A Ramarolahy

P Castany

F Prima

P Laheurte

I Péron

T Gloriant

Microstructure and mechanical behavior of superelastic Ti-24Nb-0.5O and Ti-24Nb-0.5N biomedical alloys

Keywords: Titanium alloys, Superelasticity, Stress induced martensite, Deformation twinning

In this study, the microstructure and the mechanical properties of two new biocompatible superelastic alloys, Ti-24Nb-0.5O and Ti-24Nb-0.5N (at.%), were investigated. Special attention was focused on the role of O and N addition on α ″ formation, supereleastic recovery and mechanical strength by comparison with the Ti-24Nb and Ti-26Nb (at.%) alloy compositions taken as references. Microstructures were characterized by optical microscopy, X-ray diffraction and transmission electron microscopy before and after deformation. The mechanical properties and the superelastic behavior were evaluated by conventional and cyclic tensile tests. High tensile strength, low Young's modulus, rather high superelastic recovery and excellent ductility were observed for both superelastic Ti-24Nb-0.5O and Ti-24Nb-0.5N alloys. Deformation twinning was shown to accommodate the plastic deformation in these alloys and only the {332} 113 twinning system was observed to be activated by electron backscattered diffraction analyses.

Introduction

Titanium alloys are widely used as biomaterials due to their low density, good mechanical properties and biocompatibility [START_REF] Geetha | Ti based biomaterials, the ultimate choice for orthopaedic implants -A review[END_REF]. A standard titanium biomaterial for orthopedic implants is the Ti-6Al-4V alloy, which was first developed for aeronautic applications. However, a perfectly biocompatible Ti-based alloy must be composed of noncytotoxic elements. In Ti-6Al-4V, aluminium ions released from the implant are associated with long term health problems like neurological pathologies while vanadium oxide, V 2 O 5 , is also known to be cytotoxic [START_REF] Eisenbarth | Biocompatibility of bstabilizing elements of titanium alloys[END_REF][START_REF] Williams | On the mechanisms of biocompatibility[END_REF]. Moreover, the high Young's modulus of the Ti-6Al-4V alloy caused by the α+β microstructure affects long-term practical performance due to the stress shielding effect [START_REF] Huiskes | The relationship between stress shielding and bone-resorption around total hip stems and the effects of flexible materials[END_REF][START_REF] Niinomi | Development of low rigidity β-type titanium alloy for biomedical applications[END_REF]. Since the β microstructure in titanium alloys exhibits a significantly lower modulus than the α+β microstructure, there is a tremendous scope for improvement in terms of alloy design for an ideal orthopedic implant. Consequently, a new generation of low modulus β-titanium alloys, which are free of vanadium and aluminium, have been made with β-stabilizing and biocompatible elements such as Nb, Ta and Zr [START_REF] Laheurte | Mechanical properties of low modulus beta titanium alloys designed from the electronic approach[END_REF][START_REF] Niinomi | Mechanical biocompatibilities of titanium alloys for biomedical applications[END_REF].

Among metastable β-titanium alloys, biocompatible Ti-Nb based alloys are very promising for biomedical applications. Indeed, these alloys possess a low elastic modulus closer to that of bone, which contributes to avoiding the harmful stress shielding phenomenon. On the other hand, promising shape memory effect and superelastic behavior can be achieved due to the α ″ martensitic transformation (orthorhombic phase) occurring in these alloys. Consequently, Nifree alloys can also be envisaged for biomedical smart devices such as coronary stents, orthopedic staples or orthodontic wires [START_REF] Miyazaki | Development and characterization of Ni-free Tibase shape memory and superelastic alloys[END_REF][START_REF] Sun | Influence of a short thermal treatment on the superelastic properties of a titanium-based alloy[END_REF].

In the Ti-Nb binary system, it was reported that Ti-(22-25)Nb (at.%) alloys exhibit a shape memory effect while a superelastic behavior is observed in Ti-(25.5-27)Nb alloys [START_REF] Kim | Mechanical properties and shape memory behavior of Ti-Nb alloys[END_REF][START_REF] Kim | Martensitic transformation, shape memory effect and superelasticity of Ti-Nb binary alloys[END_REF]. For the specific Ti-24Nb composition, the α ″ martensitic transformation start (M s ) was estimated to be at 338 K, which is consistent with the observation of a fully α ″ microstructure at room temperature after water quenching from the β phase domain. It is well known that the superelastic behavior in β-titanium alloys results from the reversible stress-induced α ″ martensitic transformation (SIM) from the parent β phase.

Consequently, an initial fully β microstructure is needed to obtain a superelastic effect. This is typically the case for the Ti-26Nb alloy composition for which the critical stress inducing the α ″ martensitic transformation from β was observed at about 170 MPa by tensile testing [START_REF] Kim | Mechanical properties and shape memory behavior of Ti-Nb alloys[END_REF].

In Ti-Nb alloys, formation of as-quenched α ″ martensite can be inhibited by the addition of oxygen [START_REF] Kim | Shape memory behavior of Ti-22Nb-(0.5-2.0)O (at%) biomedical alloys[END_REF]. For example, it was shown that addition of 1 at.% of O decreases Ms by about 160 K in the Ti-22Nb alloy composition. Furthermore, it was also recently shown that the formation of as-quenched α ″ martensite is suppressed by adding oxygen in Ti-Nb-Ta-Zr (TNTZ) alloys [START_REF] Besse | Mechanisms of deformation in Gum Metal TNTZ-O and TNTZ titanium alloys: a comparative study on the oxygen influence[END_REF]. Similarly, formation of as-quenched α ″ martensite was suppressed by addition of nitrogen in Ti-(18-24)Nb-4Zr-2Ta (at.%) [START_REF] Tahara | Effect of nitrogen addition and annealing temperature on superelastic properties of Ti-Nb-Zr-Ta alloys[END_REF].

The objective of this work is to investigate the microstructure and the mechanical properties of two new biocompatible superelastic alloys containing oxygen and nitrogen: Ti-24Nb-0.5O and Ti-24Nb-0.5N (at.%). Special attention will be focused on the role of O and N addition on α ″ formation, supereleastic recovery and tensile strength by comparison with the Ti-24Nb and Ti-26Nb (at.%) alloy compositions taken as references.

In superelastic Ti-Nb based titanium alloys, large ductility is often observed due to various mechanisms of plastic deformation that can be activated depending on the composition and thus on the stability of the β phase. Indeed, in β-type alloys, plastic deformation can be accommodated by both dislocation slip and twinning [START_REF] Hanada | Transmission electron-microscopic observations of mechanical twinning in metastable beta titanium alloys[END_REF][START_REF] Furuhara | Crystallography of alpha precipitated on dislocations and deformation twin boundaries in a beta titanium alloy[END_REF][START_REF] Bertrand | Twinning system selection in a metastable beta-titanium alloy by Schmid factor analysis[END_REF]. In this study, deformation twinning of the two superelastic Ti-24Nb-0.5X (X = O and N) alloys is more particularly investigated.

Experimental

Ti-24Nb, Ti-26Nb, Ti-24Nb-0.5O and Ti-24Nb-0.5N alloy compositions (at.%) were synthesized by the cold crucible levitation melting technique (CCLM) under high vacuum in this study. For the synthesis, ultra-pure raw metals, 99.999% pure titanium and 99.9% pure niobium were used, and oxygen and nitrogen were added by using 99.8% pure TiO 2 powder and 99.5% pure TiN powder, respectively. The CCLM process, which limits the contact between the melt and the cold copper crucible, was shown to be an efficient method to melt biocompatible Ti-based alloys possessing a high melting point and to obtain uniform chemical composition without contamination [START_REF] Bertrand | Synthesis and characterisation of a new superelastic Ti-25Ta-25Nb biomedical alloy[END_REF].

Ingots obtained from melting were next homogenized at 1223 K for 72 ks under high vacuum, followed by water quenching. The ingots were then cold rolled at room temperature until a 90% reduction in thickness was achieved. From this cold-rolled state, tensile specimens were machined to obtain normalized shapes: 3 mm width, 0.5 mm in thickness and a gage length of 15 mm. The specimens were finally solution-treated in the β-phase domain at 1123 K for 1.8 ks under vacuum in order to restore a fully recrystallized microstructure from the coldrolled state after water quenching.

The microstructures of the solution treated alloys were characterized by X-ray diffraction (XRD) with a Philips PW3710 diffractometer using Cu Kα 1 radiation (λ=0.154060 nm) and observed by optical microscopy (Leica, OM). Before observations, the samples were first mechanically polished on silicon carbide abrasive papers followed by a final polishing step with a colloidal silica suspension (particle size: 50 nm) and next chemically etched with a 5% HNO 3 , 5% HF and 90% H 2 O solution (vol.%).

The mechanical properties were evaluated from conventional tensile tests until rupture carried out on an Instron 3369 machine with a strain rate of 10 -4 s -1 . An extensometer was used to precisely measure the deformation of the specimens. The superelastic behavior was characterized from cyclic tensile tests with strain increments of 0.5% followed by stress release up to an elongation of 5%. All tensile tests were performed with the tensile direction parallel to the rolling direction.

The microstructures after deformation of both Ti-24Nb-0.5O and Ti-24Nb-0.5N alloys were observed by electron backscattered diffraction (EBSD) in a scanning electron microscope (SEM) and by transmission electron microscopy (TEM). These observations were carried out on tensile specimens after being strained up to 6%. SEM-EBSD analyses were performed in a Jeol JSM 6400 microscope equipped with a TSL EBSD system. Surface samples were prepared by a similar route to the samples for optical microscopy. TEM observations were made using a JEOL 2000FX microscope operating at 200 kV. Specimens for TEM observations were prepared by a conventional twin-jet polishing technique.

Results and discussion

Microstructure

Fig. 1 shows typical optical micrographs displaying the microstructures of the four solutiontreated alloys investigated in this study. The Ti-24Nb alloy (a) seems to present a dual phase microstructure, composed of equiaxed β grains with thin α ″ martensite needles inside. On the contrary, the Ti-26Nb (b), Ti-24Nb-0.5O (c) and Ti-24Nb-0.5N (d) alloys seems to possess a fully β-grain microstructure with a grain dimension varying between a few tens and a few hundreds of micrometers. X-ray diffraction analyses carried out on each alloy are presented in Fig. 2. The XRD pattern of the Ti-24Nb alloy confirms the presence of the selfaccommodated α ″ martensite (a), which composed the majority of the microstructure.

However, a small amount of residual parent β phase was detected according to the relative intensities of the peaks of both phases. As only residual β phase is detected, the equiaxed microstructure observed by optical microscopy is thus the signature of the high temperature β phase microstructure due to the solution treatment, which was done in the β domain before quenching. From this XRD pattern, each α ″ peak could be clearly identified and the cell parameters of this C-centered orthorhombic phase (space group: Cmcm) were determined and are reported in Table 1. The XRD patterns obtained from Ti-26Nb (b), Ti-24Nb-0.5O (c) and Ti-24Nb-0.5N (d) alloys show that these alloys present a β-phase microstructure with, however, a very small amount of residual α ″ martensite phase in Ti-24Nb-0.5N. For the latter, a fully β microstructure can be considered but the martensite start is probably close to the room temperature. From the XRD patterns, the cell parameter for each body-centered cubic β-phase (space group: Im3m) was calculated and the values are presented in Table 1.

This study clearly shows that the addition of O or N inhibits the formation of the selfaccommodated α ″ martensite during quenching. This effect was previously mentioned in binary Ti-Nb alloys when O is added [START_REF] Kim | Shape memory behavior of Ti-22Nb-(0.5-2.0)O (at%) biomedical alloys[END_REF] and this study shows a similar behavior with N addition. It is also worth noting that nitrogen is a weaker inhibitor of the α ″ formation than oxygen. A small addition of O or N (0.5 at.%) is thus shown to be sufficient to modify the microstructure of the Ti-24Nb alloy from a near fully α ″ microstructure into a fully β microstructure.

Tensile tests

Conventional and cyclic tensile tests were performed at room temperature to characterize the mechanical properties of the four alloys. The conventional tensile curves until rupture are presented in Fig. 3 and the cyclic tensile curves between 0% and 5% strain are presented in Fig. 4. All the mechanical characteristics determined in this study from tensile tests are reported in Table 2. In this table we also report the mechanical characteristics of the Ti-6Al-4V ELI alloy used in orthopedics (http://www.aerospacemetals.com/titanium.html-10/2011) and those reported from the literature concerning the solution-treated superelastic binary Ti-26Nb alloy [START_REF] Kim | Mechanical properties and shape memory behavior of Ti-Nb alloys[END_REF].

The conventional tensile curves (Fig. 3) show an increase of both yield strength (black arrows) and ultimate tensile strength (white arrows) with the addition of O or N in comparison with both binary Ti-24Nb and Ti-26Nb alloys. This strengthening is due to the fact that both O and N, as interstitial elements, are well known to involve a solid-solution hardening effect in alloys. As seen in Table 2, the yield and ultimate tensile strengths of both Ti-24Nb-0.5O and Ti-24Nb-0.5N alloys are considerably increased in comparison with the superelastic binary Ti-26Nb. The mechanical strength is almost two times higher in the case of the Ti-24Nb-0.5O alloy and reaches a similar order of magnitude in comparison with the orthopedic Ti-6Al-4V ELI alloy.

O and N are also usually known to decrease the ductility of alloys but in the present case the ductility seems not to be altered by the addition of interstitial elements by comparison with the binary Ti-26Nb alloy. In the case of the Ti-24Nb-0.5O alloy, the measured elongation at rupture (22%) exceeds even that of Ti-6Al-4V.

The Ti-24Nb, Ti-26Nb and Ti-24Nb-0.5N stress-strain curves show a double yielding behavior. As self-accommo-dated α ″ martensite is present after quenching in Ti-24Nb, the plateau on the stress-strain curve of this alloy is mainly due to the reorientation of martensite variants. Concerning the Ti-26Nb and Ti-24Nb-0.5N alloys, as the microstructure is fully β, the plateau is due to the formation of α ″ stress-induced martensite (SIM) during straining. A similar plateau induced by SIM transformation is also observed in the superelastic β Ti-26Nb alloy elaborated by [START_REF] Kim | Mechanical properties and shape memory behavior of Ti-Nb alloys[END_REF]. On the contrary, the stress-strain curve of Ti-24Nb-0.5O does not possess such a plateau.

Oxygen is observed to possess a higher strengthening effect than nitrogen in this study. In fact, the strengthening is also dependent on the microstructure, as stress-induced martensitic transformation occurs in superelastic alloys. As previously mentioned, nitrogen was shown to be a weaker inhibitor of α ″ formation than oxygen. Consequently, it can be supposed that the α ″ formation intervening at lower stress in Ti-25Nb-0.5N limits the solid-solution hardening effect of N. The Young's modulus values (incipient modulus defined by the slope of the tangent at zero stress) were found to be very low in this study: 50, 54 and 43 GPa for Ti-26Nb, Ti-24Nb-0.5O and Ti-24Nb-0.5N, respectively. Thus, these alloys possess elastic modulii approaching that of bone (∼30 GPa) as is the case with many superelastic β-type Ti-based alloys.

Consequently, addition of interstitial elements seems not to change the Young's modulus significantly. With this alloy family, the elastic modulus is often observed to be more than twice as low as that of Ti-6Al-4V alloy as is the case with the Ti-24Nb-0.5O and Ti-24Nb-0.5N alloys of the present study. It has to be mentioned that the excellent yield strength/elastic modulus ratio (0.012) obtained for the Ti-24Nb-0.5O alloy is much higher than in the case of the Ti-6Al4V alloy (0.007). Consequently, this biocompatible Ti-24Nb-0.5O alloy, combining high strength, low modulus and good ductility, is very competitive with the Ti-6Al-4V alloy for use as bone substitute in orthopedics.

The superelastic behavior of the three Ti-26Nb, Ti-24Nb-0.5O and Ti-24Nb-0.5N alloys can be well characterized by cyclic tensile tests at room temperature. Superelastic recovery strains (ε se ) were measured at 3% strain (6th load-unload cycle) as shown in Fig. 4 and values are reported in Table 2. The measured ε se values are found to be higher for both Ti-24Nb-0.5N (2%) and Ti-24Nb-0.5O (2.2%) alloys than for interstitial-free solution-treated β-type superelastic alloys. It is also confirmed by these cyclic tensile tests that Ti-26Nb and Ti-24Nb-0.5N possess a double yielding while Ti-24Nb-0.5O does not. The presence of a plateau for Ti-24Nb-0.5N is clearly due to the reversible SIM transformation as commonly observed in superelastic β-titanium alloys. In the present case, the critical stress inducing SIM transformation, σ c , was evaluated to be at about 130 MPa (Fig. 4c), which is a rather low value, confirming that for this alloy the martensite start, Ms, is close to room temperature. For Ti-24Nb-0.5O, the absence of a plateau may suggest that no SIM transformation occurs in this alloy.

Microstructure after deformation

The two superelastic Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys were observed by TEM on tensile specimens after being strained up to 6%. Fig. 5 shows a typical bright field TEM image (a) and the corresponding electron diffraction pattern (b) of the Ti-24Nb-0.5N microstructure after deformation. In the bright field image, the typical needle-like morphology of the α ″ martensite phase is observed. The diffraction pattern exhibits typical diffraction spots related to the presence of both a body-centered cubic β matrix (higher intensity spot network) and C-centered orthorhombic α ″ martensite needles (lower intensity spot network). This observation confirms that the plateau observed on the tensile curve for this alloy composition is definitely related to the stress-induced α ″ martensitic transformation. Fig. 6 shows a typical bright field TEM image (a) and the corresponding electron diffraction pattern (b) representing the microstructure of the deformed Ti-24Nb-0.5O alloy. Once again, needle-like α ″ martensitic precipitates are observed in β grains, which is confirmed in the related electron diffraction pattern. Consequently, although no plateau was observed on tensile test, stress-induced martensite transformation is also a mechanism occurring in this Ti-24Nb-0.5O alloy composition as was the case with the Ti-24Nb-0.5N alloy.

A multifunctional β-type Ti-23Nb-0.7Ta-2Zr-1.2O alloy composition, called "gum metal", showing a superelastic behavior without a plateau on tensile test, has been reported [START_REF] Saito | Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism[END_REF]. It was claimed that this alloy deforms through unconventional dislocation-free mechanisms implying "giant faults" and stress-induced martensite was excluded from the deformation mechanisms. The fact that no SIM transformation occurs in "gum metal" was confirmed by our group in a recent study in which cyclic tensile tests and TEM observations were carried out [START_REF] Besse | Mechanisms of deformation in Gum Metal TNTZ-O and TNTZ titanium alloys: a comparative study on the oxygen influence[END_REF].

In a recent work [START_REF] Tahara | Lattice modulation and superelasticity in oxygen-added β-Ti alloys[END_REF], it was shown that the addition of 1 at.% O in a Ti-23Nb-1.0O (at.%) alloy composition produces nanosized modulated domains in the β matrix and it was suggested that these domains could suppress long-range martensitic transformation.

In the Ti-24Nb-0.5O alloy composition, the oxygen content is half that of the "gum metal" composition. Consequently, the oxygen content in the present alloy is not high enough to suppress the martensitic transformation as this phase was clearly observed by TEM. In a recent work concerning Ti-24Nb-4Zr-8Sn-(0.08-0.40)O (wt%) alloys [START_REF] Obbard | The effect of oxygen on alpha'' martensite and superelasticity in Ti-24Nb-4Zr-8Sn[END_REF], it was established that oxygen raises the critical stress required to induce martensitic transformation; in other word suppression of the martensitic transformation becomes stronger with increasing oxygen content. The absence of the plateau in Ti-24Nb-0.5O can thus be explained by the fact that the stress required to induce the martensite transformation is close to the yield strength situated at about 665 MPa. This hypothesis is confirmed by the existence of hysteresis between loading and unloading on the cyclic tensile curve (Fig. 4b). This behavior is typically characteristic of the reversible β/α ″ transformation. In the "gum metal" alloy such hysteresis was not observed by cyclic tensile tests and perfect superimposition between loading and unloading was obtained [START_REF] Besse | Mechanisms of deformation in Gum Metal TNTZ-O and TNTZ titanium alloys: a comparative study on the oxygen influence[END_REF]. It has to be mentioned that hysteresis is still observed in the plastic domain in the present alloy, which constitutes an original behavior never reported in the literature.

In β-type Ti alloys, plastic deformation is known to be accommodated by dislocation glide and, in most cases, by twinning formation. Dislocation glide in a metastable β alloy has recently been studied by our group and dislocations were observed to glide in the {110},{112} or {123} planes with a/2 111 Burgers vectors by TEM [START_REF] Castany | Dislocation mobility in gum metal beta-titanium alloy studied via in situ transmission electron microscopy[END_REF].

Concerning the twinning formation, two twinning systems are commonly observed in metastable β titanium alloys: {112} 111 and {332} 113 [START_REF] Hanada | Transmission electron-microscopic observations of mechanical twinning in metastable beta titanium alloys[END_REF][START_REF] Hanada | Transmission electron-microscopic observations of mechanical twinning in metastable beta titanium alloys[END_REF][START_REF] Bertrand | Twinning system selection in a metastable beta-titanium alloy by Schmid factor analysis[END_REF]. In the present study, twinning formation in Ti-24Nb-0.5N

and Ti-24Nb-0.5N alloys was more particularly investigated. For that, EBSD analyses were performed on the two alloys after tensile tests interrupted after 6% strain. The results are presented in Fig. 7.

In this figure, inverse pole figure maps ((a) and (c)) show that bands of a few micrometers in thickness inside the β grains were formed in both deformed Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys. These bands correspond to mechanical twins intervening in both alloys. In some grains, two twin variants can be observed in the same grain. In Fig. 7, we also present examples of misorientation profiles (along the AB line) in each alloy ((b) and (d)). In all cases, the angle measurements between the parent grain and the twin were found to be about 50.5°.

This misorientation corresponds to a Σ11 CSL (coincidence site lattice) boundary characteristic of the {332} 113 twinning system. This twinning system was confirmed by analysis of common poles of the two crystals on 332 and 113 pole figures. Such analysis was carried out on several grains for the two investigated alloys and only {332} 113 twins were found.

Thus, the plastic deformation of the present superelastic alloys is accommodated by both dislocation glide and twinning. For the latter, only the {332} 113 twinning system was observed to be activated. Additionally, in the case of the Ti-24Nb-0.5O alloy, it seems that SIM also occurred in the plastic region. This point needs, however, to be elucidated and insitu experiments are under development. All the deformation mechanisms occurring in superelastic Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys contribute to the very good ductility observed.

Conclusion

In this study, the microstructure and the mechanical behavior of two new superelastic Ti-24Nb-0.5X (X = O, N) biomedical alloys were investigated.

By tensile tests, high mechanical strength, low Young's modulus and excellent ductility were observed for both β-type alloys. By cyclic tensile tests, superelastic behavior was evidenced and rather high superelastic recoveries were obtained. Tensile tests carried out on the Ti-24Nb-0.5N alloy clearly show a double yielding behavior due to the reversible stress-induced α ″ martensitic transformation (SIM). In the Ti-24Nb-0.5O alloy, such a double yielding behavior was not observed because the critical stress inducing the SIM transformation was

shown to occur at a value close to the yield strength.

The plastic deformation of both alloys was shown to be accommodated by dislocation glide and twinning which contribute to the high ductility observed. For the latter, only the {332} 113 twinning system was observed to be activated.

All the properties obtained with these superelastic Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys seem to be particularly useful for their use as implants or prostheses in medicine

Fig. 1 .FigFig. 3 .Fig. 4 .

 134 Fig. 1. Optical micrographs of the solution-treated Ti-24Nb (a), Ti-26Nb (b), Ti-24Nb-0.5O (c), and Ti-24Nb-0.5N (d) alloys, respectively.

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. Typical bright-field TEM image (a) and the corresponding electron diffraction pattern ([100] β zone axis) (b) from the deformed Ti-24Nb-0.5N alloy.

Table 1 .

 1 Microstructure and lattice parameters of the β (cubic) and α ″ (orthorhombic) phases in Ti-24Nb, Ti-26Nb, Ti-24Nb-0.5O and Ti-24Nb-0.5N alloys.

			Lattice parameters (nm)
	Alloys (at %)	Phases	Parent (β)		Martensite (α'')
			aβ	aα ''	bα ''	cα ''
	Ti-24Nb	near fully α"	-	0.31752 0.47879 0.46339
	Ti-26Nb	β	0.32786	-	-	-
	Ti-24Nb-0.5O	β	0.32882	-	-	-
	Ti-24Nb-0.5N	β	0.32837	-	-	-

Table 2 .

 2 Mechanical properties of the studied Ti-24Nb, Ti-26Nb, Ti-24Nb-0.5O and Ti-24Nb-0.5N alloys. Mechanical characteristics (from the literature) of the Ti-6Al-4V ELI alloy and the solution-treated superelastic Ti-26Nb alloy are also indicated.

	Alloys	YS [MPa]	UTS [MPa]	ER [%]	E [GPa]	ε e (or ε se ) [%]	σ c [MPa]
	Ti-24Nb (α'')	300	445	33	60	0.5	no SIM
	Ti-26Nb (β) this work	290	420	22	50	1 (ε se )	80
	Ti-24Nb-0.5O (β)	665	810	22	54	2.2 (ε se )	?
	Ti-24Nb-0.5N (β)	665	665	13	43	2 (ε se )	130
	Ti-6Al-4V ELI (α+β)	790	860	15	114	0.7	no SIM
	Ti-26Nb (β) from Kim et al., 2004	350	420	17	50	1.5 (ε se )	170
	YS : yield strength	UTS : ultimate tensile strength	ER : elongation at rupture

E : Young's modulus ε e (or ε se ) : elastic (or superelastic) recovery σ c

: critical SIM stress

Acknowledgment

This work was supported by the French National Research Agency (No. ANR 08MAPR 0017).


S Holé

ECT sensor design using machine learning techniques

Within the framework of image reconstruction by Electrical Capacitance Tomography (ECT) sensing, we investigate the relevance of the sensor structure embodied in both the number and the size of the electrodes. While most of the studies in the literature exhibit sensors with an assumed arbitrary structure, we consider that these two properties possess a significant impact on the sensor performances. In our study, the emphasis is about the detection of a single bubble with random size and position. We propose to determine the architecture of the sensor that leads to the most accurate image reconstruction. To achieve this objective, we propose to determine the image from a set of independent measurements using LS-SVM models selected with a sophisticated validation method. Indeed, this way to proceed ensures a faster image computation than inverting an underdetermined set of linear equations. By doing so, the computational burden is reduced since it leads to a direct calculation instead of an iterative optimization. Va

I. INTRODUCTION

Electrical Capacitance Tomography (ECT) is a wellknown and a widely used sensing technique. Generally, an ECT sensor is formed by a set of electrodes arranged circularly. The sensing technique is founded on measuring the electrical capacitances appearing between all the possible couples of electrodes. Typically, an ECT sensor can be used to inform about the spatial configuration (size, position) of a non-conducting material which permittivity is different from the one of the environment.

The image of a cross-section of the environment of interest can be obtained using the available measurements. Usually, the image pixels correspond to nodes from as p a t i a ld i s c r e t i z a t i o ns c h e m et h a ti n v o l v e sal i n e a r knowledge-based model. This model describes the relationship between the measurements and the permittivity at each node. The more the space discretization grid is fine, the better the image resolution. Constructing the image leads to invert an underdetermined set of linear equations [START_REF] Yang | Image reconstruction algorithms for electrical capacitance tomography[END_REF].

The literature dedicated to ECT sensors presents various applications. While both the number and the size of electrodes have great importance regarding the sensor performances, they are usually set arbitrarily and seldom discussed [START_REF] Chen | Image reconstruction for an electrical capacitance tomography system based on a least-squares support vector machine and a self-adaptive particle swarm optimization algorithm[END_REF]. Except for few works [START_REF] Wang | Identification of two-phase flow regimes based on support vector machine and electrical capacitance tomography[END_REF], the overall in-dependent measurements are readily taken into account. One can wonder if all the measurements are relevant or just few of them are sufficient to achieve the best image reconstruction given the sensor structure. Moreover, one might ask: what should be the optimal structure of the sensor for a given application?

In the present study, we propose to pay attention to the design of ECT sensors. We assume that tuning the number and the size of electrodes allows to optimize the sensor performances for a given application. More precisely, we are interested in determining the sensor architecture that best predicts the size and the position of a bubble lying in the sensor environment. To achieve this goal, we propose to implement non linear black box models using LS-SVM to establish the relationship between the sensor measurements and the bubble size and position.

II. ECT SENSOR AND PROBLEM STATEMENT

Figure 1 illustrates the structure of a typical ECT sensor considered in this study. It is formed by eight electrodes arranged circularly. The environment delimited by the sensor is characterized by a permittivity ε 1 .In Figure 1, the gap between electrodes is relatively small. This may not be always the case since the size and the number of electrodes have a significant impact on the sensor performances. From the electrical point of view, each electrode can be polarized independently. When electrode i is polarized, the measurement of the charges on electrode j allows the capacitance C ij between these two electrodes to be determined. Due to the circular symmetry of the ECT sensor, there are at most S = 1 2 P (P -1) independent capacities for a given P-electrode sensor.

In the present study, we are interested in optimizing the structure of ECT sensors for image reconstruction by setting the number and the size of the electrodes to their optimal values for a given sensing task. We assume that an ECT sensor is fully defined by giving the number and the size of the electrodes. Indeed, this couple sets the complete sensor structure. The basic motivation for this study is to lead to the most accurate and computationally cost effective image reconstruction.

We consid er tha t a non cond ucting single bubble with permittivity ε 2 lies in the closed region circumscribed by the sensor electrodes. This bubble has unknown size and position respectively pointed by its radius R and its coordinates x and y.Hence,accordingtothisformulation,the image reconstruction consists in estimating as accurately as possible both the bubble radius and coordinates.

Instead of dealing with an image reconstruction founded on inverting an undetermined set of equations, we assume that a black-box inverse model that predicts the size and the position of the bubble exists and can be trained using data generated by numerical ments. This approach allows an almost instantaneous image reconstruction which makes it highly attractive for a real-time implementation. The black-box models are implemented using the LS-SVM technique.

III. TRAINING AND VALIDATING LS-SVM MODELS

A. LS-SVM model description

In spite of several and efficient techniques for non linear static modeling, such as neural networks, Least Square Support Vector Machines (LS-SVM) are attractive candidates thanks to various interesting properties: they are linear-in-their-parameters models, their training algorithm consists in a quadratic minimization and they have a built-in regularization mechanism [START_REF] Vapnik | The Nature of Statistical Learning Theory.S p r i n g e r[END_REF]. As a result, these properties confer to them the ability to build models with high generalization capabilities by avoiding overfitting and controlling model complexity. Given a set D formed by N measurements of S variables and a scalar z: D = (t i , z i ) ∈ R S × R, i = 1, ...N ,w ea r ei n t e r e s t e d in nonlinear models of the form:

ẑ = w T ϕ (t) + b ( 1
)
where t is the vector of input variables, ẑ is the model output, w and b are unknown parameters and φ(•), usually introduced by a kernel function, is a nonlinear mapping from the original feature space to a high dimensional feature space. The LS-SVM training procedure consists in minimizing the cost function defined by:

J (w, e) = 1 2 w 2 + C N ∑ i=1 e i ( 2
)
subject to the equality constraints e i = z iẑi with e i the prediction error taking into account the target output z i and the predicted output ẑi . N is the size of the training set and C is the regularization parameter. This optimization problem can be cast into a dual form with unknown parameters α and b, α being the vector of the Lagrange multipliers [START_REF] Bertsekas | Nonlinear Programming[END_REF]. The parameters can be computed by resolving the following system of linear equations:

K + 1 2C I N 1 N 1 T N 0 α b = z 0 (3)
with

1 N = [1, 1, ..., 1] T , α = [α 1 , α 2 ,. . . ,α N ]
T and I n is the identity matrix. K is the kernel matrix: its component k ij is defined as the dot product between the φ(t i ) and the φ(t j ) mappings. In our study, we use the gaussian kernel function. It introduces an additional parameter σ which is its standard deviation. Parameters σ and C are called the hyperparameters of the LS-SVM model. Their values can be optimized using a validation procedure. Hence, the expression of the model becomes:

ẑ = N ∑ i=1 α i K (t i , t) + b (4)
where α and b are the solutions of the system given by (3).

B. LS-SVM model selection procedure

Since LS-SVM models are linear in their parameters models, the solution of the training phase is unique and can be computed straightforwardly using the set of linear equations given by relation 3. This holds when the hyperparameters C and σ are known with fixed values. Usually, these hyperparameters are unknown and must be computed prior to the training phase. A suitable way to proceed consists in selecting the couple (C, σ ) that best validates the LS-SVM model. In practice, the generalization capabilities of such black box model are estimated by computing the validation error. Various validation techniques exist in the literature [START_REF] Hastie | The Elements of Statistical Learning[END_REF]. The most popular techniques are probably the cross validation method and the Leave-One-Out (LOO) technique. Since afi n es e a r c hi sd e s i r a b l et ob e s to p t i m i z et h em o d e l performance, the computational burden can rapidly become untractable when using either methods. In order to reduce substantially the computational time of the selection procedure without compromising its efficiency, we propose to estimate the validation error using the Virtual Leave-One-Out (VLOO) method. This method, first proposed for linear models [START_REF] Belsley | Regression Diagnostics: Identifying Influential Data And Sources Of Collinearity[END_REF] and later extended to nonlinear models [START_REF] Laurent | Leverage, local influence and curvature in nonlinear regression[END_REF], allows an estimation of the validation error to be computed by performing only one training involving the whole available data. This estimation is exact when dealing with linear-in-theirparameters models, such as LS-SVM models, while it remains an approximation for models which are nonlinear with respect to their parameters. More recently, af r a m e w o r kw a sd e s c r i b e d [START_REF] Cawley | Preventing over-fitting during model selection via bayesian regularization of the hyperparameters[END_REF]t oi m p l e m e n tt h eV L O O method for LS-SVM models. For a given LS-SVM model, the VLOO error is computed as:

VLOO = 1 N N ∑ i=1 α i (M -1 ) ii 2 (5)
where N is the size of the training set, and (M -1 ) ii is the i-th diagonal element of the inverse of matrix M that appears in relation 3. Thus, the VLOO permits a fast and exact estimation of the validation error which consists in a great benefit when optimizing the values of the hyperparameters according to a grid search.

IV. OPTIMAL SENSOR FOR ONE-BUBBLE

DETECTION

Whereas the number of electrodes is a discrete value beginning from 2 and going typically to few units, their size is a continuous value. As a result, investigating the influence of electrode size can lead to a large number of cases to study while evaluating the influence of the number of electrodes is more straightforward. However, various numerical experiments showed that it is convenient to express the electrodes size as a ratio of the sensor perimeter they cover and that three types of electrodes size must be distinguished: pinpoint, contiguous and intermediate size electrodes.

• Pinpoint electrodes: they cover a very small ratio of the sensor perimeter, typically 1%

• Contiguous electrodes: they cover the major sensor perimeter, typically 99%

• Intermediate size electrodes: they cover any other ratio of the sensor perimeter. We considered three values: 25%, 50% and 75%

Basically, there are three models: each of them predicts ac o m p o n e n to ft h et r i p l e t( x, y, R). The search of the optimal sensor architecture leads to a combinatorial optimization. We proceed according to a grid search by considering all the sensors with a number of electrodes in {2, 3, 4, 5, 6} and electrodes size in {1%, 25%, 50%, 75%, 99%}. To generate the data, we intend to detect air bubbles with relative permittivity ε 2 = 1i na no i lfl o w having a relative permittivity ε 1 = 3. For each given sensor, a set of data dedicated to training and validation is generated. It is formed by 500 examples. Each example is computed by considering a bubble with both random radius and coordinates. We implement a knowledgebased model that allows to compute the capacitances between the sensor electrodes given a spatial configuration. Since the knowledge-based model is a set of differential equations, its implementation is performed using a finite elements modeling approach with the Gmsh software [10] as a mesh generator and the GetDP software [11] as as o l v e r . Afi r s ta t t e m p tt ob u i l di n v e r s em o d e l st op r e d i c t bubbles configuration with a linear LS-SVM leads to poor generalization capabilities. Therefore, the use of nonlinear models is necessary.

Figure 2 illustrates the main result of our study. It shows the validation error for the prediction of radius R (top graph), coordinate x (middle graph) and coordinate y (bottom graph). Obviously, a 2-electrode sensor is not satisfactory to predict the bubble configuration: it provides only 1 measurement while there are 3 unknown parameters. Beginning from 3 electrodes, all the sensors provide enough information to estimate accurately the bubble size and position with some differences depending on electrodes size.

Regarding the coordinates x and y,ad e e p e ra n a l y s i s of the results shows that the most accurate model is obtained with 3 electrodes of intermediate size 50%. The VLOO error computed using relation 5 is less than 3% of the sensor radius. Increasing the number of electrodes leads systematically to worse models. For intermediate size electrodes, the error increases almost monotonically. However, the behavior of sensors with contiguous electrodes is quite different. At least 5 electrodes are required to achieve a satisfactory prediction accuracy. A physical explanation can be found in the penetration of the electric field in the sensor. Contrarily to intermediate size electrodes, and with contiguous electrodes, the electric field is indeed concentrated at the junction of electrodes. Therefore, the impact of the bubble on measurements is limited.

Predicting the bubble size is clearly less critical than the bubble coordinates. A 2-electrode sensor of intermediate size 50% allows to predict radius R with an error less than 1% of the sensor radius. Similarly to the coordinates, the radius prediction with a contiguous electrodes sensor requires at least 5 electrodes.

V. C ONCLUSION

This study was dedicated to investigate the influence of both the number and the size of an ECT sensor electrodes on its performances. Our goal was to determine the optimal sensor structure for a given application. We have implemented LS-SVM models validated by a sophisticated procedure to build nonlinear inverse models that predict the size and the position of a single bubble using independent measurements.

Regarding the number of electrodes, our results show that increasing this parameter do not lead systematically to an improvement of the sensor performances. The best accuracy is achieved with a sensor formed by 3 electrodes. This result is counterintuitive.

Albeit to a lesser extent, the electrodes size also has an impact on the sensor performances. The most accurate sensor we obtained uses intermediate size electrodes that occupy 50% of the sensor perimeter. The overall results show that the sensor performances are roughly stable with pinpoint or intermediate size electrodes (between 25% and 75% of the sensor perimeter). However, the use of contiguous electrodes necessitates the involvement of more electrodes to achieve satisfactory performances.

The main result of our study: the use of a 3-electrode sensor, associated to an image reconstruction via a direct calculation for a single bubble localization and sizing, leads to a computational cost effective sensing approach. This makes it highly attractive for real-time implementations.

2 Figure 1 .

 21 Figure 1. An 8-electrode ECT sensor