Vincent Wagner
email: vincent.wagner@enit.fr

Emmanuel Duc

Study of Ti-1023 milling with toroidal tool

Keywords: Ti-1023, Milling, Tool wear, Cutting conditions, Toroidal tool

The Ti-1023 beta alloy titanium is used in aeronautics for the production of structure parts such as landing gear and as most of titanium alloys its machinability is poor explained by its physical and mechanical properties. This work presents the machinability results carried out for Ti-1023 milling in using toroidal tool. The aim of this research is the understanding of tool wear mechanisms and to establish the relation between cutting conditions, milling tool geometry, and tool life. A section is also devoted to the chip analysis. The wear modes are also analyzed and defined for different cutting tool geometries where several steps occur. This study is focus on material removal rate, tool life, and volume.

and a high chemical reactivity with the most of cutting tool materials [START_REF] Ezugwu | Titanium alloys and their machinability-a review[END_REF]. Wagner compares the mechanical and the physical properties of Ti-64 and Ti-5553 [START_REF] Wagner | Improvment of machinabiltiy of refractory titanium: the ti-5553[END_REF]. Identical physical behavior for both materials but a large difference in mechanical behavior has been noted and may explain the poor machinability. For example, the Ti-5553 titanium alloy, its low machinability is mainly due to its concentration in alphagene elements [START_REF] Arrazola | Machinability of titanium alloys (ti6al4v and ti555.3)[END_REF]. Concerning the Ti-64, some studies show also its low machinability. However, few studies concern the machinability of Ti-1023 titanium alloy. But its chemical composition compared to the common titanium alloys as the Ti-64 can induce a difference. Machai notes that a slower cutting speed (V c < 50 m/min) is recommended in order to avoid catastrophic failure [START_REF] Machai | Machining ob beta-titanium allloy ti-10v-2fe-3al under conditions: cooling with carbon dioxide snow[END_REF]. For higher cutting speeds (V c > 100 m/min), the flank wear increases rapidly.

The toroidal cutter can be used in many sort of operations : contouring and surfacing. Its cylindrical inserts allow also milling of complex surfacing. Moreover, compared to a cylindrical tool with a constant cutting engagement, the high cutting edge angle reduces the chip cross section and consequently decreases the cutting forces. An highest tool life could be then expected. In addition to increase the productivity, a new tool cutter positioning can be used [START_REF] Gilles | Optimum orientation of a torus milling cutter: Method to balance the transversal cutting force[END_REF]. The considered tool inclination is towards the back of the tool. In this case, all inserts work simultaneously and generate a continuous cutting phenomenon.

However, the advantages of toroidal tool used in machining of titanium alloys are not yet demonstrated and their use can be optimized. This work presents a study of the employment of toroidal tool during the Ti-1023 milling. In first section, an optimal range of cutting conditions in using the French method of Tool Material Couple (COM) is defined [START_REF]Working zones of cutting tools couple tool material part 1 general presentation[END_REF]. A part of this study, will be devoted to the study of the chip formation with torus tool. The last section presents a tool wear study where are presented the tool damages during milling of Ti-1023. The last section points the relation between the cutting speed and the tool life.

Titanium alloy presentation

For all experiments completed in this study, the workpiece material is a Ti-1023 titanium alloy containing iron (2 % of weight), aluminum (3 % of weight), vanadium (10 % of weight), carbon (0,01 % of weight), and the presence of oxygen, hydrogen, and nitrogen. The titanium is a structural β titanium alloy and presents some interests for applications such as landing gears. The β-alloys compared to αβ alloys exhibit higher strength to density values and good strength-toughness combinations. In order to make tests in a homogenous material, the superficial coat has been removed.

The material composition of titanium alloys modifies the phase volume fraction with the phase stability. The titanium alloys can be divided in three categories:

-The quasi-α where the stable β phase is very low (less of 5 %), -the α + β alloys where the β phase represent 5 to 10 % of the volume, -the β alloys, -the quasi-β where the α phase is very low. This classification can be made according to Al eq and Mo eq which are the equivalent in aluminum and molybdenum of α and β phases. They are respectively defined with the (1) and [START_REF] Wagner | Improvment of machinabiltiy of refractory titanium: the ti-5553[END_REF].

Al eq = Al%m + Sn%m 3 + Zr%m 6 + 10 × (O 2 %m + C%m + 2N%m) (1)
Mo eq = Mo%m + 1.5V %m + Cr%m 0.6

+ D F e%m 3.5 + Cu%m 1, 3 + Nb%m 3.6 (2)
As described by [START_REF]Combres Y Metallarugy and recyclability of titanium and its alloys[END_REF], the type of titanium alloy can be defined based on a graph (Fig. 1).

The mechanical properties of the Ti-1023 are partially explained with its composition. The aluminum is an alphagene component which increases the alloy's hardness, the mechanical, and the fatigue resistance. However, this element affects its ductility. At room temperature, the iron increases the mechanical resistance, the ductility, and the creep. The component with the highest concentration, the vanadium, is a betagene element increasing the ductility but raises the corrosion alloy resistance. Its low density Fig. 1 Type of titanium alloy according to Al eq and Mo eq combined with its good mechanical properties, the toughness and the fatigue resistance are the main causes of the Ti-1023 alloy use for the critical parts on airplane. The beta transus temperature is approximately 800 • C, the Ti-1023 alloy presents a better forgeability compare to the Ti-64 where the beta transus temperature is approximately 1830 • C.

Concerning the alloys used in this study, a first forging operation is carried out beyond 800 • C and a second forging operation is made under the beta-transus temperature (750 • C). The alloy properties improve its sensibility to heat treatments. In order to verify the alloy, a micrography has been carried out. A sample has been removed, embedded in a resin and a chemical etching (Kroll) has been made. With this forging process, the titanium alloy microstructure observed is a regular microstructure. It's divided in a primary α phase and a matrix which is composed of β phase stabilized by the precipitations of secondary phase α (Fig. 2). Concerning the tool 1, the reference of tool holder is R300-032A25-12H with a HSK63 attachment and the insert used is R300-1204E-PL S40T. The cutting tool geometry is defined in Table 2. For the tool 2, the tool holder reference is A251.40.R04.12RS with RPHX 1204M4EN M31 CTC 5240 inserts (Table 1). Each insert has been embedded in a resin and cutted in order to measure the cutting tool geometry (Fig. 3). For both tool holders, the lead angle, the axial rake angle, and the index angle are equal to zero. Consequently, the axial rake angle and the radial rake angle are similar to the insert rake angle.

Fig. 2 Ti1023 microstructure

All tests have been carried out with external lubricant (Quakercool 8 %) and with low pressure (<12 bars). The three cutting force components were measured using a KISTLER quartz three-components dynamometer type 9257B and PC-based force acquisition system using Pimento software. The precision of acquisition was evaluated to be able to measure cutting forces with an error inferior to 3.5 %. The dynamometer was mounted on a numerical controlled milling device Huron KX15. The cutting power was measured with Siemens system, and they were treated with Matlab software.

The forces recorded by the dynamometer are represented as shown in Fig. 4. For a point of the tool cutting edge, F is the resultant cutting forces with its three components: F a , F r , and F t . F a is the axial force, F r is the radial force, and F t is the tangential force. The wear of the inserts was measured with a multi-sensor device Werth at regular intervals by interrupting machining operations.

Cutting conditions definitions

In industry, a practical method used to characterize the machinability of a particular couple of tool-material is to use the specific cutting energy (W c ). This method is presented in the next section. It shows the ratio between the cutting process in considering the cutting power and the material removal rate [START_REF] Arrazola | Machinability of titanium alloys (ti6al4v and ti555.3)[END_REF].

W c = P Q = P a e a e V f ( 3
)
where P is the cutting power (W), Q is the material removal rate (mm 3 /min), a e is the radial depth (mm), a p is the axial depth (mm), and V f is the feed (mm/min). Cutting power can be defined according two methods. The first is the measurement of the spindle power using a wattmeter [START_REF] Maitre | Relations between cutting energies and tools damage[END_REF]. The second method consists to use a dynamometer platform where the cutting power is calculated by the following relation ( 4):

P = -→ F . -→ v (4)
with -→ F : the cutting forces (N) and -→ v : the velocity (m/s). According to some observations made by [START_REF] Maitre | Relations between cutting energies and tools damage[END_REF], Deshayes and coworkers note a better definition of the cutting parameters in using dynamometer platform allowing to exclude cinematic effects of lathe [START_REF] Deshayes | Analysis of an equivalent tool face for the cutting speed range prediction of complex grooved tools[END_REF]. The specific cutting forces and consequently spindle power remain stable over a range of cutting speed contrary to the power measurements.

Deshayes and coworkers compare the energies formulas for cutting power, specific cutting energy, and specific cutting force in oblique turning [START_REF] Deshayes | Analysis of an equivalent tool face for the cutting speed range prediction of complex grooved tools[END_REF]. However, for milling operations, the cutting forces depend on the tool insert position. In this study, all cutting energies will be calculated according to the maximum cutting forces [START_REF] Gilles | Optimum orientation of a torus milling cutter: Method to balance the transversal cutting force[END_REF].

W = F V c W c = F c V c t W r = F r V f t W a = F a V c t (5)
In using a toroidal cutter, the instantaneous chip crosssection depends of insert angular position, axial depth, insert radius, and axial depth of cut. Contrary to the contouring operations with cylindrical tool where the instantaneous chip cross section varies with the angular insert position and the feed. In order to define a model of cutting forces, [START_REF] Gilles | Optimum orientation of a torus milling cutter: Method to balance the transversal cutting force[END_REF] bases his model on the average chip thickness calculated by the following relations [START_REF]Working zones of cutting tools couple tool material part 1 general presentation[END_REF][START_REF]Combres Y Metallarugy and recyclability of titanium and its alloys[END_REF][START_REF] Maitre | Relations between cutting energies and tools damage[END_REF].

where f z is the feed rate (mm/tooth), a p is the axial depth (mm), r is tool radius (mm), and θ is the insert position ( • C) However, this method does not allow to define the specific pressure for each insert position. For this study, the chip thickness will be defined with the following relation [START_REF] Deshayes | Analysis of an equivalent tool face for the cutting speed range prediction of complex grooved tools[END_REF].

e(θ ) = π 2 ϕ o f z sin(θ )cos(ϕ)rdϕ (9)
4.1 Cutting speed influence Figure 6 shows the specific cutting energies (W ) for both tools. In first, whatever the cutting speed, the specific cutting energy for the tool 2 is the highest. The cutting tool geometry for the second tool is the sharpest. Consequently, there is a better tool penetration into the workpiece material and the cutting forces are reduced.

Concerning the evolution of the specific cutting energy for both tools, the influence of cutting speed is very low. For the tool 1, the specific cutting energies remain stable. The second tool's values is also stable when V c < 50 m/min. Once this value passed, the specific cutting energy increases Based on the standard of the couple tool material, the range of an optimal cutting speeds is defined between two limits called V cmin and V cmax . The lowest cutting speed named V cmin corresponds to the bottom limit. It's defined as the cutting speed where the specific energy remains minimum and with a minimum variation. For Ti-1023 machining and as for other titanium alloy [START_REF] Wagner | Experimental study of coated carbide tools behaviour: Application for ti-5-5-5-3 turning[END_REF], it's very difficult to define this limit. Concerning the tool 1, the V cmin appears when V c = 35 m/min and for tool 2 the V cmin can be defined at 30 m/min. The maximal cutting speed (V cmax ) corresponds to the maximum cutting speed admissible by the tool. Generally, it's defined as the cutting speed where the tool wear becomes uncontrollable. When the cutting speed is higher than V cmax , the specific energy is not considered as constant. Between the two limits, the process behavior must be controlled in different points. The tool life must have a physical and measurable value. The tool does not have a significant wear or a non-expected wear behavior. The disturbance phenomena such as chattering, dynamic tool deflections, or other machine tools dynamics phenomenon must be limited. In this case, the V cmax is defined when some tool wear appear. Concerning tool 1, the upper limit is V cmax = 65 m/min and V cmax = 70 m/min for the tool 2.

The evolution of the specific energy according to the cutting speeds observed for conventional material [START_REF] Maitre | Relations between cutting energies and tools damage[END_REF] does not appear clearly with some specific alloys. Indeed, the increase of cutting speed reduces the cutting forces with the thermal softening. Consequently, the specific cutting energy decreases. However, during the machining of Ti-1023, this evolution does not appear. Wagner and coworkers show also a low influence of cutting speed on specific cutting energy during the Ti-5553 machining [START_REF] Wagner | Experimental study of coated carbide tools behaviour: Application for ti-5-5-5-3 turning[END_REF]. These phenomena are explained by the high mechanical properties of titanium alloys in high temperatures combined with the low thermal properties of titanium alloys. They are often used for the rotating parts of aeronautical engine where the temperatures are high and where they must keep their mechanical properties. Moreover, the low thermal properties of titanium alloys compared to the cutting tool material restrict the temperature diffusion into workpiece material [START_REF] Ezugwu | Titanium alloys and their machinability-a review[END_REF]. Consequently, the cutting temperatures during titanium alloys machining are very high but a major proportion of the thermal energy is transferred into the tool. The thermal energy is then too low to get the thermal softening as observed in traditional material. In order to get this thermal softening, some machining tests have been made in hot workpiece [START_REF] Baili | An experimental investigation of hot machining with induction to improve ti-5553 machinability[END_REF]. The tests show a low reduction of cutting force when the workpiece temperature is equal to 750 • C. These tests point the high strength of titanium alloys in high temperatures. Moreover, the high temperatures conducted into the tool promote the diffusion and consequently the tool wear. In order to get this thermal effect, some tests with highest cutting speeds can be made. However, the rapid tool wear observed when the cutting speed is among 65 and 120 m/min prevent these tests.

The specific cutting energies observed for the Ti-1023 alloy machining are often very high compared to the traditional materials and confirms its poor machinability. These observations corroborate with the conclusions form [START_REF] Donachie | A technical guide[END_REF] and [START_REF] Zlatin | Procedures and precautions in machining titanium alloys[END_REF] which note that the β-titanium alloys are the most difficult to machine among all titanium alloys. This low machinability is mainly to the mechanical, chemical, and physical properties of titanium alloys.

The specific cutting energies observed with tool 2 are always the highest. To explain these phenomena, there is a difference of coating, which can modify stresses at the tool chip interface [START_REF] Moufki | Modelling of orthogonal cutting with a temperature dependent friction law[END_REF] and consequently the cutting forces. The technique of coating deposit can also influence the roughness [START_REF] Padley | Single layer and multilayer wear resistant coating of (ti,al)n: a review[END_REF]. A rough surface leads to high frictions and may override the coating performance. As shown previously, the cutting tool geometry and especially the rake angle of the both tools is different and can induce some highest cutting forces.

Concerning the Ti-5553 dry turning [START_REF] Wagner | Improvment of machinabiltiy of refractory titanium: the ti-5553[END_REF] and for steel machining [START_REF] Ranganath | A model to calibrate and predict forces in machining with honed cutting tools or inserts[END_REF], some works show the influence of the cutting edge radius and the influence of the rake angle on cutting forces. The first study on the effects of cutting edge radius is made by Albrecht [START_REF] Albrecht | New developments in the theory of the metalcutting process -part i. the ploughing process in metal cutting[END_REF]. He states the presence of a stagnation point located below the cutting edge radius. This point is the limit where the workpiece material is divided into the chip and into the final part. Based on an experimental study, Albrecht concludes that the phenomena at the tool chip interface and under the edge radius are independent. When the cutting speed increases, the edge radius effect decreases due to a reduction of the built-up edge. However, the cutting edge radius can be a major part of the cutting forces especially when the uncut chip thickness is lower than the cutting edge radius. As described in Fig. 7, the stagnation point is located at h s . This height was studied attentively because the most of models concerning the effect of the edge radius are based on this height. In milling, Vogler and coworkers study the chip formation during milling with a frequency analysis [START_REF] Kapoor | Microstructure-level force prediction model for micro-milling of multi-phase materials[END_REF]. They note a lower frequency compared to teeth excitation when the uncut chip thickness is lower to h s , and they conclude that this frequency is due to the chip formation. In first, the workpiece material is upset until the uncut chip thickness is enough to form a chip. Accordingly, there is an higher cutting force and at a lower frequency. This difference is due to a better edge penetration in the workpiece material. A high rake angle generates also an increasing of the shear angle as illustrated by the model of [START_REF] Oxley | Mechanics of machining: an analytical approach to assessing machinability[END_REF]. During these tests, the highest rake angle is observed on the second tool. But, the angle at the vicinity of the cutting edge is very low. The effect of edge honed on cutting forces is then non negligible. Concerning the Ti-5553 Fig. 7 Influence of the cutting edge radius on cutting process turning and based on the model developed by [START_REF] Ren | Mechanics of machining with chamfered tools[END_REF], Wagner states the influence of the honed cutting edge on the cutting forces [START_REF] Wagner | Improvment of machinabiltiy of refractory titanium: the ti-5553[END_REF]. He observes an increasing of cutting forces according to the cutting edge radius. This effect is amplified when the uncut chip thickness is lower than the cutting edge radius and its effect is reduced when the uncut chip thickness is two time the cutting edge radius. Moreover, the cutting tool geometry for the second tool is composed of two rake angles (α = 6 • C and α = 15 • C). Consequently, the effect of cutting edge radius is also amplified by the low rake angle. Based on the analysis of the cutting tool geometry effect, it's possible to use the effective rake angle as defined by [START_REF] Manjunathaiah | A study of apparent negative rake angle and its effect on shear angle during orthogonal cutting with edge radiused tools[END_REF]. Indeed, these equations show the relation between the relative size of the undeformed chip thickness to the tool edge radius, the effective rake angle and the separation point between chip and final part. Consequently, for the same cutting edge radius, a decrease of the uncut chip thickness and effective rake angle reduce the average rake angle and the cutting forces increase.

The feed rate effect on cutting process

Based on the French standard [START_REF]Working zones of cutting tools couple tool material part 1 general presentation[END_REF], the range of feed rate can be defined with an identical method. The minimal feed rate (f zmin ) is characterized when the specific energy is at its lowest level and when the tool wear can be controlled. The maximal feed (f zmax ) is defined when the specific energy increases or when some disturbance phenomena occur. Figure 8 shows the evolution of specific cutting energies according to feed rate for two cutting speed (V c = 40 and 60 m/min) in order to validate the low effect of cutting speed.

Concerning the tool 1, the specific cutting energy begins with the highest value and decreases with the feed rise. The range where W c is stabilized and where no disturbance phenomena appeared ranges from f z of 0.1 to 0.3 mm/tooth for both cutting speeds. The evolution of the specific cutting energy of the second tool is different. The values start smaller compared to the first tool and decreases from 0.075 to 0.125 mm/tooth. When V c = 40m/min, the specific cutting energy is stabilized. But after 0.175mm/tooth, some vibrations appear and the tool wear cannot be controlled. If V c = 60m/min, the specific cutting energy increases from a feed of 0.15 mm/tooth and decreases. The region where W c is stabilized ranges from f z of 0.1 to 0.125 mm/tooth.

The reduction of the specific cutting energy is due to an increase of the instantaneous uncut chip thickness. Indeed, when the feed rate is lower than the radius of the cutting edge, the edge radius also generates the chip and some high specific pressures are produced. In order to avoid this phenomenon, the minimum feed rate is often chosen as equal to the value of the edge radius when straight edge is used. However for toroidal tool, use a minimal feed rate equal to the edge radius is more difficult. Indeed, the uncut chip thickness is different along the edge. Figure 9 shows the evolution of the maximal uncut chip thickness at the vicinity of depth of cut according to axial and radial depth for an insert radius equal to 6 mm. When the radial and the axial depths decrease the maximal uncut chip thickness decreases.

For both tools, the uncut chip thickness is often smaller than the cutting edge radius. Consequently, the cutting edge radius effect is amplified. Moreover, the low rake angle at the vicinity of cutting edge for the second tool can increase the cutting forces.

Chip analysis

During machining tests, a difference in specific cutting pressure and consequently on the cutting forces has been observed. This is due to the cutting tool edge geometry (rake angle, rake angle close to the cutting edge, and the cutting tool edge radius). The cutting process generated by each Fig. 9 Evolution of uncut chip thickness for toroidal tool according to insert position and axial depth tool and for each cutting condition can be also observed in analyzing the chip formation. This analysis is divided in two steps. The first step is the analysis of the chips morphology. The second level is the comparison of the chips microstructure obtained with different feeds and at several axial levels of chips. The state of the art is often focused on chip issues of turning operations where the uncut chip thickness does not depend of time. However, in milling and especially with a toroidal tool, the uncut chip thickness depends on insert angular position and axial depth. Consequently, each chip is analyzed in its totally length (total work period of insert) and for two levels (a p = 4 and 2 mm). The first level corresponds to the highest uncut chip thickness a the top of the chip which is the limit of axial depth. In order to analyze the second level, each chip embedded in resin is polished to remove 2 mm of height.

Figure 10 shows the evolution of the chips morphology for different feeds. This study is focused on the chip obtained with the tool 2 because the same conclusions can be made with the first tool. In order to compare the different morphologies, each chip is compared to the theoretical chip. In first, the chip morphology is mainly influenced by the feed. When it is the lowest, the chip is the smallest and when the feed increases the recolted chip become similar at the theoretical chip. The second difference is the shape of each chip. For the lowest feeds (f = 0.05 and 0.1 mm/tooth), the chip seems to be totally wavy. Contrary to the chip obtained with the highest feed where the waves occur only at the bottom part of the chip. The second difference is observed at the free surface which is not in contact with the tool rake face. When the feed is f = 0, 05 mm/tooth, the surface is slight contrary to the chips generated with the highest feeds where some bands occurs.

The different chips morphologies can be explained with the analysis of the chip formation. As described previously, two cutting process which depend of feed, axial depth and insert radius, occur during machining with toroidal tools. The cutting process when the uncut chip thickness is lower than the cutting tool edge radius and the cutting process when the uncut chip thickness is higher than the cutting edge radius. These two cutting processes modify also the chip formation and the chip morphology. When the feed is the lower, the uncut chip thickness is always lower than the edge radius whatever the insert position and the axial position. The chip is formed only by plastic deformation and the chip is also wavy by the insert. These phenomena are amplified by the low chip thickness. The shear bands are also not observed on the free surface, which show the different chip formation. When the feed increases, the chip formation is also modified. The shear bands appear on the free surface of the chip. A study of chip microstructure was carried out on order to complete this analysis. To reveal the titanium alloy microstructure, a chemical attack has been made on each chip. As seen in the previous section, there is a relation between the feed and the chip microstructure and consequently with the chip formation.

Figure 11 shows the evolution of titanium alloy microstructure for several feeds and for two levels of the chip. When the feed is the lowest (f = 0.05mm/tooth), the shear bands don't appear clearly whatever the height analysed and the chip seems to be continuous. Consequently as described previously, the chip is mainly formed by plastic deformations and not by catastrophic shearing. The material is also ploughed by the cutting edge. When the feed increases, the chip formation is modified. Some shear bands appear especially when the uncut chip thickness is the highest (highest feed and highest a p ). According to [START_REF] Barry | The mechanisms of chip formation in machining hardened steels[END_REF], the transition from continuous to saw tooth chip formation is favored by the increase of the cutting speed, the undeformed chip thickness, and the decrease in the tool rake angle. For this analysis, these modifications occur directly on each chip at several angular positions and at several axial depths. When the feed increases, the uncut chip thickness becomes higher than the cutting tool edge radius and the chip formation changes. On a same chip, the type of chip can be different. When the insert comes into contact with the workpiece material, the uncut chip thickness is the highest. The chip is also a saw tooth chip. The uncut chip thickness reduces with the insert angular position and consequently the chip become continuous. The same conclusions can be made at a lowest axial depth but the transition between continuous chip and angular chip is modified.

The chips collected present some shear bands when the uncut chip thickness is the highest (Fig. 13a). All primary shear zones start at the chip base and with the same radius. This type of bands has been observed by [START_REF] Komanduri | New observations on the mechanism of chip formation when machining titanium alloys[END_REF]. The chip formation in machining of titanium alloys has been studied since many years. Several theories have been developed and discussed. The first is based on the theory of adiabatic shear catastrophic. During the cutting process, most of the deformation is concentrated in the primary shear plane. It appears a high rate of work hardening of the material. Moreover, most of the deformation energy is converted into heat energy. Consequently, there is a substantially increase of the temperature in the primary shear band. Due to this heat, the workpiece mechanical properties are then reduced. This phenomenon is called thermal softening. There is a catastrophic shear or adiabatic shear when the thermal softening effect exceeds the rate of hardening. According to the second theory, the segmented chip formation is the result of a crack initiation at the free surface and its spread to cutting edge in the primary shear plane. In their works, Poulachon and coworkers describe the chip formation for steel as four successive phases [START_REF] Poulachon | A contribution to the study of the cutting mechanisms during high speed machining of hardened steel[END_REF]. In despite of the number of models and of theories, the development of a new material with complex behavior, it is not always possible to predict phenomena and especially for the cutting process domain. Concerning the titanium machining, Komanduri explains the chip morphology in several steps [START_REF] Komanduri | New observations on the mechanism of chip formation when machining titanium alloys[END_REF]: (1) represents the undeformed surface of a chip. ( 2) is the α section of the primary shear plane. This zone is a part of the catastrophic shear zone, which is separated from the previous segment.

(3) is the zone where an intense shear band is formed by catastrophic shearing during the failure stage of the segment formation. ( 4) is the zone in contact with the rake face where some intense shearing occurs. ( 5) is the primary shear plane where some intense localized deformations appear. As described by, the primary shear plane is curved. ( 6) is the machined surface.

When the uncut chip thickness is too low, this crack does not appear due to some low strains, stresses, and thermal effect (Fig. 13b).

The feed used must be a compromise in order to limit the effect of edge radius when the uncut chip thickness is very low and to increase the effect of stress when the uncut chip thickness is high. However, the type of wear observed during these tests seems mainly due to some high frictions at the tool chip interface. Indeed, some edge failures occurred explain by the high stresses at the tool chip interface. The increase of feed generates an excessive uncut chip thickness and consequently some highest cutting forces and highest temperatures at the tool chip interface.

To conclude, it seems very difficult to define an optimal range of cutting conditions for Ti-1023 milling only based on workpiece behavior during machining. For example, the thermal energy provides by the cutting process is not enough to get thermal softening. This effect is amplified by the poor thermal conductivity of Ti-1023. The temperature is then evacuated into the tool and the cutting temperature is always too low in order to benefit thermal softening. Concerning the cutting speed tests, the minimal value has been defined for the lowest specific pressure for both tools. The maximal cutting speed will is defined according to the tool wear and no according to cutting process as defined in French Standard [START_REF]Working zones of cutting tools couple tool material part 1 general presentation[END_REF]. The specific pressure is stable, and the tool wear appears quickly. To conclude, the optimal cutting condition for Ti1023 is equal to conventional cutting speeds find in other titanium alloys (35m/min < V c < 80m/min) [START_REF] Koning | Applied research on the machinability of titanium and its alloys[END_REF].

Tool wear

This section presents the results of the tool wear study. The aim is the highlighting of the tool wear modes during the Ti-1023 milling with toroidal tool and the relationship between cutting conditions, tool life, and productivity. The Table 2 shows the evolution of the first tool after each test (150 mm). The second tool is presented Table 3.

Tool wear evolution

For both tools, the tool wear is divided in several steps. The first can be assimilated to a crater wear, which occurs like a concave section on the rake face generated by the chip sliding. The titanium machining is an abrasive process [START_REF] Wagner | Experimental study of coated carbide tools behaviour: Application for ti-5-5-5-3 turning[END_REF]. The contact between chip and tool on the rake face combined with high temperatures generates tearing of some coating particles. Concerning the Ti-64 turning, Corduan and coworkers observe a delimitation of coating into two layers before that coating is totally torn off [START_REF] Corduan | Wear mechanisms of new tool materials for ti-6ai-4v high performance machining[END_REF]. They note a correlation between the cutting conditions and the tool failures. Freeman has studied the tool life of steel cutting grades (containing other material than tungsten) and straight WC/Co grades of cemented carbide in machining of two commercially available titanium alloys [START_REF] Freeman | The machining of titanium and some of its alloys[END_REF]. They show that plastic deformations occurred, especially for high cutting speeds and then shearing on rake face can form a crater. According to [START_REF] Machai | Machining ob beta-titanium allloy ti-10v-2fe-3al under conditions: cooling with carbon dioxide snow[END_REF], some high frictions occur on the rake face of tool. Consequently, the heat is concentrated at the tool chip interface and the chemical reactivity of titanium generates some welding and accelerates the adhesion process.

The tool wear on the rake face during Ti1023 machining is located close to the main cutting edge. Moreover, its shape is similar to the uncut chip thickness (Fig. 14). Some marks appear on rake face in the direction of the chip flow angle.

Concerning the second step of the tool wear, a notch occurs always at the vicinity of the axial depth. According to [28], the side flow of the chip and the burrs on the workpiece was responsible of the formation of serious notch wear. The notch is caused by a fracture process or a chemical reaction [START_REF] Turkes | Linear analysis of chatter vibration and stability for orthogonal cutting in turning[END_REF]. This type of tool wear is due too some high cutting temperatures on rake face explain by the mechanical and the thermal properties which lead to very high cutting temperatures (∼1000 • C. The hardness of the cutting tool materials decreases for high temperatures which aggravates the abrasive wear. For a toroidal tool, the maximum of cutting temperature occurs at the vicinity of axial depth where the uncut chip thickness is the highest. Moreover, the softer cutting edges due to the high temperature under the concentrated cutting forces near to the cutting edge and at the limit of axial depth generate some deflections and deformations. Fig. 14 Tool wear pattern and instantaneous uncut chip thickness

The last cause is the intermittent milling process, which generate mechanical impacts and thermal shocks.

In order to verify the notch position, some tests have been made with some different axial depths (Table 4). The notch is always generated at the vicinity of axial depth where the uncut chip thickness is the highest. The reduction of axial depth leads to lowest uncut chip thickness and consequently lowest cutting forces and temperatures.

The control of the tool wear can be carried out according to classical tool wear criteria (Fig. 15). Several parameters on the flank face or on the rake face can be considered. The flank wear width which is measured in zone B in the cutting edge plane perpendicular to the main cutting edge. According to the standard [START_REF]Working zones of cutting tools couple tool material part 1 general presentation[END_REF], the width of the flank wear land is measured from the position of the original main cutting edge. However, the flank wear criteria, as V b = 0.3mm, has never been reached. Indeed, the flank wear has been not observed during the tests. Concerning the rake face, the crater depth KT defined the maximum distance between the crater bottom and the original face in region B (Fig. 15). However, an optical analysis has been made and during these test, this type of tool wear has not been observed. Moreover, the critical wear seems to be the notch wear for this configuration.

Figure 16 shows the evolution of the tool wear (crater width and notch width) and the spindle power according to the machining length. The notch wear and the crater wear were measured at their highest widths. As shown in Fig. 16, there is a correlation between the tool wear and the spindle power where two levels occur. Firstly, the power is stable and the value is between 2,000 and 1,900 W mainly due to a constant tool wear. After the ninth test, the power increases of 20 % and becomes stable around 2,200 W. The power evolution and notch width evolution are identical. The notch appears at the end of tool life and it is the critical factor to monitor the tool life. These tests point the low influence of crater wear on cutting process on the contrary to the notch wear that seems to be more influential. The increasing of cutting power is due to a modification of cutting tool geometry (rake angle). Moreover, the highest modification is located at maximal uncut chip thickness and its effect is then amplified.

The notch wear occurs also in flank wear but its effect on cutting power is not negligible. The tool will be then considered as worn when notch width exceeds 0.3 mm.

To summarize, the first damage is a crater wear close to the cutting edge where the coating is turn off. This stage represents 40 % of the total tool life. The cutting geometry is also damaged but the cutting process seems to be stable. However, the coating is turn off and the tool seems to be less resistant. A notch appears at the limit of axial depth where the crater width is the highest. The tool wear will be monitor with the notch width.

Cutting conditions influence on tool life and tool performance

This section presents the influence of the cutting conditions on tool life and tool performance.

Based on French standard or in Taylor's model, the tool life can be defined according to time [START_REF] Wagner | Experimental study of coated carbide tools behaviour: Application for ti-5-5-5-3 turning[END_REF].

a F p a e D f D f E z T G V c = C (10)
But for the industrial, the productivity or the tool performance must be measured with material removal rate [START_REF] Baili | An experimental investigation of hot machining with induction to improve ti-5553 machinability[END_REF]. It is particularly important during roughing operations where the aim is to remove material in a short time.

Q = S 1, 000V c π D Zf z (11)
In order to link productivity and tool life, it seems important to use material removal rate combined to volume of material removed (V) [START_REF] Donachie | A technical guide[END_REF].

V = SL ( 12
)
where V is the volume (mm 3 ), S is the cross chip section (mm 2 ), and L is the length machined by one tool (mm).

Influence of cutting conditions

The analysis is made on four major axes : the tool life (min), the machining length (mm), the volume of material removed (mm 3 ) and the material removal rate (mm 3 /min). Figure 17 show the evolution of the tool life in minute according to the cutting speed and the feed rate. Fig. 17 Influence of the cutting conditions on the tool life when f z = 0.2mm/tooth (a) and when f z = 0.3mm/tooth When the cutting speed increases, the tool life raises whatever the feed rate. However, the influence of the cutting speed on the tool life changes for the highest values (V c = 60 and 70 m/min). When V c = 60 m/min, the tool life attains 7 min and when feed is higher than 0.3mm/tooth the tool life decreases immediately (2 min when f z = 0.3mm/tooth and V c = 70m/min). The same conclusions can be made for the machining length and for the volume. According to [START_REF] Donachie | A technical guide[END_REF], the material removal rate increases linearly according to the cutting speeds.

As described in the previous section, the most influential parameter on the tool wear is the cutting speed. According to the analytical model from [START_REF] Oxley | Mechanics of machining: an analytical approach to assessing machinability[END_REF], an elevation of the cutting speed leads to an highest strain rate and an highest chip velocity. Accordingly, the cutting temperature at the tool chip-interface increases. With a thermocouple placed inside the tool close to the cutting edge, Wagner and coworkers observe the tool temperature [START_REF] Wagner | Experimental study of coated carbide tools behaviour: Application for ti-5-5-5-3 turning[END_REF]. They note an increase of the cutting temperatures with highest cutting speeds. Consequently, the tool life is affected by the cutting speed explains by the highest temperatures at the tool chip interface which induces the diffusion phenomena. There is a linear relation between the cutting forces and the uncut chip thickness. An increase of feed generates some highest cutting forces and some highest cutting temperatures and generates an highest diffusion. Moreover, the feed rate influences the mechanical impact between the insert and the workpiece material and may induce a shorter of tool life and generates some thermal cracks.

Influence of radial depth

The influence of the radial depth on the tool life has been also tested. The different works about titanium milling recommend several techniques to increase tool life. For example, the contouring operation must be made in down milling where the cutting tool is fed with the direction of rotation. In up milling, the tool is rotating at the opposite direction of the feed. Consequently, the instantaneous uncut chip thickness starts lower to increase to maximal uncut chip thickness. The edge radius mainly forms the chip and the cutting temperatures generate by rubbing may be important. In down milling, the chip thickness starts bigger and a mechanical impact between insert and material occurs. However, this technic allows having a longer tool life because the cutting temperatures and the cutting forces are reduced. Several tool manufacturers purpose to use a radial depth lower than 30 % of tool radius generating a reduction of engagement curve. But the relation between radial depth and tool life for Ti-1023 milling with toroidal tool is still unknown. Some tests were performed with four radial depths (ae = 8.5, 12.5, 17.5, and 25 mm). Figure 18 shows the influence of radial depth in tool life and tool performances. An increasing of radial depth reduces the tool life and the cutting length while radial depth is lower than the tool radius (12.5 mm). For the highest values, the same tool life could be achieved (T=5.4 min). The difference appears especially in volume and in material removal rate. The volume increases with the rising of radial depth. This difference is explained by the highest cross section with highest radial depth (a e = 8.5 mm: S = 42.5 mm 2 ; a e = 12.5 mm : S = 42.5 mm 2 ; a e = 17 mm : S = 60.5 mm 2 ; a e = 25 mm : S = 60.5 mm 2 ). In considering only the tool life, the best condition is ae = 8.5 mm where the highest values are observed. However, in the tool performance viewpoint, the better condition is ae = 25mm. In this case, a maximal volume is removed and with the highest material removal rate.

The better performance of tool when radial depth of cut is higher than tool radius is due to the evolution of cross chip section. However, an highest radial depth can modify the cutting process and explained the identical tool life gotten with ae = 17 and 25 mm. In milling operation, a cyclic variation due to the interrupted cutting generates some stresses and temperatures. A relation between thermal crack apparition and the cyclical variations of the cutting temperature in milling is pointed by [START_REF] Palmai | Cutting temperature in intermittent cutting[END_REF][START_REF] Ferrasi | Metal cutting[END_REF]. The thermal cycle for a tool is defined for continuous and interrupted cutting (Fig. 19). The plot a represents the cutting temperature Fig. 19 Cyclic variation during cutting temperature for interrupted and continuous cutting [START_REF] Palmai | Cutting temperature in intermittent cutting[END_REF] during continuous cutting where the temperature is rapidly stabilized. For interrupted cutting as milling operations, the curves are drastically different. During the cutting time t1 where cutting tool is in contact with the workpiece material, the tool heats to T 1 temperature. The idle period begins at the end of t1 and, during the period t2, the tool cools to attain the temperature T ′ 1 at the end of time t1 + t2. The total cycle is repeated for each insert revolution. For different materials, some metal cracks occur in tool and results of cyclic variations in stress and temperature [32]. The idle period and the cutting period are affected by the cutting conditions, the workpiece material, the tool material properties (thermal and mechanical properties), and the type of cooling (lubricant, air, and cryogenic system). According to [32], this cyclic variation of temperature causes the thermal cracking due to the very high chemical reactivity of titanium alloy, it has tendency to weld to the cutting tool during machining which leads to chipping. This phenomenon is amplified with friction and high temperature near to cutting edge when insert penetrates the workpiece material.

In addition of this cyclic variation, some mechanical impact occurs between inserts and the workpiece material before each thermal and mechanical solicitation. The impact intensity is function of workpiece material, chip cross section but also function of the position of tool center compared to the machined surface. The relative position of tool modify also the length of cut and consequently the cutting and the idle periods. The last point noted by [START_REF] Wagner | Improvment of machinabiltiy of refractory titanium: the ti-5553[END_REF] is the impact position of tool edge.

Figure 20 represents the tool wear when radial depth is equal to 25 mm and when the radial depth is equal to 12.5 mm. The crater wear and the notch wear observed in the previous section do not appear. The tool wear is an edge chipping. Moreover, the spindle power measured when a e = 250mm is between 1090 and 2020W. Compared to lower radial depth (ae = 8.5mm, ae = 12.5mm), the spindle power is stable in despite of number of teeth in material higher.

It appears that the most important influential phenomenon on the tool life is the impact between the insert and the workpiece. Its intensity is function of the workpiece material properties but also of the chip cross section. The position of tool according to machined surface can modify the impact intensity. To reduce it, then it is necessary to decrease the radial engagement and therefore the cutting section. The most unfavorable configuration is a radial depth equal to the tool radius. The impact is then the more important and where the cyclic variation of temperature is negative. The relationship between the tool life and the volume of material removed is not optimal in this case. If the radial depth is higher than the tool radius, the cyclic variation of temperature is damaging. Moreover, the contact is made on a portion of the insert very close to the cutting edge and which is not the machining section. Consequently, the impact and the solicitations which occur during the cutting period appear on two sections of cutting edge. The last case is a radial depth equal to the tool diameter. The energy of impact is then limits but the thermal solicitations are most important. Indeed, the idle time and the cutting time are also same. The cutting tool material is then weakened and some chippings appear.

Influence of axial depth

The last parameter tested is the axial depth (Fig. 21). In previous section, the notch position depends of axial depth. Moreover, the notch width is the lowest when axial depth Fig. [START_REF] Manjunathaiah | A study of apparent negative rake angle and its effect on shear angle during orthogonal cutting with edge radiused tools[END_REF] Evolution of volume and material removal rate according to axial depth decreases. Compared to radial depth influence, a lowest axial depth expands the tool life. However, the gain of workpiece material removal rate is too low to grow the tool performance. Indeed, the volume decreases with an highest axial depth.

When axial depth decreases, the maximal uncut chip thickness is reduced. Consequently, the stresses and temperature on insert are lowest and the tool life rises.

Conclusion

This work presents a study of Ti-1023 milling with toroidal tool. In first section, the optimal cutting conditions for two tools have been defined. The tool material couple method describes in French standard has been used. The tests show a range of optimal cutting conditions in classical range find in common titanium machining. The chip formation and the chip morphology are also analyzed and show different phenomena on a same chip. In a second section, a tool wear analysis is made. The tool wear can be divided in two steps : a crater wear close to cutting edge and a notch wear at the axial depth limit. For industry, tool life is not the most important point. It seems important to define the best cutting condition and cutting engagement in order to increase productivity. Consequently, the last analysis is based on tool life but also material removal. Based on the results obtained in this works, the following conclusions can be drawn for milling:

-the relative position between the tool and the workpiece influences the tool life where the worst position is a radial depth equal to tool radius where impact between insert and workpiece is high and damage cutting edge appears, -to increase tool life, the radial depth must be the lowest according to minimal uncut chip thickness, -to raise the material removal rate, the radial depth must be highest than tool radius.

Fig. 3 Fig. 4

 34 Fig. 3 Cutting tool geometry for the tool 1 (a) and the tool 2 (b)

Fig. 5

 5 Fig.[START_REF] Gilles | Optimum orientation of a torus milling cutter: Method to balance the transversal cutting force[END_REF] Geometry and parameters used for uncut chip cross section[START_REF] Gilles | Optimum orientation of a torus milling cutter: Method to balance the transversal cutting force[END_REF]

Fig. 6

 6 Fig. 6 Evolution of specific cutting energy according to cutting speed

Fig. 8

 8 Fig. 8 Evolution of specific pressure according to feed rate for tool 1 (a) and tool 2 (b)

Fig. 10

 10 Fig. 10 Evolution of chip morphology according feed

Fig. 11 Fig. 12

 1112 Fig. 11 Evolution of chip formation for several feeds

Fig. 13

 13 Fig. 13 Chip formation for the highest uncut chip thickness (a) and for the lowest uncut chip thickness (b)

a p = 2 Fig

 2 Fig. 15 Tool wear criteria

Fig. 16

 16 Fig. 16 Evolution of tool wear, spindle power according to test number

Fig. 18 Fig. 20

 1820 Fig. 18 Influence of radial depth on tool life (Vc=70 m/min and fz=0,2 mm/dt)

Table 1

 1 Tools geometry and grade of tools Some contouring operations were carried out using two tool holders and two different inserts. The cutting tool geometry and the type of grade are presented in Table

	Description	Tool 1	Tool 2
	Commercial nuance ST40	CTC5240
	Coating	TiCn-Al2O3-TiN (CVD) TiN-TiB2 (CVD)
	Substrate	CW + Cobalt 21%	CW + Cobalt 10 %
	Insert rake angle	10 • C	10-20 • C
	Edge preparation	Honing 0.06 mm	Honing 0.07 mm
	3 Experimental setup

Table 2

 2 Evolution of tool wear for Tool2 (V c = 60m/min, f z = 0.3mm/tooth, a e = 12.5mm, a p = 4mm)

	Test 1	Test 2	Test 3
	Test 4	Test 5	Test 6
	Test 8	Test 8	Test 9
	Test 10	Test 11

Table 3

 3 Evolution of tool wear for Tool 1 (V

	Test 1	Test 2	Test 3
	Test 4	Test 5	Test 6

c = 60 m/min, f z = 0.2 mm/tooth, a e = 12.5 mm, a p = 4 mm)

Table 4

 4 Evolution of notch limit according to axial depth

Acknowledgments

The authors would like to express their thanks to Aubert & Duval for his support during this experimental work.


Imane Demnati

David Grossin

Christèle Combes

Christian Rey
email: christian.rey@ensiacet.fr

Plasma-Sprayed Apatite Coatings: Review of Physical-Chemical Characteristics and Their Biological Consequences

Keywords: Hydroxyapatite, Plasma-sprayed coatings, Thermal decomposition, Characterization

Hydroxyapatite (HA) is widely used as a coating on orthopedic medical implants to improve anchoring to bone tissue and facilitate bone repair. The use of plasma-sprayed HA coatings on metal implants has raised as much controversy as interest over the last decade. Although faster and stronger fixation and enhanced bone growth have been reported, the long-term performance of HA-coated implants has been questionned This ambivalent behaviour is related to the thermal decomposition of HA during the plasma spraying of HA powder, which has several observable consequences for the physical-chemical and biological properties of the coatings. This review presents the current knowledge on the characteristics of plasma-sprayed HA coatings and their biological consequences. Various schemes of chemical reactions and physical-chemical processes that can occur during HA-particle plasma spraying, HA-coating formation, or after implantation of HA coatings are presented. Standard and emerging methods developed for the characterization of HA coatings are reviewed, with a focus on Raman spectroscopy, which allows high-quality imaging. Finally, several promising trends that have been explored in recent years to improve the mechanical resistance of coatings and their adhesion on the substrate, to improve the coating microstructure and crystallinity, or to confer a therapeutic activity (e.g., antibacterial activity) on plasma-sprayed HA coatings are reviewed.

Introduction

Plasma spraying hydroxyapatite (HA) remains the most commonly used process for giving biological properties to orthopedic and dental metal implants, especially osteoconductive capabilities and efficient primary biointegration into bone tissue. Plasma spraying techniques have the ability to construct thick, adherent, and biologically active HA coatings [START_REF] Leeuwenburgh | Calcium phosphate coatings[END_REF][START_REF] Sun | Material fundamentals and clinical performance of plasma sprayed hydroxyapatite coatings: a review[END_REF][START_REF] De Groot | Plasma sprayed coatings of hydroxylapatite[END_REF][START_REF] Legeros | Calcium phosphates in plasma-sprayed HA coatings[END_REF][START_REF] Heimann | Thermal spraying of biomaterials[END_REF]. However, it is well established that HA decomposes during plasma spraying, leading to the formation of various phases [START_REF] Leeuwenburgh | Calcium phosphate coatings[END_REF][START_REF] Sun | Material fundamentals and clinical performance of plasma sprayed hydroxyapatite coatings: a review[END_REF][START_REF] De Groot | Plasma sprayed coatings of hydroxylapatite[END_REF][START_REF] Legeros | Calcium phosphates in plasma-sprayed HA coatings[END_REF][START_REF] Heimann | Thermal spraying of biomaterials[END_REF][START_REF] Ranz | Développement et caractérisation de dépôts d'apatite obtenus par projection plasma sur prothèses orthopédiques[END_REF][START_REF] Gross | Amorphous phase formation in plasma-sprayed hydroxyapatite coatings[END_REF][START_REF] Gross | Identification and mapping of the amorphous phase in plasma-sprayed hydroxyapatite coatings using scanning cathodoluminescence microscopy[END_REF][START_REF] Chou | Phase transformation during plasma spraying of hydroxyapatite-10-wt%-zirconia composite coating[END_REF][START_REF] Carayon | Study of the Ca/P atomic ratio of the amorphous phase in plasma-sprayed hydroxyapatite coatings[END_REF][START_REF] Demnati | Hydroxyapatite coating on titanium by a low energy plasma spraying mini-gun[END_REF][START_REF] Demnati | Développement et caractérisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie: application aux implants biomédicaux[END_REF][START_REF] Cheang | Addressing processing problems associated with plasma spraying of hydroxyapatite coatings[END_REF]. This decomposition has several drawbacks for the behavior of coatings after implantation, such as degradation, release of particulates, and delamination due to a loss of cohesion and poor metal-coating bonding. Plasma-sprayed coatings are usually composed of large crystalline HA particles embedded into a relatively soluble amorphous phase.

Several physical-chemical properties of plasma-sprayed HA coatings have been standardized with regard to their chemical composition (Ca/P ratio), phase composition, and crystallinity [START_REF]Implants for surgery-Hydroxyapatite-Chemical analysis and characterization of christallinity and phase purity[END_REF]. Most of these parameters are based on X-ray diffraction (XRD) analyzes. However, several additional parameters that are more difficult to determine, such as crystalline phase distribution within the coating, the amount of amorphous phase, and the oxyapatite (OA) fraction, are not considered in the standards although they could be related to the coating behavior in vitro and in vivo.

This review presents the current knowledge on the characteristics of plasma-sprayed HA coatings and their biological consequences. The reactions involved in the thermal decomposition of HA are summarized, and the consequences of these reactions for the coating characteristics are discussed. The biological consequences and recent developments are also reviewed.

Thermal decomposition of apatites

The thermal decomposition of HA occurs in two steps: an irreversible decomposition reaction in the range 200-800 °C and a reversible decomposition above 1000 °C.

Irreversible decomposition

Generally, HAs used in plasma spraying are prepared in aqueous media either by using a double decomposition reaction between soluble calcium and phosphate salts, for example ammonium phosphate and calcium nitrate, or by neutralization of a calcium hydroxide suspension with a phosphoric acid solution [START_REF] Cheang | Addressing processing problems associated with plasma spraying of hydroxyapatite coatings[END_REF]. After their synthesis, HAs may contain several impurities such as counter ions as well as HPO 4

2-and carbonate ions related to the weak dissociation constant of HPO 4 2-and contamination of the solutions with atmospheric CO 2 .

According to several studies, the chemical composition of such apatites can be represented as:

Ca 10-x-y+z (PO 4 ) 6-x-y (HPO 4 ) x (CO 3 ) y (OH) At temperatures between 850 and 1000 °C, the mineral phases coexisting with HA depend only on the Ca/P ratio of the apatite (Fig. 1).

Reversible high-temperature decomposition

Several phase diagrams have been provided for the CaO-P 2 O 5 system, with water vapor not considered in most cases [START_REF] Elliott | Structure and chemistry of the apatites and other calcium orthophosphates[END_REF][START_REF] Riboud | Composition et stabilité des apatites dans le système CaO-P2O5-oxyde de fer-H2O[END_REF]. However, it is well established that the decomposition of stoichiometric HA begins with dehydroxylation and the formation of OA related to the loss of water [START_REF] Trombe | Contribution à l'étude de la décomposition et de la réactivité de certaines apatites hydroxylées, carbonatées ou fluorées alcalino-terreuses[END_REF]: Ca 10 (PO 4 ) 6 (OH) Various values of this decomposition temperature, which increases with the water partial pressure, have been reported [START_REF] Riboud | Composition et stabilité des apatites dans le système CaO-P2O5-oxyde de fer-H2O[END_REF][START_REF] Gross | Oxyapatite in hydroxyapatite coatings[END_REF]. At higher temperature (1570 °C), a liquid phase, which is believed to correspond to a eutectic mixture is obtained. However some reports mention the existence of CaO resulting from the decomposition of HA within the liquid phase [START_REF] Heimann | Thermal spraying of biomaterials[END_REF].

Foreign phases in plasma-sprayed coatings

The plasma spraying processes used by several medical manufacturers consist of the acceleration and heating at very high temperatures of HA crystalline particles introduced in a plasma flame produced by an electrical discharge between electrodes. The partly (or totally) melted HA particles impact the substrate, where they rapidly cool, resulting in a coating microstructure with a lamellar structure. These thermal events lead to a partial decomposition of the apatite structure and several foreign phases are generally observed in the coatings after cooling (Table 1). The possible formations of these phases are shown in Table 1. Multiple thermal events have to be considered: the rapid heating of in-flight particles, the quenching of the HA particles upon impact with the target, and the partial re-heating and cooling in multiple torch passes. The first layer is different from subsequent ones since it is deposited on a metal (most generally titanium or one of its alloys), which has good thermal conductivity. The other layers impact thermally insulating splats that have already been deposited, and thus their quenching is much less efficient and is associated with local re-heating and slow cooling. The formation of some of the phases found in coatings can be easily explained by the thermal stability of apatite. The amorphous calcium phosphate (ACP) phase, which is generally the most abundant foreign phase found in coatings, is considered to originate from the fast quenching of the liquid phase upon reaching the target, as schematized in Fig. 2. This amorphous phase is more abundant in coatings made with one pass of the plasma torch (Fig. 2). The TTCP and TCP phases are considered to correspond to phases formed during the heating of the HA particles, inside the grain, in areas where the temperature is in the narrow gap between the maximum stability temperatures of the apatite phase and the liquid phase. Another possibility is that these phases are formed during the cooling period if the rate of cooling of the liquid fraction is slow enough to allow the nucleation of these phases. In coatings formed by multiple torch passes, re-heating and a slower cooling rate could favor the re-formation of apatite from the decomposition products, as it is the most stable phase, having a large domain of thermal stability.

OA is considered by most researchers to exist in solid solution with HA [START_REF] Riboud | Composition et stabilité des apatites dans le système CaO-P2O5-oxyde de fer-H2O[END_REF][START_REF] Trombe | Contribution à l'étude de la décomposition et de la réactivité de certaines apatites hydroxylées, carbonatées ou fluorées alcalino-terreuses[END_REF][START_REF] Gross | Oxyapatite in hydroxyapatite coatings[END_REF]. It likely exists in deeper zones of the in-flight particles where the temperature is lower (in the range 1050-1350 °C). It seems reasonable to consider that the OA content varies within the flying particles (Fig. 2). Rehydration of the OA fraction during cooling in air, which can be considered as re-hydroxylation, has been shown to be active at temperatures as low as 400 °C [START_REF] Trombe | Contribution à l'étude de la décomposition et de la réactivité de certaines apatites hydroxylées, carbonatées ou fluorées alcalino-terreuses[END_REF]. The formation of CaO, which is often observed in coatings rich in amorphous phase obtained at high power, remains, however, unexplained by phase diagrams. Some researchers consider that the liquid phase may in fact contain CaO. This assumption is supported by experimental data on the fast quenching of totally melted HA particles, which shows only an amorphous phase and CaO [START_REF] Ranz | Développement et caractérisation de dépôts d'apatite obtenus par projection plasma sur prothèses orthopédiques[END_REF]. In this case, however, one may wonder why CaO is not observed systematically in all coatings containing an amorphous phase but is only seen in some of them. Several hypotheses can be proposed: CaO forms only nanocrystals which are difficult to detect; CaO reacts with water and/or CO 2 from the atmosphere once the coating has been formed, giving poorly crystalline or amorphous phases; the solubility of CaO in the melted phosphate varies with temperature. CaO is the most refractory phase in coatings with a melting point of 2613 °C.

Most coatings obtained in argon plasma do not show any variation in their chemical composition and after plasma spraying their Ca/P ratio is the same as that of the initial apatite powder. However, very high powers and/or reductive plasmas, containing hydrogen, may induce a loss of phosphorus and are associated with an increase in the Ca/P ratio.

The global chemical phase composition and crystallinity of the coatings are determined, in most countries, according to ISO standards, based on the analysis of XRD patterns. However, certain characteristics of the coatings which have to be considered for their biological behavior have not been standardized, such as OA content, amorphous phase content (not only crystallinity, which is a different parameter), and phase distribution.

The most important parameter is certainly the amorphous phase content. Old standards (AFNOR S 94-068) distinguished between crystallinity and amorphous phase content. Although a large amount of amorphous phase affects the crystallinity of the coating, other parameters are also involved in the crystallinity, such as the re-formation of poorly crystalline apatite during cooling or re-heating in multiple torch passes and heterogeneous solid solution formation between OHAs and HAs, which increase the level of strain and contribute to the XRD background. The determination of the real amount of amorphous phase by XRD is rather tedious and imprecise; however, Raman spectroscopy allows a clear identification of the amorphous component, which is distinct from the crystalline apatite phase. The phase distribution is also crucial for the evolution of the coating after implantation.

Phase distribution in coatings

Plasma-sprayed coatings have been shown to be very inhomogeneous. Several methods have been proposed for evaluating the distribution of phases in coatings, including optical microscopy [START_REF] Gross | Amorphous phase formation in plasma-sprayed hydroxyapatite coatings[END_REF], scanning electron microscopy (SEM), backscattered electron imaging, and scanning cathodeluminescence microscopy [START_REF] Gross | Identification and mapping of the amorphous phase in plasma-sprayed hydroxyapatite coatings using scanning cathodoluminescence microscopy[END_REF]. A few studies have used microdiffraction [START_REF] Chou | Phase transformation during plasma spraying of hydroxyapatite-10-wt%-zirconia composite coating[END_REF]. The amorphous phase appears as the major foreign phase in most commercial HA coatings. It has been identified in close proximity to the metal surface, whose high thermal conductivity facilitates quenching of the molten phase. TCP and TTCP phase formation resulting from the decomposition of OA between 1050 °C and 1550 °C should occupy the same areas; however, these phases are generally present in very small amounts in commercial HA coatings and their distribution is not well known [START_REF] Carayon | Study of the Ca/P atomic ratio of the amorphous phase in plasma-sprayed hydroxyapatite coatings[END_REF]; in some coatings they do not appear at all. Several explanations can be given, such as the existence of supercooled liquid in the narrow window of stability of TCP and TTCP, which would favor the formation of OHA, whose nucleation would be facilitated by its abundance in the coating. The reformation of apatite from these phases, which should be in close proximity in multiple coatings, also seems likely [START_REF] Demnati | Hydroxyapatite coating on titanium by a low energy plasma spraying mini-gun[END_REF]. The OA fraction is also an important characteristic of the coatings, but as its XRD pattern is very close to that of apatite, it is very difficult to identify.

Information about the morphological changes at the surface and interface of coatings can be obtained from SEM observations, but a detailed structural chemical analysis of the coatings in these regions is still lacking. Raman scattering allows local investigations of the coating's heterogeneity. It is a unique tool providing information on the structure and shortrange order of amorphous and crystalline phases [START_REF] Yan | Characterization of chemical inhomogeneity in plasma-sprayed hydroxyapatite coatings[END_REF][START_REF] Podlesak | Advanced microstructural study of suspension plasma sprayed hydroxyapatite coatings[END_REF][START_REF] Weinlaender | Raman microprobe investigation of the calcium phosphate phases of three commercially available plasma-flame-sprayed hydroxyapatite-coated dental implants[END_REF]. Moreover, Raman imaging based on different strategies from a given dataset may provide a powerful means of studying structural changes and visualizing the distribution of phases within heterogeneous plasma-sprayed HA coatings. All mineral phases present in a coating can potentially be distinguished by Raman spectroscopy in the n 1 PO 4 domain. An example of micro-Raman spectroscopic imaging collected across an HA coating cross-section is shown in Fig. 3 The ACP is characterized by a broad Raman line at 950 cm -1 and all crystalline phases show lines distinct from that of apatite at 961 cm -1 (Fig. 4). OA, difficult to distinguish by other methods, shows very specific Raman scattering lines at about 951 cm -1 and 966 cm -1 , which have been assigned respectively to the vibration of phosphate ions close to a vacancy and close to an O 2-ion [START_REF] Trombe | Contribution à l'étude de la décomposition et de la réactivité de certaines apatites hydroxylées, carbonatées ou fluorées alcalino-terreuses[END_REF]. CaO is the only phase which cannot be detected in this domain. The most accurate mapping of the phases in HA coatings can be obtained by Raman imaging. Imaging of the phase distribution using Raman scattering involves curve fitting in the n 1 PO 4 domain, extraction of the characteristic line(s) associated with specific phases, and mapping of their relative amounts [START_REF] Demnati | Développement et caractérisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie: application aux implants biomédicaux[END_REF]. It should be noted that without standardization, the data represent only the relative integrated intensities of lines associated with a specific phase and their relative variation, not their content in the coating. Figure . 3 shows the distributions of the amorphous component (Fig. 3(a)), HA (Fig. 3(b)), and OA (Fig. 3(c)) in a cross section of a HA coating obtained using a low-energy plasma spraying device [START_REF] Demnati | Hydroxyapatite coating on titanium by a low energy plasma spraying mini-gun[END_REF]. The lamellar structure of the plasma-sprayed coating appears clearly on all these images; however, the limits of individual splats are difficult to distinguish. Heterogeneous structures much smaller than the particles are also observed, suggesting rupturing of grains and splitting on impact. The amorphous phase appears in close proximity to the metal surface but not directly on the surface, probably because the plastic deformation of the solidifying liquid phase on the impact results in a rejection of this phase to the periphery of the splat, whereas the solid core, containing mostly HA, reaches the surface. The domains rich in OA are relatively narrow and dispersed. Two types of OA-rich domain can be distinguished as those found within the amorphous phase, which may correspond in part to a recrystallization process, and those contiguous to HA domains, possibly formed by dehydration during particle flight. HA-rich domains can be found directly in contact with the metal. It is clear that HA-rich domains exist in zones with low amorphous content and inversely.

In addition to a heterogeneous distribution of phases in terms of thickness in the deposit cross section, there is also a heterogeneous distribution on the surface itself and very different Raman spectra are collected depending on the location (Fig. 5) [START_REF] Demnati | Développement et caractérisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie: application aux implants biomédicaux[END_REF]. Therefore, Raman micro-spectroscopy can be used as a routine non-destructive tool to obtain rapid analysis of the composition of apatite coatings and grains. This technique could be adapted for the chemical characterization of coatings applied directly on biomedical implants. Figure 5. Raman spectra at various points on the surface of a HA coating showing its heterogeneity.

The heterogeneities of plasma-sprayed coatings affect their biological behavior.

Physical-chemical properties of HA coatings and their effect on biological behavior

Several physical-chemical properties of the coatings affect biological behavior, namely the solubility of mineral phases, hydration characteristics, and propensity to hydrolyze or transform spontaneously into more stable compounds.

Solubility of mineral phases and biodegradation

The biodegradation of calcium phosphates has often been related to their solubility, and this relationship has been confirmed by experimental data [START_REF] Legeros | Biodegradation and bioresorption of calcium phosphate ceramics[END_REF]. According to their solubility product, the phases found in plasma-sprayed HA coatings can be arranged in order of decreasing biodegradability (Table 1). Among these phases, one may however distinguish those which should spontaneously dissolve, because their solubility product is higher than the ion product of body fluids, and those for which the dissolution is mediated by cellular activity. The phases belonging to the former type, namely CaO, ACP, and a-TCP, cannot remain as long as they are in contact with body fluids. Due to the spraying process, however, some phases can be embedded in the accumulated splats, where they are not accessible to body fluids or cells, allowing them to remain unaltered until cracks or dissolution paths reach them.

Hydration and hydrolysis reactions

Several phases formed at high temperature are not stable in aqueous media and may transform more or less rapidly into other compounds. This is the case of CaO, which hydrolyzes into Ca(OH) 2 . This reaction leads to a 40% increase in volume and may lead to a rupture of the coating. In addition, in the presence of phosphate ions from body fluids, a precipitate may form with the calcium ions released from the very soluble Ca(OH) 2 .

The amorphous phase can also hydrolyze into apatite. This reaction first involves a hydration of the amorphous phase [START_REF] Legeros | Biodegradation and bioresorption of calcium phosphate ceramics[END_REF]. It has been suggested that this hydration process could also lead to fracture of the coating [START_REF] Somrani | Thermal evolution of amorphous tricalcium phosphate[END_REF]. Once hydration has occurred, ACP transforms rather rapidly into apatite [START_REF] Weng | Environmentally induced cracks and detachment of plasma-sprayed hydroxyapatite coatings under ultrasonic treatment[END_REF]. Several investigations have shown that the ACP content decreases and the crystallinity increases after contact with body fluids or their model solutions (Ringer solution or simulated body fluid (SBF) solution) due to the hydrolysis of ACP into hydroxylated apatite.

a-TCP and TTCP hydrolyzes have also been observed and, like ACP, they lead to the formation of hydroxylated apatite [START_REF] Somrani | Physico-chemical and thermochemical studies of the hydrolytic conversion of amorphous tricalcium phosphate into apatite[END_REF][START_REF] Monma | Properties of hydroxyapatite prepared by the hydrolysis of tricalcium phosphate[END_REF]. These reactions can be rapid at body temperature (a few tens of minutes when the phases are in a powder state) and they are involved in the setting of several hydraulic mineral cements for orthopedic applications.

Although b-TCP has been shown to hydrolyze into apatite in aqueous media, the reaction at body temperature seems extremely slow and it is generally not mentioned; however, when considering months of implantation, it might be involved.

The transformation of OA into HA by hydrolysis requires the diffusion of water molecules into the apatite structure. Surface hydrolysis may be responsible; however, the diffusion of water molecules inside the lattice at body temperature would be much too slow. Another possibility is a dissolutionreprecipitation process, as OA has been claimed to be more soluble than HA [START_REF] Gross | Oxyapatite in hydroxyapatite coatings[END_REF]. However, very often, no accurate phase analysis of coatings is conducted and thus this question remains open. These reactions and their effects on coatings are summarized in Fig. 6. Figure 6. Scheme of reactions at the coating aqueous medium interface.

Biological role and biological consequences of HA coating composition

The main role of a coating is to improve bone integration and the stability of the interface between the implant and the tissue. This role is essential in the first months after implantation, as most failures of implants are related to poor stability in the biological tissue. However, as bone remodels continuously, it seems important for some biological activity to occur on the implant surface during its life span. Several studies have shown the positive effect of coatings on bone integration during the first months after implantation [START_REF] Coathup | A comparison of bone remodelling around hydroxyapatite-coated, porous-coated and grit-blasted hip replacements retrieved at post-mortem[END_REF][START_REF] Aebli | In vivo comparison of osseointegration of vacuum plasma sprayed titanium-and hydroxyapatite-coated implants[END_REF][START_REF] Moroni | Histomorphometry of hydroxyapatite coated and uncoated porous titanium bone implants[END_REF]. In the long term, however, the difference between coated and uncoated implants generally tends to diminish and coated implants have sometimes been reported to deteriorate [START_REF] Porter | Bone bonding to hydroxyapatite and titanium surfaces on femoral stems retrieved from human subjects at autopsy[END_REF].

The preferential dissolution of foreign mineral phases weakens coating integrity and cohesion. The events associated with mechanical stress may lead to delamination and scaling and to the release of particles [START_REF] Porter | Bone bonding to hydroxyapatite and titanium surfaces on femoral stems retrieved from human subjects at autopsy[END_REF][START_REF] Bauer | Hydroxyapatite-coated femoral stems-Histological analysis of components retrieved at autopsy[END_REF][START_REF] Denissen | Behavior of calcium phosphate coatings with different chemistries in bone[END_REF]. Potentially, such a release of small particulates may lead to inflammatory reactions, which do not seem to have been observed in most cases. Nevertheless, the actual tendency is to prefer coatings with a high crystallinity to avoid weakening by dissolution of the heterogeneously distributed amorphous phase. Another precaution is to ensure that either very large particles or small degradable particles, which would have little inflammatory power, are generated during degradation.

However, the presence of hydrolyzable phases has also been seen as an advantage as it allows the formation of a neoformed layer of carbonated apatite analogous to the mineral phase of bone. The development of such layers is considered an essential step in the colonization of the implant surface by osteoblast cells and bone reconstruction. Another cellular effect of the hydrolysis reaction, especially in the case of CaO and TTCP, is the local alkalinization of the surrounding media, which could inhibit osteoclast activity.

Most studies have shown that a weakness of plasmasprayed coatings is the attachment to the metal surface. Although chemical reactions with metal oxide layers cannot be excluded, their role in the adherence of the coating seems marginal and the main parameters to consider are the surface roughness and the thickness and composition of the coating, as discussed in previous studies [START_REF] Sun | Material fundamentals and clinical performance of plasma sprayed hydroxyapatite coatings: a review[END_REF][START_REF] Cheang | Addressing processing problems associated with plasma spraying of hydroxyapatite coatings[END_REF]. This mechanical weakness plays a role in degradation, as underlayers rich in a soluble amorphous phase can be uncovered. A second problem with HA-coated dental implants is the possible promotion of bone infection. Recent developments have addressed some of these drawbacks.

Recent developments

Three types of development can be distinguished: that aimed at improving the mechanical properties and adhesion to the substrate, that claiming a therapeutic activity, and that aimed at improving microstructure and crystallinity.

Mechanical properties and adhesion

Several approaches have been made to improve the mechanical resistance of coatings and their adhesion to the substrate. The most successful incorporate ZrO 2 [START_REF] Chang | Plasma spraying of zirconia-reinforced hydroxyapatite composite coatings on titanium: Part I Phase, microstructure and bonding strength[END_REF] or other ceramic oxides and use graded co-spraying of HA and titanium [START_REF] Chen | Characterization of functionally graded hydroxyapatite/ titanium composite coatings plasma-sprayed on Ti alloys[END_REF] or titanium oxide. In these coatings, the proportion of apatite co-deposited increases progressively from the metallic substrate to the surface. However, the implantation of coatings with ZrO 2 has led to poor biological results with lower adhesion of bone to the coating [START_REF] Lee | Comparison of plasma-sprayed hydroxyapatite coatings and zirconia-reinforced hydroxyapatite composite coatings: in vivo study[END_REF]. Other coatings have apparently not been tested in vivo. There are still improvements to be made in this domain.

Coatings with therapeutic activity

The second field of development concerns the biological activity of coatings. Two approaches have been proposed: the association of coatings with growth factors or active molecules and the use of active mineral ions. The first one is rather limited as the surface area of plasma-sprayed HA coatings is low. The second one seems more promising and several biologically active elements or their combinations have been proposed: Ag + , Cu 2+ , and Zn 2+ ions for their antibacterial properties, Sr 2+ for its anti-osteoporotic activity, and silicon oxide to improve bone formation.

Improvement of microstructure and crystallinity

The third type of development is more specifically related to the physical-chemical properties of the coatings and their biological consequences. Increases in coating crystallinity may result from the choice of plasma-spraying conditions and several reports have been made. The use of mini-or even micro-torches working at low energy has been promoted [START_REF] Demnati | Développement et caractérisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie: application aux implants biomédicaux[END_REF]. These systems need smaller particulates but they produce coatings with high crystallinity and have several other advantages such as better homogeneity of composition within the particle and within the coating and thus a lower amount of metastable phases along with better control of coating microporosity. The use of phases other than HAs, especially fluorapatites, fluorhydroxyapatites, and more recently chlorapatites [START_REF] Demnati | Développement et caractérisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie: application aux implants biomédicaux[END_REF], has also been proposed. Unlike HA, fluor-and chlor-apatites melt without decomposition. Although some of these coatings have been tested with some success, they have not yet been commercially applied. The improvement of crystallinity can also be obtained by post-treatments: thermal and hydrothermal treatments and treatments in solutions have been applied [START_REF] Yang | In vitro and in vivo biological responses of plasmasprayed hydroxyapatite coatings with posthydrothermal treatment[END_REF]. There is however a risk of deterioration of the mechanical integrity of the coating.

Conclusion

Plasma-sprayed coatings are largely used in many countries. Since the first experiments, considerable progress has been made, especially in the understanding of the characteristics of coatings and their biological behavior. Several means of improvement have been explored; however, the problems of coating adhesion and mechanical strength remain. Other coating processes have been investigated, especially low-temperature coating, which offers more flexibility concerning, in particular, the types of apatite used (e.g., non-stoichiometric, biomimetic, and carbonated apatites), but without dethroning, at present, plasma spraying.

Figure 1 .

 1 Figure 1. Thermal decomposition of apatite. Reversible and irreversible reactions as functions of temperature.

Figure 2 .

 2 Figure 2. Sequence of events during deposition of a semi-molten HA particle impinging on a roughened substrate.

Figure 3 .Figure 4 .

 34 Figure 3. Representation of micro-Raman spectroscopic images of a HA coating cross section showing the phases distributions: ACP (a); HA (b) and OA (c).

  2 « Ca 10 (PO 4 ) 6 O + H 2 O This reaction is reversible and depends on the water vapor pressure. It seems to begin at about 850 °C in vacuum. Pure OA is very difficult to obtain and generally the prepared samples contain residual OH -ions. Between 1050 °C and Ca 10 (PO 4 ) 6 O « Ca 3 (PO 4 ) 2 + Ca 4 (PO 4 ) 2 O

	1550 °C, oxy-hydroxyapatite (OHA) begins to decompose into
	tricalcium phosphate (TCP) and tetracalcium phosphate
	(TTCP):

Table 1 .

 1 Phases found in plasma-sprayed HA coating and solubility products of various calcium phosphates.

	Phase	Formula	Formation	-log Ksp at 25°C
			Decomposition
	Calcium oxide	CaO	of HA and TTCP (volatilization of	-
			P)
			Reaction of CaO
	Calcium hydroxide		with humidity
	and calcium	Ca(OH)2	and CO2 (on	-
	carbonate		contact with air or
			body fluids)
	Amorphous calcium phosphate (ACP)	Variable	Melting of HA and quenching	-

Acknowledgments

Part of the work reported in this review was carried out within the framework of a European Community funded project, Brite-Euram II, and a Midi-Pyrénées region MNT ERA-Net funded project, NANOMED2.


Pierre Donat-Bouillud
email: pierre.donat-bouillud@ens-rennes.fr

Florent Jacquemard
email: florent.jacquemard@inria.fr

Masahiko Sakai
email: sakai@is.nagoya-u.ac.jp

Towards an Equational Theory of Rhythm Notation

Trees are classical representations of hierarchical structures in symbolic music, in particular for rhythm notations, where the durations are defined by a hierarchy of subdivisions; see e.g. [START_REF] Rizo | Symbolic music comparison with tree data structures[END_REF] chapter 3 for a survey. Structures called rhythm trees have been integrated since a long time into CAC environments such as Patchwork and OpenMusic [START_REF] Agon | Representation and rendering of rhythm structures[END_REF][START_REF] Bresson | OpenMusic: visual programming environment for music composition, analysis and research[END_REF], for programming rhythmic objects.

Term rewriting [START_REF] Baader | Term Rewriting and All That[END_REF] and tree automata and transducers [START_REF] Comon | Tree Automata Techniques and Applications[END_REF] are well established formalisms for transforming and reasoning on trees. With solid theoretical foundations, they are used in a wide range of applications including automatic reasoning, natural language processing, and foundations of web data processing. In this work, we consider a tree structured representation of rhythm suitable for defining a set of rewrite rules (i.e. oriented equations) preserving rhythms, while enabling the simplification of notations. This set can be seen as an axiomatization of rhythm notation which can be applied to reasoning on equivalent notations in assisted composition.

Ranked Tree Representation of Rhythm. We propose a tree-structured representation of rhythms based on a finite signature made of ranked symbols. Let us assume given a countable set of variables X , and a finite set containing the first prime integers P = {2, 3, 5, 7, . . .}. A tree t is either a single node labeled with a variable x ∈ X or one of the following constant symbols n, r, s, 1, or it is made of one root node labeled with p ∈ P and p subtrees t 1 , . . . , t p . In the latter case, t is denoted p(t 1 , . . . , t p ). For 1 ≤ i ≤ p, the previous cousin of t i is either t i-1 if i > 1, or the last children of the previous cousin of t if i = 1 and if this tree exists. In other terms, it is the subtree whose root is the node immediately at left of the root of t i , at the same level.

Intuitively, n represents a note, r a rest, p(t 1 , . . . , t p ) a p-tuplet and the symbols s (as slur) and 1 are used to sum up durations. Formally, we associate a duration value to every tree t as follows. If t is not a subtree then we assume that is has been associated a duration which can be e.g. one beat (Figure 1), several beats or a whole bar (the actual value of this duration is arbitrary and not relevant for our purpose). If t is the subtree t i in t 0 = p(t 1 , . . . , t p ) (with 1 ≤ i ≤ p) and t 0 has been associated the duration d 0 , then the duration d associated to t is d 0 /p plus the duration associated to its previous cousin if it exists and is labelled by 1, and the duration associated to the previous leaf in depth-first traversal (dft) if t i = s.

A tree represents a sequence of durations associated to some of its leafs. Formally, let us call a leaf principal if it is not labeled with 1 and if the next leaf wrt dft is not labeled with s. The rhythmic value of a tree is the sequence of the durations associated to its principal leaves, enumerated in dft.

This tree format has similarities with the rhythm trees (RT) of [START_REF] Agon | Representation and rendering of rhythm structures[END_REF] but there are however important differences. In particular, although RT represent durations with integers labeling nodes, we only use the tree structure (the labels in P are not needed) and labels in a finite (and small) set for leaves. This make this representation more amenable to purely syntactic processing when RT needs arithmetic. This permits also to enforce certain subdivisions in rhythms, following the analogous of a schema for XML data defined e.g. according to the metre or user's constraints. Rewrite Rules. We define rewrite rules which, applied to trees, do not change the rhythmic values. They can therefore be used to characterize equivalent notations of the same rhythm.

The application of a rule of the form ℓ → r to a tree t works by replacing by r subtrees of t matching ℓ. Some rules apply to successive cousins (the cousin relation begin denoted by ;) like: 1; n → n; s, 1; s → s; s, 1; r → r; r, r; s → r; r, and the normalization rules, with variables: 1; 2(x 1 , x 2 ) → x 1 ; x 2 , and 1; 1; 3(x 1 , x 2 , x 3 ) → x 1 ; x 2 ; x 3 etc. Other more standard rewrite rules [START_REF] Baader | Term Rewriting and All That[END_REF] are applied to subtrees, such as 2(r, r) → r, 3(r, r, r) → r, etc, 2(s, s) → s, 3(s, s, s) → s, etc and 2(n, s) → n, 3(n, s, s) → n, etc. We also consider rules for redefining subdivisions:

2(x 1 , x 2 ) → 3(2(1, 1), 2(x 1 , 1), 2(1, x 2 )), 2(x 1 , x 2 ) → 5(2(1, 1), 2(1, 1), 2(x 1 , 1), 2(1, 1), 2(1, x 2 )), . . . , 3(x 1 , x 2 , x 3 ) → 2(3(1, x 1 , 1), 3(x 2 , 1, x 3 )),. . .
Applying the above rules to the tree of Figure 1(d), we obtain the following rewrite sequence (the positions of application of rewrite rules are marked by circles). Some additional symbols may be used for dealing with notation details regarding e.g. dots, tuplet beaming, ratio notations for tuplets (n : m meaning n in the time of m) [START_REF] Gould | Behind Bars: The Definitive Guide to Music Notation[END_REF]. Our tree representation has been integrated in OpenMusic [START_REF] Bresson | OpenMusic: visual programming environment for music composition, analysis and research[END_REF] with conversion functions to and from RT of [START_REF] Agon | Representation and rendering of rhythm structures[END_REF]. We are also planning conversion to GUIDO and MusicXML.

Duration values are well defined only for trees satisfying constraints like: every 1 must be the previous cousin of some subtree, successive 1-labeled cousins must be associated the same duration. The subset of such trees is a context-free tree language [START_REF] Comon | Tree Automata Techniques and Applications[END_REF]. An interesting research problem is whether this language is invariant under application of the rewrite rules.

The above rewrite sequence can be seen as a notation simplification for a given rhythm. In a context of assisted composition (CAC), this approach can permit suggesting a user various notations of the same rhythmic value, with different complexities. Using some standard measures for tree structures (involving depth, number of symbols...) enables to quantify this process. Some other research questions are concerned with divergence in the application of the rewrite rules, the definition of appropriate rewrite strategies, canonical forms for rhythm notations and the decision of equivalence (i.e. equality of rhythmic values) of rhythm notations.

Fig. 1 .

 1 Fig. 1. Some trees with their corresponding rhythm notation and rhythmic values.

  Fig.1(d)