Y Ayed

• G Germain

High-pressure water-jet-assisted machining of Ti555-3 titanium alloy: investigation of tool wear mechanisms

Keywords: Tool wear, High pressure, Cutting forces, Titanium alloy, Machining

The main objective of this study is to investigate uncoated tungsten carbide tool wear mechanisms for high-pressure waterjet machining of the Ti555-3 titanium alloy. A comparative study has been undertaken (i.e. conventional versus assisted machining) based on numerous experimental tests. These tests have been accompanied by the measurement of the cutting forces and flank wear. It is concluded that the high-pressure water-jet assistance can greatly increase tool life compared to conventional machining, for all cutting conditions. The gain in tool life depends on the severity of the cutting condition. The analyses performed for each test (i.e. SEM, EDS and 3D profilometer) made it possible to monitor the tool wear and to investigate the main wear mechanisms. Based on these analyses, adhesion wear appears to be the most influential mechanism and it is accelerated by an increase in water-jet pressure. Monitoring of the wear profile made it possible to study the evolution of crater wear and material chipping during machining.

Introduction

The machining of titanium alloys is often challenging due to their excellent mechanical properties and their high chemical reactivity with most cutting tool materials [START_REF] Phapale | Tool wear investigation in dry and high pressure coolant assisted machining of titanium alloy ti6al4v with variable α and β volume fraction[END_REF]. The past decades have seen the development of new nearbeta alloys that have better mechanical properties, improved fracture toughness and good hardenability [START_REF] Ghosh | Microstructure-fracture toughness correlation in an aircraft structural component alloy Ti-5Al-5V-5Mo-3Cr[END_REF][START_REF] Nie | High temperature deformation and creep behavior of Ti-5Al-5Mo-5V-1Fe-1Cr alloy[END_REF]. However, these alloys such as Ti-10V-2Fe-3Al and Ti-5Al-5V-5Mo-3Cr raise significant machinability problems [START_REF] Wagner | The relationship between the cutting speed, tool wear, and chip formation during Ti-5553 dry cutting[END_REF], leading to severely reduced cutting conditions. This in turn leads to lower productivity [START_REF] Bai | Wear mechanisms of WC/Co tools when machining high-strength titanium alloy TB6 (Ti-10V-2Fe-3Al)[END_REF].

The machinabilty of these alloys has already been the subject of some research. A comparative study concerning the machining of Ti555-3 (near beta) and Ti-6Al-4V (α +β) titanium alloys has been carried out by Arrazola et al. [START_REF] Arrazola | Machinability of titanium alloys (Ti6Al4V and Ti555.3)[END_REF]. This study showed that the machinability of the Ti555-3 alloy is reduced by about 44% compared to the Ti-6Al-4V (the reference material). Moreover, the specific feed force and the specific cutting force are 35 and 20% higher, respectively. Also, for a wear criterion of V B = 0.3 mm, with a cutting speed ranging from 40 to 90 m/min and a machining time of 15 min, the optimum cutting speed was determined to be 45 m/min for the Ti555-3 alloy and 80 m/min for the Ti-6Al-4V alloy. A similar study has been conducted by Nouari et al. [START_REF] Nouari | Experimental investigation on the effect of the material microstructure on tool wear when machining hard titanium alloys: Ti-6Al-4V and Ti-555[END_REF] based on orthogonal cutting tests. This research showed that Ti-6Al-4V alloy is more sensitive to thermal softening; however, Ti555-3 alloy is more sensitive to strain rate hardening. The authors demonstrated that the main tool wear modes are adhesion and diffusion. Moreover, the cutting forces are 20 to 40% higher for the Ti555-3 (depending on the tool rake angle). According to Bai et al. [START_REF] Bai | Wear mechanisms of WC/Co tools when machining high-strength titanium alloy TB6 (Ti-10V-2Fe-3Al)[END_REF], adhesion, diffusion and abrasion are the main tool wear mechanisms when machining the Ti-10V-2Fe-3Al titanium alloy. This lower machinablility has been explained by [START_REF] Ugarte | Machining behaviour of Ti-6Al-4V and Ti-5553 alloys in interrupted cutting with PVD coated cemented carbide[END_REF] as the result of higher mechanical properties, cutting temperatures, adhesion tendency and chip segmentation frequency.

In order to meet this challenge, various solutions have been proposed. Among them is the development of new tool coatings and more efficient lubricants. Good performance has been obtained using self-lubricating T iB 2 PVD coatings [START_REF] Chowdhury | Wear behaviour of coated carbide tools during machining of Ti6Al4V aerospace alloy associated with strong built up edge formation[END_REF]. This improves tool life by approximately 60% compared to uncoated and 70% compared to T iAlN coated tools. This is mainly due to the formation of a B 2 O 3 phase that acts as a lubricant. The impact of the lubricant type on tool wear when machining the Ti-6Al-4V alloy has been studied by Moura et al. [10]. This study demonstrated that the use of a solid lubricant, notably MoS 2 , results in an increase in tool life, reduces cutting forces and minimizes frictional effects. This is due to the ability of the solid lubricant to penetrate the cutting zone. However, in order to achieve better results, other strategies must be employed. At present, the most interesting and promising approach consists of adding assistance to machining such as cryogenic assistance, laser assistance and high-pressure water-jet assistance.

Laser-assisted machining has been employed to enhance the machinablity of titanium alloys such as Ti-10V-2Fe-3Al, Ti-6Cr-5Mo-5V-4Al and Ti-6Al-4V [11][START_REF] Rashid | The response of the high strength Ti-10V-2Fe-3Al beta titanium alloy to laser assisted cutting[END_REF][START_REF] Ayed | Experimental and numerical study of laser-assisted machining of Ti6Al4V titanium alloy[END_REF]. This technique has been shown to reduce cutting forces up to 60%. However, the influence of the assistance on tool wear remains debatable. As opposed to this technique, cryogenicassisted machining consists of cooling the tool/chip interface with a liquefied gas such as liquid nitrogen and CO 2 . The use of this technique has been effective in limiting adhesive wear, improving surface roughness and increasing tool life [START_REF] Machai | Machining of β-titaniumalloy Ti-10V-2Fe-3Al under cryogenic conditions: cooling with carbon dioxide snow[END_REF][START_REF] Bordin | Analysis of tool wear in cryogenic machining of additive manufactured Ti6Al4V alloy[END_REF][START_REF] Lv | Tool wear in milling Ti40 burn-resistant titanium alloy using pneumatic mist jet impinging cooling[END_REF]. Recent studies [START_REF] Shokrani | Investigation of the effects of cryogenic machining on surface integrity in cnc end milling of Ti-6Al-4V titanium alloy[END_REF][START_REF] Ayed | Impact of supply conditions of liquid nitrogen on tool wear and surface integrity when machining the Ti-6Al-4V titanium alloy[END_REF] showed a great improvement in surface integrity when machining Ti-6Al-4V titanium alloy.

Under certain cutting conditions, the best results in terms of tool life and productivity have been obtained using high-pressure water-jet assistance (HPWJA). This technique consists of spraying a high-pressure water jet between the tool rake face and the chip. On the one hand, the mechanical action of the jet breaks the chips and on the other hand it cools the tool and the workpiece.

In this context, several studies have been conducted especially for refractory alloys. Due to the high-pressure lubricant jet, the temperature drops drastically and consequently slows down certain tool wear mechanisms [START_REF] Courbon | Investigation of machining performance in high-pressure jet assisted turning of Inconel 718: an experimental study[END_REF][START_REF] Kramar | Capability of high pressure cooling in the turning of surface hardened piston rods[END_REF]. Machado et al. [START_REF] Machado | Tool performance and chip control when machining Ti6Al4V and Inconel 901 using high pressure coolant supply[END_REF] reported that the machining of the Ti6Al4V alloy using high-pressure lubrication improves carbide tool life by up to 300% and ensures chip fragmentation. This achievement is the result of the reduction in diffusion wear due to the drop in cutting temperature. Similarly to carbide tools, the tool life of PCD tools can be enhanced by approximately 21 times (depending on cutting the conditions) [START_REF] Da Silva | Tool life and wear mechanisms in high speed machining of Ti-6Al-4V alloy with PCD tools under various coolant pressures[END_REF]. The main tool wear mechanisms identified by the authors are adhesion and attrition. As HPWJA provides efficient lubrication of the tool/chip interface, the cutting temperature is drastically reduced. This effects surface integrity by inducing a softening effect of the machined surface [START_REF] Ezugwu | Surface integrity of finished turned Ti-6Al-4V alloy with PCD tools using conventional and high pressure coolant supplies[END_REF][START_REF] Silva | Surface integrity and tool life when turning of Ti-6Al-4V with coolant applied by different methods[END_REF]. A comparative study carried out by Da Silva et al. [START_REF] Da Silva | Advances in the turning of titanium alloys with carbide and superabrasive cutting tools[END_REF] has shown that neither the use of a CBN tool nor the use of a ceramic tool gives satisfactory results when machining Ti-6Al4V using HPWJA. The main tool wear mechanisms in this case are chipping, notching and cutting edge fracture. The best results have been obtained using carbide and PCD tools. As stated by Ayed et al. [START_REF] Ayed | Degradation modes and tool wear mechanisms in finish and rough machining of Ti17 titanium alloy under high-pressure water jet assistance[END_REF][START_REF] Ayed | Tool wear analysis and improvement of cutting conditions using the high-pressure water-jet assistance when machining the Ti17 titanium alloy[END_REF], tool life can be increased up to nine times when machining Ti17 titanium alloy using HPWJA. Under these conditions, and in order to boost productivity, the cutting speed could be increased by about 30%. In addition, this study demonstrated the existence of an optimum fluid pressure (100 bar) resulting in the best performance in terms of tool wear and productivity. EDS and SEM analyses of uncoated carbide tools have shown that adhesion is the main tool wear mechanism for all of the tested cutting conditions.

This study aims to investigate the impact of high-pressure water-jet-assisted machining and the cutting parameters on the machinability of the difficult-to-cut Ti555-3 (Ti-5Al-5V-5Mo-3Cr) titanium alloy. To this end, instrumented experimental tests accompanied by different analyses have been carried out.

Experimental setup

The Ti555-3 near-beta titanium alloy was selected for this study. It has improved mechanical properties compared to the Ti-10V-2Fe-3Al alloy and is mainly intended for landing gear and other aeronautical structural components. It also has high hardenability, high fracture toughness and a good strength to weight ratio [START_REF] Kar | Processing-microstructure-yield strength correlation in a near β ti alloy, Ti-5Al-5Mo-5V-3Cr[END_REF][START_REF] Matsumoto | Hot forging characteristic of Ti-5Al-5V-5Mo-3Cr alloy with single metastable β microstructure[END_REF]. The microstructure of the as-delivered material, etched with Kroll's reagent, is shown in Fig. 1. It displays a two-phase microstructure: lamellar and globular α phase within the β matrix.

Longitudinal turning tests have been carried out on Leadwell LTC25iL CNC turning lathe supplying a maximum power of 24 kW. Based on the recommendations from previous studies [START_REF] Bai | Wear mechanisms of WC/Co tools when machining high-strength titanium alloy TB6 (Ti-10V-2Fe-3Al)[END_REF] and due to the high chemical reactivity of titanium alloys, uncoated tungsten carbide inserts (ISO designation CNMG 12 04 12-23 H13A) have been used. An emulsion-based cutting fluid (5% concentration) has been used for the conventional lubrication tests and HPWJA tests at 100 bar (15 l/min) and 250 bar (26 l/min). Instrumented experiments have been carried out to track and to analyse tool wear evolution. A Kistler dynamometer (9257B) has been used to measure the evolution of the cutting forces. Flank wear has been measured using a stereo microscope on the basis of the standard ISO3685. The tool wear rejection criterion used is VB = 0.3 mm (VB represents the average flank wear). For a more comprehensive investigation of the tool wear mechanisms, a SEM and a 3D profilometer have been used. Figure 2 illustrates the experimental procedure.

On the basis of previous research concerning the machining of titanium alloys [START_REF] Arrazola | Machinability of titanium alloys (Ti6Al4V and Ti555.3)[END_REF][START_REF] Ayed | Tool wear analysis and improvement of cutting conditions using the high-pressure water-jet assistance when machining the Ti17 titanium alloy[END_REF], the cutting conditions, summarized in Table 1, have been selected.

Cutting speed influence

First of all, in order to highlight the impact of the waterjet assistance on tool life, the cutting conditions were fixed (V c = 60 m/min, a p = 1.5 mm and f = 0.1 mm/rev). Two tests were carried out under conventional and HPWJA conditions (P = 100 bar). As showed by Fig. 3, the contribution of HPWJA can be clearly observed with an increase in tool life of up to a factor of 8 (from 4 to 34 min). Based on these results, a significant gain in terms of productivity could be obtained by improving the cutting conditions. To this end, a slight increase in the cutting speed has been tested (V c = 68 m/min, V c = 75 m/min, V c = 80 m/min).

Figure 3 demonstrates that the cutting speed could be further increased to 75 m/min while maintaining an acceptable tool life of up to 11 min. Beyond this speed, HPWJA is no longer efficient. The acceleration of wear as a consequence of an increase in cutting speed is mainly related to the activation of different tool wear mechanisms. During the cutting process, heat is generated in the cutting zone due to plastic deformation and friction. The amount of heat generated is increased by an increase in the cutting speed. As reported by Moura et al. [START_REF] Moura | An investigation of cutting forces and cutting temperatures during laser-assisted machining of the Ti-6Cr-5Mo-5V-4Al beta titanium alloy[END_REF], at the cutting edge, the temperature may reach 1000 • C. Hence, the high temperature levels accelerate diffusion and adhesion (plucking mechanism) [START_REF] Moura | An investigation of cutting forces and cutting temperatures during laser-assisted machining of the Ti-6Cr-5Mo-5V-4Al beta titanium alloy[END_REF]. In the case of HPWJA, the lubricant jet insures good cooling of the cutting zone, thereby slowing tool wear evolution. However, the progressive increase of the cutting speed leads to a rapid temperature increase that the lubricant jet cannot evacuate quickly. This finally leads to a reduction of the HPWJA efficiency (80 m/min). This part of the study has been achieved by a comparative analysis of the impact of the lubricant-jet pressure on tool life. Figure 4 illustrates the evolution of the tool wear curves. It shows that a pressure increase causes a considerable decrease in tool life.

For both cases, observations of the tool using a stereo microscope revealed the formation of a crater at the cutting face. To get more information about tool wear evolution and mechanisms, cutting tools have been analysed using a 3D profilometer and a SEM. Figures 5 and6 show different stages of crater formation.

For a pressure of 100 bar, tool wear evolves gradually and crater wear takes place only after 15 min of machining. However, for a pressure of 250 bar, excessive tool wear Fig. 3 Influence of the cutting speed and the high-pressure water-jet assistance on tool life (a p = 1.5 mm, f = 0.1 mm/rev) Fig. [START_REF] Wagner | The relationship between the cutting speed, tool wear, and chip formation during Ti-5553 dry cutting[END_REF] Influence of the cutting speed and coolant pressure on tool life (a p = 1.5 mm, f = 0.1 mm/rev) is seen, notably at the level of the tool rake face. The crater becomes larger and deeper as shows by Fig. 7. The progressive formation of the crater weakens the cutting tool. Hence, machining up to 10 min has caused the cutting edge to break and substantial material chipping.

The SEM observations, illustrated in Fig. 8, show the existence of adherent layers on the tool rake and flank faces. The parallel scratches on the tool flank face are a sign of abrasion wear. Micro-cracks have also been detected and they are mainly due the combination of tool stiffness degradation and high cutting forces.

An EDS examination of the adherent layers has confirmed the activation of an adhesion mechanism by the detection of titanium on the tool rake face as illustrated by Fig. 9. For the case of conventional machining, titanium deposits are uniformly distributed along the contact area. However, for the case of high-pressure water-jet assistance, the amount of these deposits has been greatly reduced which is mainly due to the reduction of the tool/chip contact length.

Under these cutting conditions, crater wear becomes more pronounced with the increase of the lubricant-jet pressure. Generally, the crater formation mechanism is often related to the activation and the acceleration of diffusion at high cutting temperatures [START_REF] Moura | An investigation of cutting forces and cutting temperatures during laser-assisted machining of the Ti-6Cr-5Mo-5V-4Al beta titanium alloy[END_REF][START_REF] Odelros | Experimental and theoretical study of the microscopic crater wear mechanism in titanium machining[END_REF]. Optimum cutting conditions and an effective lubrication must avoid overheating of the cutting zone and thus slowdown crater formation. However, this hypothesis is inconsistent with the results found. Hence, it could be concluded that the formation of crater wear in HPWJA must be due to another mechanism. Similar results have been obtained when machining the Ti17 titanium alloy under the same lubrication conditions [START_REF] Ayed | Degradation modes and tool wear mechanisms in finish and rough machining of Ti17 titanium alloy under high-pressure water jet assistance[END_REF][START_REF] Ayed | Tool wear analysis and improvement of cutting conditions using the high-pressure water-jet assistance when machining the Ti17 titanium alloy[END_REF]. It seems that crater formation in this case is due to an adhesion/plucking mechanism. Indeed, during machining, adherent layers are formed and stick to the surface of the tool under the action of the high contact pressure and temperature. The mechanical action of the water jet causes the tearing off and plucking of these adherent layers. The increase of the water-jet pressure intensifies this phenomenon; the tool/chip contact length is more and more reduced.

The evolution of the mean arithmetic (average value) cutting force and the axial force (F c , F a ) during machining is presented in Fig. 10. It can be seen that for 100 bar, the cutting force evolves gradually. However, for 250 bar, a sharp and rapid evolution can be observed. This could be linked to flank wear, crater wear and material chipping at the cutting edge which causes a local change of the tool geometry, cutting angles and contact length.

Feed rate impact

Feed rate variation is often related to notch wear, the question now arises about the impact of high-pressure water-jet assistance on tool wear, notably at high feed rates.

At a cutting speed of 60 m/min and a pressure of 100 bar, the feed rate has been varied from 0.1 to 0.3 mm/rev. The experimental results are plotted in Fig. 11. It shows a sharp decrease in tool life (at least 50%); tool life does not exceed 4 min at 0.3 mm/rev. Also, for high feed rates, a large notch occurs and weakens the tool insert. In this context, it is necessary to point out that under these conditions and with conventional lubrication, machining of the Ti555-3 alloy is almost impossible (only 5 min at 0.1 mm/rev). Machining is accompanied by high temperature generation and smoke that ends by the collapse of the cutting edge after a few seconds.

Figure 12 emphasizes the influence of the water-jet pressure at 100 bar and 250 bar. Tests have been stopped at VB = 0.23 mm to preserve the highly damaged cutting edge for further analysis. As well as the last tests, the increase of pressure leads to a more degradation of the tool.

Monitoring of tool wear evolution using a stereo microscope is illustrated in Fig. 13. It shows that after only 30 s, severe wear has taken place. A notch and a crater have been formed; material chipping has been observed on For a more exhaustive tool wear analysis, tool inserts have been observed using a SEM and scanned using a 3D profilometer. Figure 14 shows the scanned surfaces of the cutting tools at the end of the tool life. Figures 15 and16 show the SEM observations of the worn tools.

These analysis have permitted a closer view of the damaged areas. Material deposits (adherent layers) and material chipping are clearly distinguishable (f = 0.2 mm/rev and f = 0.3 mm/rev) on the flank face, the rake face and the notch. For a feed rate of 0.3 mm/rev, the tool seems to have undergone very severe thermo-mechanical loads, given the significant deterioration of the cutting edge. Indeed, a large notch of approximately 400 μm wide and material chipping along the cutting edge have occurred in only 4 min.

In order to explain the major notch wear mechanisms, researchers proposed many assumptions. The combination of very complex factors makes it a complicated task, which explains the big differences between the proposed hypotheses. However, experimental tests have shown that materials that have strong strain hardening behavior such as titanium alloys and nickel-base alloys are more subjected to notch wear; strain hardening, strain rate sensibility and thermal softening seem to be key contributing factors [START_REF] Kong | Investigation on notch wear mechanism in finish turning pure iron material with uncoated carbide tools under different cooling/lubrication conditions[END_REF]. Adhesion is the most likely activated mechanism. In fact, material chipping is mainly caused by the cyclic formation of adhesive contacts and their frequency [START_REF] Chandrasekaran | Chip flow and notch wear mechanisms during the machining of high austenitic stainless steels[END_REF][START_REF] Kasim | Wear mechanism and notch wear location prediction model in ball nose end milling of Inconel 718[END_REF].

To explain the formation mechanisms of the notch, we assume that it occurs with the combination of all cited factors (mechanical and chemical affinity factors). For the different wear tests carried out at this stage, the water-jet assistance significantly improves tool life. The drop of the cutting temperature ensured by the high-pressure lubrication slows down some wear mechanisms (notably diffusion and adhesion). However, with the increase of the axial force (due to feed rate increase), the cutting edge is increasingly loaded in particular at the level of the depth of cut (DOC). This generates a significant stress concentration, heating, material adhesion and chipping.

Depth of cut influence

A final series of tests have been carried out in order to investigate the impact of the depth of cut on tool wear using HPWJA. The depth of cut has been varied from 1.5 to 2.5 mm (f = 0.1 mm/rev, V c = 60 m/min). The results of these tests are presented in Fig. 17.

First of all, it could be noted that the Ti555-3 titanium alloy is almost not machinable under these cutting conditions, using conventional lubrication; the tool life does not exceed 2 min at a p = 2.0 mm. However, the use of the high-pressure water-jet assistance significantly increases tool life. Indeed, for a p = 1.5 mm, tool life has been multiplied by a factor of 8 and for a p = 2 mm, it has been multiplied by a factor of 10 (from 2 to 21 min).

As the cutting forces increase when the depth of cut increases, it leads to a considerable increase in temperature in the cutting zone. In this very confined zone, which is subjected to very high cutting pressures, the evacuation of heat by the lubricant fluid remains very difficult. Tool wear mechanisms are then accelerated; the combined effect of temperature and mechanical actions will eventually cause plastic deformation of the cutting edge and subsequently its collapse [START_REF] Astakhov | The assessment of cutting tool wear[END_REF]. This has been confirmed by analyses using the SEM and the 3D profilometer. Figure 18 illustrates the plastic deformation at the cutting edge (conventional lubrication); the cutting edge has been deformed by approximately 0.15 mm.

The evolution of the cutting forces is provided by Fig. 19. Hence, it can be deduced that the initial specific cutting force (K c = F a p f ) is approximately 2800 MPa and it reaches 3520 MPa at the end of the tool life. This further explains the plastic deformation phenomenon which has occurred. On the bases of the work by Astakhov [START_REF] Astakhov | The assessment of cutting tool wear[END_REF], to prevent thermal softening of tungsten carbide, the temperature should not exceed 800 • C for such a loading level. However, this temperature is far exceeded during machining of titanium alloys even for a lower cutting conditions [START_REF] Kikuchi | The use of cutting temperature to evaluate the machinability of titanium alloys[END_REF]. In the case of water-jet assistance, a large part of the generated heat is evacuated.

The efficient lubrication provided by the high-pressure water jet avoids the plastic deformation wear mechanism. Figure 20 shows the rake face of the tool at the end of the tool life (19 min). It shows the zones of material adhesion and a large crater; it also shows micro-cracks on the border between the crater and the cutting edge. The gradual formation of the crater under the action of the adhesion mechanism (accelerated by the water jet) weakens the tool. Meanwhile, wear on rake and flank faces leads to an increase in the cutting forces. At this time, micro-cracks begin to forme; this will end by the rupture of the cutting edge.

Discussion

For conventional machining in the different tested configurations, tool life has not exceeded 5 min. Under these conditions, the wear could be due to different mechanisms occuring simultaneously, including adhesion, diffusion, abrasion and plastic deformation. Moreover, the impact of these mechanisms differs according to the cutting conditions (roughing or finishing). During roughing machining conditions (high depth of cut) with conventional lubrication, the tool is exposed to high cutting forces combined with a high generation of heat due to plastic deformation and friction. Thus, in the cutting zone, the temperature becomes very high; the specific cutting force exceeds 2800 MPa which leads to the plastic deformation of the cutting edge [START_REF] Astakhov | The assessment of cutting tool wear[END_REF]. The excessive heat generation also accelerates adhesion and diffusion mechanisms [START_REF] Moura | An investigation of cutting forces and cutting temperatures during laser-assisted machining of the Ti-6Cr-5Mo-5V-4Al beta titanium alloy[END_REF][START_REF] Odelros | Experimental and theoretical study of the microscopic crater wear mechanism in titanium machining[END_REF]. Similarly, the increase of the cutting speed and feed rate resulted in a very rapid degradation of tool life; the resulting temperature rise plays a key role in accelerating tool wear. Moreover, tool wear is accelerated due to the high chemical [START_REF] Courbon | Investigation of machining performance in high-pressure jet assisted turning of Inconel 718: an experimental study[END_REF] Cutting force evolution reactivity of the Ti555-3 alloy, its low thermal conductivity and its high sensitivity to strain rate. To cope with the increase in cutting temperature, high-pressure water-jet assistance has been used. Under these conditions, tool life has been increased up to a factor of 8. The tool wear analysis has revealed that the main tool wear mechanisms are adhesion and abrasion. It has been noted that the deposed material layers are pulled out by the lubricant jet, thus accelerating adhesion wear. A crater is progressively formed causing a considerable degradation in tool stiffness; similar results have been obtained when machining the Ti17 titanium alloy [START_REF] Ayed | Degradation modes and tool wear mechanisms in finish and rough machining of Ti17 titanium alloy under high-pressure water jet assistance[END_REF]. A feed rate increase involves thermal and mechanical effects (high stress and high strain rate); HPWJA permits to slow down wear mechanisms due to thermal effects. However, increasing mechanical effects cause the acceleration of notch wear.

On the basis of the experimental results, the Taylor wear model parameters (1) have been identified for a coolant

Conclusion

The titanium alloy Ti555-3 is a difficult-to-cut material; using emulsion lubrication and for a machining time of 15 min, the maximum cutting speed does not exceed 45 m/min [START_REF] Arrazola | Machinability of titanium alloys (Ti6Al4V and Ti555.3)[END_REF]. In order to enhance productivity, HPWJA has been used to asses an eventual productivity gain; a wide range of cutting conditions have been tested (cutting speed, feed rate and depth of cut). The results obtained are both interesting and promising and can be summarized in the following conclusions:

-The use of high-pressure water-jet assistance has significantly increased tool life. Under certain conditions, tool life increased by more than a factor of 8. It is shown that very severe cutting conditions that are not possible in conventional machining become feasible using HPWJA. -Notch wear is strongly affected by feed rate due to the high strain rate hardening sensibility of the material. -Different experimental means have been used to monitor tool wear and to determine the major wear mechanisms. The use of 3D profilometer made it possible to track the evolution of crater wear. The analysis of SEM images and EDS analyses have been useful for determining the nature of the deposits on the tool. -Plastic deformation, adhesion and abrasion are the main wear mechanisms under conventional lubrication conditions.

-Adhesion remains the major wear mechanism when using the high-pressure water-jet assistance. The increase of lubricant-jet pressure lead to a tool life decrease due to the acceleration of adhesion mechanism.

Fig. 1

 1 Fig. 1 Microstructure of the Ti555-3 alloy

Fig. 2

 2 Fig. 2 Experimental setup

Fig. 5

 5 Fig. 5 Tool wear evolution (P = 100 bar)

Fig. 6 Fig. 7 Fig. 8 Fig. 10 Fig. 9 Fig. 12

 67810912 Fig. 6 Tool wear evolution (P = 250 bar)

Fig. 13

 13 Fig. 13 Observation of tool wear evolution on the tool rake face (a p = 1.5 mm, f = 0.3 mm/rev, V c = 60 m/min, P = 100 bar)

Fig. 14 Fig. 15 Fig. 17

 141517 Fig. 14 Tool wear at the end of tool life (a p = 1.5 mm, V c = 60 m/min, P = 100 bar)

Fig. 18

 18 Fig. 18 Plastic deformation of the cutting edge (conventional lubrication, a p = 2 mm, T = 2 min)

Fig. 20

 20 Fig. 20 Tool wear: a p = 2 mm, P = 100 bar, T = 19 min

Table 1

 1 Cutting conditions

	The cutting conditions
	Cutting parameters	Cutting speed V c (m/min)	60, 68, 75, 80
		Feed rate f (mm/rev)	0.1, 0.2, 0.25, 0.3
		Depth of cut a p (mm)	1.5, 2, 2.5
	Tool geometry	Nose radius r ε (mm)	1.2
		Cutting edge angle (κ r )	95
		Rake angle γ ( • )	7
		Clearance angle α ( • )	6

Acknowledgements Authors would like to thank the company Timet (Ugine-France) for the material support given to this study.


Viet-Duc Le
email: viet-duc.le@ensam.eu

Nicolas Saintier

Franck Morel

Daniel Bellett

Pierre Osmond

Investigation of the effect of porosity on the high cycle fatigue behaviour of cast Al-Si alloy by X-ray micro-tomography

Keywords:

Porosity generated by the casting process has a detrimental effect on the high cycle fatigue strength of cast aluminium alloys. The current study presents an investigation using the non-destructive X-ray micro-tomo-graphy technique of the effect of the size and the population of casting pores on the fatigue strength of the AlSi7Mg0.3 alloy. This alloy was obtained by the lost foam casting process, which leads to a "high" volume fraction of porosity. Observations by micro-tomography, realized on specimens containing fatigue cracks, highlight an important role of the pore distance to the specimen surface in addition to the pore size. In the second part, the local mechanical response around real 3D pores is investigated thanks to finite element models using an elasto-plastic material behaviour law for the aluminium matrix. A critical volume approach based on the Dang Van criterion to predict the fatigue limit is introduced and shows good agreement with the experimental data. The effect of the pore geometry on the Dang Van equivalent stress distribution around individual pores is also investigated.

Introduction

Cast aluminium alloys are widely used for automotive applications due to their high strength-to-weight ratio and their good castability which allows the manufacturing of components with very complex geometries. However, traditional casting processes usually generate a variety of defects such as pores and oxides. These defects, particularly shrinkage pores, have a detrimental effect on the fatigue strength of cast aluminium alloys [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF][START_REF] Mcdowell | Microstructure-based fatigue modeling of cast a356-t6 alloy[END_REF][START_REF] Wang | Fatigue behavior of a356-t6 aluminum cast alloys. Part I. Effect of casting defects[END_REF].

The effect of the pore size on the fatigue strength of cast aluminium alloys has been widely studied in the literature [START_REF] Roy | Multiaxial kitagawa analysis of a356-t6[END_REF][START_REF] Roy | Multiaxial fatigue behaviour of a356-t6[END_REF][START_REF] Houria | Influence of casting defect and {SDAS} on the multiaxial fatigue behaviour of a356-t6 alloy including mean stress effect[END_REF][START_REF] Koutiri | A probabilistic model for the high cycle fatigue behaviour of cast aluminium alloys subject to complex loads[END_REF][START_REF] Brochu | High cycle fatigue strength of permanent mold and rheocast aluminum 357 alloy[END_REF][START_REF] Wang | Fatigue behavior of a356/357 aluminum cast alloys. part {II} -effect of microstructural constituents[END_REF][START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF][START_REF] Le | Simulation of the kitagawa-takahashi diagram using a probabilistic approach for cast al-si alloys under different multiaxial loads[END_REF]. In this body of work the pore size has been typically measured via the observation of polished samples or by the inspection of fatigue failure surfaces. However, these techniques only supply two-dimensional information about the pore geometry at the crack initiation site. Because of this limitation, very few studies have investigated the effect of the free surface (i.e. the spatial position of the pores with respect to the specimen surface) and effect of the 3D pore geometry. For instant, in the work of Serrano et al. [START_REF] Serrano-Munoz | Influence of surface and internal casting defects on the fatigue behaviour of a357-t6 cast aluminium alloy[END_REF] artificial defects were introduced on the surface (surface defects) and in the bulk (internal defects) of a cast Al-Si-Mg alloy. The authors observed that the specimens with surface defects had a fatigue strength at × 5 10 6 cycles that was significantly lower (i.e. 70% lower) than that of the specimens with internal defects. Con- cerning the effect of the pore geometry, in the work of Houria [START_REF] Houria | Etude exprimentale et modlisation de la dure de vie en fatigue d'un alliage d'aluminium de fonderie a356-t6 sous chargement multiaxial[END_REF] the fatigue strength of specimens with natural casting pores is compared with the those with artificial semi-spherical surface defects. The author observed that for the same value of the area square root ( area ) measured on the failure surfaces, there is no difference in the fatigue strength between the artificial defects and natural ones. Besides these experimental approaches, the fatigue behaviour of cast aluminium alloys has been also investigated by means of the finite element method (FEM). Gao et al. [START_REF] Gao | A micro-cell model of the effect of microstructure and defects on fatigue resistance in cast aluminum alloys[END_REF] and Fan et al. [START_REF] Fan | Cyclic plasticity at pores and inclusions in cast alsi alloys[END_REF] studied the cyclic behaviour of cast aluminium alloys at the scale of silicon particles in the eutectic zones. Recently, based on X-ray micro-tomography measurements, several studies concerning the influence of casting pores have been conducted using 3D FEM simulations of pores. For instance, Ferri et al. [START_REF] Ferri | Fatigue crack propagation: In situ visualization using X-ray microtomography and 3d simulation using the extended finite element method[END_REF][START_REF] Ferri | 3d characterisation of the nucleation of a short fatigue crack at a pore in a cast al alloy using high resolution synchrotron microtomography[END_REF] and Dezecot et al. [START_REF] Dezecot | In situ 3d characterization of high temperature fatigue damage mechanisms in a cast aluminum alloy using synchrotron X-ray tomography[END_REF], studied crack initiation and crack propagation mechanisms from casting pores in the low cycle fatigue regime, thanks to high resolution in-situ micro-tomography observations. In the work by Nicoletto et al. [START_REF] Nicoletto | X-ray computed tomography vs. metallography for pore sizing and fatigue of cast al-alloys[END_REF][START_REF] Nicoletto | Characterization of microshrinkage casting defects of alsi alloys by X-ray computed tomography and metallography[END_REF] and Tijani et al. [START_REF] Tijani | Quantitative evaluation of fatigue life of cast aluminum alloys by non-destructive testing and parameter model[END_REF][START_REF] Tijani | Detection and influence of shrinkage pores and nonmetallic inclusions on fatigue life of cast aluminum alloys[END_REF], stress concentration factors at casting pores in an aluminium alloy were calculated considering a purely linear elastic behaviour. Vanderesse et al. [START_REF] Vanderesse | Effect of porosity on the fatigue life of a cast Al alloy[END_REF] defined the pore influence zone, the regions in which the von Mises stress is higher than the material 0.2% yield stress, to correlate with the location of the crack starting point.

In summary, to the authors knowledge the effect of the distance between a casting pore and the free surface has not yet been clearly investigated in the literature. Also, the results obtained from FEM simulations of real 3D pore geometries in the High Cycle Fatigue (HCF) regime have been limited to the estimation of stress concentration factors. No real approach to estimate the high cycle fatigue strength based on real 3D pore geometries has been proposed.

In the first part of this paper, the influence of the pore population on the high cycle fatigue behaviour of a cast Al-Si alloy is investigated using the micro-tomography technique. More exactly, the role of the spatial position of each pore to the free surface on the fatigue strength is studied. This aspect is evaluated by the extensive use of micro-tomography in order to precisely determine the characteristics of the critical defect population compared to the whole defect population. In the second section, FEM simulations are presented to analyse the stress distribution around casting pores. In these simulations, an elasto-plastic material behaviour law including non-linear isotropic and kinematic hardening is used for the aluminium matrix. This law is identified thanks to the results obtained from cyclic hardening tests. Based on the results of the FEM simulations, a critical volume approach based on the Dang Van criterion is proposed to estimate the high cycle fatigue strength. The effect of the 3D pore geometry is also investigated via the comparison of the Dang Van equivalent stress for natural casting pores and artificial pores with different simplified geometries.

Materials and fatigue tests

In two previous publications [START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF][START_REF] Le | Simulation of the kitagawa-takahashi diagram using a probabilistic approach for cast al-si alloys under different multiaxial loads[END_REF], the authors have presented the results of experimental work investigating the high cycle multiaxial fatigue behaviour of three cast aluminium alloys, referred to as alloys A, B and C. The fabrication steps of the three alloys are as follows.

• Alloy A: gravity die casting followed by a standard T7 heat treat- ment • Alloy B: lost foam casting followed by a standard T7 heat treatment • Alloy C: lost foam casting followed by Hot Isostatic Pressing (HIP) and then a T7 heat treatment

In the present study, the effect of porosity on the fatigue strength is investigated only on alloy B. However, alloy C, subjected to an HIP post-treatment and hence considered to be porosity-free, is used to identify the cyclic hardening behaviour that is used in FEM simulations of alloy B. It is then considered that the HIP treatment has no effect on the microstructure of the material apart from closing the pores. The micrography of the two materials suggest that the HIP treatment has effectively a low impact on the microstructure (similar SDAS) These aspects will be further detailed in the following sections.

Alloy B and high cycle fatigue behaviour

This AlSi7Mg0.3 alloy, was produced by lost foam die casting in the form of plates, 200 mm × 150 mm × 18 mm in size. The material was then subjected to a T7 post-casting heat treatment. This casting process leads to a material with relatively high porosity volume fraction (0.28% estimated by micro-tomography measures). Fig. 1 , a Young's modulus of = E 68 GPa, and an elongation at fracture of = A 0.8%. Monotonic tensile tests were con- ducted on cylindrical specimens with a gauge length of 30 mm and a gauge diameter of 5 mm. Concerning the properties of the aluminium matrix, alloy B has an average SDAS of 77 μm and a micro-hardness of 99 HV025 with a standard deviation of 9 HV025. Plain fatigue specimens with a circular cross-section of diameter 7 mm and a gauge length of 15 mm were used (Fig. 5(a)). The gage length of the specimens was mirror polished in order to evaluate the effect of microstructure and defects on the fatigue strength and eliminate the impact of specimen machining.

The fatigue tests were conducted using a Rumul resonance testing machine with a ± 10 kN load cell, at room temperature. Constant- amplitude uniaxial stress-controlled tests were conducted with a stress ratio of R = -1 and a frequency of approximately 80 Hz. The staircase technique with a stress amplitude step of 10 MPa was used to determine the fatigue strength at × 2 10 6 cycles. The tests were stopped when the presence of a fatigue crack of approximately 3 mm in length was de- tected via a drop in the resonant frequency. The specimens that survived × 2 10 6 cycles in the staircase procedure were retested at a stress amplitude one step higher for another × 2 10 6 cycles. This was repeated until failure of the specimen at less than × 2 10 6 cycles. For this alloy (alloy B) 21 specimens were tested and the results are shown in Table 1. The number of cycles to failure corresponding to the final stress level, N f , and the defect size at the crack initiation site observed on the failure surfaces, area , are also reported. The main advantage of this testing method is that the fatigue behaviour of the material, including the average fatigue strength and the associated scatter, can be determined using the staircase data (i.e. the black coloured data in the staircase table). The mean fatigue strength is determined to be = Σ 62 a D , MPa and the standard deviation is estimated to be 12 MPa. The fatigue strength of each specimen was also estimated by the LOCATI method (see Section 4.3 below).

Alloy C and identification of the cyclic hardening behaviour

In order to properly evaluate the local stress/strain fields by FEM simulations around the defects of alloy B, an elasto-plastic constitutive model is required to simulate the mechanical response of the aluminium microstructure (i.e. the α phase dendrites and the eutectic zones). Because of the high fraction of porosity in alloy B, it was decided to use the cyclic hardening behaviour of alloy C. Note that alloy C, has the same chemical composition as alloy B and that it is obtained by lost foam casting, followed by Hot Isostatic Pressing (HIP) and then a conventional T7 heat treatment. The maximum defect size, area , ob- served by optical microscopy in alloy C is smaller than 50 μm, compared to an average defect size of 470 μm in alloy B. Hence, alloy C is considered to be porosity-free. Alloy C has a tensile strength of

= σ 275 MPa UTS , a yield strength of = σ 250 MPa Y 0.2%
, a Young's modulus of = E 74 GPa and an elongation at fracture of = A 2.3%. The fa- tigue strength corresponding to a failure probability of 50% under uniaxial loading conditions with a load ratio of R=-1 and at × 2 10 6 cycles is = Σ 126 a D , MPa. In particular, SDAS and micro micro-hardness of alloy C are close to these of alloy B. It is therefore expected that the macroscopic cyclic hardening behaviour of alloy C will be the same as that of the aluminium matrix of alloy B (α-Al phase and eutectic phase).

Imposed total strain cyclic hardening tests on specimen of alloy C have been conducted with a load ratio of R = -1 at a constant strain rate of 0.004 s -1 . Three imposed strain amplitude levels have been applied: 0.30%, 0.42% and 0.50%. Fig. 2a-c shows the cyclic behaviour for these three imposed strain levels for different numbers of cycles.

The cyclic hardening behaviour of alloy C is decomposed using 4 hardening variables, X R , ,Σ ,Σ

a a eqa eqm , ,
as introduced in the work of Feaugas et al. [START_REF] Feaugas | Ratchetting process in the stainless steel AISI 316l at 300 k: an experimental investigation[END_REF]. These variables are given by Eq. ( 1) and schematically given in Fig. 2(d) for each full cycle. The classical von Mises yield function is used to describe the yield surface. The yield stress is determined using a plastic strain threshold of × - 2 10 4 .

= = = = + + + - + - + - X R Σ Σ a X X a R R eq a eq m | | 2 | | 2 , Σ |Σ | 2 , Σ |Σ | 2 max min max min (1)
The evolution of these variables as a function of the number of cycles is shown in Fig. 3 for different imposed strain amplitudes.

It can be observed that the stabilisation of the cyclic behaviour is reached very quickly (after approximately 10 cycles) and that the evolution of the isotropic hardening is negligible for all applied strain levels. It can therefore be concluded that the cyclic behaviour of the alloy C is not characterized by significant cyclic hardening or softening.

In order to model the cyclic hardening behaviour of alloy A, the Lemaitre and Chaboche [START_REF] Lemaitre | Mécanique des matériaux solides -3ème édition[END_REF] non-linear isotropic and kinematic hardening constitutive equations were chosen. The isotropic and kinematic hardening are described by the following expressions.

= + ⎛ ⎝ -⎞ ⎠ R R Q e 1 1 bp 0 (2) Table 1
Fatigue test result of alloy B presented in the form of stair-case table. O = specimens survived 2 × 10 6 cycles. X = specimens failure at a

N f cycles. = ∊ - X C DXp ̇2 3 ̇ṗ (3)
These parameters were identified by the Levenberg Marquardt optimization algorithm implemented in the ZeBuLoN finite element code. The optimised numerical values are shown in Table 2.

Fig. 4 shows the comparison between the experimental stress-strain curves and the model predictions for the first cycle a good agreement can be seen. In the FEM simulations presented in the next sections, all the simulations have been carried out for only 1 cycle.

Non-destructive testing by X-ray microtomography on alloy B

Measurements using the X-ray micro-tomography technique were conducted at the PLACAMAT (attached to the University of Bordeaux and CNRS), in France on a microtomograph X GE V/TOME/SX. A spatial resolution of 5 μm × 5 μm × 5 μm was used with a 160 kV (max) X-ray source. The obtained data were reconstructed and analysed with the Avizo® software.

Because of the large size of the fatigue specimens (diameter = 7 mm, gauge length = 15 mm), it was not possible to carry out in-situ fatigue tests. Consequently, it was decided to scan each specimen only once, after being tested in fatigue, in the zone containing the principal fatigue crack. Note that the crack length is approximatively 3 mm on surface. The scanned volume covers the complete cross section of the specimen with a length of 8 mm so that a large amount of critical configuration could be captured. Fig. 5(b) shows schematically the size and the position of the volume observed by micro-tomography on the specimens tested in fatigue.

In total, 7 alloy B specimens taken from the 21 specimens tested in fatigue by the staircase technique (see Table 1 in Section 2) were observed by micro-tomography. Table 3 regroups the results of the fatigue tests on these specimens.

In order to distinguish the cracks and the pores causing crack initiation in the tomography images, different grey level thresholds were used. Fig. 6 shows 3D renderings of a principal crack and the pore at the origin of crack initiation.

The size, in terms of volume, and the spatial position of all pores in the scanned volume have been subsequently measured. Note that pores with a size smaller than 15 μm < V ( 15 μm)

1/3
were disregarded in the analysis. Effectively this limit corresponds to the numerical noises produced in the image scanning or image post-treating process. It is also expected that the influence of such small pores on the fatigue strength of the cast aluminium alloy is likely to be negligible [START_REF] Roy | Multiaxial kitagawa analysis of a356-t6[END_REF][START_REF] Roy | Multiaxial fatigue behaviour of a356-t6[END_REF][START_REF] Brochu | High cycle fatigue strength of permanent mold and rheocast aluminum 357 alloy[END_REF][START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF].

Defect population

The volume and the maximum feret diameter are measured for each pore. In order to obtain a representative distribution of the pore size, the data obtained from the 7 scanned specimens have been regrouped and treated as a unique data set. The pore size distributions from all the observed specimens are presented in Fig. 7. These distributions correspond hence to a total scanned volume of 2.16× 10 3 mm 3 .

The distributions show that the fraction in number of large pores is very small and that the lognormal distribution is not a good fit for the large pores. It can also be seen that the feret diameter of the largest pores is 5 to 10 times greater than the volume cube root (for comparison, this factor is 1.24 for a perfectly spherical pore). This implies that the largest casting pores have very torturous shapes. In order to characterize the pore tortuosity, the sphericity of the pores, defined by Eq. ( 4), is analysed. A small sphericity value correspond to the pores with high tortuosity. The maximum value of the sphericity is 1 corresponding to a perfectly spherical pores.

= π V A Sphericity ( 6
) 1/3 2/3 ( 4
)
where V is the pore volume and A is the pore surface area. Fig. 8 shows the relationship between the sphericity and the pore volume cube root.

It can be seen that as a general rule the pore volume decreases when the sphericity increases. In other words, the large pores are more tortuous than the small pores. This is because of the presence of different kinds of porosity (i.e. shrinkage and gaseous cavities).

Effect of the distance from the pore to the specimen surface on fatigue crack initiation

In previous work concerning the HCF behaviour of cast aluminium alloys conducted by the Arts et Métiers ParisTech research team [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF][START_REF] Koutiri | A probabilistic model for the high cycle fatigue behaviour of cast aluminium alloys subject to complex loads[END_REF][START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF], it has been shown that the principal fatigue cracks leading to final failure of the specimens initiated systematically from pores located on the specimen surface or sub-surface. No crack initiation from internal pores has been observed.

In order to better understand the role of the distance from a pore to the specimen surface on crack initiation process, the size (in terms of V 1/3 ) and the distance to the specimen surface for all of the pores in the scanned volumes are analysed. Note that the distance to the surface is defined as the shortest distance from the pore centroid to the specimen

Table 2

The optimised elasto-plastic parameters obtained for alloy C. surface.

Fig. 9 shows the "distance to surface versus pore size" and "sphericity versus pore size" diagrams for the 7 scanned specimens (the specimen numbers are indicated in Table 3) as well as diagrams superposing the results for all 7 specimens.

In these figures each point corresponds to a pore. The open points represent the pores not associated with crack initiation. The large filled points are the pores associated with principal crack initiation and the small filled points represent the secondary cracks. The distinction between the principal crack and the secondary cracks is based on the crack size observed by micro-tomography. Only secondary cracks located far enough away from the principal crack are reported so that it can be assumed that the presence of the principal crack did not affect initiation of the secondary cracks. Table 4 summarizes the size and the distance to specimen surface of the pores causing crack initiation and of the largest pore observed in each specimen.

It can be seen that for specimens 11, 12, 17 and 19 the largest pores in the observed volumes do not result in crack initiation. In these specimens, the largest pores are located closer to the center of the observed volumes while the pores causing crack initiation are near the specimen surface and are smaller. Furthermore, it is generally observed that all the pores initiating the fatigue cracks are located near the specimen surface with a distance to the surface that is less than 500 μm. By examining the raw images from the micro-tomography scans, it is observed that there is no sub-surface crack initiation. All of the pores causing crack initiation intersect with the specimen surface, as illustrated in Fig. 10. There are two possible explanations for this phenomenon. Firstly, a higher stress concentration is likely to be generated at a surface or sub-surface pore compared to the same pore morphology located in the bulk material [START_REF] Tijani | Quantitative evaluation of fatigue life of cast aluminum alloys by non-destructive testing and parameter model[END_REF][START_REF] Tijani | Detection and influence of shrinkage pores and nonmetallic inclusions on fatigue life of cast aluminum alloys[END_REF][START_REF] Borbély | A finite element modelling study of strain localization in the vicinity of near-surface cavities as a cause of subsurface fatigue crack initiation[END_REF]. The second possible reason is that due to environmental effects cracks initiating from surface pores have a higher crack growth rate compared to those initiating from internal pores. It has been reported that crack propagation plays a significant role in the high cycle fatigue life of cast aluminium alloys [START_REF] Buffière | Experimental study of porosity and its relation to fatigue mechanisms of model alsi7mg0.3 cast al alloys[END_REF]. For cracks initiated from internal pores propagation occurs essentially in a vacuum while for cracks initiating from a surface pore, the propagation takes place in an air environment. In the litterature, the beneficial effect of the vacuum on the fatigue life in metallic materials was shown [START_REF] Petit | Environmentally assisted fatigue in the gaseous atmosphere[END_REF]. For cast aluminium alloys Serrano et al. [START_REF] Serrano-Munoz | Influence of surface and internal casting defects on the fatigue behaviour of a357-t6 cast aluminium alloy[END_REF] showed that the difference of the crack growth rate between the crack propagation in vacuum and in air is by a factor of 10.

Concerning the role of the pore sphericity, it can be observed that 6 pores at the origin of fatal cracks (out of a total of 7 pores) have a

Table 3

Summary of 7 specimens tested in fatigue and subsequently scanned via micro-tomography. O = the specimen survived 2 × 10 6 cycles. X = specimen failure at a N f cycles.

Stress amplitude Σ a [MPa] Specimen number sphericity that is less than 0.2. However, because these pores are also among the largest pores close to the specimen free surface, it is difficult to make a conclusion concerning a threshold value of the sphericity on the initiation of fatal cracks.

Finite element simulations of real 3D casting pores in alloy B

Based on the micro-tomography analyses, the numerical geometries of all pores located in the scanned volumes were obtained. Finite element models of the 3D casting pores have been then built and the local mechanical response surrounding the pores is investigated. A fatigue modelling approach is then developed to predict the fatigue strength of the specimens from the results of the FE simulation for critical pores.

Finite element model

The FE simulations are described in Fig. 11a-d. After generating a surface model of the pores using the Avizo® software, the defects were placed in a cylindrical volume with a diameter of 7 mm and a height of 4-6 mm (Fig. 11b). The size of the simulated volume is chosen to achieve reasonable computation time and to be large enough to avoid border effects (at the upper and lower faces of the cylinder) in the stress field surrounding the pores. The pores are spatially positioned in the modelled volume in their the real positions, identified from the microtomography observation. The finite element mesh size is approximatively 8 μm the pore surface and 200 μm at the outer surfaces of the cylindrical volume. Linear tetrahedral elements designated "C3D4" are used in the simulations. It is important to note that each simulated volume contains only one pore. The effect of the interaction between pores in the mechanical response is not considered in this study. Fig. 11d shows the boundary conditions applied to the simulated volumes. The numerically applied stress Σ a is equal the experimentally applied stress level corresponding to failure of the specimens (see Table 3). Fig. 11e shows an example of a critical pore and the stress field in the axial direction (σ 33 ) around the pore, without the presence of the fatigue crack. Note that the axial direction is the loading direction and is noted as e 3 for both the FE models.

Local mechanical response and a proposed fatigue modelling approach

Based on the critical distance theories developed by Susmel et al. [START_REF] Susmel | The theory of critical distances as an alternative experimental strategy for the determination of kic and kth[END_REF] (the line and point methods) and Bellett et al. [START_REF] Bellett | The fatigue behaviour of three-dimensional stress concentrations[END_REF] (the volume method), El may et al. [START_REF] El Maym | Etude de la tenue en fatigue d'un acier inoxydable pour l'aronautique en milieu marin corrosif[END_REF] proposed a critical volume approach to predict the fatigue strength of the X12CrNiMoV12-3 steel containing corrosion pores on the specimen surface. The approach uses the "average value" of the equivalent stress determined using the Crossland criterion, in a spherical volume of radius r. For the cast Al-Si alloy under investigation, this approach presents two main advantages. Firstly, the effect of the mesh size on the pore surface is strongly reduced. Because of the small pore size (V 1/3 > 150 μm), the FE mesh cannot be adequately refined to achieve a full convergence of the local stresses. The "average" stress is much less dependent on the element size. The second advantage is that this approach takes into account not only the crack initiation condition but also micro-crack propagation. Because of the important role of crack propagation on the HCF life of cast aluminium alloys, it can be argued that the use of the average value is more reasonable for engineering fatigue design than the local value at the pore surface.

In the present study, the "average" Cauchy stress tensor at gauss point i centered in a sphere of radius r and volume V r is firstly calcu- lated. The Cauchy stress tensor is "averaged" in the volume V r using Eq. ( 5).

∫ 〈 〉 = σ t V σ t dV ( ) 1 ( ) i r r V r r (5)
This calculation has been implemented in a finite element code by using the numerical transformation of Eq. ( 5), given by following expression:

∑ 〈 〉 = ∈ σ t V σ t J W ( ) 1 ( )det( ) i r r k V k k k r ( 6
)
where σ t ( )

k
is the Cauchy stress tensor for time t at gauss point k located in the volume V r . det J ( ) k and W k are the Jacobian determinant (relating to the differential area of the deformed element to the differential area of the master element) and weight function of the Gaussian quadrature rule at gauss point k. From the average Cauchy stress tensor, other stresses such as the principal stresses or the von Mises equivalent stress can be calculated.

Fig. 13 shows an example of the evolution of the average values of ∊ σ , 33 33 , the von Mises equivalent stress and the accumulated plastic strain at a "hot-spot" (point P on the pore illustrated in Fig. 12) at loads corresponding to the macroscopic high-cycle fatigue strength.

It can be seen that without averaging, the local value of the von Mises stress at point P is approximately 210 MPa and the total strain amplitude is ∊ ≈ 0.3% a 33,

. For comparison, the cyclic hardening tests conducted on alloy C (i.e. the porosity-free alloy with a microstructure similar to alloy B) at this strain level had fatigue lives lower than 10 4 cycles. Hence, the high strain level at the "hot-spot" can explain early failures in the aluminium microstructure around pores and thus the rapid crack initiation (often less than 10 4 cycles) that has been widely observed in cast aluminium alloys even in the HCF regime [START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF][START_REF] Buffière | Experimental study of porosity and its relation to fatigue mechanisms of model alsi7mg0.3 cast al alloys[END_REF]. It is also important to note that by using an averaging methodology, the plastic strain is lower and the mechanical response becomes essentially elastic for volumes of radius = r 40-60 μm. From the average Cauchy stress tensor calculated at all gauss points surrounding the pore and at each step t in the loading cycle, a fatigue approach based on the Dang Van criterion [START_REF] Dangvan | Sur la resistance a la fatigue des metaux[END_REF] is proposed. The classical formulation of the Dang Van criterion is given by:

̂+ < ∈ τ n t ασ t β max(max [|| ( , )|| ( )]) n t T H ( 7
)
where τ n t ( , ) is the mesoscopic shear stress vector acting on the slip plane n for time t and σ t ( )

H
is the mesoscopic hydrostatic pressure. T is the time period of the loading cycle. α and β are two material para- meters. Based on this formulation, an Dang Van equivalent stress is defined and the calculation of the average value of this stress is given by Eq. ( 8)

̂〈 〉 = 〈 〉 + 〈 〉 ∈ σ τ n t α σt max(max [|| ( , ) || ( ) ]) DV i r n t T r H r , (8)
The average resolved shear stress 〈 〉 τ n t ( , ) r and the average hy- drostatic stress 〈 〉 σ t ( ) H r are calculated from the average Cauchy stress tensor 〈 〉 σ t ( ) i r . The Dang Van coefficient α is taken to be equal to 0.205. This value was identified from the fully reversed tension-compression fatigue strength and the fully reversed fatigue strength in torsion of alloy C (i.e. the porosity-free alloy) [START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF]. Finally, the Dang Van fatigue indicator is given by Eq. ( 9). Note that when this parameter is greater than or equal to one ( ⩾ DV 1.0), failure is predicted.

= 〈 〉 ∈ DV σ β max i DV i r Ω , (9)
where = β 72 MPa is equal to the torsional fatigue strength of alloy C (R = -1) [START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF]. Ω is the set of all gauss points surrounding the pore. This set contains the gauss points of all elements with at least one node on the pore surface.

In order to determine the optimum radius r for the averaging volume V r which results in the best prediction of the macroscopic fatigue strength, the critical pore cumulated probability of occurrence (i.e. pores at the origin of the crack initiations) is calculated. Fig. 14 shows the evolution of this probability as a function of the Dang Van fatigue danger indicator, DV. The danger indicator, DV is calculated for two different sized averaging volumes = V r : 40 r μm and r = 60 μm. It can be seen that the r = 40 μm curves passes through the point where the Dang Van indicator equals one ( = DV 1) and the accumu- lated probability of occurrence is 50%, while for the r = 60 μm curve this value is approximately 95%. Thus, the radius r = 40 μm is chosen to best predict the macroscopic fatigue strength.

Table 4

Summary of pore size (V 1/3 ) and the distance to the specimen surface (Dist.) of the largest pore observed in each specimen and the pore causing fatigue crack initiation. Fatigue cracks are classified as either principal cracks (Prin. crack) or secondary cracks (Sec. crack). All values are in μm.

Sp. N°P ri. crack

Sec. crack 1 Sec. crack 2 Largest pore

V 1/3 Dist. V 1/3 Dist. V 1/3 Dist. V 1/

Prediction of the macroscopic fatigue strength and comparison with the experimental data

The proposed fatigue approach is used to predict the macroscopic fatigue strength of the 7 specimens shown in Table 3. The aim is to determine the macroscopic applied stress amplitude Σ a for which the Dang Van fatigue indicator is equal to one ( = DV 1) for a averaging volume with a radius of = r 40 μm, around the pore initiating the principal fatigue crack (see Eq. ( 9)).

Fig. 15 shows the comparison between the predicted and experimentally determined fatigue strength of each specimen. The experimental fatigue strengths at × 2 10 6 cycles are determined by the fol- lowing procedure [START_REF] Houria | Influence of casting defect and {SDAS} on the multiaxial fatigue behaviour of a356-t6 alloy including mean stress effect[END_REF]:

• For specimens loaded at several load levels before failure, the fa- tigue strength at × 2 10 6 cycles is determined by Eq. ( 10)

= + × - - - N Σ Σ 2.10 (Σ Σ ) a D a n f a n a n , , 1 6 ,
, 1
f f f (10)
where Σ a D , is the fatigue strength, Σ a n , f is the applied stress amplitude level at the final step.

• For specimens that failed during the first load level, the Basquin equation (Eq. ( 11)) is used to determine the fatigue strength at It can be seen in Fig. 15(a) that the proposed approach predicts a decrease in the fatigue strength with an increase in the equivalent pore size V 1/3 that is in agreement with approaches at the macroscopic scale [START_REF] Mcdowell | Microstructure-based fatigue modeling of cast a356-t6 alloy[END_REF][START_REF] Houria | Influence of casting defect and {SDAS} on the multiaxial fatigue behaviour of a356-t6 alloy including mean stress effect[END_REF][START_REF] Le | Simulation of the kitagawa-takahashi diagram using a probabilistic approach for cast al-si alloys under different multiaxial loads[END_REF]. The difference between the numerical predictions and the experimental data is generally less than 15% (Fig. 15(b)).

In the next section, the effect of pore morphology on the local mechanical response is studied. The Dang Van equivalent stress is assessed for real casting pores and compared to several simplified pore geometries.

Effect of pore morphology

Two simplified pore geometries are considered: a "spherical pore" and an "oblate spheroidal pore" with a major axis equal to 4 times the minor axis. The factor of 4 is arbitrarily chosen. These defects are introduced in the center of cylindrical matrix (diameter = 7 mm, height = 4 mm). Fig. 16 shows the geometry of these two pores. Note that the loading direction is e 3 .

Concerning the real casting pores, in order to obtain a statistically representative result, 87 critical and non-critical pores were modelled. These pores are the largest pores observed by micro-tomography, located near the specimen surface (with a distance to the surface of less than 500 μm).

The normalized Dang Van equivalent stress "averaged" in spheres V r of radius of 40 μm, defined by Eq. ( 12), is used to compare the local mechanical response of the different pores.

where Σ a is the macroscopic applied stress and Ω is the set of the gauss points surrounding the pore. Fig. 17 shows the comparison of the normalized average Dang Van stress as a function of the equivalent pore size V 1/3 for the real casting pores and the simplified pore geometries. The linear regression line in log-log space and the 95% confidence bands for the real casting pores are also shown.

It can be seen that the slope of the power trendline fitted to the real casting pores is higher than the slope of the trendline for the spherical pores. In other words, the effect of pore size on the normalized Dang Van stress is more pronounced for the real casting pores. This conclusion suggests that the effect of pore size on the fatigue strength is more detrimental for casting pores than the spherical pores. This prediction is confirmed by the recent work of Dezecot et al. [START_REF] Dezecot | 3d characterization and modeling of low cycle fatigue damage mechanisms at high temperature in a cast aluminum alloy[END_REF]. These authors showed via high resolution micro-tomography observations conducted on low cycle fatigue test of a cast aluminium alloy, that spherical pores (degassing pores) are less detrimental than tortuous pores (microshrinkage pore) of the same size. Concerning the "oblate spheroidal pores", it is observed in Fig. 17 that the slope of the trendline is much closer to the slope of the real casting pores. However, in order to better correlate the slope of the real casting pores, simplified geometries with shaper shapes will be simulated in future work.

Conclusion

The X-ray micro-tomography technique was used to investigate the fatigue behaviour of an AlSi7Mg0.3 aluminium alloy obtained by the lost foam casting process. It was observed that, in addition to the pore size, the distance to the specimen surface plays an significant role in the initiation and propagation of fatigue cracks. All pores causing the initiation of principal fatigue cracks were located close to the specimen surface (with the distance to the surface less than 500 μm) even though larger internal pores were observed in the volume.

From the numerically scanned pore geometries obtained by the micro-tomography observations, 3D finite element simulations were conducted to investigate the local mechanical response surrounding the real pores. An elasto-plastic material behaviour model was used to describe the aluminium matrix. In order to identify the elasto-plastic constitutive equations, including non-linear isotropic and kinematic hardening, cyclic hardening tests were conducted on the same porosityfree cast Al-Si alloy (obtained via a HIP treatment). The behaviour model was identified by an inverse optimization algorithm.

The finite element simulations showed that stress concentrations at the pores are very high and lead to local plasticity. In order to predict the fatigue strength, a critical volume approach was proposed that is base on the averaged values of the Dang Van criterion in a spherical volume of radius 40 μm. The radius of the critical volume has been determined via an inverse methodology. Good agreement between the predictions and the experimental data is obtained with errors of less than 15%). The influence of the pore geometry/morphology was also investigated. The proposed approach predicts a pore size sensitivity to the fatigue strength more significant for the real casting pores than for the spherical pores. For the oblate spheroidal pores with = × r r 4 maj ax min ax . . . . , a similar pore size sensitivity to the fatigue strength comparing to the real pores is suggested. The approach used to model the fatigue strength of Alloy B can also be applied to Alloy A (AlSi7Cu05Mg03). However, the elasto-plastic constitutive equations must be re-identified because Alloy A contains copper, which modifies the mechanical response of the material microstructure.

Fig. 1 .

 1 Fig. 1. Microstructural heterogeneities present in the cast aluminium alloys under investigation AlSi7Mg0.3: (a) typical microstructure of cast aluminium alloys; (b) zoomed view [10].

Fig. 2 .

 2 Fig.2. Stress-strain loops corresponding to a total strain amplitude of 0.30% (a), 0.42% (b), 0.50% (c) and schema of a uniaxial tension-compression loading cycle showing the isotropic and kinematic hardening contributions (d).

Fig. 3 .

 3 Fig. 3. Evolution of the stress amplitude, of the mean stress, of the kinematic hardening and of the isotropic hardening as a function of the cycle number for alloy C.

Fig. 4 .

 4 Fig. 4. Comparison of the experimental cyclic hardening behaviour for alloy C and the simulation for the cycle N = 1.

Fig. 5 .

 5 Fig. 5. (a) Fatigue specimen; (b) Scanned volume by micro-tomography on the fatigue tested specimens.

Fig. 6 .

 6 Fig. 6. Two 3D renderings of a principal crack observed on the specimen number 11, tested in fatigue. The pore causing crack initiation is in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

  Histogram of the cube root of the volume (V 1/ 3 ) and the best t lognormal curve Lognormal distribution (b) Histogram of the feret diameter and the best t lognormal curve

Fig. 7 .

 7 Fig. 7. Pore size distributions for alloy B including zoomed views in the large pore domain.

Fig. 8 .

 8 Fig. 8. Sphericity as a function of the volume cube root of the observed pores.

  surface [μm] Distance to surface [μm] Distance to surface [μm] Distance to surface [μm]

Fig. 9 .

 9 Fig.9. Size, sphericity and distance to the specimen surface of pores. Open spots: pores not associating to any crack; big full spots: pores associating to the principal cracks; small full spots: pores associating to the secondary cracks.

Fig. 9 .

 9 Fig. 9. (continued)

Fig. 10 .

 10 Fig.10. An example of crack initiation from the critical pore of the specimen number 11 that intersect the specimen surface, observed by X-ray micro-tomography.

Fig. 11 .

 11 Fig. 11. Description of the FE models: (a) meshed surface of a pore, (b) meshed volume, (c) cut volume showing the pore and the volume mesh surrounding the pore, (d) boundary conditions, (e) -e e 1 2 cut plan showing an image obtained by micro-tomography of the critical pore of the specimen number 14 and the σ 33 stress field surrounding the pore.

Fig. 12 .

 12 Fig. 12. A section through a pore of the specimen number 14 in thee e 2 3 plane showing the 3D stress field as well as the position of a "hot-spot", marked point P and the volume V r in which the average Cauchy stress tensor is assessed. The applied macroscopic stress amplitude is = Σ 70 MPa a 33,

  where C and b are the Basquin constants. The Basquin slope parameter was identified for alloy B in previous work to be =b 5[START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF].

Fig. 13 .Fig. 14 .

 1314 Fig. 13. The mechanical response at point P for volumes of different radii V r . (a) Stress-strain curves in the loading direction. (b) von Mises stress vs accumulated plastic strain curves. The applied macroscopic stress amplitude is = Σ 70 MPa a 33,

Fig. 15 .

 15 Fig. 15. Comparison between the predicted and the experimentally determined fatigue strengths of the 7 specimens shown in Table 3. (a) Fatigue strength amplitude as a function of pore size, (b) the predicted fatigue strength versus the experimental fatigue strength. The dotted lines show the margins of error of ± 15%.

Fig. 17 .

 17 Fig. 17. Comparison of the local mechanical response of real casting pores and simplified pore geometries.

Fig. 16 .

 16 Fig. 16. Two simplified geometries: (a) spherical pore (sphericity = 1); (b) oblate spheroid pore with = × r r 4 maj min (sphericity ≈ 0.7). The loading direction is e 3.

Acknowledgement

This work was financially supported by PSA Peugeot Citroën.