Emmanuel Bertrand
email: emmanuel.bertrand@univ-nantes.fr

Philippe Castany

Yang Yang

Edern Menou

Laurent Couturier

Thierry Gloriant

Origin of {112}<111> antitwinning in a Ti-24Nb-4Zr-8Sn superelastic single crystal

Keywords: Metastable  titanium alloys, Twinning, Antitwinning, Stress-induced martensitic transformation, Schmid factor analysis

{112}〈111〉 𝛽 twins are observed in a superelastic  Ti-24Nb-4Zr-8Sn (wt.%) single crystal after tensile test. A careful Schmid factor analysis shows that these twins are formed in the antitwinning sense regarding the classical {112}〈111〉 𝛽 twinning system of bcc structures.

These are then {112}〈111〉 𝛽 antitwins. Moreover, a full stress-induced martensitic (SIM) transformation of  phase into " martensite is evidenced from in situ synchrotron X-rays diffraction. This transformation is fully accomplished before the onset of plastic deformation and, in turn, the formation of twins. From crystallographic reconstruction, {112}〈111〉 𝛽 antitwins are shown to be passively formed from the reversion, during the reverse SIM

Introduction

Metastable  titanium alloys can be designed with highly-biocompatible elements such as Nb, Ta, Mo, Zr [START_REF] Niinomi | Fatigue performance and cyto-toxicity of low rigidity titanium alloy, Ti-29Nb-13Ta-4.6[END_REF][START_REF] Geetha | Ti based biomaterials, the ultimate choice for orthopaedic implants -A review[END_REF][START_REF] Cimpean | Osteoblast cell behavior on the new beta-type Ti-25Ta-25Nb alloy[END_REF][START_REF] Castany | Deformation mechanisms and biocompatibility of the superelastic Ti-23Nb-0.7Ta-2Zr-0.5N alloy[END_REF][START_REF] Gordin | Potentiality of the "Gum Metal" titanium-based alloy for biomedical applications[END_REF] and exhibit a low Young's modulus which makes these alloys particularly suitable for medical applications [START_REF] Bertrand | Synthesis and characterisation of a new superelastic Ti-25Ta-25Nb biomedical alloy[END_REF][START_REF] Laheurte | Mechanical properties of low modulus β titanium alloys designed from the electronic approach[END_REF][START_REF] Bertrand | Investigation of the martensitic transformation and the damping behavior of a superelastic Ti-Ta-Nb alloy[END_REF][START_REF] Hao | Effect of Zr and Sn on Young's modulus and superelasticity of Ti-Nb-based alloys[END_REF][START_REF] Santos | Improvement of microstructure, mechanical and corrosion properties of biomedical Ti-Mn alloys by Mo addition[END_REF][START_REF] Zhao | Microstructures and mechanical properties of metastable Ti-30Zr-(Cr, Mo) alloys with changeable Young's modulus for spinal fixation applications[END_REF]. Numerous deformation mechanisms are known to operate in metastable  titanium alloys: stress-induced martensitic (SIM) transformation [START_REF] Castany | In situ synchrotron X-ray diffraction study of the martensitic transformation in superelastic Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys[END_REF][START_REF] Gao | Influence of texture and transformation strain on the superelastic performance of a new Ti-20Zr-3Mo-3Sn alloy[END_REF][START_REF] Obbard | Mechanics of superelasticity in Ti-30Nb-(8-10)Ta-5Zr alloy[END_REF][START_REF] Tahara | Cyclic deformation behavior of a Ti-26 at.% Nb alloy[END_REF][START_REF] Ijaz | Effect of Sn addition on stress hysteresis and superelastic properties of a Ti-15Nb-3Mo alloy[END_REF], mechanical twinning [START_REF] Bertrand | Twinning system selection in a metastable β-titanium alloy by Schmid factor analysis[END_REF][START_REF] Ramarolahy | Microstructure and mechanical behavior of superelastic Ti-24Nb-0.5O and Ti-24Nb-0.5N biomedical alloys[END_REF][START_REF] Lin | Transmission of {332}〈113〉 twins across grain boundaries in a metastable β-titanium alloy[END_REF][START_REF] Cho | Study on formation mechanism of {332}<113> deformation twinning in metastable β-type Ti alloy focusing on stress-induced α" martensite phase[END_REF][START_REF] Min | Mechanism of twinning-induced plasticity in β-type Ti-15Mo alloy[END_REF][START_REF] Yang | Plastic deformation via hierarchical nano-sized martensitic twinning in the metastable β Ti-24Nb-4Zr-8Sn alloy[END_REF] and dislocation slip [START_REF] Castany | In situ TEM study of dislocation slip in a metastable β titanium alloy[END_REF][START_REF] Mompiou | Conventional vs harmonicstructured β-Ti-25Nb-25Zr alloys: A comparative study of deformation mechanisms[END_REF][START_REF] Castany | Dislocation mobility in gum metal beta-titanium alloy studied via in situ transmission electron microscopy[END_REF][START_REF] Besse | Mechanisms of deformation in gum metal TNTZ-O and TNTZ titanium alloys: A comparative study on the oxygen influence[END_REF]. These deformation mechanisms lead to peculiar mechanical properties such as superelastic or shape memory effects [START_REF] Bertrand | Deformation twinning in the full-α" martensitic Ti-25Ta-20Nb shape memory alloy[END_REF][START_REF] Castany | In situ synchrotron X-ray diffraction study of the martensitic transformation in superelastic Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys[END_REF][START_REF] Kim | Martensitic transformation, shape memory effect and superelasticity of Ti-Nb binary alloys[END_REF][START_REF] Tahara | Effect of nitrogen addition and annealing temperature on superelastic properties of Ti-Nb-Zr-Ta alloys[END_REF][START_REF] Kim | Texture and shape memory behavior of Ti-22Nb-6Ta alloy[END_REF][START_REF] Gao | Influence of texture and transformation strain on the superelastic performance of a new Ti-20Zr-3Mo-3Sn alloy[END_REF][START_REF] Ijaz | Effect of Sn addition on stress hysteresis and superelastic properties of a Ti-15Nb-3Mo alloy[END_REF][START_REF] Tahara | Cyclic deformation behavior of a Ti-26 at.% Nb alloy[END_REF][START_REF] Obbard | Mechanics of superelasticity in Ti-30Nb-(8-10)Ta-5Zr alloy[END_REF] and transformation induced plasticity (TRIP) or twinning induced plasticity (TWIP) effects [START_REF] Bignon | Modelling martensitic transformation in titanium alloys: The influence of temperature and deformation[END_REF][START_REF] Brozek | A β-titanium alloy with extra high strain-hardening rate: Design and mechanical properties[END_REF][START_REF] Danard | Microstructure design and in-situ investigation of TRIP/TWIP effects in a forged dual-phase Ti-10V-2Fe-3Al alloy[END_REF][START_REF] Lilensten | On the heterogeneous nature of deformation in a straintransformable beta metastable Ti-V-Cr-Al alloy[END_REF][START_REF] Sun | A new titanium alloy with a combination of high strength, high strain hardening and improved ductility[END_REF].

SIM transformation occurs from the  phase (bcc) to the " phase (C-centered orthorhombic).

It has to be noticed that stress-induced " martensite is formed as single untwinned variant in  Ti-based alloys [START_REF] Obbard | Mechanics of superelasticity in Ti-30Nb-(8-10)Ta-5Zr alloy[END_REF][START_REF] Castany | Deformation mechanisms and biocompatibility of the superelastic Ti-23Nb-0.7Ta-2Zr-0.5N alloy[END_REF][START_REF] Ramarolahy | Microstructure and mechanical behavior of superelastic Ti-24Nb-0.5O and Ti-24Nb-0.5N biomedical alloys[END_REF] contrarily to other superelastic alloys such as NiTi alloys wherein pairs of twinned variants are formed [START_REF] Laplanche | Sudden stress-induced transformation events during nanoindentation of NiTi shape memory alloys[END_REF][START_REF] Otsuka | Physical metallurgy of Ti-Ni-based shape memory alloys[END_REF][START_REF] Gall | The role of texture in tension-compression asymmetry in polycrystalline NiTi[END_REF][START_REF] Gao | A multivariant micromechanical model for SMAs Part 1. Crystallographic issues for single crystal model[END_REF]. The crystallographic correspondence between these two phases is given by the modified Burgers Orientation Relationship (BOR): {110} 𝛽 //(001) 𝛼 ′′ and 〈111〉 𝛽 //[110] 𝛼 ′′ . Its reversibility generally depends on the metastability of the alloy [START_REF] Castany | Design of strain-transformable titanium alloys[END_REF]: a SIM transformation occurring during plastic deformation is no or few reversible and leads to TRIP effect [START_REF] Bignon | Modelling martensitic transformation in titanium alloys: The influence of temperature and deformation[END_REF][START_REF] Brozek | A β-titanium alloy with extra high strain-hardening rate: Design and mechanical properties[END_REF][START_REF] Danard | Microstructure design and in-situ investigation of TRIP/TWIP effects in a forged dual-phase Ti-10V-2Fe-3Al alloy[END_REF][START_REF] Lilensten | On the heterogeneous nature of deformation in a straintransformable beta metastable Ti-V-Cr-Al alloy[END_REF][START_REF] Sun | A new titanium alloy with a combination of high strength, high strain hardening and improved ductility[END_REF], while a SIM transformation occurring before plastic deformation is generally highly (completely in the most favourable case) reversible and leads to superelasticity [START_REF] Castany | In situ synchrotron X-ray diffraction study of the martensitic transformation in superelastic Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys[END_REF][START_REF] Gao | Influence of texture and transformation strain on the superelastic performance of a new Ti-20Zr-3Mo-3Sn alloy[END_REF][START_REF] Ijaz | Superelastic properties of biomedical (Ti-Zr)-Mo-Sn alloys[END_REF][START_REF] Kim | Martensitic transformation, shape memory effect and superelasticity of Ti-Nb binary alloys[END_REF][START_REF] Tahara | Effect of nitrogen addition and annealing temperature on superelastic properties of Ti-Nb-Zr-Ta alloys[END_REF]. Superelasticity is generally evidenced by the occurrence of a stress plateau on tensile curves followed by a full recovery when the stress is removed. The presence of SIM phase in superelastic alloys can then only be detected under stress due to its reversibility when the stress is released. Residual '' phase can be observed in highly deformed samples as plasticity impedes the reversion of martensite [START_REF] Kim | Effect of Ta addition on shape memory behavior of Ti-22Nb alloy[END_REF][START_REF] Kim | Shape memory characteristics of Ti-22Nb-(2-8)Zr(at.%) biomedical alloys[END_REF], but a full reversible SIM transformation can only be evidenced by in situ techniques [START_REF] Castany | In situ synchrotron X-ray diffraction study of the martensitic transformation in superelastic Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys[END_REF][START_REF] Gao | Influence of texture and transformation strain on the superelastic performance of a new Ti-20Zr-3Mo-3Sn alloy[END_REF][START_REF] Obbard | Mechanics of superelasticity in Ti-30Nb-(8-10)Ta-5Zr alloy[END_REF][START_REF] Ijaz | Superelastic Behavior of Biomedical Metallic Alloys[END_REF][START_REF] Tahara | Cyclic deformation behavior of a Ti-26 at.% Nb alloy[END_REF][START_REF] Tahara | Lattice modulation and superelasticity in oxygen-added β-Ti alloys[END_REF].

Observed twinning systems generally have a low magnitude of shear and a low magnitude of shuffle [START_REF] Christian | Deformation twinning[END_REF]. In body centred cubic (bcc) metals, the operating twinning system is then {112}<111>. Even if its magnitude of shear is high (0.707 from Tobe et al. [START_REF] Tobe | Origin of {332} twinning in metastable β-Ti alloys[END_REF]), the absence of shuffle makes {112}<111> twins more favorable in comparison to other theoretical twinning systems with lower magnitude of shear but involving shuffle [START_REF] Christian | Deformation twinning[END_REF]. It can also be noticed that there is another theoretically predicted twinning system with the same twinning plane and the same twinning direction than the classical {112}<111> twinning system, which involves a lower value of shear (0.354 from Tobe et al. [START_REF] Tobe | Origin of {332} twinning in metastable β-Ti alloys[END_REF]) but a high magnitude of shuffle [START_REF] Christian | Deformation twinning[END_REF]. This twinning system with shuffle will be referred as "non-classical" {112}<111> twinning in the present paper in contrast to the "classical" {112}<111> system without shuffle. This non-classical system is then unfavourable in comparison with the classical one that not involves any shuffle and is thus not experimentally observed. These two twinning systems share exactly the same twinning plane and shear plane, but share the same twinning direction with a different sense [START_REF] Christian | Deformation twinning[END_REF]. The classical one is in the twinning sense while the non-classical one lies in the opposite sense, i.e. the antitwinning direction regarding the classical one. The secondary twinning elements are also different between these two systems. All the crystallographic features of these two {112}<111> twinning systems will be detailed in the section 3.4.

The classical {112}<111> twinning system is sometimes observed in metastable  titanium alloys, including TRIP/TWIP [START_REF] Danard | Microstructure design and in-situ investigation of TRIP/TWIP effects in a forged dual-phase Ti-10V-2Fe-3Al alloy[END_REF][START_REF] Lilensten | On the heterogeneous nature of deformation in a straintransformable beta metastable Ti-V-Cr-Al alloy[END_REF] and superelastic compositions [START_REF] Bertrand | Twinning system selection in a metastable β-titanium alloy by Schmid factor analysis[END_REF], but a peculiar {332}<113> twinning mode is mainly observed in these alloys [START_REF] Bertrand | Twinning system selection in a metastable β-titanium alloy by Schmid factor analysis[END_REF][START_REF] Castany | Design of strain-transformable titanium alloys[END_REF][START_REF] Castany | Deformation mechanisms and biocompatibility of the superelastic Ti-23Nb-0.7Ta-2Zr-0.5N alloy[END_REF][START_REF] Cho | Study on formation mechanism of {332}<113> deformation twinning in metastable β-type Ti alloy focusing on stress-induced α" martensite phase[END_REF][START_REF] Danard | Microstructure design and in-situ investigation of TRIP/TWIP effects in a forged dual-phase Ti-10V-2Fe-3Al alloy[END_REF][START_REF] Lilensten | On the heterogeneous nature of deformation in a straintransformable beta metastable Ti-V-Cr-Al alloy[END_REF][START_REF] Lin | Transmission of {332}〈113〉 twins across grain boundaries in a metastable β-titanium alloy[END_REF][START_REF] Min | Mechanism of twinning-induced plasticity in β-type Ti-15Mo alloy[END_REF][START_REF] Ramarolahy | Microstructure and mechanical behavior of superelastic Ti-24Nb-0.5O and Ti-24Nb-0.5N biomedical alloys[END_REF][START_REF] Yang | Plastic deformation via hierarchical nano-sized martensitic twinning in the metastable β Ti-24Nb-4Zr-8Sn alloy[END_REF][START_REF] Sun | A new titanium alloy with a combination of high strength, high strain hardening and improved ductility[END_REF]. Classical {112}<111> twins, when observed, are then a secondary and minor twinning mode occurring in addition to preponderant {332}<113> twins [START_REF] Bertrand | Twinning system selection in a metastable β-titanium alloy by Schmid factor analysis[END_REF][START_REF] Danard | Microstructure design and in-situ investigation of TRIP/TWIP effects in a forged dual-phase Ti-10V-2Fe-3Al alloy[END_REF][START_REF] Lilensten | On the heterogeneous nature of deformation in a straintransformable beta metastable Ti-V-Cr-Al alloy[END_REF]. {332}<113> twinning exhibits a much lower magnitude of shear (0.354 from Tobe et al. [START_REF] Tobe | Origin of {332} twinning in metastable β-Ti alloys[END_REF]) than classical {112}<111> twinning but is not observed in other bcc structures due to its high magnitude of shuffle. Occurrence of this peculiar twinning system can be attributed to a low shear modulus in {011} planes along <011> directions leading to a lattice instability of the bcc structure [START_REF] Tobe | Origin of {332} twinning in metastable β-Ti alloys[END_REF]. This model is consistent with the observation of {332}<113> twins in TWIP and TRIP/TWIP metastable  Ti alloys where no SIM occurs before twinning [START_REF] Danard | Microstructure design and in-situ investigation of TRIP/TWIP effects in a forged dual-phase Ti-10V-2Fe-3Al alloy[END_REF][START_REF] Lilensten | On the heterogeneous nature of deformation in a straintransformable beta metastable Ti-V-Cr-Al alloy[END_REF][START_REF] Lin | Transmission of {332}〈113〉 twins across grain boundaries in a metastable β-titanium alloy[END_REF][START_REF] Qian | In-situ microstructural investigations of the TRIP-to-TWIP evolution in Ti-Mo-Zr alloys as a function of Zr concentration[END_REF]. However, this explanation cannot be applied to twins in  Tibased superelastic alloys because a full transformation into SIM " phase occurs before twinning. Then, {332}<113> twins clearly result from twinning operations in '' phase in these alloys. Recent in situ characterization of {332}<113> twins in a Ti-27Nb (at.%) showed that superelastic alloys undergo a complex deformation sequence: (1) A full stress-induced martensitic transformation into '' occurs followed by (2) mechanical twinning of '' phase and then (3) the reversion into  phase when the stress is released leading to formation of {332}<113> twins [START_REF] Castany | Reversion of a parent {130}<310>α'' martensitic twinning system at the origin of {332}<113>β twins observed in metastable beta titanium alloys[END_REF]. {332}<113> twins in  phase are then passively formed from the reversion of twins formerly produced into stress induced " martensite. Further observations of transitional structures exhibiting residual " martensite along {332}<113> twin boundaries also support this mechanism of twin reversion [START_REF] Lai | On the mechanism of {332} twinning in metastable β titanium alloys[END_REF][START_REF] Chen | Transitional structure of {332}<113> β twin boundary in a deformed metastable β-type Ti-Nb-based alloy, revealed by atomic resolution electron microscopy[END_REF].

As twins in  phase can be linked to twins in SIM " phase, twinning of " phase has to be considered. Twinning systems of '' martensite are usually observed in titanium alloys with initial '' microstructures. {111}'' type I and <211>'' type II twinning modes preserve the orientation of the corresponding  phase by application of the BOR and are then considered as reorientation twinning systems rather than mechanical ones [START_REF] Bertrand | Deformation twinning in the full-α" martensitic Ti-25Ta-20Nb shape memory alloy[END_REF][START_REF] Chai | Self-accommodation in Ti-Nb shape memory alloys[END_REF][START_REF] Inamura | Composition dependent crystallography of α"-martensite in Ti-Nb-based β-titanium alloy[END_REF]. Plastic twinning systems are also observed in '' microstructures. The {130}<310>" system is identified in some '' microstructures and its reversion into  phase leads to {332}<113> twins [START_REF] Bertrand | Deformation twinning in the full-α" martensitic Ti-25Ta-20Nb shape memory alloy[END_REF][START_REF] Tahara | Plastic deformation behaviour of single-crystalline martensite of Ti-Nb shape memory alloy[END_REF].

{103}<301>" twins are similarly observed with the same relationship with {332}<113> twins [START_REF] Tahara | Plastic deformation behaviour of single-crystalline martensite of Ti-Nb shape memory alloy[END_REF]. {110}〈110〉 𝛼 ′′ twins are also evidenced at the surface of superelastic specimens [START_REF] Ping | Stress-induced α" martensitic (110) twinning in β-Ti alloys[END_REF]. Contrarily to the previous ones, this later twinning system corresponds to the classical {112}<111> twinning system of  phase [START_REF] Yang | Stress releaseinduced interfacial twin boundary ω phase formation in a β type Ti-based single crystal displaying stress-induced α" martensitic transformation[END_REF]. Among these twinning systems of " phase, {130}<310>" is the most observed one and was also recently evidenced in TRIP/TWIP alloys [START_REF] Qian | In-situ microstructural investigations of the TRIP-to-TWIP evolution in Ti-Mo-Zr alloys as a function of Zr concentration[END_REF]. But clear criterions to explain the activation of twinning systems in " are still not elucidated.

This paper focuses on the deformation behavior of a Ti-24Nb-4Zr-8Sn (Ti2448) superelastic single crystal during a tensile test. {112}<111> mechanical twins are observed in this alloy after 4.5% of applied strain. Schmid factor analysis will show that these twins correspond to the non-classical {112}<111> twinning system operating in the antitwinning direction.

Moreover, these antitwins in  phase will be shown to be due to the reversion of twins occurring in " phase. The role of SIM transformation on the observation of {112}<111> antitwins in  phase will be then investigated in terms of twinning shear and shuffle as well as variant selection of martensite.

Materials and methods

Experimental methods

A Ti-24Nb-4Zr-8Sn (wt.%) single crystal orientated along a <110> direction is used for this study. Experimental details are given in [START_REF] Yang | Stress releaseinduced interfacial twin boundary ω phase formation in a β type Ti-based single crystal displaying stress-induced α" martensitic transformation[END_REF]. X-ray diffraction (XRD) Laue back reflection confirmed that the <110> direction is obtained within a 1° range and the absence of metastable " and  phases. Flat tensile specimens with 3x15x0.5 mm 3 gauge dimensions were machined from the single crystal. An inverse pole figure (IPF) was taken from the single crystal using Electron backscattering diffraction on a Scanning Electron Microscope (SEM-EBSD) and showed that the surface plane of these specimens is {353}. A Jeol JSM 7100F SEM machine working with an accelerating voltage of 20 kV and equipped with a TSL-EBSD system (HKL-Channel 5 analysis software) was used. Specimen was previously prepared with the same procedure than for metallographic observation detailed below. Cyclic tensile tests were conducted at room temperature by strain increments of 0.5% followed by stress release up to an elongation of 5.5% using an INSTRON 3369 tensile machine with a strain rate of 10 -4 s -1 . An extensometer was used to ensure the accuracy of strain measurement. Tensile direction (TD) was chosen along the <110> direction of the single crystal in order to maximize the transformation strain [START_REF] Jabir | Crystallographic orientation dependence of mechanical properties in the superelastic Ti-24Nb-4Zr-8Sn alloy[END_REF]. A specimen deformed up to 4.5% of applied strain is used for twin characterization. Samples were prepared for metallographic observation by mechanical polishing with increasing grades of SiC abrasive papers, final polishing with colloidal silica suspension and final etching in a solution composed of 8% HF, 15% HNO3 and 77% H2O (vol%). TEM specimens were also cut from tensile specimens and were twin-jet electropolished at 253-263 K in a solution of 6% perchloric acid and 94% methanol (vol. %). Electropolishing was stopped before perforation and final steps of thinning were performed by two-steps ion milling using successively 5 keV and 1 keV Ar ions with a Fischione 1010 Model. Observations were conducted on a JEOL 2100 microscope operating at 200 kV.

In situ synchrotron X-ray diffraction (SXRD) during cyclic tensile test was conducted on beam line ID31 at the European Synchrotron Radiation Facility (ESRF, Grenoble, France) with a 5 kN micro-tensile machine. The high-resolution ID31 beamline offers a high-energy incident X-ray beam (= 0.040002106 nm) coupled with a nine-channel multi-analyzer detector. The transmitted diffracted beams were collected over the 2 angular range 4-28 ° with a scanning step of 0.005 °. The in situ cyclic tensile test under synchrotron radiation was realized on a single crystal specimen with the same geometry than for tensile tests (gauge width of 3 mm and thickness of 0.5 mm) with strain increments of 0.5 % until 5.5 %. SXRD scans were therefore obtained after each strain increment for both loading and unloading conditions. More details can be found in [START_REF] Yang | Stress releaseinduced interfacial twin boundary ω phase formation in a β type Ti-based single crystal displaying stress-induced α" martensitic transformation[END_REF][START_REF] Castany | In situ synchrotron X-ray diffraction study of the martensitic transformation in superelastic Ti-24Nb-0.5N and Ti-24Nb-0.5O alloys[END_REF].

Variant selection

Each variant of " martensite can be obtained from  phase orientation by application of BOR. The transformation of a  crystal into " martensite leads to 6 equivalent lattice correspondence variants (CV) given in Table 1. Under uniaxial applied stress, activated variants are expected to be the ones leading to the highest transformation strain along the stress direction [START_REF] Inamura | Composition dependent crystallography of α"-martensite in Ti-Nb-based β-titanium alloy[END_REF][START_REF] Inamura | Crystallographic orientation and stress-amplitude dependence of damping in the martensite phase in textured[END_REF][START_REF] Jabir | Crystallographic orientation dependence of mechanical properties in the superelastic Ti-24Nb-4Zr-8Sn alloy[END_REF]. Table 1. Lattice correspondence variants (CV) derived from  and " crystals and transformation lattice strain of each CV along the [101] 𝛽 tensile axis.

Variant [100] 𝛼" [010] 𝛼" [001] 𝛼" Strain CV1 [100] 𝛽 [011] 𝛽 [01 ̅ 1] 𝛽 1.51% CV2 [100] 𝛽 [01 ̅ 1] 𝛽 [01 ̅ 1 ̅ ] 𝛽 1.51% CV3 [010] 𝛽 [101] 𝛽 [101 ̅ ] 𝛽 4.31% CV4 [010] 𝛽 [101 ̅ ] 𝛽 [1 ̅ 01 ̅ ] 𝛽 3.99% CV5 [001] 𝛽 [110] 𝛽 [1 ̅ 10] 𝛽 1.51% CV6 [001] 𝛽 [1 ̅ 10] 𝛽 [1 ̅ 1 ̅ 0] 𝛽 1.51%
Twinning systems are usually described by their twinning elements: K1, 𝜂 1 , K2, 𝜂 2 and P [START_REF] Christian | Deformation twinning[END_REF] as shown on Fig 1 . Variant selection during mechanical twinning of  and " phases can be determined using a Schmid factor analysis adapted to this deformation mechanism [START_REF] Bertrand | Twinning system selection in a metastable β-titanium alloy by Schmid factor analysis[END_REF][START_REF] Bertrand | Deformation twinning in the full-α" martensitic Ti-25Ta-20Nb shape memory alloy[END_REF].

Schmid factor, written 𝑚, is defined similarly to slip as:

𝑚 = cos 𝜆 cos 𝜑
where  is the angle between tensile direction (TD) and 𝐾 1 ⃗⃗⃗⃗ and the angle between TD and 𝜂 1 ⃗⃗⃗ . Contrarily to dislocation slip, values of  and  are taken between 0° and 180° in order to obtain 𝑚 values ranging between -0.5 and 0.5. Using this convention, a twin variant can be activated in tension if its associated Schmid factor is positive. The sign of 𝑚 plays then a key role in the present study. So, signs of K1 and 𝜂 1 have to be chosen carefully with respect to Fig. 1, as detailed in Appendix A of [START_REF] Bertrand | Deformation twinning in the full-α" martensitic Ti-25Ta-20Nb shape memory alloy[END_REF]. Twinning elements of each system are identified in section 3.4.

Fig. 1 The four twinning elements contained in the shear plane (P).

In the present study, two twinning systems for each phase will be considered: {112}〈111〉 𝛽 and {110}〈110〉 𝛼 ′′ . This notation without minus sign in index of 𝜂 1 will be used for the generic denomination of each system. Conversely, signs will be specified when the twinning (t) or the antitwinning (at) sense will be considered: {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 and {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 correspond then to twinning systems when the sign of 𝜂 1 indicates the twinning sense and {112}〈111 ̅ 〉 𝛽 𝑎𝑡 and {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 correspond to twinning systems when the sign of 𝜂 1 corresponds to the antitwinning sense. In this paper, the twinning sense is always defined as the twinning direction of the classical twinning system of bcc structures, whatever the phase considered.

Results and discussion

Preliminary results: tensile tests and in situ SXRD characterization

The main experimental results are summarized in this section. A full description of phase transformations and mechanical behavior of the present Ti2448 single crystal is available in a previous study [START_REF] Yang | Stress releaseinduced interfacial twin boundary ω phase formation in a β type Ti-based single crystal displaying stress-induced α" martensitic transformation[END_REF]. Finally, the key feature for the present paper is that a full martensitic transformation from  phase to " phase is observed on loading, which means that when plastic deformation occurs, the microstructure is only composed of " martensite.

Deformation sequence

It has first to be noted that Schmid factor analysis is required to distinguish the twinning direction from the antitwinning direction of observed twins. In this section, twinning systems are then labeled without signs and twinning/antitwinning directions will be determined in section 3.3.

The microstructure of the Ti2448 single crystal after 4.5% strain is first investigated by optical microscopy. As the single crystal is observed after unloading and according to SXRD results, most of the martensite is reversed into  phase. Fig. 3a shows that two orientations of twins cross the single crystal. These twins were previously characterized by TEM observations and identified as {112}〈111〉 𝛽 twins with the presence of a thin layer of interfacial  phase along the twin boundaries [START_REF] Yang | Stress releaseinduced interfacial twin boundary ω phase formation in a β type Ti-based single crystal displaying stress-induced α" martensitic transformation[END_REF]. Another example of such a twin is shown on the Fig. 3b. The corresponding diffraction pattern in Fig. 3c is taken along a <110> zone axis which is common to both  matrix and {112}〈111〉 𝛽 twin. Typical spots of a single variant of  phase are also clearly visible. As formerly reported, this single variant of  phase lies at the twin interface and impedes any direct observation of its atomic structure [START_REF] Yang | Stress releaseinduced interfacial twin boundary ω phase formation in a β type Ti-based single crystal displaying stress-induced α" martensitic transformation[END_REF]. Some dislocations are also visible in Fig. 3b. Both dislocation glide and {112}〈111〉 𝛽 twinning are then activated during plastic deformation. As mechanical twinning involves unrecoverable plastic deformation, it must occur beyond 4% of strain, i.e. when the microstructure is only composed of " martensite. As a consequence, twinning actually occurs in " phase according to the {110}〈110〉 𝛼 ′′ twinning system as previously evidenced [START_REF] Yang | Stress releaseinduced interfacial twin boundary ω phase formation in a β type Ti-based single crystal displaying stress-induced α" martensitic transformation[END_REF]. The deformation sequence can thus be summarized as shown on Fig. 4. After elastic deformation of the  phase, when the strain is increasing:

- phase is first transformed into stress-induced " martensite. Transformation strain along TD is calculated for each possible CV of martensite (Table 1) and leads to the variant selection of CV3, which gives the highest value of transformation strain.

-At higher strain, twinning and dislocation glide of " phase occur. According to [START_REF] Yang | Stress releaseinduced interfacial twin boundary ω phase formation in a β type Ti-based single crystal displaying stress-induced α" martensitic transformation[END_REF],

{110}〈110〉 𝛼 ′′ twins are formed into martensite.

-Then, when the stress is removed, the parent crystal and twins reverse each into  phase, leading to the presence of {112}〈111〉 𝛽 twins into  phase.

As a conclusion, the observed {112}〈111〉 𝛽 twins actually correspond to the reversion during reverse SIM transformation of {110}〈110〉 𝛼 ′′ twins activated into the stress-induced " martensite.

Fig. 4 Deformation sequence of the Ti-24Nb-4Zr-8Sn single crystal under loading and after unloading.

3.3. Schmid factor analysis and twinning system identification Fig. 5a represents the stereographic projection corresponding to the  phase single crystal orientation where the TD is chosen as [101] 𝛽 and the surface plane is (3 ̅ 5 ̅ 3) 𝛽 as specified in section 2.1. Using this convention, CV3 is expected to be formed under stress (Table 1). Fig. 5b is then the stereographic projection of the corresponding " single crystal obtained by application of the /" orientation relationship for the CV3. This orientation of " phase is assumed to be the parent crystal orientation with TD along the [010] 𝛼 ′′ direction when twins are activated. The notation of each twin variant is summarized in Table 2. For each variant, the twinning plane K1 is given, as well as the index of 𝜂 1 along the twinning direction (𝜂 1𝑡 ) or along the antitwinning direction in the opposite sense (𝜂 1𝑎𝑡 ). There are 12 twin variants in  phase and 2 variants in '' phase due to their respective symmetry. Comparison of twins orientations in Fig. 3a and traces of {112} planes in Fig. 5a shows that the observed twins in  phase corresponds to V2 and V8 twin variants. Moreover, traces of {112} twinning planes corresponding to V2 and V8 twin variants correspond exactly to the traces of {110}" planes of both variants of {110}〈110〉 𝛼 ′′ twins in the CV3 of " martensite. This point confirms that the CV3 of martensite is the one really formed during deformation as expected from calculations of transformation strains. Indeed, if another CV of martensite was formed, the positions of {110}" planes would be different and would correspond to two other {112} planes whom traces will have different orientations than the ones observed on the optical micrograph.

(112) [1 ̅ 1 ̅ 1] [111 ̅ ] V2 (121) [1 ̅ 11 ̅ ] [11 ̅ 1] V3 (211) [11 ̅ 1 ̅ ] [1 ̅ 11] V4 (1 ̅ 12) [11 ̅ 1] [1 ̅ 11 ̅ ] V5 (1 ̅ 21) [111 ̅ ] [1 ̅ 1 ̅ 1] V6 (2 ̅ 11) [1 ̅ 1 ̅ 1 ̅ ] [111] V7 (11 ̅ 2) [1 ̅ 11] [11 ̅ 1 ̅ ] V8 (12 ̅ 1) [1 ̅ 1 ̅ 1 ̅ ] [111] V9 (21 ̅ 1) [111 ̅ ] [1 ̅ 1 ̅ 1] V10 (112 ̅ ) [1 ̅ 1 ̅ 1 ̅ ] [111] V11 (121 ̅ ) [1 ̅ 11] [11 ̅ 1 ̅ ] V12 (211 ̅ ) [11 ̅ 1] [1 ̅ 11 ̅ ] V1" (110) [11 ̅ 0] [1 ̅ 10] V2" (1 ̅ 10) [1 ̅ 1 ̅ 0] [110]
Schmid factor analysis is used considering both twinning and antitwinning systems for  and " phases: {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 and {112}〈111 ̅ 〉 𝛽 𝑎𝑡 ; {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 and {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 (Table 3). Of course, values of Schmid factors for twinning direction (mt) have the same absolute values than for antitwinning direction (mat), only the sign is opposite. As the observed twins are obtained during a tensile test, activated variants are expected to have a high and positive 𝑚 value. However, the observed V2 and V8 variants have surprisingly a negative 𝑚 value with high absolute values, close to -0.5 when {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 is considered. Therefore, the consideration of the classical {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 twinning system, i.e.

twinning in the twinning sense, leads to an apparently incoherent result with observed twins corresponding to a compression of the crystal.

As the positions of {110} 𝛼 ′′ planes and 〈110〉 𝛼 ′′ directions of V2'' and V1'' are very close to {112} 𝛽 planes and 〈111〉 𝛽 directions of V2 and V8 respectively (Fig. 5a and5b), similar 𝑚 values close to -0.5 are also found for {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 twinning (Table 3). Whatever  or " phase is considered, Schmid factor calculations for twin variants in the twinning sense (mt)

show that the observed twin variants lead to a compression of the single crystal along its tensile direction. This is of course not observed as the single crystal is elongated during the present tensile test. Therefore, {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 or {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 twins cannot be activated. This means that the observed twins are not the classical {112}〈111〉 𝛽 twinning system of bcc structures, and, in turn, are the non-classical {112}〈111 ̅ 〉 𝛽 𝑎𝑡 twinning system with a shear in the antitwinning sense regarding the classical {112}〈111〉 𝛽 mode. If twinning is then considered to occur in the antitwinning sense but in the same plane, Schmid factors (mat) will have high and positive values for the observed twin variants (Table 3). Considering the activation of this non-classical {112}〈111 ̅ 〉 𝛽 𝑎𝑡 system, the crystal will be elongated along the tensile direction with the two observed twin variants. Therefore, this careful Schmid factor analysis allows to determine that the observed {112}〈111〉 𝛽 twins does not correspond to the classical twinning mode of bcc structures but to the non-classical {112}〈111 ̅ 〉 𝛽 𝑎𝑡 twinning system shearing in the antitwinning sense. {112}〈111〉 𝛽 twinning in the antitwinning sense was never observed, as its activation energy is much higher than the one in the twinning sense due to a much higher value of atomic shuffle as detailed in the next section.

Twinning elements

As stated in the introduction, the classical {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 mode and the non-classical {112}〈111 ̅ 〉 𝛽 𝑎𝑡 mode in the antitwinning sense have the same principal twinning elements but different secondary twinning elements. Consequently, the same feature will be found for twinning systems in " phase due to the orientation relationship between  and " phases. Principal and secondary twinning elements of each twinning system can be calculated using the Bilby and Crocker theory and can be found in the following references for {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 and {112}〈111 ̅ 〉 𝛽 𝑎𝑡 [START_REF] Christian | Deformation twinning[END_REF][START_REF] Tobe | Origin of {332} twinning in metastable β-Ti alloys[END_REF] and for {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 and {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 [START_REF] Ping | Stress-induced α" martensitic (110) twinning in β-Ti alloys[END_REF]. These elements are given in Table 4 for each considered twinning system and in Table 5 for each twin variant. As each twinning element is commensurable, these twinning systems are all of compound type.

Fig. 6 shows these twinning elements on stereographic projections in  phase (Fig. 6a) and in '' phase (Fig. 6b) and the associated dichromatic diagrams with respect of the orientation relationship in  phase (Fig. 6c) and '' phase (Fig. 6d). The pole of 1 is on the top of the stereographic projections and the pole of the shear plane P is normal to the surface of projections. 1 directions of the  and " twinning systems point both to the right of the projections whereas 1 directions of antitwinning systems point to the left. Secondary twinning elements K2 and 2 are also shown on these figures for each system. Shear values are calculated using the formula 𝑠 = 2cotan(2𝜙) where 𝜙 is the angle between 𝜂 1 and the secondary twinning direction in the twinned crystal 𝜂 2 ′ (Fig. 1). As it is acknowledged that activated twinning systems exhibit low shear and shuffle [START_REF] Christian | Deformation twinning[END_REF][START_REF] Tobe | Origin of {332} twinning in metastable β-Ti alloys[END_REF], dichromatic diagrams of  and " phases are used to highlight shear and shuffle values of each twinning system (Fig. 6c and Fig. 6d). Twinning direction points to the right and antitwinning direction points to the left. The obtained shear and shuffle values are summarized in Table 4. They are measured as follows:

. K1 1 K2 2  K1 1 K2 2 𝑉1 𝛽 𝑡 (112) [1 ̅ 1 ̅ 1] (1 ̅ 1 ̅ 2) [111] 𝑉1 𝛽 𝑎𝑡 (112) [111 ̅ ] (332 ̅ ) [113] 𝑉2 𝛽 𝑡 (121) [1 ̅ 11 ̅ ] (1 ̅ 21 ̅ ) [111] 𝑉2 𝛽 𝑎𝑡 (121) [11 ̅ 1] (32 ̅ 3) [131] 𝑉3 𝛽 𝑡 (211) [11 ̅ 1 ̅ ] (21 ̅ 1 ̅ ) [111] 𝑉3 𝛽 𝑎𝑡 (211) [1 ̅ 11] (2 ̅ 33) [311] 𝑉4 𝛽 𝑡 (1 ̅ 12) [11 ̅ 1] (11 ̅ 2) [1 ̅ 11] 𝑉4 𝛽 𝑎𝑡 (1 ̅ 12) [1 ̅ 11 ̅ ] (3 ̅ 32 ̅ ) [1 ̅ 13] 𝑉5 𝛽 𝑡 (1 ̅ 21) [111 ̅ ] (121 ̅ ) [1 ̅ 11] 𝑉5 𝛽 𝑎𝑡 (1 ̅ 21) [1 ̅ 1 ̅ 1] (3 ̅ 2 ̅ 3) [1 ̅ 31] 𝑉6 𝛽 𝑡 (2 ̅ 11) [1 ̅ 1 ̅ 1 ̅ ] (2 ̅ 1 ̅ 1 ̅ ) [1 ̅ 11] 𝑉6 𝛽 𝑎𝑡 (2 ̅ 11) [111] (233) [3 ̅ 11] 𝑉7 𝛽 𝑡 (11 ̅ 2) [1 ̅ 11] (1 ̅ 12) [11 ̅ 1] 𝑉7 𝛽 𝑎𝑡 (11 ̅ 2) [11 ̅ 1 ̅ ] (33 ̅ 2 ̅ ) [11 ̅ 3] 𝑉8 𝛽 𝑡 (12 ̅ 1) [1 ̅ 1 ̅ 1 ̅ ] (1 ̅ 2 ̅ 1 ̅ ) [11 ̅ 1] 𝑉8 𝛽 𝑎𝑡 (12 ̅ 1) [111] (323) [13 ̅ 1] 𝑉9 𝛽 𝑡 (21 ̅ 1) [111 ̅ ] (211 ̅ ) [11 ̅ 1] 𝑉9 𝛽 𝑎𝑡 (21 ̅ 1) [1 ̅ 1 ̅ 1] (2 ̅ 3 ̅ 3) [31 ̅ 1] 𝑉10 𝛽 𝑡 (112 ̅ ) [1 ̅ 1 ̅ 1 ̅ ] (1 ̅ 1 ̅ 2 ̅ ) [111 ̅ ] 𝑉10 𝛽 𝑎𝑡 (112 ̅ ) [111] (332) [113 ̅ ] 𝑉11 𝛽 𝑡 (121 ̅ ) [1 ̅ 11] (1 ̅ 21) [111 ̅ ] 𝑉11 𝛽 𝑎𝑡 (121 ̅ ) [11 ̅ 1 ̅ ] (32 ̅ 3 ̅ ) [131 ̅ ] 𝑉12 𝛽 𝑡 (211 ̅ ) [11 ̅ 1] (21 ̅ 1) [111 ̅ ] 𝑉12 𝛽 𝑎𝑡 (211 ̅ ) [1 ̅ 11 ̅ ] (2 ̅ 33 ̅ ) [311 ̅ ] 𝑉1 𝛼 ′′ 𝑡 (110) [11 ̅ 0] (11 ̅ 0) [110] 𝑉1 𝛼 ′′ 𝑎𝑡 (110) [1 ̅ 10] (1 ̅ 30) [310] 𝑉2 𝛼 ′′ 𝑡 (1 ̅ 10) [1 ̅ 1 ̅ 0] (1 ̅ 1 ̅ 0) [1 ̅ 10] 𝑉2 𝛼 ′′ 𝑎𝑡 (1 ̅ 10) [110] (130) [3 ̅ 10]
-the plane located two atomic layers above the twinning plane of the figure is assumed to undergo simple shear during the twinning operation. This necessary shear is applied to the Shuffle is an important parameter to take into account to explain twinning of bcc structures.

Whereas shear of {112}〈111 ̅ 〉 𝛽 𝑎𝑡 is very low, this system is generally not activated in bcc structures due to its high shuffle magnitude (1/2 [11 ̅ 1]). Conversely, no shuffle is necessary for classical {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 , making it easier to activate in spite of its higher shear value [START_REF] Christian | Deformation twinning[END_REF].

The case of " phase is more complex because several dichromatic diagrams can be obtained from its twinning elements due to its double-layered structure. Investigation by high resolution transmission electron microscopy may reveal the real structure of the " twin interface. But this phase disappears when the stress is released and makes this kind of observation difficult. Several configurations were then tested and the most probable configuration, i.e. the one with minimal shear and shuffle magnitudes, is shown on Fig 6d . In this condition, the shear is lower for {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 than for {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 and the atomic shuffle is the same for both twinning systems. Thus, {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 is easier to activate thanks to its low shear and low shuffle magnitudes. Then, as the shuffle value is the same for twinning and antitwinning direction in " phase, the twinning direction has to be the one involving the lowed shear. As a consequence, twinning and antitwinning senses are inverted in " phase in comparison with  phase.

Finally, observation of twins in the antitwinning sense in  phase is due to the activation of twins in SIM " because of lower shear in the antitwinning sense than in the twinning sense of the prior  phase. In other words, it was firstly assumed that twinning senses are the same in both  and " phases, while not. Consequently, what is denoted "antitwinning" in " phase actually corresponds to twinning sense in this phase, but corresponds to antitwinning sense in  phase. Despite this, subscripts (t) and (at) in the notation of twinning systems in " phase will not be changed in order to keep a coherent notation in the whole paper. Actually, they correspond to the sense of twinning of  phase. whereas it is not the case for {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 (Fig. 7d). Therefore, only {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 can accommodate a tensile strain when CV3 is selected as in the present single crystal. This result can be now generalized to any variant as detailed hereafter.

The same procedure can be done directly into the  phase for the variant selection of both or V12would be expected to be activated (Fig. 7f). However, the superimposition of Fig. 7f with Fig. 7b shows that the CV3 of " martensite is activated, and, regarding the possible twin variants of {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 possibly formed from the reversion of twins in " (Fig. 7a), only V2 and V8 can be produced.

  Fig.2ashows the cyclic tensile curve of the [101] 𝛽 single crystal strained from 0 to 5.5% and the corresponding in situ SXRD diffraction profiles under stress (Fig2b) and when the stress is removed (Fig2c). The (110) peak is represented by black circles in SXRD profiles, while (002)", (111)" and (020)" peaks of martensite are labeled with white circles, black squares and stars, respectively. A stress plateau is observed from 0.7% to 3% of applied strain on the tensile curve and corresponds to SIM transformation as shown in SXRD patterns (Fig2b). From 4.5% of applied strain, no more  phase is detected on loading, highlighting a full SIM transformation. " phase disappears and  phase is restored upon unloading until a strain of 4%. Irreversible strain is observed from 4.5% only and residual '' phase is observed upon unloading. These observations support that plastic deformation mechanisms such as dislocation glide and mechanical twinning do not occur extensively below this strain level.

Fig. 2

 2 Fig. 2 Cyclic tensile curve of the Ti-24Nb-4Zr-8Sn single crystal (a). In situ synchrotron X-

Fig. 3

 3 Fig. 3 Optical micrograph of the Ti-24Nb-4Zr-8Sn single crystal after being strained up to

Fig. 5

 5 Fig. 5 Stereographic projections with respect to the orientation of Fig 3(a):  phase (a) and

Fig. 6

 6 Fig. 6 Stereographic projections showing twinning (T in subscript) and antitwinning (AT in subscript) elements for  phase (a) and '' phase (b) and associated dichromatic diagrams (c and d, respectively). Filled circles correspond to atoms in the plane of the figure and empty circles correspond to atoms above the plane of the figure.

  whole twinned crystal and is represented by blue arrows on Fig 6c and Fig 6d; -the plane located one atomic layer above the twinning plane needs additional shuffle to reach the twinning configuration. This shuffle, represented by red arrows, is applied to every white atom of Fig 6c and Fig 6d.

Fig. 7

 7 Fig. 7 Correspondences between variants of martensite, twin variants in martensite and twin variants in  phase (a); IPF showing variant selection of " martensite as a function of TD (b); IPFs of positive Schmid factor values of {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 (c) and {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 (d) in CV3 of " martensite; IPFs of maximum Schmid factor values of {112}〈111 ̅ 〉 𝛽 𝑎𝑡 and {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡

  Fig. 7e and Fig 7f, respectively. The criterion of selection is then the maximum positive 𝑚

  Finally, this point highlights that the classical {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 twinning system can never be formed in  phase due to the CV selection of martensite occurring during a tensile test, whatever the tensile direction. While this CV selection promotes the formation of non-classical {112}〈111 ̅ 〉 𝛽 𝑎𝑡 twins in the antitwinning sense through the activation and the reversion of {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 twins in " martensite. Therefore, a careful examination of activated {112}〈111〉 𝛽 twin variants by Schmid factor analysis allows to determine if a SIM transformation has occurred: If the observed {112}〈111〉 𝛽 twins are in the twinning sense of classical bcc structures, twins are directly formed into the  phase, while {112}〈111〉 𝛽 twins in the antitwinning sense are the proof of the reversion of twins previously formed into a SIM " phase. 4. Conclusions A single crystal of metastable  Ti-24Nb-4Zr-8Sn alloy was studied. In situ SXRD during cyclic tensile test shows that full martensitic transformation is achieved when plastic deformation begins. Deformation twins are observed post mortem after tensile test and are identified as {112}〈111〉 𝛽 twins. Careful Schmid factor analysis of the observed {112}〈111〉 𝛽 twins shows that they are {112}〈111 ̅ 〉 𝛽 𝑎𝑡 antitwins with a twinning direction opposite to that of classical {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 twins of bcc structure, i.e. along the antitwinning direction. This unusual observation is explained by the stress-induced martensitic transformation occurring during tensile test. Observed twins in  phase are actually the result of the reversion during the reverse SIM transformation of activated {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 twins in " martensite. The magnitude of shuffle drops then from 1/2 [11 ̅ 1] 𝛽 if these twins were directly formed in  phase, to 1/5 [1 ̅ 10] 𝛼 ′′ after SIM transformation, making it easier to activate in " phase. Moreover, variant selection of stress-induced martensite always leads to a martensitic crystal which is favorably oriented for {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 and unfavorably oriented for {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 twinning system, according to Schmid factor analysis. Finally, this former feature demonstrates that the occurrence of " SIM transformation promotes the observation of {112}〈111 ̅ 〉 𝛽 𝑎𝑡 antitwins and totally impedes the formation of classical {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 twins of bcc structures due to a combination of variant selection of stress-induced martensite and occurrence of reverse martensitic transformation.

Table 2 .

 2 Principal twinning elements of the 12 variants of {112}〈111〉 𝛽 and 2 variants of {110}〈110〉 𝛼 ′′ considering the twinning sense 𝜂 1𝑡 and antitwinning sense 𝜂 1𝑎𝑡 .

	Variant	K1	1t	1at
	V1

Table 3

 3

	. Schmid factors of the 12 variants of {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 and {112}〈111 ̅ 〉 𝛽 𝑎𝑡 for a [101] 𝛽
	tensile direction and the 2 variants of {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 and {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡 for a [010] 𝛼 ′′ tensile
	direction.
	Variant	𝑚 𝑡	𝑚 𝑎𝑡
	V1	0,000	0,000
	V2	-0,471	0,471
	V3	0,000	0,000
	V4	0,236	-0,236
	V5	0,000	0,000
	V6	0,236	-0,236
	V7	0,000	0,000
	V8	-0,471	0,471
	V9	0,000	0,000
	V10	0,236	-0,236
	V11	0,000	0,000
	V12	0,236	-0,236
	V1"	-0,463	0,463
	V2"	-0,463	0,463

Table 4 .

 4 Twinning elements for twinning systems in the twinning and antitwinning senses in  and " phases and associated shear (s) and shuffle magnitudes.

	Twinning system	K1	1	K2	2	P	s	shuffle
								magnitude
	{112}〈1 ̅1 ̅1〉 𝛽 𝑡 	(121)	[1 ̅ 11 ̅ ]	(1 ̅ 21 ̅ )	[111]	(101 ̅ )	0.707	0
	{112}〈111 ̅〉 𝛽 𝑎𝑡 	(121)	[11 ̅ 1]	(32 ̅ 3)	[131]	(101 ̅ )	0.354	1/2 [11 ̅ 1]
	𝑡  {110}〈11 ̅0〉 𝛼 ′′ {110}〈1 ̅10〉 𝛼 ′′ 𝑎𝑡	(110) (110)	[11 ̅ 0] [1 ̅ 10]	(11 ̅ 0) (1 ̅ 30)	[110] [310]	(001) (001)	0.821 0.259	1/5 [11 ̅ 0] 1/5 [1 ̅ 10]

Table 5 .

 5 Twinning elements of the 12 variants of {112}〈1 ̅ 1 ̅ 1〉 𝛽 𝑡 , {112}〈111 ̅ 〉 𝛽 𝑎𝑡 and the 2 variants of {110}〈11 ̅ 0〉 𝛼 ′′ 𝑡 and {110}〈1 ̅ 10〉 𝛼 ′′ 𝑎𝑡

Acknowledgements

The authors acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation facilities and would like to thank Yves Watier for assistance in using the beamline ID31 (presently ID22). The authors also acknowledge the THEMIS platform of the University of Rennes for providing access to TEM facilities. The authors finally acknowledge Prof. Yulin Hao and Dr. Haoliang Wang in Institute of Metal Research Chinese Academy of Sciences for providing single crystal. This work was financially supported by the National Science Foundation for Young Scientists of China (Grant No. 51801199).

Compliance with Ethical Standards:

The authors declare that they have no conflict of interest.