Yingchao Li

Bashir S Shariat

Hong Yang

Satyajit Sarkar

Junsong Zhang

Yunzhi Wang

Denis Favier

Yinong Liu

A mechanical criterion for Lüders-type deformation of polycrystalline NiTi

Keywords: NiTi, Shape memory alloy, Martensitic phase transformation, pseudoelasticity, Lüders deformation

NiTi shape memory alloys are known to exhibit a Lüders-type deformation behaviour associated with its stress-induced martensitic transformation. Such phase-transformation-caused Lüders deformation presents a different metallurgical mechanism to the dislocation-related Lüders deformation of mild steels. To explain this phenomenon, many hypotheses have been proposed in the literature. However, all these explanations seem to contradict some experimental evidence in one way or another. In this work, a new explanation for the occurrence of the Lüders-type deformation in NiTi is proposed and a model is derived based on the criterion of conservation of sample length changes produced by the stress-induced transformation, the crosshead movement of the testing machine and the elastic contraction. The model is verified by tensile experiment and digital image correlation measurement. The new model is able to describe the upper-lower yielding phenomenon and the plateau stress of the Lüders-type deformation behaviour.

Introduction

NiTi shape memory alloys (SMAs) are known to exhibit a Lüders-type deformation behaviour [START_REF] Shaw | On the nucleation and propagation of phase transformation fronts in a NiTi alloy[END_REF][START_REF] Miyazaki | Lüders-like deformation observed in the transformation pseudoelasticity of a Ti-Ni alloy[END_REF], analogous to the Lüders deformation behaviour of mild steels [START_REF] Hall | Yield point phenomena in metals and alloys[END_REF]. The Lüders-type deformation in NiTi occurs during stress-induced martensitic transformation [START_REF] Tan | Lüders-like deformation associated with stress-induced martensitic transformation in NiTi[END_REF] or martensite reorientation [START_REF] Liu | Lüders-like deformation associated with martensite reorientation in NiTi[END_REF], and for both the forward B2B19' transformation upon loading and the reverse B19'B2 transformation upon unloading [START_REF] Shaw | On the nucleation and propagation of phase transformation fronts in a NiTi alloy[END_REF]. The Lüders-type deformation has been observed to occur with an upper-lower yielding followed by a stress plateau with a strain span of typically 5~7% [START_REF] Sittner | On the origin of Lüders-like deformation of NiTi shape memory alloys[END_REF]. Such upper-lower yielding behaviour presents a typical situation of mechanical instability. Similar Lüders-type deformation behaviour has also been observed for stress-induced martensitic transformations in other alloy systems, such as NiFeGa (B210M14ML 10 phase transformations) [START_REF] Efstathiou | Full-field strain evolution during intermartensitic transformations in single-crystal NiFeGa[END_REF], CuZnAl (β18R6R) [START_REF] De Castro Bubani | Improvements in the mechanical properties of the 18R↔ 6R high-hysteresis martensitic transformation by nanoprecipitates in CuZnAl alloys[END_REF][START_REF] Yaacoub | Martensite variant localization effects on fatigue crack growth-The CuZnAl example[END_REF] and FeMnNiAl (BCCFCC) [START_REF] Sidharth | Relationship between functional fatigue and structural fatigue of iron-based shape memory alloy FeMnNiAl[END_REF].

The Lüders-type deformation behaviour of NiTi has been observed to occur only in tension, and not in compression [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF][START_REF] Reedlunn | Tension, compression, and bending of superelastic shape memory alloy tubes[END_REF], shear [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF][START_REF] Sun | Phase transformation in superelastic NiTi polycrystalline micro-tubes under tension and torsion--from localization to homogeneous deformation[END_REF], or biaxial tension [START_REF] Grolleau | Assessment of tension-compression asymmetry of NiTi using circular bulge testing of thin plates[END_REF]. Some studies reported the observation of strain localisation in simple shear [START_REF] Huang | An experimental study of NiTi alloy under shear loading over a large range of strain rates[END_REF] and torsion of thin wall tubes [START_REF] Reedlunn | Axial-torsion behavior of superelastic tubes: Part I, proportional isothermal experiments[END_REF] and wires [START_REF] Sedlák | 3D spatial reconstruction of macroscopic austenite-martensite transition zones in NiTi wires induced by tension and twisting using diffraction/scattering computed tomography[END_REF], but in all these cases the stress response increases monotonically with strain without the signature upper-lower yielding and the stress plateau of the Lüders deformation. In bending mode, the localized Lüders deformation appears in the tension side but not in the compression side [START_REF] Reedlunn | Tension, compression, and bending of superelastic shape memory alloy tubes[END_REF]. In this regard, the Lüders-type deformation behaviour is a general phenomenon for stress-induced martensitic transformations and yet appears to be a special occurrence only to some specific mechanical conditions such as tension.

The Lüders deformation of NiTi has also been studied by various means in addition to mechanical testing. Video cameras [START_REF] Shaw | On the nucleation and propagation of phase transformation fronts in a NiTi alloy[END_REF][START_REF] Li | The initiation and growth of macroscopic martensite band in nano-grained NiTi microtube under tension[END_REF][START_REF] Feng | Experimental investigation on macroscopic domain formation and evolution in polycrystalline NiTi microtubing under mechanical force[END_REF][START_REF] Shaw | Initiation and propagation of localized deformation in elasto-plastic strips under uniaxial tension[END_REF], infrared cameras [START_REF] Shaw | On the nucleation and propagation of phase transformation fronts in a NiTi alloy[END_REF][START_REF] Shariat | Computational and experimental analyses of martensitic transformation propagation in shape memory alloys[END_REF][START_REF] Shariat | Controlled initiation and propagation of stress-induced martensitic transformation in functionally graded NiTi[END_REF][START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline Ti-50.8 at.% Ni thin tube under tension. Investigation via temperature and strain fields measurements[END_REF][START_REF] Louche | Heat source processing for localized deformation with non-constant thermal conductivity. Application to superelastic tensile tests of NiTi shape memory alloys[END_REF] and digital image correlation (DIC) technique [START_REF] Reedlunn | Tension, compression, and bending of superelastic shape memory alloy tubes[END_REF][START_REF] Huang | An experimental study of NiTi alloy under shear loading over a large range of strain rates[END_REF][START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline Ti-50.8 at.% Ni thin tube under tension. Investigation via temperature and strain fields measurements[END_REF][START_REF] Elibol | Virtual Extensometer Analysis of Martensite Band Nucleation, Growth, and Strain Softening in Pseudoelastic NiTi Subjected to Different Load Cases[END_REF][START_REF] Shariat | On the Lüders band formation and propagation in NiTi shape memory alloys[END_REF][START_REF] Delpueyo | Applying Full-Field Measurement Techniques for the Thermomechanical Characterization of Shape Memory Alloys: A Review and Classification[END_REF][START_REF] Furgiuele | Functional and structural fatigue of pseudoelastic NiTi: global vs local thermo-mechanical response[END_REF] have been used to observe the macroscopic behaviour of Lüders bands in situ during mechanical testing. Synchrotron X-ray diffraction [START_REF] Young | Phase volume fractions and strain measurements in an ultrafine-grained NiTi shape-memory alloy during tensile loading[END_REF][START_REF] Bian | An in-situ synchrotron study of the B2→ B19′ phase transformation in a Ni-Ti alloy subjected to uniaxial monotonic tension[END_REF][START_REF] Gao | In-situ synchrotron diffraction study of the localized phase transformation and deformation behavior in NiTi SMA[END_REF][START_REF] Sedmák | Grain-resolved analysis of localized deformation in nickel-titanium wire under tensile load[END_REF], neutron diffraction [START_REF] Šittner | In situ neutron diffraction studies of martensitic transformations in NiTi polycrystals under tension and compression stress[END_REF] and electron backscatter diffraction (EBSD) [START_REF] Mao | EBSD studies of the stress-induced B2-B19′ martensitic transformation in NiTi tubes under uniaxial tension and compression[END_REF] techniques have also been used to elucidate the phase transformation and microscopic metallurgical processes of the Lüders-type deformation. Transmission electron microscopic (TEM) examination has been carried out to study the detwinning process of martensite variants during Lüders-type deformation [START_REF] Liu | On the deformation of the twinned domain in NiTi shape memory alloys[END_REF].

From these experiments, a broad empirical knowledge base of the Lüders-type deformation of NiTi has been established. Among the many characteristics of the Lüders-type deformation in NiTi, the following needs specific mentioning here. The Lüders-type deformation initiates by the sudden appearance of a narrow and localized strain band and proceeds by its propagation through the length of the sample over a stress plateau. The initiation of the band is associated with a stress drop, as commonly observed. Associated with the stress drop is an elastic strain reduction of the austenite part outside the newly formed Lüders band, as reported by Elibol and Wagner [START_REF] Elibol | Virtual Extensometer Analysis of Martensite Band Nucleation, Growth, and Strain Softening in Pseudoelastic NiTi Subjected to Different Load Cases[END_REF] and later by Shariat et al. [START_REF] Shariat | On the Lüders band formation and propagation in NiTi shape memory alloys[END_REF]. During the propagation of the Lüders band, the strain inside the band remains relatively constant at a high value corresponding to the end of the global stress plateau whereas the strain outside the band also remains constant at a low value corresponding to the beginning of the stress plateau [START_REF] Tan | Lüders-like deformation associated with stress-induced martensitic transformation in NiTi[END_REF][START_REF] Liu | Lüders-like deformation associated with martensite reorientation in NiTi[END_REF][START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline Ti-50.8 at.% Ni thin tube under tension. Investigation via temperature and strain fields measurements[END_REF]. Upon the arrival of the Lüders band, the local strain increases rapidly from the low level to the high level [START_REF] Huang | An experimental study of NiTi alloy under shear loading over a large range of strain rates[END_REF][START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline Ti-50.8 at.% Ni thin tube under tension. Investigation via temperature and strain fields measurements[END_REF][START_REF] Young | Phase volume fractions and strain measurements in an ultrafine-grained NiTi shape-memory alloy during tensile loading[END_REF][START_REF] Sedmák | Grain-resolved analysis of localized deformation in nickel-titanium wire under tensile load[END_REF]. Shariat et al. conducted a DIC measurement and reported that the local strain rates at the initiation and during the propagation of a Lüders band are orders of magnitude higher than the nominal strain rate applied by the testing machine [START_REF] Shariat | On the Lüders band formation and propagation in NiTi shape memory alloys[END_REF].

The martensitic transformation associated the Lüders-type deformation in shape memory alloys represents clearly a different metallurgical process from the dislocation-related Lüders deformation of mild steels [START_REF] Corona | Buckling of steel bars with Lüders bands[END_REF][START_REF] Weidner | Deformation Processes in TRIP/TWIP Steels: In-situ Characterization Techniques[END_REF]. This requires a new understanding of the mechanism of the phenomenon. Despite the many studies, most of the published work in the literature focus on the characterisation of the phenomenon and only a few have devoted their attention to explaining the underlying mechanisms of the phenomenon. Miyazaki et al. first reported the Lüders-type deformation in NiTi and attributed the phenomenon to the effect of local stress concentrations at grain boundaries in the small-grain matrix of the alloy used [START_REF] Miyazaki | Lüders-like deformation observed in the transformation pseudoelasticity of a Ti-Ni alloy[END_REF]. This may support the self-catalytic nature of the martensitic transformation and thus the strain localisation, but clearly cannot differentiate between tension and compression/shear deformation modes. Shaw and Kyriakides [START_REF] Shaw | On the nucleation and propagation of phase transformation fronts in a NiTi alloy[END_REF] attributed the phenomenon to a higher stress required for martensitic transformation nucleation than that for martensite growth, in analogy to the phenomenon of nucleation supercooling for solidification. This explanation is challenged by a few considerations. For example, the Lüders-type deformation also occurs during stress-induced martensite reorientation, where no phase transformation (thus nucleation) is involved. For stress-induced transformations, the Lüders-type deformation is not observed in shear or compression [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF][START_REF] Liu | On the nucleation and propagation of stress-induced martensitic transformation in NiTi[END_REF] or in biaxial tension [START_REF] Grolleau | Assessment of tension-compression asymmetry of NiTi using circular bulge testing of thin plates[END_REF] where martensite nucleation is required for the transformation. In addition, when a Lüders band propagates through a polycrystalline matrix, concurrent nucleation and growth of martensite occur in every grain continuously throughout the propagation of the band from one end of the sample to the other. In this regard, this explanation is also unsupported.

Mao et al. used EBSD method to characterise the Schmid factors of various martensite variants in a polycrystalline NiTi and calculated their critical resolved shear stresses for stress-induced transformation. They compared these critical resolved shear stresses between tension and compression and explained the absence of Lüders deformation in compression on the basis of these differences [START_REF] Mao | EBSD studies of the stress-induced B2-B19′ martensitic transformation in NiTi tubes under uniaxial tension and compression[END_REF]. In this work, the occurrence of Lüders deformation in tension is attributed to the self-catalytic effect of martensite, i.e., the stress field caused by the formation of one martensite variant in one grain triggers the formation of more martensite variants in an avalanche manner in the surrounding austenite grains. This will lead to a continuously radiating and expanding transformation region, as opposed to disconnected pockets, but how it is in the form of a band is not explained [START_REF] Mao | EBSD studies of the stress-induced B2-B19′ martensitic transformation in NiTi tubes under uniaxial tension and compression[END_REF]. Sittner et al. [START_REF] Sittner | On the origin of Lüders-like deformation of NiTi shape memory alloys[END_REF] took a micromechanics modelling approach and provided a similar explanation for the tension-compression disparity for the occurrence of Lüders deformation in polycrystalline NiTi. Their modelling suggests that a unique partitioning of stresses, strains, and phase fractions among grains for textured polycrystalline NiTi exists under tension that allows it to deform via stress-induced martensitic transformation with very small deformation modulus over the stress plateau (often on an engineering stress-strain curve), thus favourable for a localised band deformation. Such partitioning does not occur in compression. Whereas a very low (and positive) deformation modulus over the apparent stress plateau for the stress-induced transformation may be favourable for a localized deformation, it does not present a strain softening (negative modulus) or a mechanical instability condition for an upper-lower yielding, which is essential for a Lüders deformation behaviour.

To overcome this conundrum, Hallai and Kyriakides [START_REF] Hallai | Underlying material response for Lüders-like instabilities[END_REF] conducted a carefully designed comparative experiment using sandwiched steel-NiTi-steel and steel-steel-steel plate samples and derived a "natural" mechanical behaviour of stress-induced martensitic transformation of NiTi when strain localization is prevented from occurring (by the steel casing). This behaviour features a continuously decreasing stress, i.e., a continuous strain softening, over the course of the transformation in the sample. Such behaviour has enabled several successful finite element simulations of the Lüders-type deformation of NiTi samples [START_REF] Jiang | Modeling of propagation of phase transformation fronts in NiTi under uniaxial tension[END_REF][START_REF] Frost | Thermomechanical model for NiTi-based shape memory alloys covering macroscopic localization of martensitic transformation[END_REF][START_REF] Jiang | Propagation of phase transformation fronts in pseudoelastic NiTi tubes under uniaxial tension[END_REF][START_REF] Rezaee-Hajidehi | Gradient-enhanced thermomechanical 3D model for simulation of transformation patterns in pseudoelastic shape memory alloys[END_REF]. However, neither the original work nor the later works have offered a metallurgical justification of such a behaviour, i.e., why a stress-induced transformation will proceed with a progressively decreasing global stress. In summary, to date, a credible explanation of the mechanism of the occurrence of the Lüders deformation in NiTi is still absent.

In this work, we propose a new explanation for the occurrence of the Lüders-type deformation in NiTi based on the consideration of the mismatch between the fast speed of stress-induced martensitic transformation and the slow speed of deformation applied by the testing machine.

The local strain rate of Lüders deformation in NiTi has been measured to be more than two orders of magnitude higher than the global strain rate applied by the testing machine [START_REF] Shariat | On the Lüders band formation and propagation in NiTi shape memory alloys[END_REF]. Such large discrepancy between the two must have direct impact on the deformation behaviour of the alloy in tension. This paper presents the argument, the mathematical formulation, and the experimental verification of this mechanism for the occurrence of Lüders-type deformation in NiTi.

Experimental method

A commercial polycrystalline Ti-50.9at.%Ni sheet from Memry Corp. was used in this study.

The sheet was 0.52 mm in thickness. The transformation behaviour of the as-received alloy was measured by differential scanning calorimetry (DSC) using a TA Instrument Discovery DSC 25 unit with a heating/cooling rate of 10 K/min. Dog-bone shaped tensile samples (Fig. 1) were fabricated by wire cutting using an electrical discharge machine (EDM). The samples have a gauge section of 2.530 mm. Tensile testing was conducted using a WDT-10 universal mechanical testing machine at 28 C (±0.5 C) in an air-conditioned environment. A slow nominal strain rate of 110 -4 s -1 was used to minimize the heating effect caused by the release of latent heat during the stress-induced martensitic transformation.

Digital image correlation (DIC) method was used to measure the strain fields on the surface of the tensile samples. The camera used was Sony α7R IV with 60.2 megapixels in a 95046336-pixel format. This gives a camera resolution of 3.76 m on the sensor and 5.25 m resolution on the sample on the image of 50×28 mm in dimension. This also implies a strain measurement resolution of 0.0175%. DIC analysis was conducted using the GOM Correlate Pro software. For this analysis, a grid size of 35 m was used, which yielded reliable data for local strain measurement. Images were captured at 1 Hz frequency.

Theoretical formulation of Lüders deformation mechanism in NiTi

The hypothesis

It is a common and fundamental knowledge that martensitic transformation occurs locally at a speed close to that of sound in the alloy [START_REF] Kaufmann | Thermodynamics and Kinetics of Martensitic Transformation[END_REF]. This is due to the need to transmit the large lattice distortion strains through the matrix to a free surface or an internal interface so to be accommodated. The speed of this transmission is the speed of the mechanical waves of atoms in the lattice. Experimentally, the martensitic transformation in a Fe-29.5Ni alloy has been measured to propagate at a speed of ~1×10 6 mm/s, corresponding to roughly 1/3 of the speed of sound in the alloy [START_REF] Bunshah | Rate of propagation of martensite[END_REF]. The same behaviour has also been observed under high-speed camera on a NiTi thin-wall tube as the sudden formation of a transformation band, of which the interface front velocity was estimated to be up to 2×10 3 mm/s [START_REF] Feng | Experimental investigation on macroscopic domain formation and evolution in polycrystalline NiTi microtubing under mechanical force[END_REF]. This is an intrinsic nature of the martensitic transformation. Recently, Shariat et al. used DIC technique to measure the local strain rates of Lüders deformation of a polycrystalline NiTi in tension and determined a local strain rate of 22310 -4 s -1 for the nucleation of a Lüders band [START_REF] Shariat | On the Lüders band formation and propagation in NiTi shape memory alloys[END_REF], which is 223 times of the global strain rate applied by the testing machine, or corresponding to a sample elongation speed several times higher than the crosshead movement speed. This implies that the testing device is unable to provide the necessary elongation required by the formation of the Lüders band. In other words, a Lüders band cannot form unless additional elongation is provided to it. With these thoughts, we hypothesize that elastic contraction of the sample outside the band is the source of this "additional" elongation required for the formation of the Lüders band. This is consistent with the observed stress drop [START_REF] Shaw | On the nucleation and propagation of phase transformation fronts in a NiTi alloy[END_REF] and elastic strain drop [START_REF] Elibol | Virtual Extensometer Analysis of Martensite Band Nucleation, Growth, and Strain Softening in Pseudoelastic NiTi Subjected to Different Load Cases[END_REF] observed at the initiation of a Lüders band. In addition, it is also known that elastic deformation occurs by the same mechanism of lattice waves of atoms, i.e., also at the speed of sound in the solid, thus has the intrinsic ability to match in speed with the martensitic transformation.

In this hypothesis, the occurrence of martensitic-transformation-associated Lüders-type deformation must have four essential conditions:

(1) The martensite must form into a continuous band and the band must reach the free surface of the sample due to the need to accommodate (relax) the large lattice strain energy of the transformation.

It is known that the martensitic transformation is associated with a lattice distortion of a finite lattice strain [START_REF] Otsuka | Physical metallurgy of Ti-Ni-based shape memory alloys[END_REF]. Due to the diffusionless nature of the transformation [START_REF] Delaey | Diffusionless transformations[END_REF], the lattice distortion of the unit cell manifests as a shape change of a martensite (M) crystal relative to the austenite (A) it replaces. In a thermally-induced martensitic transformation, these local shape changes of the martensite crystals are self-accommodated in variants of different orientations [START_REF] Miyazaki | The shape memory mechanism associated with the martensitic transformation in Ti-Ni alloys-II. Variant coalescence and shape recovery[END_REF]. In a stress-induced martensitic transformation, the selection of variants is dictated by the need to produce a maximum strain in the loading direction, thus there is no freedom for self-accommodation (in fact, there is still a secondary self-accommodation among the variants in the lateral directions to maintain a nil lateral shape distortion and loading axis alignment). In this case, the formation of one martensite in one grain with an elongated dimension relieves itself immediately from bearing the load, thus redistributing the load to the adjacent grains in parallel to the loading axis and preferably triggering the transformation there. This has been referred to as the self-catalytic or the avalanche effect of martensitic transformations. It dictates the martensite to form into a continuous region (band), as opposed to scattered and disconnected islands, until a free surface is reached, so to maintain the matrix continuity and to avoid excessively large and prohibitive internal elastic lattice distortions. From a mechanics viewpoint, in tension, this localized martensite region forms into a narrow band across the entire cross-section of the sample in series configuration with the untransformed part of the sample in the loading direction (same for compression). In the case of simple shear, the band runs across the sample in parallel configuration with the shear direction.

(2) The localized transformation band (Lüders band) must have a finite length (Note: this refers to the dimension of the band along the gauge length of a tensile sample, or the direction of the tensile load. This dimension has also sometimes been intuitively referred to as the width of the band when the band is newly formed and is narrow and thin. In this discussion, we use "length" so to be mathematically more consistent with the mechanics of the situation) in a polycrystalline matrix, so to be able to accommodate sufficient number of grains for coordination among the strain tensors of the variants in neighbouring grains across their boundaries.

(3) The localized transformation band must have a discrete strain output. This is apparently true considering the transformation crystallography and it is a collective manifestation of the crystallographic changes of the martensite variants in a polycrystalline matrix.

(4) The local strain rate (in the direction of loading) of this localized transformation band, which is a collective manifestation of the speed of the martensitic transformation, is high, in fact, much higher than the nominal strain rate applied by the testing machine.

The combination of the four conditions above produces a fast elongation rate by the sample upon the occurrence of the Lüders-type deformation. This implies that the testing machine must provide such fast elongation rate to allow the Lüders-type deformation to occur, i.e., meeting this elongation rate is an essential condition for Lüders-type deformation. (

The same may also be expressed in differential form, i.e.,

which states that the rate of sample length elongation applied by the testing machine equals the sum of those provided by the activity (formation or propagation) of the Lüders band and the change of elastic strain in the sample.

To derive the model, the following variables are defined:

is the local strain of the austenite, and is the local strain inside the Lüders band.

is the global strain over the full gauge length based on direct DIC measurement at momont t (i.e., it is a function of time t).

is the machine crosshead movement speed, which can be obtained from the experiment.

is the length of the original austenite corresponding to the active length of the Lüders band ( ). The active length of a Lüders band is the full length of the band during its initiation and is the front transition zone of a band during its propagation. can be measured from the experiment, but cannot be controlled.

is the length of the original austenite that has transformed to the Lüders band of length . It is a function (variable) of time t (since the Lüders band grows with time), i.e., it is an instantaneous length at t. It is also the length of the original austenite at the stress-free state (no elastic strain). It can be computed as , where can be measured from the experiment and controlled during the experiment.

is the initial gauge length of the sample at the stress-free state, which can be obtained from the experiment. Thus is the original length of the section of the sample excluding the active section of the Lüders band, which participates in elastic contraction during Lüders deformation.

is the local strain rate within the newly formed Lüders band, which can be measured from the experiment, but cannot be controlled.

is the strain rate of the elastic contraction of the sample in the section excluding , which can be measured from the experiment, but cannot be pre-determined or controlled.

Some notes should also be added on the notation of the parameters: (vi) Three different strain terms are used in this discussion, including (i) nominal strain, (ii) global strain and (iii) local strain. These are all engineering strains. Consequently, there are three corresponding strain rates. The nominal strain refers to the crosshead displacement divided by the original gauge length of the sample. In the case of localized heterogeneous deformation, this strain does not represent any actual strain in the sample.

The global strain is the strain measured by DIC over full gauge length of the sample. This is in principle the same as the nominal strain, but more practically accurate without the side effects of the shoulders of the dog-bone shaped sample and the griping etc. The global strain is used in the model analysis, whereas the nominal strain is used only in discussion to express the argument on the effect of the machine speed. The local strain is DIC data processing, the local strain is an average strain within that dimension.

Model development

Characterisation of Lüders-type deformation

To develop the analytical solution of the conceptual model expressed in Eqs. ( 1) and ( 2), some parameters need to be determined experimentally. Point P1 is at the initiation site of the Lüders band and P2 is far away in the austenite region. Fig. 2(c) shows the strain-time curves for the same test, including the global strain (the black curve) and the two local strains at points P1 and P2 on the sample (shown in Fig. 1) as determined by DIC method. It is seen that the local strains at both P1 and P2 increased linearly with time, apparently corresponding to the elastic deformation of the sample, until 190 s when the P1 local strain increased suddenly, demonstrating the burst formation of the Lüders band. The local strain rate at P1 reached 22310 -4 s -1 , which was 223 times higher than the applied global strain rate (110 -4 s -1 ). The local strain at P2 remained relatively unchanged until ~520 s when it also increased suddenly, signalling the arrival of the Lüders band front.

The local strain rate at P2 when the Lüders band front swept through reached 2610 -4 s -1 , which was 26 times higher than the applied global strain rate. It is also to be noted that the magnitude of strain increases at P1 and P2 is the same, representing the strain of the Lüders band. This strain rise corresponds to ( ) in global strain on the stress-strain curve.

Analytical solution of the model

To derive the solutions of the three terms in the model expressed in Eq. ( 1), let us examine the sample length changes between any two points in time during the Lüders-type deformation process, i.e., from to . During this period, the crosshead displacement can be computed as:

(

where and are the global strains of the sample at and respectively, is the speed of the movement of the crosshead of the testing machine, and is the generic variable time. They are measured directly from the experiment.

To solve the second term (note: it is important to reiterate that this is not the change of the length of a Lüders band, but the sample elongation caused by the formation or expansion of a Lüders band), let us examine the change of from to , as schematically expressed in Fig. 2 The third term is the elastic contraction of the sample, as seen in Fig. 2(d). This elastic contraction occurs to the entire length of the sample, except the active part of the Lüders band ( with the corresponding length of the original austenite being ) which transforms and produces the discussed above, i.e., the sample length that participates in elastic contraction is . Thus, the elastic contraction can be expressed as:

(5)

In this equation, and represent the average (elastic) strains of the length of the sample excluding the active region of the transformation . Apparently, here the elastic moduli of the austenite (A) and the martensite (M) are assumed the same [START_REF] Alonso | Characterizing transformation phenomena and elastic moduli of austenite and oriented martensite of superelastic thin NiTi wire through isothermal dynamic mechanical analysis[END_REF]. Thus, represents the elastic strain change (contraction is designated negative)

of the sample length . The active length evolves with time during the formation of the Lüders band and remains constant during its propagation [START_REF] Shariat | On the Lüders band formation and propagation in NiTi shape memory alloys[END_REF]. It is believed to be determined or influenced by three intrinsic factors, including (1) the transformation crystallography, which determines the discrete strain jump between the two phases that the transition zone needs to accommodate, (2) metallurgical condition of the matrix (e.g., grain size, texture), which needs to rally and organize sufficient grains (martensite variants) to form a reasonably continuous strain field in the context of continuum mechanics to accommodate the strain transition, and (3) the cross-section geometry of the sample (e.g., wire, rod, thin plate or thick plate) which a Lüders band transition zone has to penetrate through. It is also clear that all parameters required in Eq. ( 5) can be determined experimentally.

Substituting Eqs. ( 3), ( 4) and ( 5) into Eq. ( 1) and solving the new equation for the elastic strain contraction gives:

Following the Hooke's law, the stress drop from to can be determined as: [START_REF] Efstathiou | Full-field strain evolution during intermartensitic transformations in single-crystal NiFeGa[END_REF] at point "1" and ends at point "4" as designated in Fig. 2(b). In this case, corresponds to point "1" on the stress-strain curve, , [START_REF] Liu | Lüders-like deformation associated with martensite reorientation in NiTi[END_REF] (the sample is in full austenite state and the strain is uniform in the sample), and . Also substituting for convenience of experimental measurement, Eqs. ( 6) and ( 7) are reduced to:

(10) (11)
When corresponds to point "4" on the stress-strain curve, Eq. ( 12) expresses the full stress drop of the Lüders-type deformation. In this case, and .

Parameter is the strain within the Lüders band at point "4". Considering the Lüders-type deformation nature, , where is the global strain at the end of the stress plateau on the stress-strain curve seen in Fig. 2(a) [START_REF] Liu | Lüders-like deformation associated with martensite reorientation in NiTi[END_REF]. Thus:

In this equation, , and are all directly measured from the global stress-strain curve of a conventional tensile mechanical testing, is the sample initial gauge length, and is length (dimension in the loading direction) of the Lüders band at point "4", which needs to be determined visually, e.g., using a visible camera, an infrared camera or by means of DIC technique. Based on Eq. ( 12), it can be recognized that the full stress drop of the Lüders-type deformation is dependent on two parameters, in addition to the three characteristic strains mentioned above: sample initial gauge length and the length of the Lüders band (which is re-denoted for generality) at its full birth prior to propagation.

It is expected that is dependent on the crystallography of the transformation, magnitude of the lattice distortion strain, grain size and texture of the austenite matrix, and the sample geometry.

Model Simulation of the stress-strain response of the Lüders-type deformation

Experimental data preparation

Fig. 3 shows a time resolved DIC strain analysis of the deformation process shown in Fig. 1(a). Fig. 3(a) plots the axial normal strain ( ) profiles along the central line of the sample at different times from point "1" to point "5" as indicated on the stress-strain curve seen in Fig. It is seen that in the initial stage of the formation of the Lüders band both the peak strain and the length (width of the strain profile) of the Lüders band increased. During this stage, small changes also occurred to the austenite part outside the band. Fig. 3(c) shows an enlarged view of the progressive decrease of the strain of the austenite at time points "1-5". This corresponds to the strain-time shown in Fig. 2(d), which is measured at one observation point.

2(b)

Fig. 3(d) plots the time evolutions of the peak strain ( ) and the length ( ) (measured as the width of the strain profile at half height) of the Lüders band. It is seen that the strain rate ( ) is the most rapid between points "1" and "2" (within 2 seconds), which is measured to be 0.0223 s -1 . This strain rate is 223 times of the applied global strain rate. This demonstrates the burst initiation of the Lüders band. This is followed by a continued growth of the peak strain inside the Lüders band to the maximum ( ) at point 3. After this, further sample elongation is provided by the expansion of the length of the Lüders band, and not further increase of the peak strain inside the Lüders band. In this is regard, the Lüders-type deformation process may also be considered in two distinctive stages at point 3. It is also to be pointed out that prior to point "3", the active length of the Lüders band contributing to elongation is the full length of the band (full width of the strain profile), i.e., following the path of the red curve in (d). After point "3", the active length of the Lüders band becomes one moving front of the band (the second front is arrested by the end of the sample in this case), i.e., following the dashed red line in (d).

Using exponential curve fitting, the and curves shown in Fig. 3(d) can be empirically expressed as:

Model simulation of the experimental data

Using the experimental data and as expressed by Eq. ( 13) and ( 14), the Lüders deformation behaviour can be analysed. First let's examine the sample elongation produced by the Lüders band, which is described by Eq. ( 4). The Lüders deformation commences at point "1". At this point, is set as 0 ( ), is taken as the time variable ( ), , , and given that , Eq. ( 4) becomes:

Using the experimental data and as expressed by Eq. ( 13) and ( 14) for and respectively, is computed as a function of time using Eq. ( 15) and plotted in Fig. 4(a). Using Eq. ( 3), is also calculated as a function of time and plotted in the figure. 3)) and at point "4", the magnitude of the stress drop reached the maximum (Fig. 2). These two points represent two different characteristic points in time of the Lüders deformation process.

The stress-strain behaviour of the Lüders deformation (the section between point "1" and "7" on the stress-strain curve shown in Fig. 2(a)) can also be calculated by the model. The stress drop is described by Eq. ( 7). As discussed above, when calculated from point "1", Eq. ( 7)

is reduced to Eq. [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF]. Further, for the stage from point "1" to point "3", (Fig.

3(d))

, which is defined by Eq. ( 13), is defined by Eq. ( 14), and with =1.0% as the global strain and =436 MPa at point "1" (both measured from the stress-strain curve shown in Fig. 2(a)), =22 GPa being the apparent modulus of the austenite section as determined directly from the stress and strain measurements shown in Figs. 2(b) and (d),

=30 mm being the sample gauge length, and as the variable, then using Eq. [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF] the stress drop is calculated as a function of and as the stress is plotted sudden formation. On the other hand, the supply of a sufficient elastic strain contraction is dependent on the sample having a sufficient gauge length for a given , as per Eq.

(5).

It has been considered that the absence of Lüders-type deformation in compression and shear is possibly caused by the generally short gauge length used in such tests. However, the following experimental evidence defies such a conclusion. Orgéas and Favier [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF] conducted a comparative tension and compression test of identical typical "dog-bone" plate samples with narrow stripe gauge section of 40 mm in length (gauge section dimension 405.62.7 mm), using a specially designed constraint device to prevent compression buckling. This is probably the highest length to width/thickness aspect ratio used for compression testing in the literature. This sample did not exhibit Lüders-type deformation in compression. This experiment indicates that the absence of Lüders-type deformation in compression cannot be argued for the lack of sufficient sample gauge length.

Shear testing is another case where short sample "gauge length" is commonly used. For simple plate shear tests, the sample usually has a very narrow "gauge width" (which is effectively the length direction for Lüders band formation) and a large length parallel to the shear direction (which is effectively the width for Lüders band formation). This gives a very small aspect ratio well below 1 for the formation of a Lüders band [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF]. Huang et al. reported the formation of a localized high shear strain band in a plate sample under simple shear [START_REF] Huang | An experimental study of NiTi alloy under shear loading over a large range of strain rates[END_REF]. The shear band is in parallel configuration with the shear (as expected for a Lüders band) but did not across the entire gauge width of the sample (thus to form a band) and the global stress-strain response showed no strain softening but a monotonic hardening behaviour. Lüders-type deformation behaviour also did not happen in this case. However, torsion testing of thin wall tubes provides a practical case of shear with high "gauge length". Many studies have demonstrated the absence of Lüder-type deformation in such samples. Reedlunn et al. [START_REF] Reedlunn | Axial-torsion behavior of superelastic tubes: Part I, proportional isothermal experiments[END_REF] conducted a complete tension-compression-torsion test of thin wall tube samples, following an earlier work by Sun and Li [START_REF] Sun | Phase transformation in superelastic NiTi polycrystalline micro-tubes under tension and torsion--from localization to homogeneous deformation[END_REF]. Both works demonstrate clearly the formation of a Lüders band under tension and the disappearance of the Lüders band when shear deformation becomes dominant.

This also suggests that absence of Lüders-type deformation in shear cannot be argued for the lack of sufficient sample gauge length.

Another interesting case is stress-induced martensitic transformation of NiTi in bending mode [START_REF] Reedlunn | Tension, compression, and bending of superelastic shape memory alloy tubes[END_REF][START_REF] Watkins | Uniaxial, pure bending, and column buckling experiments on superelastic NiTi rods and tubes[END_REF][START_REF] Jiang | Bending of pseudoelastic NiTi tubes[END_REF]. Watkins et al. conducted a four-point bending test of pseudoelastic NiTi rod and thin wall tube samples [START_REF] Watkins | Uniaxial, pure bending, and column buckling experiments on superelastic NiTi rods and tubes[END_REF]. The rod sample exhibited a unique deformation localization behaviour, with enhanced strains in the outer layers on both the tensile side and the compressive side. Such behaviour is different from the typical Lüders-type deformation expected, apparently due to the self-constraint between the higher strained surface and lower strained core regions in bending, preventing the "free" Lüders-type deformation. The tube sample, due to the emptiness underneath the tube wall along the bending axis and thus the absence of self-constraint to deformation, is more comparable to uniaxial tensile and compressive tests. This sample exhibited a Lüders-type localized deformation on the tensile side and uniform deformation on the compression side. The compressive side can be effectively considered a test of compression with a long gauge length. Thus, this test provides another evidence of the absence of Lüders deformation in compression with long gauge length, echoing Orgéas and Favier's work discussed above [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF].

The evidence discussed above demonstrates that the absence of Lüders-type deformation in compression and in shear cannot be explained by the lack of sufficient gauge length for elastic contraction, but by other reasons, for example metallurgical, texture or crystallographic reasons. In fact, the suppression of Lüders-type deformation by such factors is also evident in the literature. One example is the "linear pseudoelastic behaviour" (total absence of Lüders deformation) of cold worked [START_REF] Zheng | The microstructure and linear superelasticity of cold-drawn TiNi alloy[END_REF][START_REF] Delobelle | Study of electropulse heat treatment of cold worked NiTi wire: From uniform to localised tensile behaviour[END_REF][START_REF] De Oliveira | Specific forward/reverse latent heat and martensite fraction measurement during superelastic deformation of nanostructured NiTi wires[END_REF] or pseudoelastically cycled NiTi under tension [START_REF] Liu | Strain dependence of pseudoelastic hysteresis of NiTi[END_REF][START_REF] Zhao | Quasi-Linear Superelasticity with Ultralow Modulus in Tensile Cyclic Deformed TiNi Strain Glass[END_REF]. This may be attributed to the effect of massive defects within the matrix introduced by the processes. In another work by Grassi et al [START_REF] Grassi | Anisotropy and Clausius-Clapeyron relation for forward and reverse stress-induced martensitic transformations in polycrystalline NiTi thin walled tubes[END_REF], a flattened Ti-50.8at%Ni sheet of a thin wall tube was used to fabricate tensile samples at various orientations from 0 to 90 relative to the tube axial direction. They found that whereas samples oriented in both the longitudinal (at 0~22.5) and the transverse (at 67.5~90) directions exhibited perfect pseudoelasticity in Lüders deformation mode, the 45-oriented sample showed a perfect pseudoelasticity in the total absence of Lüders deformation behaviour. This experiment gives direct evidence to the effect of metallographic texture.

To put this into the context of this model, considering that the sample elongation change conservation criterion expressed in Eq. ( 2) is non-negotiable, it implies that the demand for elastic strain contraction is negligible or nil for compression, shear or in the cases of heavily defected matrices and certain metallographic textures. This could be a result of being small so that . This, obviously, is subjective to future experimental verification. In this regard, Eq. ( 2) is a necessary condition for Lüders-type deformation of NiTi, but not a sufficient condition.

6,2 Lüders band deformation via martensite reorientation

Lüders band behaviour is also observed for tensile deformation via martensite reorientation of polycrystalline NiTi [START_REF] Liu | Lüders-like deformation associated with martensite reorientation in NiTi[END_REF]. The model presented above is also fully valid in this case. This may be explained in two aspects. First, the model is built purely on the consideration of the mechanical behaviour, with no reference to a phase transformation or otherwise. Second, the mechanical behaviour is characterized by two distinctive features, i.e., that the strain change across the Lüders band interface is a finite-discrete value and that the occurrence of this discrete strain change is intrinsically very fast (at the speed of sound in the medium) in the local scene. The macroscopic strain of stress-induced martensite reorientation is essentially the same as that of the stress-induced martensitic transformation, because they both are the manifestation of the same transformation crystallographic lattice distortion. The fast speed of propagation of the transformation interface is out of the need to accommodate such large crystallographic lattice distortions (as explained in section 3.1). Such need is also the same for martensite reorientation and for stress-induced martensitic transformation.

Conclusions

This study proposed a new explanation for the occurrence of the Lüders-type deformation of stress-induced martensitic transformation in polycrystalline NiTi. The explanation is based on the criterion of real-time conservation of sample dimension changes from three contributions, including the dimension change applied by the testing machine, the dimension change produced by the Lüders band formation or expansion, and the elastic strain contraction of the sample. Based on this criterion and the experimental evidence presented, the following conclusions are reached:

(1) The Lüders band of the stress-induced martensitic transformation of polycrystalline NiTi initiates suddenly within a finite width and a high local strain rate that can be orders of magnitude higher than the applied strain rate in a typical tensile testing. This leads to a sample elongation rate faster than the applied elongation rate by the testing machine. This causes an elastic strain contraction of the sample to produce a negative elongation to compensate for the elongation shortfall provided by the testing machine. The elastic strain contraction causes the stress drop marking the initiation of the Lüders band.

(2) Based on the criterion of real-time sample elongation conservation, a mathematical model with experimental input is established. This model is able to describe the full stress-strain response of the Lüders-type deformation of NiTi, including the initial stress drop and strain reduction, and the stress plateau that follows. The reproduced behaviour matches perfectly with the experimental measurement. According to this criterion, the initial "upper-lower yielding behaviour of the Lüders-type deformation is defined by the condition that the rate of sample elongation from the Lüders band is higher than that from the testing machine, i.e., , whereas the stress plateau section is defined by .

(3) The sample elongation conservation-based model describes a necessary condition for the occurrence of the Lüders-type deformation, but not a sufficient condition. Other factors may take effect to prevent the Lüders-type deformation behaviour, e.g., metallographic and crystallographic textures, high density of defects, and compression and shear.

(4) Considering that the criterion is a necessary condition for Lüders-type deformation, the absence of Lüders-type deformation in compression (including solid bending) or shear (including torsion) suggests that in these deformation modes the rate of sample dimension change produced by the stress-induced martensitic transformation is slow and conforms to , so that no elastic strain contraction ( ) is required for sample dimension change compensation. This is a logical deduction from the criterion, and it obviously requires experimental verification, e.g., by DIC analysis.

(5) It is also worth noting, though not directly verified in this study, that the criterion and the model proposed above are equally valid for the case of Lüders-type deformation via martensite reorientation of NiTi, given that the model is built solely on mechanical considerations without any reference to a phase transformation and that the mechanical characteristics of deformation via martensite reorientation are practically identical to those via stress-induced martensitic transformation.

Acknowledgement

The authors wish to acknowledge the financial support from the Australian Research Council in grants DP180101955 and DP190102990 for this study. YW acknowledges the support from the US National Science Foundation under Grant DMR-1923929, which has facilitated this

3. 2

 2 Mathematical model Based on this hypothesis, a mathematical model is developed based on the criterion of sample length change conservation in real time among three contributions: sample length change (elongation) produced by the formation (or expansion) of a Lüders band ( ), sample length change (elongation) caused by the crosshead displacement of the testing machine ( ), and the sample length change (reduction as a negative value) by the elastic contraction of the rest of the sample ( ). This relationship can be expressed for any time interval between and during the deformation as:

  (i) Capital-case symbols are used for testing constants (such as sample initial gauge length ). (ii) Lower-case symbols are used to express instantaneous values (such as length of the Lüders band and the length of the austenite section that transforms into a new Lüders band ). (iii) Numeral subscripts are used to indicate global strains at different stages of the deformation process (such as and for the global strains at point 1 and point 2 on the stress strain curve, respectively). (iv) Roman numeral subscripts are used to designate a specific point in time. (v) Capital letter subscripts are used to indicate local strains at specific locations of the sample along the length (such as for the instantaneous local strain inside a Lüders band and for the local strain in the austenite region of the sample outside the Lüders band), considering non-uniformity of the local strains along the length of the sample due to the Lüders deformation behaviour.

Fig. 1

 1 shows the DIC field maps of the axial strain of a sample at two different points in time during the pseudoelastic Lüders deformation. Fig. 1(a) corresponds to the time when the Lüders band was just initiated. Fig. 1(b) shows the Lüders band in the propagation stage. Some parameters defined above are indicated in the figure, including the current gauge length of the sample at time , the current length of the Lüders band at time and the active length of the Lüders band (where is the length of the original austenite corresponding to the active length of the Lüders band. It is noted that the active length of a Lüders band is the whole length of the band during the initiation stage of the band, but only the moving front edge(s) of the Lüders band during its propagation. In addition, two observation points on the sample for local strain analysis are also indicated in the figure.

Figure 1 .

 1 Figure 1. DIC strain maps of a sample at two different stages of the Lüders-type pseudoelastic tensile deformation. (a) Initiation of a Lüders band. (b) Lüders band propagation stage.

Fig. 2 (

 2 Fig. 2(d) is an enlarged view of the strain curves in the section identified by the small box d in (c), which corresponds to the section of the stress drop seen in (b). The numerals indicated in (d) correspond to those in (b) in real time. It is evident that corresponding to the progressive stress drop, the strain at P2, which represents the elastic strain in the austenite region outside the newly formed Lüders band, decreased continuously within the same time period, as expected based on Hooke's law.

Figure 2 .

 2 Figure 2. The phenomenon of Lüders-type deformation of NiTi. (a) A tensile pseudoelastic stress-strain curve of a 30 mm long NiTi dog-bone shaped sample deforming in Lüders band

Fig. 3 (

 3 b) shows the strain field image of the sample as at the time corresponding to point "5". The emerging strain peak in the profiles indicates the formation of a Lüders band.

Figure 3 .

 3 Figure 3. Time-resolved strain analysis of the initiation and propagation of a Lüders band based on DIC measurement. (a) Local strain profiles along the centre line of a sample at different times during Lüders band deformation. (b) DIC image of the axial strain map of the sample as at the time corresponding to point "5". The dashed box indicates the area of the profiles shown in (a). (c) Changes of the elastic strain of the austenite outside the Lüders band during its formation and propagation. (d) Evolutions of the length and the peak strain of the Lüders band. The numerals correspond to those identified in Fig. 2.

  Then is computed, as shown in the figure. It is seen that in the early stage of the Lüders deformation increased much faster than . This excess of elongation ( ) is the cause of the elastic strain contraction, thus the stress drop, of the Lüders deformation. The raise of ( ) slowed down gradually and reached a constant value at point "4", indicating that the elongation rate from the Lüders band equals that produced by the testing machine. This represents the stress plateau stage of the Lüders deformation.

Figure 4 .

 4 Figure 4. Model simulated tensile stress-strain behaviour of the Lüders-type deformation of NiTi. (a) Comparison between the sample elongation produced by the Lüders band and that provided by the testing machine . (b) Comparison of the experimentally measured and the model calculated stress-strain curves of the sample.

international collaboration.

where is the apparent elastic modulus of the sample length , and in this analysis .

For the parameters identified above, and are the instantaneous local strains of (inside) the Lüders band at and , and are the lengths of the Lüders band at and , and is the original austenite length corresponding to the active length of sample during the Lüders deformation, i.e., the evolving length of the Lüders band during its formation or the active front of the Lüders band during its propagation. These parameters can be determined experimentally by local strain measurement (e.g., DIC). is the local strain in the austenite section outside the Lüders band at . and are the global strains at and respectively on the stress-strain curve. Thus , and can be determined by global stress-strain curve measurement. is the original gauge length of the sample, and is the elongation speed of the testing machine. Both are setting parameters of the experiment, i.e., also available from the experiment.

In addition, from Eqs. ( 4) and [START_REF] Liu | Lüders-like deformation associated with martensite reorientation in NiTi[END_REF], the rates of sample elongations produced by the Lüders band and by the elastic contraction can also be obtained by differentiating the equations against time.

In this case, , and are of the chosen time point and thus function as known values for differentiation, whereas and for are the time variables for differentiation, thus:

4.3 Formation and propagation of a Lüders band as defined by the model Eqs. ( 6) and ( 7) are applicable to the entire Lüders-type deformation process. Depending on the values of and , the entire process may be divided into two distinctive stages, i.e., the Lüders band formation stage and the propagation stage. The Lüders band propagation stage is defined as and , which also implies . According to Eq. ( 9), therefore, . Further, according to Eq. ( 2), .

The Lüders band formation stage is characterized by and . It commences in Fig. 4(b).

For the section after point "3" to point "7", the Lüders band starts to expand on one front only, thus its active length remains unchanged as roughly half of the length at point "3", as expressed in Fig. 3(d). In this case, represents point "3", and =0.7 mm as determined by DIC measurement (Figs. 1(b) and 3(d)), thus , =6.42% is the Lüders band strain at point 3 (Fig. 3(d)) (or it can also be calculated using Eq. ( 14)), is the time function of given in Eq. ( 14), =1.05% is the global strain at point "3" (Fig. 2(b)), =0.92% is the elastic strain of the austenite section at point 3 (Fig. 2(d)), is the time function of given in Eq.( 13), =1.55 mm is the length of the Lüders band at point "3" (Fig. 3(d)), and =401 MPa is the global stress at point "3" (Fig. 2(b)), then using Eq. ( 7)

the stress drop for the section after point "3" is calculated and as the stress is also plotted in Fig. 4(b), together with the actual experimental stress-strain curve. It is evident that the two curves match nearly perfect.

Interpretation of other empirical observations based on this model

Stress-induced martensitic transformation under other loading conditions than tension

As presented above, Lüders-type deformation in NiTi has been observed empirically to occur in tension and not in compression or shear. The model presented above needs to be cross-examined with regard to those experimental observations.

The model declares one criterion of sample elongation conservation among all contributions on real time, including machine crosshead movement, Lüders band deformation and elastic strain contraction. Whereas the overspeed of the elongation production by the Lüders band over the machine crosshead movement is the root course for the elastic strain contraction, the elastic strain contraction, which implies a stress relaxation, is the prerequisite condition of deformation softening for the occurrence of the Lüders-type localized deformation.

The amount (rate) of the elastic strain contraction required is determined by the speed of the elongation generated by the Lüders band formation , which in turn can be decomposed into three aspects: the strain increment of the Lüders band, the speed of this strain increment , and the initial (active) length of the Lüders band (or ) at its


Grégory Lefèvre

Determination of isotopic ratio of boron in boric acid solutions by ATR-FTIR spectroscopy

Keywords: Boron isotopes, Solution analysis by infrared spectroscopy, Enriched boric acid

Due to the low atomic mass of boron, its different isotopes (B-10 and B-11) leads to a measurable impact on vibrational spectra. Indeed, a shift in the asymmetric stretching of the dissolved species has been found using attenuated total reflectance infrared spectroscopy, and those of acid species is higher than those of the borate ion (41 cm -1 vs. 20 cm -1 , respectively).

A calibration from pure B-10 to pure B-11 boric acid has been performed, leading to an accuracy higher than 0.5% on the B-10 percentage of 0.1 M solution of mixtures of the two isotopes.

Article Highlights

-The isotope of boron has an impact on the infrared spectra of solutions -A difference of 14 cm -1 has been found using attenuated total reflectance infrared spectroscopy -The isotopic ratio of boron in boric acid can be measured after a calibration step

Introduction

Measuring the isotopic ratio of boron, which has two stable isotopes, B-10 and B-11, has an interest in several domains, whose geochemistry and nuclear industry. In geological and environmental sciences, the isotopic composition of minerals and waters is related to the fractionation mechanisms. The isotopic variation in nature is ca. from -30 ‰ to +60 ‰ in  11 B (i.e. a few percent), so very accurate analysis methods are required in this field. Mass spectrometry family methods have been applied for decades. The main advantage is their accuracy, but they remain burdensome with complex protocols [START_REF] Tanimizu | A rapid and precise determination of boron isotopes ratio in water and carbonate samples by multiple collector ICP-MS[END_REF], [START_REF] Smith | Measurement of boron concentration and isotope ratios in biological samples by inductivey coupled plasma mass spectrometry with direct injection nebulization[END_REF]. In nuclear industry, boric acid is used to absorb neutron, due to the high cross-section absorption of B-10. Thus, it can be used in radiation shielding [START_REF] Cataldo | New composites for neutron radiation shielding[END_REF]. Moreover, the primary circuit of pressurized water reactor (PWR) can contain more than 400 m 3 of a solution up to 4 g/L of boron. This element is added as boric acid, and should be neutralized by the addition of bases, as LiOH or KOH to reduce the corrosion rate of the cooling circuits. The use of enriched boric acid (EBA) (i.e. with a higher fraction of B-10 than in natural boron, 19.9 %) would allow to use lower concentrated solutions, with a benefit for the pH control of the cooling solution. Thus, German Konvoi PWR operated with EBA at 28% B-10 enrichment [START_REF] Nordmann | Key emerging issues and recent progress related to plant chemistry/corrosion in PWR/VVER/CANDU reactors[END_REF], while in the new generation PWRs as EPR, the enrichment reaches 37% [START_REF] Cosse | Enriched boric acid as an optimized neutron absorber in the EPR primary coolant[END_REF]. Due to the absorption of neutrons by B-10 ( 10 B(n,alpha) 7 Li reaction), its concentration decreases and a depletion of natural boron in B-10 could reach 6.83% in a typical cycle [START_REF] John | Boron-10 depletion in pressurized water reators[END_REF], what would correspond to a fraction of B-10 decreasing from 19.8% to 18.8%. In this field, the measurement of isotopic composition of boric acid with an accuracy not as high as for the geochemistry studies, but with simpler methods, would be useful.

Analytical methods to determine isotopic ratio are mostly mass spectrometry and nuclear analytical techniques (as Particle Induced Gamma ray Emission spectroscopy [START_REF] Raja | Application of PIGE method for quantification of total boron in neutron absorbers and shielding materials and isotopic composition in in-house prepared enriched boron carbide samples[END_REF], Charged Particle Activation Analysis [START_REF] Ikebe | Study of charged particle activation analysis (II): Determination of boron concentration in human blood samples[END_REF] or Neutron Activation Analysis [START_REF] Matusmoto | Prompt gamma-ray neutron activation analysis of boron-10 in biological materials[END_REF]). The use of simplest instruments, as ICP-OES or AAS, could be considered since a significant isotopic shift in atomic emission or absorption lines has been shown. Two chemical reasons can explain an effect of isotopy on emission or absorption energy: either elements are light enough and the mass difference of their isotopes is relatively large (which is the case for boron) or elements are heavy but with a difference in nuclear charge distribution of the isotopes (for example mercury) [START_REF] Zolfonoun | Application of multivariate curve resolution-alternating least squares for the determination of boron isotope ratios by inductively coupled plasma-optical emission spectrometry[END_REF]. Thus, ICP-OES equipped with a high spectral resolution detector can be used to determine the B-10 content with a good accuracy (<0.001%) [START_REF] Zolfonoun | Application of multivariate curve resolution-alternating least squares for the determination of boron isotope ratios by inductively coupled plasma-optical emission spectrometry[END_REF]. More recently, emission spectroscopy with a double laser-induced plasma was tested, leading to an accuracy around one percent [START_REF] Choi | Boron isotopic analysis using molecular emission from double-pulse laser-induced plasma in aqueous boric acid solution[END_REF]. Atomic absorption spectrometry has been also tested, leading to a measurement of B-10 content with an error of about 1% [START_REF] Thangavel | Determination of boron isotope ratios by Zeeman effect background correction-graphite furnace atomic absorption spectrometry[END_REF].

For the same reason (large mass difference between B-10 and B-11), the energy of vibration modes, at the basis of infrared spectroscopy, is significantly dependent on the isotope. Thus, the asymmetric stretching of gaseous BF 3 differs by ca. 100 cm -1 with a lower wavenumber for 11 BF 3 [START_REF] Zhang | Online Determination of Boron Isotope Ratio in Boron Trifluoride by Infrared Spectroscopy[END_REF], as expected from the "ball and spring" model. The substitution of B-10 by B-11 in numerous crystalline inorganic borates has been followed by infrared spectroscopy and the impact on the different vibration modes has been detailed. Borate can form either planar triangle or tetrahedral unit but for both geometries, it is expected that symmetric stretching is not shifted by substitution, while for asymmetric stretching  3 (B-10) /  3 (B-11)  1.04. The interpretation of experimental spectra recorded for crystalline borates remains difficult and depends on the compound, since environment of borate is diverse in these conditions. On the contrary, this calculation should be applicable to the solution species. Spectra of H 3 BO 3 and B(OH) [START_REF] Nordmann | Key emerging issues and recent progress related to plant chemistry/corrosion in PWR/VVER/CANDU reactors[END_REF] -recorded by attenuated total reflectance (ATR) IR spectroscopy can be found in several works [START_REF] Peak | ATR-FTIR spectroscopic studies of boric acid adsorption on hydrous ferric oxide[END_REF], [START_REF] Tournié | Impact of boron complexation by Tris(hydroxymethyl)aminomethane on the initial dissolution rate of borosilicate glasses[END_REF]. Asymmetric stretching is located at 1410 cm -1 for trigonal geometry, and 945 cm -1 for tetrahedral geometry. The application of the factor 1.04 would lead to a shift of around 40 cm -1 , which is enough to have a good separation between the peaks coming from the two isotopes.

In this work, results of ATR-IR spectroscopy on solutions with different isotopic ratios are presented. The possibility to determine B-10 content has been evaluated and ways of optimization are given.

Experimental

Boric-11 acid and boric-10 acid were purchased to Cambridge Isotope Laboratories, Inc. The isotopic enrichment measured by ICP-MS (data from the supplier) was >99.95% for B-11, and 99% for B-10.

The ATR-IR spectra were collected in the range 4000-650 cm -1 with a dry-air-purged Thermo Scientific Nicolet IS50 FT-IR equipped with an MCT-B detector. Spectral resolution was 4 cm -1 and spectra were averaged from 256 scans. The crystal used was a single reflection diamond/ZnSe crystal with an incidence angle of 45°C (Smart PIKE). An automatic baseline correction between 900 and 1800 cm -1 has been applied. The solutions have been analyzed using a drop of 25µL deposited on the top of the ATR element, following the routine procedure.

The procedure has been repeated 3 times, and the spectrum used for calibration was the average of the three spectra. The ATR-IR spectra of solutions have not been systematically corrected using the advanced ATR correction algorithm [START_REF] Nunn | Advanced ATR Corrextion Algorithm[END_REF] developed by Thermo Fisher Scientific in OMNIC software, since in the purpose to develop a calibration method for isotopic measurement, the use of the raw spectra has been considered as more straightforward.

The determination of the peak locations has been performed by smoothing/derivative tools in OMNIC software or OriginLab.

Results and discussion

ATR-IR spectra of boric acid and borate solutions from pure B-10 or B-11 chemicals are presented in Fig. 1. Boric acid (Fig. 1A) is characterized by two bands around 1420 cm -1 and 1150 cm -1 . According to literature where spectra of natural boron solutions have been interpreted [START_REF] Peak | ATR-FTIR spectroscopic studies of boric acid adsorption on hydrous ferric oxide[END_REF], the peak at the highest wavenumber corresponds to the asymmetric stretching  a (BO). In H 3 BO 3 , the D 3h symmetry is expected to lead to a doubly degenerated band.

However, a careful look at this band in this figure or in published results (Fig. 4 in [START_REF] Peak | ATR-FTIR spectroscopic studies of boric acid adsorption on hydrous ferric oxide[END_REF] and Fig. 3 in [START_REF] Su | Coordination of adsorbed boron: a FTIR spectroscopic study[END_REF]) shows an asymmetry. From a double derivative analysis, a split of 25 cm -1 or 29 cm -1 was found for B-11 and B-10, respectively. Thus, it appears that the H 3 BO 3 molecule is slightly deformed. The wide band around 1150 cm -1 is assigned to BOH deformation. This latter band is found also in the spectra of borate anion (Fig. 1B), since this chemical group is common to both species. The other characteristic band of borate anion is located around 950 cm -1 and corresponds to the asymmetric stretching. The addition of a hydroxyl group to B(OH) 3 0 molecule leads to a red shift of this mode by almost 200 cm -1 . The symmetric stretching expected at 745 cm -1 is not visible in the IR spectra, as expected for a T d symmetry, but has been shown in Raman spectra (Table 1). The comparison of the spectra for the two isotopes shows a clear difference for  a (BO) of the two species. A red shift of 41 cm -1 is obtained for the heavier isotope in B(OH) 3 0 , in agreement with the value based on ball and spring model. A lower red shift (20 cm -1 ) is obtained for B(OH) 4 -. In purpose to use infrared spectroscopy to determine the isotopic ratio, the analysis of boric acid has been chosen since an important shift of  a (BO) has been shown. In Figure 2, spectra of B(OH) 3 solutions with different isotopic ratios are presented. An isosbestic point at 1429 cm -1 is clearly identified, as expected for a spectrum formed by a pair of compounds. From these spectra, a calibration protocol to determine the isotopic ratio of an unknown sample should be established. Since the maximal intensity for  a (BO) is known for B-10 and B-11 (respectively 1446 cm -1 and 1405 cm -1 ), measuring the ratio of the intensity at these two wavenumbers should be a quick and easy way to calculate the isotopic ratio (without needing to know the total concentration). Thus, we should determine the calibration equation between the intensity ratio and the isotopic ratio. For a pure solution of B-10 measured using an ATR system, Beer-Lambert law becomes for a given wavenumber  1 :

(1) With I, the intensity of the spectrum,  B10 , the molar absorption coefficient for B-10, C, the molar concentration, and de, the equivalent path length (assumed to not depend on the isotopic ratio of the solution).

For a mixture of both isotopes:

(2) For a given spectra of the mixture, it is possible to calculate the ratio of the absorbance at two different wavenumbers  1 and  2 :

(3)

We can define x, the molar fraction of B-10 in the mixture as So Eq. ( 3) becomes: [START_REF] Nordmann | Key emerging issues and recent progress related to plant chemistry/corrosion in PWR/VVER/CANDU reactors[END_REF] To use the spectra to determine the isotopic composition, we need a relation between the ratio of the absorbance at two different wavenumbers and x. After development, we get the following relation: Moreover, from eq. ( 1), for pure solutions with the same concentrations, r( 2 ) can be written as: [START_REF] Matusmoto | Prompt gamma-ray neutron activation analysis of boron-10 in biological materials[END_REF] Thus, from the spectra of pure B-10 and B-11 solutions with the same concentrations, the parameters for eq. ( 5) can be calculated (Table 1).

In Figure 3, the two ratios I(1446) / I(1405) and I(1405) / I(1446) have been plotted versus the isotopic ratio, with a good consistency between experimental values and equation ( 5). The isotope purity of boric-10 acid was only 99% (x = 0.99), but its spectra has been used as a pure B-10 solution in the formula ( 7) with a possible impact on the calculated curves. To be able to use these data to determine the isotopic ratio of an unknown sample, the equation ( 5) can be rewritten as follows: [START_REF] Zolfonoun | Application of multivariate curve resolution-alternating least squares for the determination of boron isotope ratios by inductively coupled plasma-optical emission spectrometry[END_REF] It can be noted that the spectra of pure isotopic species would be required to apply this approach, what can be a limitation. To extend the use of this method, it is possible to fit the three parameters in the equation ( 10) using data of the calibration curve, either for a full range from pure isotopes, or for a limited range. In the first case, , and have been determined by the fit of the experimental values (Table 2). A slight difference appears, with a positive effect on the accuracy, as this will be shown later. To simulate the calibration on a restricted range of isotopic fraction (but with the same concentrations step), only the data between 10% and 80% have been arbitrary chosen. In this case, using the spectra of pure isotopes to calculate the parameters in equation ( 5) is no longer possible, so the approach by fit is mandatory. The parameters obtained in this case (Table 2) are close to the previous values, indicating that the values of the calibration curve follow closely the theoretical equation. Once the parameters have been obtained, they have been used to calculate the boron isotopy in a reverse way. In the Table 3 the values calculated using the three methods described above are presented. To illustrate the accuracy, the absolute error is plotted on Figure 4. The first approach is based on only two experimental spectra (corresponding to the pure isotopes), and the uncertainty on these measurements may be responsible to the poor accuracy, with a difference which reaches about -1% between experimental value and calculated one (the average of absolute values of difference for each point is 0.48%). The fit of the whole data set (11 spectra) has a visible impact, with a maximum difference of -0.5% (the average decreases to 0.16%). Using only a part of the experimental data decrease the accuracy, with the same maximum, but an average increasing at 0.18%. These results show that plotting a calibration curve is more accurate than using the theoretical curve calibrated on pure isotopic solutions and should be preferred if accuracy is the key element. On the contrary, using the method based on the spectra of the pure isotope is simpler but can lead to an error around one percent on the isotopic composition.

Conclusions

These results confirmed that ATR-IR spectroscopy of solutions is a method able to determine the isotopic ratio of dissolved boron. Two limits of this method have been shown. The first one is the accuracy of the result obtained, to be compared to the accuracy needed depending on the application. This aspect would need some specific works, out of the scope of this study that aims to get a proof-of-concept of the ATR-IR spectroscopic approach. It depends on the calibration step, the statistical treatments, the sampling method (an online measurement is possible and can improve the background treatment) and the instrument. Thus, better performances than those presented in these first experiments could be expected in an operational system, by optimizing the calibration domain around the target isotopic ratio, or by increasing the boric acid concentration (using a first pre-concentration step for example).

The second limit is the minimum concentration of boric acid that is required to be able to determine the isotopic ratio. In a classical ATR configuration, the available analytic range will remain above several millimolar, and the accurate measurement of the ratio of intensities would be difficult below 0.01 M. Recording more scans or using multiple-reflection ATR element would only slightly improve the signal/noise ratio. To reduce this limit, a possibility would be to modify the ATR element surface to concentrate boric species inside the volume probed by the evanescent wave. Thus, the analyte can be concentrated in a layer deposited on the ATR element surface if the material presents a high affinity toward it. This approach has been successful in the analysis of organic molecules by IR chemical sensors based on ATR elements coated with a hydrophobic polymer. ATR elements as LDPE-coated silver halide [START_REF] Mizaikoff | Infrared fiber-optical chemical sensors with reactive surface coatings[END_REF] or fluorocarbon-coated ZnSe [START_REF] Dobbs | Mid-infrared chemical sensors utilizing plasma-deposited fluorocarbon membranes[END_REF] were used to quantify chlorinated hydrocarbons in gas phase or in aqueous solution, respectively. A coating consisting in a solid with a high affinity towards boric acid or borate could be tested, with the possible difficulty of the interference with other adsorbed species, especially anionic ones (for example carbonate which are ubiquitous in water and have a high affinity for several solids).

Fig. 1 .

 1 Fig. 1. Spectra of isotopes B-10 and B-11 of 0.1 M boric acid and borate at (A) pH 5.5 and

Fig 2 .

 2 Fig 2. Spectra of boric acid for different B-10 isotopic ratios from 0 to 1 by increment of 0.1.

Fig. 3 .

 3 Fig. 3. Ratios of the absorbance spectra intensities at 1405 and 1446 cm -1 as a function of B-

Fig. 4 .

 4 Fig. 4. Difference between the calculated B-10 percentage and the experimental one from the

Table 1 .

 1 Stretching vibrations of boric acid/borate species.

	Species		Vibration		Wavenumbers (cm -1 )
			mode
				10 B	11 B	Difference
	B(OH) 3	0	 s (BO)	877 a	877 a	0
	B(OH) 4	-	 a (BO)  s (BO)	1446 745 a	1405 745 a	41 0
	a reference [18]	 a (BO)	968	948	20

Table 2 .

 2 Calibration parameters obtained from (1) spectra of pure isotopes samples, (2) fit of the whole calibration curve or (3) fit of an arbitrary restricted calibration range.

	Parameter
	Method (1)	3.24	3.70	0.222
	Method (2)	3.26	3.62	0.215
	Restricted range (3)	3.28	3.68	0.220

Table 3 .

 3 Experimental and calculated B-10 percentages, based on the use of parameters determined by (1) solutions of pure isotopes, (2) fit of the whole calibration curve and (3) fit of an arbitrary restricted calibration range, where some data have been excluded (excl.).

	Exp.	0	10	20	30	40	50	60	70	80	90	99
	(1)	0.0	9.6	19.0 29.5 39.6 49.2 59.6 69.8 79.9 90.4 100.1
	(2)	0.2	10.0 19.5 30.1 40.2 49.7 60.0 70.0 79.8 90.0 99.3
	(3)	excl. 10.0 19.5 30.0 40.1 49.6 60.0 70.1 80.0 excl. excl.

Acknowlegdements

Nazaryan Melsik, Jiawen Li and Ruopei Sun (IRCP) are thanked for their help in the improvement trials of the anaytical method.


Hina Verma

Karine Le Guen

Mohammed H Modi

Rajnish Dhawan

Philippe Jonnard

Philippe Jonnard Mutilayer

Mutilayer for Imaging Biological samples in the Water Window Range

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Mutilayer for Imaging Biological samples in the Water Window Range

Hina Verma 1 , Karine Le Guen 1 , Mohammed H. Modi 2,3 , Rajnish Dhawan 4 , Philippe Jonnard 1 1 Sorbonne Université, Faculté des sciences et Ingénierie, UMR CNRS, Laboratoire de Chimie Physique -Matière et Rayonnement, Paris, France; 2 Soft X-ray Applications Lab, Raja Ramanna Centre for Advanced Technology, Indore, India; 3 Homi Bhabha National Institute, Bhabha Atomic Research Centre, Mumbai, India; 4 Thin Films Lab, Synchrotrons Utilization Section, Raja Ramanna Centre for Advanced Technology, Indore, India.

X-ray at 0.

Region of Intermixing

Grazing Incidence X-ray reflectivity measured at 8048 eV.

Resonant soft X-ray reflectivity measured near Sc L-edge ( 400 eV)

Water Window

Multilayer Mirror

Table-top water-window X-ray microscopy arrangement 1

Rigrous Intermixing between Mg & Sc Layers

Sc L-edge is encircled in red* 1. M. Berglund et al., 2000. Journal of Microscopy, 197, 268-273 (2000) ? 2. P. Jonnard et al., 2019. Journal of Applied Physics, 126, 195301 (2019)