France[END_REF]. Details on this filtering may be found in [START_REF] Phd | Développement d'un capteur de suies pour application automobile -Etude des paramètres clés affectant sa réponse[END_REF] Figure 10: High sampling frequency acquisition (5 kHz) for OP1 and OP4 at 60 V, stationnary state (about 50 minutes after beginning). Mean value of conductance is substracted (113µS for OP1 and 106 µS for OP4), "Wavelet filtered" signals.

After this filtering, a step detection algorithm, based on threshold detection of the signal derivative, counts the positive and negative steps. Finally, only steps above a threshold in absolute value (0.02 µS) are retained.

The stationary part of the signal was investigated. Figure 6 shows the conductance curves used for counting jumps.

We have studied samples lasting 200 seconds, the beginning of which is indicated by a black arrow in Figure 10. Two experiments with different concentrations of the same particles (OP1 and OP1b) and two experiments with different types of particles (OP1 and OP4) have been compared. The presented data are averaged over 1 second (Figure 11)

Effect of soot concentration on steps number (OP1, OP1b)

Subsequently, in order to determine the effect of particles concentration, OP1 and OP1b were compared, at identical polarization (30 V). These two modes produce the same type of particles but at different concentrations of about 30% (table 1). The steps distributions (number of steps per second as a function of steps height) calculated from Figure 10 First, we can notice that the number of positive steps is approximately equal to the number of negative steps, which is normal since the sensor is in a stationary state (as much creations as destructions). Finally, the number of steps for OP1 is about 30% higher than for OP1b, which is close the ratio of the concentrations between these two modes. The average height of these steps can be calculated and is represented in Figure 12a. The average height of the steps with OP1 or OP1b at 30 V is around 0.04 µS, which can therefore be considered as the order of magnitude of the electrical conductance of one "OP1" bridge at 180°C. This value is compatible with the counting of bridges made from a scanning microscopy photograph (about 5000) of a previously charged sensor (OP1, 30V), for which electrical conductance between electrodes is about 200 µS at 180°C, leading to an average conductance of 0,04µS for one soot bridge.

Therefore, the number of steps seems to be proportional to the soots concentration, while the average step is independent on this concentration, but it depends on the soot nature as discussed hereafter.

Influence of particle type (OP1, OP4)

With particles of different nature (OP1 and OP4) and identical measuring conditions (same polarization voltage, 60V), we observe a significant difference between the high sampling frequency conductance, although these two OPs have similar soot mass concentration: OP1 steps have a greater amplitude (Figure 10). This observation is confirmed by the step distribution (Figure 11b) and especially by the calculated average height of the steps (Figure 12b), where a more large difference between the two operating modes is observed. For OP1, 60V is quite higher than the optimal polarization voltage, contrary to OP4, explaining a quite larger jump frequency due to bridges destruction. For the average jump height, since we have previously seen that this parameter is independent on concentration, this information enables us to discriminate between two different types of particles: this difference being due to both the geometry of the particles and their electrical conductivity.

-Conclusions

This study is based on the properties of a resistive particle sensor exposed to different types of particles with different polarization voltages.

Experimental results (shape of the load curve, noise measurement, optical measurements) can be explained by a phenomenon of creation -destruction of particles bridges between the two electrodes. The creation is favored by the electric field whereas the destruction is due to combustion initiated by the Joule effect due to the polarization voltage. A simplified thermal modeling validates this assumption. Another model, based on the resolution of differential equations of creation -destruction of bridges, explain the shape of the load curves, including the "percolation time" as well as some observed oscillations.

Finally, high frequency electrical conductivity acquisitions allowed observing these creations -destructions in real time, while wavelet-based processing and filtering makes it possible to count them. An adequate statistical study (measurement of the average height of the steps of electrical conductance) enables to characterize the nature of soots independently of their concentration. Such a model and signal exploitation open the possibility of measuring the number of particles independently of their mass concentration, which is required by the most recent emission standards.

Figure 1 :

 1 Figure 1: Conductance variation versus time for different polarization voltages under a fixed operation point, OP1 (from [8]).

Figure 2 :

 2 Figure 2: Exploded view of resistive soot sensor.

Figure 3 :

 3 Figure 3: Evolution of the standard deviation of conductance signal noise as a function of the polarization voltage for 3 operating points (OP1, 4 and 5).

Figure 4 :

 4 Figure 4: Evolution of the conductance, at ambient, of a pre-loaded sensor (under OP1 with a polarization voltage of 30V) subjected to a voltage ramp from 30 V to 80 V at 24 V/min. Plain line corresponds to measurements made under a flow of 450 l/h air and dashed line to a flow of 450 l/h nitrogen (and low residual O2 concentration) with gases at 180°C.

Figure 5 : 3 . 2

 532 Figure 5 : Thermal modeling in a soot bridge (relating to OP1 condition) with various thermal conductivities "k" of soot; a) : Temperature profile between electrodes, b): Maximal temperature (middle of the bridge) versus polarization voltage

Figure 6 :

 6 Figure 6: Sensor loading curves (conductivity versus time). Soot flow at 180°C, with different polarization voltages for OP1 and OP4. Arrows indicate the area of high frequency signal processing (&3.3)

Figure 7 :

 7 Figure 7 : Schematic model of soot loading considering bridges building and destruction between electrodes

Figure 8 :

 8 Figure 8 : Simulation of normalized conductance (Nimax / Nmax = G / Gmax) versus time with hypothesis of no bridge destruction and no influence of electrical field due to bridge construction, for differents values of imax

Figure 9 :

 9 Figure 9: Simulation of conductance for different values of imax, with: a): k=1 (construction) and h=10 (destruction) and b): reverse situation (k=10, h=1)

  are shown in Figure 11 a.

Figure 11 :

 11 Figure 11: Frequency histogram of the heights of the steps of conductivity per second, calculated from experimental results, a): OP1 and OP1b modes, 30 V, b): OP1 and OP4 modes, 60 V.

Figure 12 :

 12 Figure 12: Average height of the steps (a) OP1 and OP1b, 30V (b) OP1 and OP4, 60V.

Table 1 : Properties of soots generated with four different Operating Points (OP)

 1

			Temperature	Maximum	Mass	Electrical	Conductivity	Optimal
		of	of starting	of size	concentration	conductivity(T=25°C)	temperature	polarization
		propane/air	oxidation	distribution			coefficient	voltage
		mixture					"a"
	unit		°C	nm	mg/m 3	S/m	°C-1	V
	OP1	0.97	525	90	94	200	2.10 -3	30
	OP1b	"	"	"	61	"	"	"
	OP4	1.00	496	70	91	8.0	5.10 -3	60
	OP5	1.02	326	50	55	0.3	0.075	>150

Acknowledgements

The authors wish to acknowledge "Arc Environnement" from Region Auvergne Rhone-Alpes for the funding and EFI Automotive, automotive supplier, for providing prototype sensors.