Shabnam Arbab Chirani

Quentin Christopher

Francois Gabriel

Thierry Gregory

Ana Anais

Khouloud, Maxime Edouard ) Mohammed

Prof Jose Carlos

De Araujo

Albert Guia

Einstein

Terezinha

Neto

Henrique Martinni

Ramos De Oliveira

Henrique Martinni Ramos

Ramos De

List of Tables

Keywords: Full-field kinematics and thermal measurements, nanostructured NiTi wires, thermal and mechanical powers, heat balance, specific latent heat, martensite fraction

for providing a great stimulating and fun environment through these years. I would like to give special thanks to some of them: to Chris, for being always available and providing information about everything, really everything. To Quentin, for his company and knowledge, from which I also learned a lot of French expressions and vocabulary. To Thierry, for being always available to discuss my endless doubts about everything and for teaching me to climb. To Ali, for all the discussions and time we spent in the lab and also for his availability and kindness when receiving me at his home in Iran. To Greg, for his kindness and advices through these

Study of thermomechanical couplings in nanostructured superelastic nickel-titanium wires

Henrique Martinni Ramos de Oliveira December 28, 2018

RESUME

Cette thèse est une étude expérimentale du comportement thermo-mécanique superélastique d'un fil nanocristallin T i -50.9 at.% N i en alliage à mémoire de forme (SMA) (diamètre 0.5 mm), après subir un cold work (CW). Les AMF sont capables d'induire des changements de température importants lorsqu'ils sont chargés mécaniquement. Ce phénomène est dû à un important couplage thermomécanique présent dans cette transformation de phase solide entre les phases Austénite (A) et Martensite (M). La chaleur latente par unité de masse (∆H) tout au long de la transformation de phase est l'énergie responsable de cette variation de température. La détermination de ∆H est généralement effectuée par calorimétrie à balayage différentiel (DSC). Cependant, pour les SMA nanocristallins, les résultats DSC obtenus ne sont pas concluants sur la détermination de cette propriété.

Dans ce travail, une méthode utilisant la corrélation d'image numérique (DIC) et les mesures de champ thermique (TFM) a été utilisée pour analyser les couplages thermomécaniques lors d'une transformation de phase induite par contrainte. Des champs cinématiques et thermiques ont été acquis lors d'essais de traction superélastiques réalisés sur des fils CW NiTi soumis à différentes températures de traitements thermiques (TTT) allant de 523 à 598 K pendant 30 min. Un tel traitement thermique à basse température favorise une boucle totalement superélastique sans plateau de contrainte et sans déformation de type Lüders. En supposant un modèle thermique uniforme, les sources de chaleur impliquées lors du chargement cyclique ont été estimées. Cette puissance thermique par unité de masse a été comparée à la puissance mécanique et intégrée au fil du temps pour obtenir l'équilibre énergétique. De plus, grâce à une analyse thermodynamique basée sur l'énergie libre de Gibbs, les valeurs de ∆H, ainsi que la fraction de martensite, ont été estimées au cours des transformations de phase A-M directe et inverse M-A. L'analyse des résultats a conduit aux conclusions suivantes: (1) Les puissances et énergies thermiques et mécaniques présentaient une dépendance significative vis-à-vis du TTT. ( 2) Malgré l'effet important des valeurs du TTT sur les réponses mécaniques et thermiques, les ∆H obtenues étaient très proches pour tous les TTT et dans la même gamme de valeurs fondée dans la littérature pour un alliage T i -50.9 at.% N i entièrement recuit testé par technique DSC. (3) Pour une deformation donnée, la fraction de martensite augmente avec l'augmentation de TTT. (4) Pour une contrainte imposée de 4,5%, la fraction de martensite augmente de 30% à 40% en augmentant le TTT de 523K à 598K.

Mots clefs: Mesure des champs cinématique et thermique, fils nanostructurées NiTi, puissances thermiques et mécaniques, chaleur latente spécifique, fraction martensitique. rate and/or strong heat transfer); (b) and in non-isothermal tension (high strain rate and/or weak heat transfer)(adapted from [START_REF] He | On non-monotonic rate dependence of stress hysteresis of superelastic shape memory alloy bars[END_REF]. . 10 1.9 (a) Macroscopic nominal stress-strain curves for different strain rates and (b) the variation in the average stress hysteresis with nominal strain rate (adapted from [START_REF] Zhang | Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips[END_REF]. . . . . . . . . . . . . . . . . . . . . . . 12 1.10 Energy dissipation (hysteresis area) in function of strain rate at three different ambient conditions (a) still air, (b) airflow at v = 2m/s and (c) airflow at v = 17m/s (dapted from He et al., 2010). . . . . . . . . . . . . 13 1.11 Schematic of a SMA based cooling process described in four steps: 1.Loading, 2.Heat transfer to heat sink, 3.Unloading and 4.Heat transfer from heat source (Adapted from Schmidt et al., 2015a). . . . . . . . . . . . . . 14 1.12 (a)Temperature variation in function of time for different cross-head speeds (in mm/min); (b) and variation of the critical stress for martensite nucleation (σ c ) as a function of temperature (adapted from [START_REF] Rodriguez | The thermal effect due to stress-induced martensite formation in -CuAlNi single crystals[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.13 Thermal energy (W ch ), mechanical energy (W m ) and temperature variations (θ) at different strain rates (adapted from Peyroux et al., 1998b). . 21 1.14 Temperature measurement and heat sources estimation during a tensile test of a NiTi tube sample (adapted from [START_REF] Schlosser | Image Processing to Estimate the Heat Sources Related to Phase Transformations during Tensile Tests of NiTi Tubes[END_REF] (T (t)) during loading -unloading at three strain rates of ε = 10 -3 s -1 , ε = 5 × 10 -3 s -1 and ε = 10 -1 s -1 (Adapted from [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF] . tion strain ∆ε tr of solution-treated NiTi single crystals (parenthesis) and calculated values (contour lines)(Adapted from [START_REF] Miyazaki | The habit plane and transformation strains associated with the martensitic transformation in Ti-Ni single crystals[END_REF]. . . 3.17 Ratio qin / εin , computed using experimental data and Eq. 3.25, as a function of a) strain and b) nominal stress for the cyclic test. . . . . . . . . . 3.18 Average temperature variation θ in function of the average transformation strain ε tr . The dashed red line stands for the temperature variation during loading (θ A-M (ε tr )) and the blue dashed line during unloading (θ M -A (ε tr )). The black curve stands for the difference between temperature variation during loading and unloading (θ A-M -θ M -A ). . . . . . . . 3.19 Ratio qin / εin computed with Eq. 3.25 and experimental data, as a function of strain a) and as a function of nominal stress for the rupture test b). . Power vs. true strain on the right ordinate axis where the red dots represent thermal power ( q) and blue dots the mechanical power (P mech ). Dots represent the loading while the circles represent the unloading. . . . 102 4.6 Thermal and mechanical powers normalized by the strain rate ( q/ ε and P mech / ε, respectively) on the left ordinate axis and Ratio between thermal and mechanical powers q/P mech , on the right ordinate axis.

General

(-(θ A-M -θ M -A )∆S c ) and friction energies (E A-M f r + E M -A f r
) between loading and unloading for all the heat treated wires. . . . . . . . . . . . . 113

GENERAL INTRODUCTION

The NiTi shape memory alloys (SMA) are part of a material class exhibiting particular properties like shape memory effect (SME) and superelasticity (SE). Such properties are due to a martensite phase transformation taking place in a solid state. These effects have large potential to be used in a variety of fields such as aeronautical, mechanical engineering and biomedical. Because of that, an increasing interest in the study of their thermomechanical behaviour has been recorded.

Phase transformation can be induced by temperature variations (Thermal Induced Phase Transformation -TIPT) and/or by mechanical loading (Stress Induced Phase Transformation -SIPT). The superelasticity is the result of the austenite to martensite (A-M) (or the martensite to austenite (M-A)) phase transformation when the material is mechanically loaded (or unloaded). The complete recovery of strain is achieved at the end of unloading. The classic stress -strain curve obtained during an uniaxial tensile test is a hysteretic loop with two stress plateaus: an upper stress plateau during loading and a lower one during unloading. Thanks to phase transformation, strain levels achieve up to 10% under tension.

Temperature variations are generally observed during a SIPT. They are due to the latent heat of phase transformation, positive (exothermic heat source) during A-M and negative (endothermic heat source) during M-A. These temperature variations are depending on these heat sources but also on the different parameters of the thermal problem (heat losses, geometry, etc.). The dependence of the plateau level on temperature is known as a thermomechanical coupling between the thermal and mechanical states. The Clausius Clapeyron relation, leading to a linear dependence between the transformation stress and the temperature, describes it. The temperature can then be used to change, locally, the mechanical response of a material. Another interest of this thermomechanical coupling is to obtain thermal powers (positive or negative) from a mechanical loading. Recent applications of this elastocaloric effect [START_REF] Tušek | Elastocaloric effect of Ni-Ti wire for application in a cooling device[END_REF][START_REF] Pataky | Elastocaloric cooling potential of NiTi, Ni2FeGa, and CoNiAl[END_REF][START_REF] Wu | Elastocaloric cooling capacity of shape memory alloys -Role of deformation temperatures, mechanical cycling, stress hysteresis and inhomogeneity of transformation[END_REF] of NiTi alloys for cooling systems use such mechanical to thermal energy conversion. One interest of these materials is to obtain, locally, important temperature variations in relatively short times.

The study of the thermomechanical behaviour for modeling the cooling potential, requires the knowledge of the latent heats of A-M and M-A phase transformations. Different experimental techniques have been used to measure these heats per unit of mass. The most known, but limited to TIPT is DSC (Differential Scanning Calorimetry). For example, DSC has been used to determine the latent heat of phase transformation in different NiTi alloy compositions [START_REF] Frenzel | On the effect of alloy composition on martensite start temperatures and latent heats in Ni-Ti-based shape memory alloys[END_REF], and in NiTi alloys submitted to General Introduction ageing heat treatments [START_REF] Xiao | Grain size effect on mechanical performance of nanostructured superelastic NiTi alloy[END_REF]. During a SIPT, calorimetric technique is a solution to obtain these energies. For example Friend and Planes 1991 measured the entropy and enthalpy of transformation in a CuZnAl SMA. Another technique, available for SIPT, is to estimate the energies form the internal heat sources. Several methods, based on different thermal models (0D, 1D or 2D), have been proposed to obtain heat source estimations from the temperature measurements of the material. They were applied to study energy balances in CuZnAl materials (Peyroux et al., 1998a;[START_REF] Chrysochoos | Energy balance of thermoelastic martensite transformation under stress[END_REF][START_REF] Balandraud | Monotone strain-stress models for shape memory alloys hysteresis loop and pseudoelastic behaviorx[END_REF], to study the homogeneous [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] or localized [START_REF] Schlosser | Image Processing to Estimate the Heat Sources Related to Phase Transformations during Tensile Tests of NiTi Tubes[END_REF] thermomechanical behaviour of NiTi tubes, and to compare to DSC results [START_REF] Delobelle | Contribuitions à l'étude thermomécanique des alliages à memoire de forme NiTi et à la réalisation par soudage e matériaux architecturés NiTi[END_REF].

The material studied in this PhD thesis was a commercial cold worked (CW) NiTi wire of d=0.5 mm diameter with a composition of T i -50.9 at.% N i. The material was submitted to a heat treatment for 30 min, at different temperatures (from 523K to 598K), which were below the recrystallization temperature (623K). Such low heat treatment temperatures promotes a fully superelastic loop without stress plateau and no Luders-like deformation. In addition, according to the material fabrication history and consequently the material microstructure (amorphous or nanostructured materials), DSC tests may show inconclusive results [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF][START_REF] Li | Cold Rolling-Induced Multistage Transformation in Ni-Rich NiTi Shape Memory Alloys[END_REF][START_REF] Chang | Transformation sequence in severely cold-rolled and annealed Ti50Ni50 alloy[END_REF].

The context of this thesis is the study of the thermomechanical couplings during a SIPT of a nanocrystalline NiTi SMA. For that the objectives of this work are the following:

• To propose a method to estimate the thermal power and energy during a SIPT of a nanostructured superelastic NiTi wire.

• To investigate the thermal and mechanical energies involved in the SIPT.

• To estimate the latent heat of phase transformation during a SIPT in a material that shows inconclusive results when submitted to a DSC test.

• To calculate the martensite fraction in function of strain and stress during a SIPT.

• Finally, to study the influence of heat treatments on the latent heat of phase transformation and martensite fraction.

For that, the present thesis report is organized in four chapters:

The first chapter presents a brief bibliography on the shape memory alloys with the generalities of the mechanism of phase transformation and the consequent thermomechanical behaviour. Also, the thermomechanical coupling during superelasticity and some engineering applications are presented. A thermodynamic framework used to analyze the involved energies on the phase transformation is shown. Finally, some specificities of NiTi alloys and the influence of its crystallographic microstructure on the thermomechanical behavior is presented.

The second chapter details the experimental aspects of the set-up. The used material, the performed heat treatments and tests are first described in this chapter. Some basic elements of the full field measurements techniques, namely Digital Image Correlation (DIC) and Infrared Thermography, as well as the experimental issues of these techniques, are presented. Data post processing used to analyze the mechanical and thermal data are also detailed in this chapter. Finally, a short investigation of the repeatability of the performed tests is presented.

The third chapter is written in a paper format ready to be submitted. It presents the first results of the thermomechanical coupling analysis on a T i -50.9 at.% N i cold worked wire submitted to a heat treatment of 598 K for 30 min. In this chapter, the method proposed to estimate thermal and mechanical power during a SIPT is presented. Further, using a thermodynamic framework based on the Gibbs free energy the specific, latent heat of phase transformation is determined. Also the estimation of the martensite fraction in function of strain and stress is presented and discussed.

The objective of the fourth chapter was to study the influence of the heat treatment temperature on the latent heat of phase transformation and on the martensite fraction. For that, the T i -50.9 at.% N i CW wire was submitted to four heat treatments at 523 K, 548 K, 573 K and 598 K for 30 min. Using the thermomechanical approach given in the previous chapter, the thermal and mechanical powers and energies associated with the SIPT, were first estimated. Then, the specific latent heat, as well as the martensite fraction in function of strain and stress, were determined and analyzed. 1.1 Shape Memory Alloys (SMA)

Shape memory alloys (SMAs) are a class of materials which presents the "ability" of recovering large amount of deformation. At "high" temperature levels the deformed material has the ability of recovering its initial form just after being stress released. This phenomenon is called Superelasticity or Superelastic effect (SE) Fig. 1.1.a. At "low" temperatures, upon heating, the material also recovers its previous shape driven by a phenomenon called Shape Memory Effect (SME) as shown in These phenomena put this type of material into the class of active materials once they have the ability to respond mechanically to non-mechanical stimuli. The stimuli may be of different natures, such as thermal, electrical or magnetic. Thanks to these unique characteristics, the SMA's have been used in a large range of applications. They are present in biomedical devices [START_REF] Mohd | A review of shape memory alloy research, applications and opportunities[END_REF], such as stents (Antherieu, 2016) controllable medical needle [START_REF] Alonso | Caractérisation par essais DMA et optimisation du comportement thermomécanique de fils de NiTi -Application à une aiguille médicale pilotable[END_REF],endodontic instruments [START_REF] Arbab-Chirani | Comparative analysis of torsional and bending behavior through finite-element models of 5 Ni-Ti endodontic instruments[END_REF], prosthetic devices [START_REF] Chee Siong Loh | New Shape Memory Alloy Actuator: Design and Application in the Prosthetic Hand[END_REF], device for ophthalmic correction [START_REF] Khmelevskaya | A shape memory device for the treatment of high myopia[END_REF]; in safety devices on automotive industry [START_REF] Gheorghita | Using Shape Memory Alloys in Automotive Safety Systems[END_REF] and in active composites on aerospace industry [START_REF] Yang | Characteristics of smart composite wing with SMA actuators and optical fiber sensors[END_REF]

Mechanism of phase transformation

The phenomena of superelasticity and shape memory effect are governed by a phase transformation between Austenite (A) and Martensite (M) phases. During phase transformation the atoms move at short distances by changing the structure of the crystalline lattice. They move cooperatively with displacement on the order of 1/10 of the inter-atomic distance. This phase transformation is characterized to be diffusionless and happens in a solid state.

Chapter 1. Bibliography On a direct phase transformation, where austenite phase transforms into martensite phase (A-M), the martensitic crystal can be created in two different structures: a self-accommodated martensite (also called twinned), which is formed shortly after the cooling of the material in the absence of applied stress; and an oriented martensite (also called detwinned), which occurs when the material, in austenite phase, undergoes the application of a stress. The reverse transformation, where the martensite transforms into austenite (M-A), can be thermally induced through heating the martensite phase, or mechanically induced by unloading the stress induced martensite. Both the direct and reverse phase transformations arise from the interaction of temperature and stress on the alloy [START_REF] Leo | Transient heat transfer effects on the pseudoelastic behavior of shape-memory wires[END_REF]. Figure 1.2 shows schematically the behaviour of the crystalline lattice when it is subjected to thermal or mechanical external stimuli.

Thermally-induced phase transformation

The temperatures responsible for the phase change are known as critical temperatures of phase transformation. In a direct thermally-induced phase transformation, the material begins to transform at the martensitic transformation start temperature (M s ) and completes its transformation when the final temperature of martensitic transformation (M f ) is reached. Similarly, in the reverse phase transformation, the material begins its transformation at the start austenitic transformation temperature (A s ) and completes it when the final austenitic transformation temperature (A f ) is achieved. In a free stress state, the austenitic structure transforms into a twinned martensite upon cooling as shown in Fig. 1.3. This mechanism is responsible for the shape memory effect.

The critical temperatures may be obtained from the analysis of physical properties as electrical resistivity, specific heat and storage modulus. Different techniques are used to

Stress-induced phase transformation

The martensitic phase transformation can also be mechanically induced. With the material initially in the austenitic phase, the martensitic transformation can be initiated by mechanically loading above the critical stress of martensitic transformation (σ Ms ). The transformation finalizes when the final critical stress of martensitic transformation (σ M f ) is achieved. Due to the reversibility nature of the phenomenon, the reverse transformation initiates at the critical starting stress of austenitic transformation (σ As ), finalizing once the final critical stress of austenitic transformation (σ A f ) is reached. Figure 1.4 presents a schematic representation of this phenomenon.

The stress induced phase transformation (SIPT) is responsible for the superelastic effect, which generates macroscopic deformation during the loading with a subsequent recovery upon unloading. This behavior takes place at sufficiently high temperatures where the austenite phase is stable, i.e. above A f .

Shape Memory Effect

Shape memory effect occurs when the SMA is deformed at a temperature below M f while the material presents a twinned martensitic structure. The imposed deformation promotes the formation of a detwinned martensite structure which can be further reversely transformed to austenite upon heating.

Figure 1.5 presents in detail a schematic representation of the shape memory effect with its characteristic mechanical behavior. Starting from the austenite phase (point A in Fig. 1.5), when cooled in a stress-free state the material transforms into twinned martensite phase (point B) after reaching a temperature below M f . Upon loading, the twinned martensitic structure starts its detwinning process at (σ s ) with the growth of favorably oriented martensite variants in the loading direction. The stress level necessary to perform the reorientation of these martensite variants is far lower than the stress to induce plastic deformation. The detwinning process is completed when the stress reaches (σ f ). After (σ f ) up to the point C, only elastic deformation is observed. Between the points C and D the material is elastically unloaded and retains an important pseudoplastic deformation. Upon heating, the reverse phase transformation starts at temperature A s (point E) and finishes when the A f temperature is achieved. With the absence of plastic deformation during the detwinning process the material's initial shape is recovered.

The reverse phase transformation (M-A) can also occur in the presence of a constant external applied stress. In other words, during reverse transformation, the SMA is capable of applying a force (F) to displace a body on a distance "d". This process is possible when the stress level achieved by the SME is high enough to produce the necessary force (F) to displace a body. Due to this ability, the SMA's are largely used as actuators in several devices.

Superelasticity

Superelasticity is the ability of the SMA to recover large deformation levels only upon releasing the applied stress. Also known as a pseudoelastic effect, this phenomenon is due to the stress induced phase transformation and occurs exclusively when the SMA is above its final austenitic transformation temperature A f . Figure 1.6 shows a typical SMA superelastic loop.

Generally, the following interpretation of this mechanical response is proposed. Upon loading, the material describes an elastic path between points A and B, until reaching its critical starting stress of martensitic phase transformation σ Ms . Between points B and C, the applied stress promotes the phase transformation inducing the apparition of detwinned martensite phase. The martensitic phase is completed at the final critical stress for martensite transformation σ M f (point C). Between the points C and D, it is usually assumed that elastic deformation of the detwinned martensite phase takes place. From the point D to E, elastic unloading of martensitic phase takes place, reaching the starting critical stress of austenitic phase transformation σ As . With further unloading, the reverse phase transformation is completed when the stress level achieves the final critical stress for austenitic transformation named σ A f . Finally in the absence of plastic deformation, a total recovery can be achieved at the end of unloading. For some polycrystalline SMA, the stress induced phase transformation may cause up to 8% strain in Chapter 1. Bibliography Strain Stress (MPa) tension reaching 12% strain in shear [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF].

Due to the higher stress level of the superelastic plateau during loading than during unloading, a mechanical hysteresis is observed on superelasticity. This hysteresis represents a mechanical energy dissipation. Because of that, the shape memory alloys are largely used as active element in damping devices.

Thermomechanical coupling in superelasticity

The thermomechanical coupling is a complex interaction between thermal and mechanical effects present on the martensitic phase transformation. During direct phase transformation (A -M) latent heat is released to the ambient, while during the reverse phase transformation (M -A) it is absorbed from the ambient. Also called enthalpy change, the latent heat is positive during direct phase transformation ∆H released > 0 and negative during the reverse phase transformation ∆H absorbed < 0. Figure 1.7 shows a schematic representation of the latent heat behavior during a stress induced phase transformation. During a SIPT the latent heat is released during loading and due to the reversible characteristic of the phenomenon it is absorbed during unloading. On a tensile test, the loading rate plays an important role on the materials temperature due to the presence of the released/absorbed latent heat. If loading/unloading is performed slowly enough, heat transfer processes will remove the additional heat without any noticeable temperature variation. However if loading/unloading is applied rapidly, the temperature will rise during loading and fall during unloading, misleading any isothermal assumption.

Moreover, the rise in temperature leads to a rise of the critical transformation stress levels. This variation happens to be linear and is described by the Clausius-Clapeyron Eq.1.1 which states this variation as a constant (C) and relates with the entropy of phase transformation per unit of volume (∆S), enthalpy of phase transformation per unit of volume (∆H), test temperature (T ) and transformation strain (ε tr ) [START_REF] Otsuka | Shape memory materials[END_REF].

C = dσ dT = - ∆S ε tr = - ∆H ε tr T (1.1)
Figure 1.8 present a schematic representation of both isothermal (slow strain rate) and non-isothermal (high strain rate) tensile stress-strain behavior. During an isothermal SIPT (low strain-rate, see Fig. 1.8a) the heat transfer is strong enough to avoid the material's temperature increase. Because of that, the direct SIPT (from point B to C in Fig. 1.8a) presents a well defined stress plateau as well as the reverse SIPT (from point E to F in Fig. 1.8a). In another situation, increasing the strain-rate in the presence of a weak heat transfer rate (see Fig. 1.8b), the SIPT takes place with a non-isothermal response. In this case, the phase transformation no longer happens in a well defined stress plateau but shows a monotonic variation of the critical transformation stress level. A monotonic increase is observed on the critical stress level during loading (from point B' to C' in Fig. 1.8b) and a monotonic decrease on the critical stress level during unloading (from point E' to F' in Fig. 1.8b). The first studies on the latent heat and the related temperature change in superelastic SMA were performed focusing on the strain-rate impact and the origin of these thermal effects [START_REF] Rodriguez | The thermal effect due to stress-induced martensite formation in -CuAlNi single crystals[END_REF][START_REF] Brown | The Thermal Effect in Pseudoelastic Single Crystals of β-CuZnSn[END_REF][START_REF] Mukherjee | Thermal effects associated with stress-induced martensitic transformation in a TiNi alloy[END_REF][START_REF] Mccormick | Intrinsic thermal-mechanical behaviour associated with the stress-induced martensitic transformation in NiTi[END_REF]. Recent studies have also continued on the investigation of this phenomenon using advanced measurement methods such as digital image correlation (DIC) and infrared images to acquire local strain and temperature fields, respectively [START_REF] Schlosser | Influence of thermal and mechanical aspects on deformation behaviour of NiTi alloys[END_REF][START_REF] Delobelle | Contribuitions à l'étude thermomécanique des alliages à memoire de forme NiTi et à la réalisation par soudage e matériaux architecturés NiTi[END_REF][START_REF] Pieczyska | Activity of stress-induced martensite transformation in TiNi shape memory alloy studied by infrared technique[END_REF][START_REF] Zhang | Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips[END_REF][START_REF] Xie | Observation on rate-dependent cyclic transformation domain of super-elastic NiTi shape memory alloy[END_REF]. Also, the study of this phenomenon has been accompanied by several constitutive models to present an appropriate description of the superelastic effect when subjected to thermomechanical coupling [START_REF] Schmidt | A Phenomenological Model for Superelastic NiTi Wires Based on Plasticity With Focus on Strain-Rate Dependency Caused by Temperature[END_REF][START_REF] Zhu | A thermomechanical constitutive model for superelastic SMA wire with strain-rate dependence[END_REF][START_REF] Yin | Temperature Variation in NiTi Shape Memory Alloy During Cyclic Phase Transition[END_REF][START_REF] Roh | Thermomechanical Modeling of Shape Memory Alloys with Rate Dependency on the Pseudoelastic Behavior[END_REF]. Some engineering applications based on the thermomechanical coupling are presented in Section 1.2.1. Also, theoretical and experimental aspects present in literature in order to help understanding this phenomenon will be further discussed in the sections 1.2.2 and 1.2.3 respectively.

Chapter 1. Bibliography

Engineering applications based on thermomechanical coupling in NiTi superelasticity

Due to the potential use of the SMA superelastic effect in engineering applications, several studies have been performed to improve knowledge on the thermomechanical coupling present on this phenomenon. Among other applications, they have been used

1.2 Thermomechanical coupling in superelasticity as damping actuators to mitigate vibration amplitudes [START_REF] Pan | The Investigation of a Shape Memory Alloy Micro-Damper for MEMS Applications[END_REF][START_REF] Amarante | Bridge hinge-restrainers built up of NiTi superelastic shape-memory alloys[END_REF][START_REF] Dieng | Use of Shape Memory Alloys damper device to mitigate vibration amplitudes of bridge cables[END_REF][START_REF] Yin | Effect of deformation frequency on temperature and stress oscillations in cyclic phase transition of NiTi shape memory alloy[END_REF][START_REF] Gur | Thermo-mechanical strain ratedependent behavior of shape memory alloys as vibration dampers and comparison to conventional dampers[END_REF] and as solid state elastocaloric cooling devices as a potential material to replace vapor compression refrigerators [START_REF] Pataky | Elastocaloric cooling potential of NiTi, Ni2FeGa, and CoNiAl[END_REF][START_REF] Ossmer | Elastocaloric heat pumping using a shape memory alloy foil device[END_REF]Qian et al., 2016b;[START_REF] Tušek | Elastocaloric effect of Ni-Ti wire for application in a cooling device[END_REF]Tušek et al., 2016a).

The influence of the thermomechanical coupling in damping application and its use in elastocaloric cooling devices will be further discussed in sections 1.2.1.1 and 1.2.1.2 respectively.

Damping Applications

The mechanical hysteresis present on a superelastic cycle represents a mechanical energy dissipation and can be used to reduce vibration amplitude in dynamic applications. The higher the mechanical hysteresis the greater the dissipated mechanical energy.

The shape of the stress-strain superelastic loop is directly related to the critical stresses to induce phase transformation (σ As , σ A f , σ Ms and σ M f ), which in turn are linear dependent on temperature. So, during loading in a tensile test, if the strain rate is high enough to increase the temperature of the material due to the released latent heat, the critical stress levels will increase. Analogously, during unloading, if the strain rate is high enough to decrease the temperature of the material due to the absorbed latent heat, the critical stress levels will decrease. This variation on the critical stress levels will modify the shape of the superelastic hysteretic loop and consequently the amount of dissipated energy. In this context, several works investigated the influence of strain rate on the superelastic hysteretic loop of shape memory alloys [START_REF] Zhang | Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips[END_REF][START_REF] He | On non-monotonic rate dependence of stress hysteresis of superelastic shape memory alloy bars[END_REF][START_REF] Kan | Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy[END_REF]. [START_REF] Zhang | Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips[END_REF] studied the influence of the strain rate on the stress hysteresis size during tensile tests of superelastic NiTi alloys performed at room temperature in stagnant air conditions. They observed temperature variations (∆T ) on the order of ∆T = +29 • C during loading and ∆T = -9 • C during unloading. These temperature variations were responsible for increasing the stress level in around ∆σ = 100M P a at the end of loading and ∆σ = 50M P a at the end of unloading. The critical stress variation associated to the temperature variation promoted an important change on the superelastic loop area as observed in Fig. 1.9a. From the stress-strain curves plotted in Fig. 1.9a the authors calculated the dissipated mechanical energies as the area inside the hysteresis loop and plotted in function of strain rate, as shown in Fig. 1.9b.

They observed a non-monotonic variation of the stress hysteresis with the strain rate reaching a peak at ε = 2.0 × 10 -3 s -1 . The stress hysteresis peak observed at this strain rate was the result of the interaction between the heat released/absorbed during the material's phase transformation and the ambient conditions. This interaction allowed the increase of the upper stress plateau and the decrease of the lower plateau (see the blue loop in Fig. 1.9a) allowing so the increase in the loop area. They also observed that for strain rates below ε = 1.1 × 10 -4 s -1 the stress hysteresis showed a constant trend Chapter 1. Bibliography once the temperature variation observed was negligible. The same constant tendency was observed for strain rates above ε = 1.1 × 10 -1 s -1 were an adiabatic behavior was observed. Finally they concluded that the mechanical hysteresis variation was governed by the effect of the temperature induced by the released/absorbed latent heat during SIPT.

a) b)

Figure 1.9: (a) Macroscopic nominal stress-strain curves for different strain rates and (b) the variation in the average stress hysteresis with nominal strain rate (adapted from [START_REF] Zhang | Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips[END_REF].

In a similar work, He et al., 2010 investigated the effect of the ambient condition on the thermomechanical behavior of SMA. They submitted SMA specimens to tensile tests at three different ambient conditions: still air and 2 different airflow speeds, all of them at the room temperature. Figure 1.10 shows the stress hysteresis in function of the strain rate for all the ambient conditions. As well as [START_REF] Zhang | Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips[END_REF], they observed a non-monotonic increase on the stress hysteresis with the strain rate for the three ambient conditions. With these results He et al., 2010 showed that independently of the ambient condition, the temperature variation due to thermomechanical coupling will always affect the stress hysteresis once the strain rate do not provide an isothermal 1.2 Thermomechanical coupling in superelasticity or adiabatic test condition. In other words, there will be always a competition between the release/absorption of heat during SIPT and its dissipation to the ambient.

Elastocaloric Cooling Devices

As previously shown, the thermomechanical coupling present on the SIPT has been studied regarding its mechanism and its effect on the mechanical response in cyclic dynamic applications (as described in section 1.2.1.1). It was not until very recently that the thermomechanical coupling also called elastocaloric effect was recognized as a potential cooling mechanism for heat-pumping applications. Its advantage over the classic vapor compression cooling technology is the use of a solid state material instead of refrigerant gases. The commonly used gases such as hydrochloroflurocarbons (HCFCs) and hydrofluorocarbons (HFCs) are a significant source of the greenhouse effect, making this technology not environmentally friendly.

The use of the SMA in elastocaloric cooling devices is based on the heat pump principle which is to transfer the energy from a heat source to a heat sink. As described by Schmidt et al., 2015a, similar to the conventional vapor compression refrigeration cycle, the SMA based cooling cycle can be divided in four phases (see Fig. 1.11):

1. Loading: The strain increases at a high strain rate leads to a near-adiabatic temperature increase of the SMA ribbon above the temperature of the heat sink.

2. Heat transfer to heat sink: At constant high strain the SMA ribbon contacts the heat sink. The heat sink absorbs heat from the SMA, the temperature of the SMA ribbon decreases and the temperature of the heat sink increases.

Chapter 1. Bibliography 3. Unloading: The strain decreases at a high strain rate, leading to a temperature decrease of the SMA ribbon below the temperature of the heat source. 4. Heat transfer from heat source: At constant low strain the SMA ribbon contacts the heat source. The SMA absorbs heat from the heat source, the temperature of the SMA ribbon increases and the temperature of the heat source decreases.

Figure 1.11: Schematic of a SMA based cooling process described in four steps: 1.Loading, 2.Heat transfer to heat sink, 3.Unloading and 4.Heat transfer from heat source (Adapted from Schmidt et al., 2015a).

This exemplary elastocaloric cooling cycle shows the feasibility of a heat pump system using shape memory alloys. In this way several works on the application of SMA in elastocaloric cooling devices have been proposed in the literature (Schmidt et al., 2015a;[START_REF] Schmidt | Elastocaloric Cooling: from Fundamental Thermodynamics to Solid State Air Conditioning[END_REF]Qian et al., 2016b;Qian et al., 2016a). In these studies an important property analyzed in regard of heat pumping is the system efficiency. To measure the efficiency of a heat pump cycle a commonly metric used is the coefficient of performance (COP) (Tušek et al., 2016b). It is defined as the ratio between the cooling/heating energy and the input work required to perform a thermodynamic cycle (Eq. 1.2).

COP

= Q W (1.2)
with Q standing for the thermal energy and W the mechanical energy.

In the work published by [START_REF] Cui | Demonstration of high efficiency elastocaloric cooling with large ∆T using NiTi wires[END_REF] it was investigated the efficiency of elastocaloric cooling SMA as solid-state refrigerant to be used in heat pumps. They obtained 1.2 Thermomechanical coupling in superelasticity COP levels in the order of 11.8 and 3.7 for NiTi wires when submitted to tension and compression respectively. These values corresponds to 83.7% and 56.5% when compared to theoretical Carnot cycle COP values (see Eq.1.3). For comparison purpose, vapor compression commercial systems show COP values up to 60% of the theoretical Carnot cycle [START_REF] Cui | Demonstration of high efficiency elastocaloric cooling with large ∆T using NiTi wires[END_REF].

COP Carnot = T c T h -T c (1.3)
with T c standing for the temperature of the cold source and T h the temperature of the hot source.

In this way, the use of SMA shows great potential as refrigerant element to replace gas in vapor compression heat pump systems. However, to master this technology, one of the key parameter is the knowledge of the thermal energy involved on the SIPT.

Thermodynamic framework of thermoelastic martensitic transformation

Several thermodynamic frameworks provide a description of the thermomechanical behavior of shape memory alloys. Some aims to account the complex state behaviour and thermomechanical coupling of polycrystaline SMA [START_REF] Mcnichols | Thermodynamics of Nitinol[END_REF][START_REF] Chrysochoos | Energy balance of thermoelastic martensite transformation under stress[END_REF]. Others are based on the pioneering work of Cohen and his collaborators [START_REF] Patel | Criterion for the action of applied stress in the martensitic transformation[END_REF][START_REF] Salzbrenner | On the thermodynamics of thermoelastic martensitic transformations[END_REF]. In all of these approaches, the martensite transformation is considered a first order diffusionless reversible phase transformation. Also equilibrium condition is analyzed at the austenite-martensite interface of an elementary volume of material.

In this context, a thermodynamic framework based on the studies developed by Ortín and[START_REF] Ortín | Thermodynamics of thermoelastic martensitic transformations[END_REF]Wollants et al., 1993 is presented. Firstly the hypothetical absence of any irreversibility is considered. Next, the particular case of a uniaxial stress induced phase transformation is presented. Finally, the sources of irreversibility are introduced and its effects on thermomechanical behaviour are discussed.

Notation and units

The units of properties involved in the thermodynamic framework is listed on Table 1.1. The units apply to the symbols even when sub or superscripts are added. This table is for reference and throughout text these symbols are still presented. Throughout this section ƥ denotes the difference of a property between martensite and the parent phases.

Chapter 1. Bibliography A thermodynamic potential is a state function that describes the behaviour of a thermodynamic system as function of its natural variables. For the thermodynamic system consisting of martensite or austenite phase, the thermodynamic potential is the specific Gibbs free energy G and the natural variables are pressure P and temperature T . According to the first and second laws of thermodynamics, Eq. (1.4) relates G to P and V for the case where only P -V reversible work is involved

G(P, T ) = U + P V -T S = H -T S (1.4)
where U is the system's specific internal energy, V is the specific volume, S is the specific entropy and H is the specific enthalpy.

The driving force for the phase transformation is the difference between the Gibbs free energy of austenite G A P,T and that of martensite G M P,T :

∆G P,T = G M P,T -G A P,T (1.5)
A thermodynamic equilibrium is met when the difference between the thermodynamic 1.2 Thermomechanical coupling in superelasticity potential of the phases is minimal for a set of values of the natural variables (P, T ) . In the two-phase system composed by martensite (M ) and austenite phases (A), this means at

G M P,T = G A P,T (1.6)
and according to Eq. (1.4), at

∆G P,T = ∆H P,T -T ∆S P,T = 0 (1.7)
with ∆H P,T and ∆S P,T being the enthalpy and entropy variation between phases at a given P, T pair. They originate from the chemical differences of austenite and martensite phases.

Uniaxially stress-induced transformation

When a stress is externally applied in the thermodynamic system, the thermoelastic martensitic transformation is said to be stress-induced. In this case a reversible mechanical work is performed on the system. According to continuum mechanics theory, the mechanical work per unit of mass performed on a representative element volume (REV) can be written as a function of the average stress and macroscopic strain tensors. If the stress is applied uniaxially the specific mechanical work is

W mech = 1 ρ σdε (1.8)
where ρ is the mass density, σ is the uniaxial stress tensor externally applied on the system and ε is the uniaxial macroscopic strain.

Ignoring elastic and plastic strain components and considering only transformation strains (ε tr ), as well as introducing f m as the martensite fraction and df m as an infinitesimal step of martensite fraction, Eq. (1.8) can be written as

W mech = 1 ρ σdε = ∂W mech ∂f m df m = W mech df m (1.9)
A linear relation between dε tr and df m is assumed:

dε tr = ∆ε tr df m (1.10)
where ∆ε tr is defined as the transformation strain for a complete transformation, at f m = 1. Equation (1.9) then becomes Chapter 1. Bibliography

W mech = 1 ρ σ∆ε tr (1.11)
Finally, the difference between the Gibbs free energy of martensite and austenite in Eq. (1.7) can be rewritten as

∆G P,T = ∆H P,T -T ∆S P,T - 1 ρ σ∆ε tr = 0 (1.12)
which expresses the thermodynamic potential of a uniaxially stress-induced thermoelastic martensitic transformation happening on thermodynamic equilibrium.

This external work displaces the thermodynamic equilibrium, that now occurs at T = T (σ). However, it does not change the fact that the transformation happens isothermally when pressure is kept constant.

Sources of thermodynamic irreversibility

In the previous section the thermodynamic potential for phase transformation was presented for the case where only reversible work is involved (either P -V or external mechanical work). However, in reality, the martensitic phase transformation does not occur isothermally, but rather in a temperature range, generally accompanied by a hysteresis between forward and reverse transformations.

These deviations from equilibrium are caused by the presence of two main nonchemical irreversible contributions in the thermodynamic potential: the frictional work, associated with the energy dissipated by the relative movements between the austenite and martensite interfaces; and the stored elastic energy, associated with the shape and volume changes during transformation between austenite and martensite.

1.2.2.3.a Frictional work E f r

Frictional work is present in thermally or stress induced transformations. It can be defined as the work performed to overcome interface motion, which is directly related to the amount and nature of the barriers opposing the movement of transformation fronts. This parcel of energy is ultimately dissipated from the system as heat during both forward and reverse transformations [START_REF] Wollants | Thermally-and stress-induced thermoelastic martensitic transformations in the reference frame of equilibrium thermodynamics[END_REF].

1.2.2.3.b Stored elastic energy E st

The stored elastic energy is generated from two sources: elastic strain energy and interfacial energy [START_REF] Ortín | Thermodynamics of thermoelastic martensitic transformations[END_REF][START_REF] Hamilton | Stress dependence of the hysteresis in single crystal NiTi alloys[END_REF]. The first can be defined as 1.2 Thermomechanical coupling in superelasticity the work performed against internal forces to modify the total strain of the system and the latter comes from the creation of new interfaces and/or changes in existing interface dimensions.

Thermodynamic potential with irreversibility contributions

These non-chemical contributions must be added to the thermodynamic potential expressed in Eq. (1.7), characterizing the thermoelastic phase transformation as it happens in reality. From now on, the subscript notation "P, T " is omitted and the following subscripts are adopted: "ch" for chemical energy contributions, "f r" for non-chemical frictional work and "st" for non-chemical stored elastic energy. Thus, for a given set of natural variables one obtains

dG = (∆H ch -T ∆S ch )df m + ∂E st + ∂E f r = 0 (1.13)
where ∆H ch and ∆S ch constitute the chemical contributions of enthalpy and entropy changes, respectively; and ∂E st and ∂E f r constitute the non-chemical contributions of stored elastic energy and frictional work, respectively.

Equation (1.13) can be rewritten as function of the martensite mass fraction as

dG = (∆H ch -T ∆S ch )df m + (∂E st /∂f m )df m + (∂E f r /∂f m )df m = 0 (1.14)
or in derivative form as

G = ∆H ch -T ∆S ch + E st + E f r = 0 (1.15)
where the superscript designates the partial derivative of a property with respect to f m . Thus,

E st = ∂E st /∂f m and E f r = ∂E f r /∂f m .
When the material is fully in the austenite phase, the martensitic fraction is written f m = 0 and when fully in the martensite phase the martensitic fration is f m = 1. During the forward transformation the infinitesimal martensite mass fraction step df m is positive and during reverse transformation it is negative.

Including the contribution of an external mechanical work in a stress-induced phase transformation occurring under the application of uniaxial stress, Eq. (1.15) can be rewritten as

G = ∆H ch -T ∆S ch + E st + E f r - 1 ρ σ∆ε tr = 0 (1.16)
Chapter 1. Bibliography

Sign convention

Considering that work performed by the system is negative and work performed on the system is positive, the sign of each non-chemical contribution in Eq. (1.15) can be defined for the forward and reverse transformations.

The stored elastic energy is work performed by the system during forward transformation and work performed on the system during reverse transformation. Therefore dE st must be positive during forward and negative during reverse transformation. This makes both E A→M [START_REF] Rodriguez | The thermal effect due to stress-induced martensite formation in -CuAlNi single crystals[END_REF].

Thermomechanical coupling in superelasticity

A thermocouple welded to the center of the specimen's gauge length (l 0 =20 mm) was used to investigate the temperature variation with the increase of the strain rate on the superelastic behavior. They observed an increase in the specimen temperature of about ∼ 4 • C for a cross-head speed of 5 mm/min and up to ∼ 14 • C when submitted to a higher cross-head speed of 500 mm/min. Peyroux et al., 1998b also investigated the thermomechanical coupling during a stress induced phase transformation on superelastic CuZnAl alloys. Using infrared and calorimetric techniques, they analyzed the temperature variation and investigated the heat sources present on this phenomenon. Mechanical and thermal energies were estimated during the stress induced phase transformation for different strain rates. They observed that, independently of the strain-rate, mechanical and thermal energy remained constant despite the increasing temperature (see Fig. 1.13). They concluded that the increasing behavior of the stress level with the strain-rate was due to the thermomechanical coupling effects rather than viscous effects.

Full field measurements

Since the pioneer work of [START_REF] Miyazaki | Lüders-like deformation observed in the transformation pseudoelasticity of a TiNi alloy[END_REF] using optical methods, many studies has been published with this kind of technique. With the advance of technology, thermal measurements also started to be performed via infrared cameras. The first observation of thermal fields recorded during a tensile test was published by Shaw and Kyriakides, 1997a. Then, many studies using simultaneous kinematics and thermal field measurements were applied to investigate the thermomechanical coupling in SMA's. For example, [START_REF] Schlosser | Image Processing to Estimate the Heat Sources Related to Phase Transformations during Tensile Tests of NiTi Tubes[END_REF] investigated the thermomechanical coupling during a stress induced phase transformation on a superelastic NiTi tube. Thanks to a thermal camera and a digital image correlation (DIC) system, they obtained the temperature and strain fields during a tensile test and quantified the amount of energy involved on the SIPT. The material showed a strong localized deformation behavior and due to the specimen's geometry (tube), the Luders-like deformation bands appeared in a helical or ring shape [START_REF] Schlosser | Influence of thermal and mechanical aspects on deformation behaviour of NiTi alloys[END_REF]. Due to the localized strain presence a temperature gradient was observed in the specimen. Finally, even with a heterogeneous temperature field, they were able to perform a 2D formulation of the thermal problem in order to quantify the heat sources on the tube. Figures 1.14 a and c show an example of a temperature and heat sources with a spatiotemporal representation in Fig. 1.14d.

These results put in evidence how the advances of technology and the development of full field measurement techniques, like infrared thermography and DIC, improved the accuracy and allowed local investigations of thermomechanical couplings in SMA.

Heat sources estimation

As mentioned before, temperature variations are observed during a SIPT. The cause of these thermal response are the local heat sources. The aim of this section is to present the method chosen to estimate heat sources from temperature measurements. A first set of assumptions is stated in order to write a general heat balance equation describing the heat transfer in solid materials (Section 1. 2.3.3.a). Then, the model is simplified into a 0D formulation which is the case when homogeneous thermal response is observed (Section 1.2.3.3.b).

Thermomechanical coupling in superelasticity

1.2.3.3.a General heat balance equation

The heat balance equation governing heat transfer is deduced from the first law of thermodynamics applied to continuum media. The spatial local form of the first law combined with the momentum and mass energy balances leads to the energy equation. This one introduces the specific internal energy e, the mass density ρ, the internal stress power per unit of volume p int , the heat flow vector q per unit area and the distributed external heat sources s e per unit volume. The given equation reads:

ρ ė = -p int + s e -div q
(1.17)

The term "div" shown in Eq.1.17 is the divergence operator applied to the heat flow vector q, defined with respect to the spatial coordinates x i . The term ė represents the material time derivative of the specific internal energy e.

The heat conduction is assumed to be governed by the standard Fourier's law and the medium considered isotropic, with that, the heat flow q is written:

q = -k grad T (1.18)
where "grad" is the gradient operator with respect to the spatial coordinates x i and the thermal conductivity k. In our analysis, thermal conductivity is assumed uniform along the sample and constant during the test.

Rewriting the Eqs. 1.17 and 1.18 with the specific heat capacity c, the local heat balance is defined as [START_REF] Louche | Analyse pas thermographie infrarouge des effets dissipatifs de la localisation dans des aciers[END_REF][START_REF] Schlosser | Influence of thermal and mechanical aspects on deformation behaviour of NiTi alloys[END_REF][START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF]:

ρc Ṫ -k lap T = s i + s e (1.19)
where "lap" stands for the Laplacian operator with respect to the x i and s i stands for the heat rate generation per unit volume of all internal origins such as: intrinsic generation, thermoelastic coupling or latent heat due to solid-solid phase transformation [START_REF] Mccormick | Intrinsic thermal-mechanical behaviour associated with the stress-induced martensitic transformation in NiTi[END_REF].

Analyzing the Eq.1.19, if the temperature is expressed as a function of the spatial position M of a particle P , i.e. T (M, t), the material time derivative of temperature is given by:

Ṫ = T ,t + v grad T (1.20)
with T ,t = ∂T /∂t as temperature time derivative and v the current velocity of the considered particle at the spatial position M . The second term of the right hand side of the Eq. 1.20 implies the necessity to know simultaneously the kinematic and thermal Chapter 1. Bibliography fields. In our approach the terms "v grad T " and "s e " are assumed to be negligible. Because of that, the general heat balance equation is finally written as:

ρcT ,t -k lap T = s i + s e (1.21) 1.2.3.3.b Simplified thermal model
The general heat balance equation Eq.1.21 can be written in a simpler form when the internal heat sources are uniformly distributed (s i (x, y, z, t) = s i (t)) in a specimen surrounded by ambient medium at a temperature T 0 and when the specimen's temperature (T ) is not too far from thermal equilibrium [START_REF] Chrysochoos | Energy balance of thermoelastic martensite transformation under stress[END_REF][START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF]. Hence, at any point of the specimen's surface, the Laplacian term can be linearised:

-k ρc lap θ(t) ≈ θ(t) τ eq (1.22)
where θ is the temperature variation (θ = T -T 0 ) and θ T .

Therefore, Eq.1.21 becomes:

ρc dθ(t) dt + θ(t) τ eq = s i (t) (1.23)
The parameter "τ eq " represents a characteristic time reflecting the heat losses (by radiation, convection and conduction) through the specimen surfaces and towards the grip zones, respectively. This parameter is experimentally estimated and is necessary to solve the Eq. 1.23. One method to estimate the τ eq is to observe the natural temperature variation (towards the equilibrium) during a hold time after a loading/unloading test. During the temperature stabilization, heat sources are assumed to be zero once no phase transformation, thermoelasticity nor plasticity are present. From that, τ eq is then estimated through the following equation:

dθ(t) dt + θ(t) τ eq = 0 (1.24) 1.3 Nanostructured NiTi wires 1.3 Nanostructured NiTi wires 1.3.1 Specificities of the NiTi alloys 1.3.1.1 General Characteristics
NiTi alloys have been studied most extensively and used in the greatest number of commercial applications when compared to other SMA. This alloy exhibits strong shape memory and superelastic effects (up to 8% recoverable strain) [START_REF] Lagoudas | Shape Memory Alloys[END_REF] which makes this material ideal for a variety of applications.

Equiatomic NiTi alloys (i.e. 50 at. % of Ni) exhibit the highest final critical temperature transformation A f , around 120 • C. Decreasing the atomic percentage of Ni in the alloy does not change the transformation temperatures. However, increasing the Ni percentage to 51 % at. the transformation temperature decreases to A f = -40 • C. This transformation temperature variation can be designed to obtain the desired behaviour (SME or SE) at the application's temperature (e.g. human's body temperature at biomedical applications).

Thermomechanical properties are strongly dependent on the alloy's composition as well as thermomechanical historic processing. In Ni-rich alloys, heat treatments may promote the formation of T i 3 N i 4 precipitates which reduces the amount of Ni on the NiTi matrix. This reduction directly affects the critical transformation temperature which may change the alloy's characteristics from SME to SE at room temperature, for example.

Crystallographic microstructure of NiTi alloys

The martensite transformation is a diffusionless phase transformation in which the atoms move cooperatively driven by a shear-like mechanism. The atomic movement inside the crystal lattice promotes an atomic rearrangement which generates a crystallographic phase change. Even though the relative atomic displacements are small (compared to inter-atomic distance) a macroscopic shape change is produced [START_REF] Otsuka | Shape memory materials[END_REF]. NiTi near-equiatomic SMAs show three different crystallographic lattices in solid state, namely: body centered cubic (BCC) also known as B2 cubic austenite; monoclinic or B19 martensite and rhomboedral or R-phase. The latter appear under certain structural and thermomechanical conditions between the austenite and martensite phases. A schematic representation of the above mentioned lattices is shown in the Fig. 1.15.

Strain localization phenomenon in NiTi alloys

The strain localization consists on the appearance and propagation of macroscopic strain bands under constant stress level (plateau-type stress-strain curve). Also known as Luders-like deformation mode, this mechanism has only been observed in NiTi alloys, not in any other polycrystalline SMAs [START_REF] Sittner | On the origin of Lüders-like deformation of NiTi shape memory alloys[END_REF]. An example of strain localization observed by full field measurements (kinematic and thermal fields) is given by Fig. 1.16. The main characteristics of this phenomenon are:

• It has been found to occur only during deformation in tension, not under another applied stress state, e.g. shear, compression, tension-torsion [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];

• It has been found to occur during tensile deformation via any process: stress induced transformation or stress martensitic reorientation;

• Whereas it has been observed in a variety of specimen geometries (thin wires, thick bars, ribbons, plates, dog-bone shaped specimens and tubes), this effect appears to be more pronounced in specimens of large length-to-thickness aspect ratios (Shaw and Kyriakides, 1997a;[START_REF] Messner | On instabilities and autocatalytic effects associated with the nucleation of martensite bands in polycrystalline SMA flat tensile specimens[END_REF].

Influence of wire manufacturing process on crystallographic structure and mechanical behaviour

NiTi wires are commonly manufactured through a sequence of hot and/or cold working which may be followed by a heat treatment above its recrystallization temperature (350 • C).

The hot/cold working is a process in which the material is deformed plastically to take the shape of the die geometry. In these thermomechanical forming processes the thickness reduction may reach values up to 60% [START_REF] Koike | Crystal-to-amorphous transformation of NiTi induced by cold rolling[END_REF][START_REF] Otsuka | Physical metallurgy of Ti-Ni-based shape memory alloys[END_REF], Chapter 1. Bibliography strongly modifying the material microstructure and directly affecting its thermomechanical properties. In most metallic alloys, these thermomechanical forming processes are used not only to obtain a final shape for the material, but also with the purpose of modifying the cast structure and produce a desired microstructure [START_REF] Ahadi | Effect of hot rolling on microstructure and transformation cycling behaviour of equiatomic NiTi shape memory alloy[END_REF]. The main differences between hot and cold working are listed in the Table 1.2 In NiTi alloys, hot working is mostly used to produce specific microstructure and induce a desired anisotropy usually oriented in the rolling/drawing direction. Also, since it is performed above the recrystallization temperature, hot working produces a material showing higher shape memory properties [START_REF] Ahadi | Effect of hot rolling on microstructure and transformation cycling behaviour of equiatomic NiTi shape memory alloy[END_REF] once dynamic recrystallization takes place during the process.

In cold working process, however, recrystallization does not take place and a strong grain size reduction is observed. Such a process causes an increase of the tensile strength and hardness, eliminates the superelastic plateau, reduces the ductility and shape memory recovery [START_REF] Mitwally | Effect of cold work and annealing on the structure and characteristics of NiTi alloy[END_REF]. It has also been published in literature that cold working process suppress martensitic transformation via grain size reduction and even the amorphization of the material's structure [START_REF] Koike | Crystal-to-amorphous transformation of NiTi induced by cold rolling[END_REF]. A heat treatment performed on a cold worked material can always recover and even optimize the suppressed properties, such as SE or SME [START_REF] Mitwally | Effect of cold work and annealing on the structure and characteristics of NiTi alloy[END_REF].

Influence of heat treatment on microstructure and mechanical behavior

It has been shown in literature that the properties of NiTi alloys can be manipulated through cold work, heat treatment or a combination of both. As previously mentioned, cold work promotes a grain-size reduction and even a partial or total amorphization of the crystalline microstructure in NiTi alloys [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF][START_REF] Koike | Crystal-to-amorphous transformation of NiTi induced by cold rolling[END_REF]. In a work published by [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF] a study was performed on the effect of non-conventional heat treatments on cold worked NiTi wires. On their approach, they submitted NiTi wires to different pulsed electric current times in order to perform local heat treatment through joule effect. From the obtained results they compared the evo-1.3 Nanostructured NiTi wires lution of the microstructure and the mechanical response of the tested wires, shown in Fig. 1.17.

Figure 1.17: Evolution of microstructure and corresponding superelastic functional properties in the Ni-Ti wires heat treated by electric pulses with pulse times t= 0, 6, 8, 10, 12, 14, 16 and 18 ms (a to h, respectively)(Adapted from [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF] Chapter 1. Bibliography Figure 1.17(a) presents the result obtained for the "as received" cold worked wire which was submitted to section area reduction of 45±5%. The stress -strain curve shows a pure elastic behavior characteristic of a strain hardened NiTi alloy. It is also possible to observe the partially amorphous microstructure revealed by the electron microscopy and transmission electron microscopy (TEM) diffraction pattern. Observing the stress -strain curve in Fig. 1.17(b) it is already noticeable a minor changing on the elastic behavior and a smaller increase on the strain level. From the TEM diffraction pattern, the weak intensity of halo suggests that recrystallization in the amorphous areas has taken place.

From Figures 1.17(c) to (h) it is clearly observed the evolution of the hysteretic superelastic loop with the increase of the intensity of heat treatment (pulsed current from 8ms to 18ms). This evolution is followed by the change of the microstructure characterized by the increase of the grain size observed on the micrographies and ratified through the explicit dots on the diffraction patterns.

In a similar work on the investigation of the grain size influence on the thermomechanical behavior of SMA Ahadi and Sun, 2014 studied the effect of the grain size on the rate -dependent thermomechanical response of cold worked NiTi sheets. In their study, they submitted a NiTi sheet to a cold work process where a 42% thickness reduction was achieved. A further heat treatment was performed at four different temperature -time configurations, namely SA = 520 • C for 6 min.; SB = 520 • C for 3 min.; SC = 520 • C for 2 min. and SD = 485 • C for 2 min. They observed an increase of the grain size with the increase of the intensity of heat treatment (temperature vs. time). The average grain size values observed were: SA = 90 nm, SB = 68 nm, SC = 42 nm and SD = 27 nm. This variation was reflected on the superelastic response where a higher hysteretic behavior was observed for the larger grain sized material, as shown at Fig. 1.18.

Nanostructured NiTi wires

A more explicit hysteretic loop implies that more phase transformation is happening on superelasticity. In this way, the more austenite phase being stress induced into martensite phase, the more latent heat of transformation can be released/absorbed. Following this reasoning, they submitted the cold rolled (CR) and the heat treated (SA, SB, SC and SD) sheets to three different strain rates in order to observe the temperature variation and further analyse its effect on the mechanical behavior. Figure 1.19 shows the (a) full -field thermografic images as well as the (b) volume -averaged temperatures T (t) for all the tensile tested sheets at the following strain rates: ε = 10 -3 s -1 , ε = 5×10 -3 s -1 and ε = 10 -1 s -1 .

Figure 1.19: (a) Full-field thermographic images and (b) volume -averaged temperature (T (t)) during loading -unloading at three strain rates of ε = 10 -3 s -1 , ε = 5 × 10 -3 s -1 and ε = 10 -1 s -1 (Adapted from [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF] Analyzing Fig. 1.19 combined with Fig. 1.18, it is easily shown that the more developed is the superelastic loop, more phase transformation is present and so, more latent Chapter 1. Bibliography heat is released. Due to this latent heat releasing, the temperature variations observed on tests increased from 17.2 • C for the SD sheet (lower heat treatment magnitude) to 27.1 • C for the SA sheet (higher heat treatment magnitude).

Chapter review

This chapter has presented, in the Section 1.1, some information about the characteristics of the shape memory alloys (SMA's). It was explained the mechanisms of phase transformation as well as the two main properties of these alloys, the shape memory effect (SME) and the superelasticity (SE).

The Section 1.2 presented the thermomechanical coupling during the superelasticity. In a first moment it was shown some engineering applications using this phenomenon. Further, a thermodynamic framework used to study the thermomechanical coupling was presented. Finally, some experimental studies on the thermomechanical coupling found in literature was shown.

In the last Section 1.3, the specificities of the NiTi SMA was presented. The influence of the manufacturing process and heat treatments in the microstructure of the material, as well as their impact on the thermomechanical coupling, was also presented.

CHAPTER

TWO EXPERIMENTAL ASPECTS

Introduction

The objective of this chapter is to present in detail all the experimental aspects of the work. Section 2.2 introduces the material and heat treatments performed to obtain different mechanical properties. Section 2.3 presents the details of the uniaxial tensile test and DSC performed on the material. Section 2.4 shows the applied tools and methods to measure the kinematic and thermal full field responses of the analyzed material. Experimental details and difficulties are also discussed. Finally, section 2.5 presents a study on the repeatability of thermomechanical measurements.

Material 2.2.1 NiTi wire sample

The wire used in all performed experiments was a commercial cold worked (noted CW) T i -50.9 at.% N i alloy with a nominal diameter of d=0.5mm provided by Fort Wayne Metals. Figure 2.1 shows a) an example of such a NiTi wire sample and b) its mechanical behavior.

Due to its fabrication history a hardened mechanical response is observed. To overcome that and to allow the intended study, heat treatments were performed on the wire. Chapter 2. Experimental Aspects

Wire Microstructure

The microstructure of the cold worked wire was analyzed using a transmission electron microscopy (TEM) technique. Figure 2.2a shows a bright field (BF) picture from a highresolution electron microscopy in which one can observe the nanosized polycrystalline microstructure of the material. Nanograins ranging from nearly 10nm to 100nm are observed. The TEM tests were performed by Professor Yinong Liu from the School of Mechanical Engineering of the University of Western Australia.

Heat treatments

Four heat treatments were performed at 523K, 548K, 573K and 598K for 30min. The objective of the heat treatments was to trigger the superelastic effect on the initially cold worked material. In the following, these tests will be reported as HT523, HT548, HT573 and HT598. Figure 2.3 shows the mechanical behavior of the initial cold worked (CW) wire and the different heat treatments.

Performed tests 2.3.1 DSC test

After heat treatments, every sample was submitted to a DSC (Differential Scanning Calorimetry) analysis to investigate their thermomechanical properties. The purpose of this thermal investigation was to obtain the characteristic phase transformation temperatures (A s , A f , M s and M f ) and the specific latent heat of phase transformation (∆H).

The specific latent of phase transformation is determined by the area between the base line and the peak of heat flow vs. temperature curve. The characteristic transformation temperatures are determined from the inflection points of the curve. Figure 2.4a shows schematically the graphical method used to obtain these properties.

Uniaxial tensile tests 2.3.2.1 Test details

All tests were performed in a tensile testing machine Eplexor Gabo (see Fig. 2.5a) and carried out in controlled room temperature. During tests, two external data acquisition systems were used: an infrared (IR) camera and a visible light camera. The first one was responsible for the temperature field measurement while the second one for the strain field measurement. Despite the bigger area filmed by the visible light camera (compared to the IR camera), the analyzed area was limited by the IR camera spacial resolution (length of l 0 = 8.2mm).

The cameras were positioned in opposition. Each filmed "side" was coated differently. A black paint on the IR "side" and a black and white speckle painting on the visible light camera "side". The schematic representation of the tensile test with both external systems is shown in Fig. 2.5b.

The tensile test consisted in deforming the wire at a constant deformation rate up to a maximum deformation and returning to its initial (zero) deformation. Also, in order to obtain the necessary data to allow the thermal analysis, a hold time was performed after each loading and unloading. In this way, the tensile test was divided in four steps: I-loading, II-hold time, III-unloading and IV-hold time (loading: up to 6%, loading rate: ε = 10 -2 s -1 and hold time: 50s each). The schematic representation of the loading path is presented in Fig. 2.5c.

Performed tests

Gripping system

A so called "V-clamp" was used to minimize the sliding effect on the sample/clamp interface. Figure 2.6a shows two clamp models, a "V-clamp" and a "Flat clamp". On the detail of Fig. 2.6a, one can observe that the "V-clamp" promotes 3 contact points on the sample, while the "Flat clamp" holds the sample over 2 contact points only. Because of that, the "V-clamp" promotes a higher gripping when compared to the "Flat clamp". Fig. 2.6b shows a sample mounted on the "V-clamp" used on tests.

Full field measurements

Kinematic and thermal full field measurements are commonly used nowadays to study the thermomechanical coupling in materials and structures. With this kind of approach it is possible to study localized phenomena such as necking [START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF], Luder's like bands [START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF][START_REF] Louche | Thermal and dissipative effects of accompanying Lüders band propagation[END_REF],Portevin-Le Chatelier effect [START_REF] Louche | Thermal observations associated with the Portevin-Le Châtelier effect in an Al-Mg alloy[END_REF], phase transformation [START_REF] Schlosser | Image Processing to Estimate the Heat Sources Related to Phase Transformations during Tensile Tests of NiTi Tubes[END_REF][START_REF] Chrysochoos | Use of full-field digital image correlation and infrared thermography measurements for the thermomechanical analysis of material behaviour[END_REF] and fracture.

Despite the easy implementation of these techniques at first glance, quantitative measurements are not so easily obtained. The smaller detail on the experimental implementation can affect the image quality and may hugely affect the post processing data.

The objective of this section is to present the thermal and kinematic full field techniques used in this work and the experimental issues involved. The kinematic measurement technique (from Digital Image Correlation -DIC), as well as the experimental aspects and difficulties of the developed tests are exposed in the following subsection 2.4 Full field measurements 2.4.1. Analogously, the thermal measurement technique with its respective experimental aspects and difficulties are discussed in subsection 2.4.2.

Kinematic measurements 2.4.1.1 DIC method

Differently from using a strain gage, which gives a single local deformation, with this kind of tool it is possible to obtain a strain field throughout a whole surface. This method consists in measuring the displacement field of an object's surface by comparing its morphology (through the analysis of its grey level) between two images prior and after deformation. The texture of the surface can be either natural or created by a painted speckle that gives a high contrast random pattern to the surface (Fig. 2.7a).

The process of displacement field determination begins by placing a virtual grid over the image, generating smaller interest zones called "subset" (Fig. 2.7b). The grid and subset sizes are typically 10 to 50 pixels, but they are strongly related to the observed texture size (speckle size). The relation between the speckled texture size and the subset size may strongly influence the measurement quality. Once the grid and subset are defined, the displacement field is calculated by the software (VIC 2D from Correlated Solutions was used in this work) through the tracking of the new position of each defined subset on the deformed image. This tracking is performed through the minimization of a correlation factor calculated from the gray level at each subset prior and after deformation. Figure 2.8 shows an example of a correlated image with its respective grid before (initial grid Fig. 2.8a) and after (deformed grid Fig. 2.8b) deformation. With this, displacement and strain fields can be calculated.

Full field measurements

When using an artificial texture (speckle painting), it is important that the texture adheres to the surface during all test. This may be an obvious statement but it is common that paints loose its elasticity after a long dry. In this case, the texture may break and get loose from sample's surface, making measurement impossible. Because of that, it is crucial to perform tests soon after the sample preparation.

The size of random pattern also plays an important role since the deformation resolution is directly affected by its size and consequently by the subset size. This can be an issue specially in the preparation of surfaces with small dimensions, which is the case in this study. For example, preparing a 100mm x 100mm sample is much easier than a wire of d=0.5mm. In the latter case each drop of paint may have the size of the wire's diameter, reducing drastically the resolution of the measured deformation.

Experimental set-up and difficulties

To obtain a good quality artificial texture some special attention is required. As the correlation requires a good contrast, using a speckled black paint over a white surface is often more suitable than the inverse. Also, on the experiment set-up, the lightening has to be adjusted to avoid reflections or shadows on the sample to maximize the field of view, especially in the case of circular section samples (e.g. tubes and wires). Finally, a cold light source is recommended when a simultaneous thermal acquisition is performed. This prevents the heat of lightening from being absorbed by the material and interfering with the measurement. NiTi wire. In Fig. 2.10a one can observe the big drop of white paint disturbing the random speckle paint pattern. The Fig. 2.10b shows a low contrast speckle paint image, which commonly produces a high noise correlation result. Finally in Fig. 2.10c, one can observe an ideal speckled painting which produces a good correlation result due to its uniform pattern and good contrast. In this study, a black paint speckle over a white paint layer was used on the wire's side of the visible light camera acquisition (as observed in Fig. 2.10c).

Thermal measurements 2.4.2.1 Infrared thermography

Infrared (IR) cameras are based on the detection of infrared light, first discovered in the early 19 th century. The first IR camera was created in 1964 and firstly used for military purposes in the beginning of the 1980 decade. The use of this device for measurement purposes is now largely applied in mechanics of material studies to observe localized phenomena such as phase transformation (Shaw and Kyriakides, 1997b;[START_REF] Schlosser | Influence of thermal and mechanical aspects on deformation behaviour of NiTi alloys[END_REF][START_REF] Delobelle | Heat Estimation from Infrared Measurement Compared to DSC for Austenite to R Phase Transformation in a NiTi Alloy[END_REF], plasticity in metallic alloys [START_REF] Louche | Thermal and dissipative effects of accompanying Lüders band propagation[END_REF][START_REF] Louche | Thermal observations associated with the Portevin-Le Châtelier effect in an Al-Mg alloy[END_REF], fatigue behavior [START_REF] Berthel | Infrared image processing for the calorimetric analysis of fatigue phenomena[END_REF] and thermophysical properties [START_REF] Delobelle | Determination of Local Thermophysical Properties and Heat of Transition from Thermal Fields Measurement During Drop Calorimetric Experiment[END_REF][START_REF] Delobelle | Contribuitions à l'étude thermomécanique des alliages à memoire de forme NiTi et à la réalisation par soudage e matériaux architecturés NiTi[END_REF].

In the electromagnetic radiation spectrum, the IR wavelengths are present approximately between 0.7µm and 1mm. The first part of this range is classically subdivided into four categories: Near Infrared (NIR) between 0.7µm ∼ 1µm, Short Wave Infrared (SWIR) between 0.9µm ∼ 1.7µm, Medium Wave Infrared (MWIR) between 2µm ∼ 5µm and Long Wave Infrared (LWIR) between 7.5µm ∼ 14µm or more. The choice of the IR bandwidth depends on the application. An infrared camera does not measure the temperature of a filmed object but the 2.4 Full field measurements radiation that emanates from it. However, this radiation may be coming from the object itself (heat generated) or from the environment around it through reflection or transmission phenomena. It means that when a radiation is incident on an object it can be either absorbed, transmitted or reflected as schematically shown in Fig. 2.12. The emissivity factor represents the capacity of a body to absorb the incident IR radiation. An ideal object that absorbs all the incident radiation is called black body and its emissivity factor is = 1. Any real object would have an emissivity factor < 1. The emissivity of a body depends on many parameters, such as: chemical composition, wavelength, observation angle of the camera towards the surface, material temperature and surface aspect (roughness, polishing, etc). A commonly used hypothesis is that the emissivity and absorptivity of a body do not depend on the wavelength and on the temperature, and thus the emissivity is a constant. Fig. 2.13 shows the emissivity factors of typical materials.

1.0 0.0 0.2 0.4 0.6 0.8 0.9 In the present work a FLIR IR camera (model A6500sc) was chosen. It was equipped with a MWIR detector with a resolution of 640x512 pixels and a maximum frame rate of 120Hz (the acquisition rate used during tests was 30Hz).

A zoom objective lens was selected in order to observe the very small specimen. The standard objective lens provided a pixel resolution of 122µm while the zoom objective lens has a pixel resolution of 16µm. This lens provided an optic increase on the field of view (FOV) of about 7.6 times compared to the standard objective lens.

Sample preparation and experimental cautions

In this study the sample used (NiTi wire) presented a polished surface and so a low emissivity factor. The samples were covered with a thin layer of a high emissivity black paint on the IR measurement side of the sample. This technique, commonly applied in thermography, allowed to obtain a gray body condition with an emissivity coefficient higher than > 0.95.

Due to the emissivity smaller than 1, the measured body may reflect the surroundings heat. Because of that, the sample's surrounding was insulated during tests. When performing the simultaneous DIC, LED lights were used to illuminate the scene (LED light promotes a colder light source when compared to incandescent light bulbs). Finally, the test operator was not positioned behind the filmed scene to avoid heat interference.

Camera calibration

The infrared camera has a matrix of detectors which translate the IR radiation into temperature. However, to reach this, some calibration procedures must be taken. The NUC procedure consists in submitting all the captors to two IR radiation levels, i.e. to film a uniform black body at two constant temperatures. Basically, the camera's software uses the two temperature levels to correct the gain (curve slope) and the offset (distance between the curves) of the measured signals.

Figure 2.16a shows the schematic representation of the random digital output of the detectors and an example of radiance image generated before the NUC procedure. Despite filming a black body surface with uniform temperature, the radiance image shows a non-uniform response.

After performing the NUC procedure, the detectors output signal are aligned and Chapter 2. Experimental Aspects show the same digital level between them. Fig. 2.16b shows the schematic representation of the aligned digital output responses and an example of radiance image after the NUC procedure. In this case, it is possible to observe the uniformity on the radiance image.

2.4.2.4.c From radiance to temperature

After the NUC procedure a temperature calibration must be performed. It consists in filming a black body in several temperature in a given range. Automatically, the camera's software draws a curve of digital level vs. temperature and find its equation. In this study, the camera was calibrated in a temperature range between 268K and 318K.

Data processing

The raw data obtained by the IR camera was processed in the MATLAB environment. A code was developed to identify the wire on every image and further obtain the value of temperature along its surface.

In the following, preliminary thermal images will be presented as temperature variation fields θ(x, y, t) = T (x, y, t) -T 0 (x, y, t) where T and T 0 are the temperature fields at the current time t and the initial time t 0 , respectively. The spatial positions in pixel are noted x for the circumferential direction and y for the longitudinal direction. As one can notice, the wire is distinguished on the image because of the difference between its temperature and the background. When tracing the circumferential temperature profile, one can observe (see Fig. 2.17b) that the image's background is around 300K (at this specific instant) while the wire is at 307K. So, by calculating the spatial derivative of the temperature profile along x direction 2 peaks are obtained as it can be seen in Fig. 2.17c. From that, the peaks named p2 and p3 represent the initial and final positions of the wire when sweeping the thermal image from left to right.

2.4.2.5.b Mean temperature estimation

On the detail of Fig. 2.18b, it is possible to observe the wire occupying 30 pixels on the image width. However, temperature is not uniform along the whole profile as shown by Fig. 2.18c. A lower temperature is observed at the extremity of curve (approximately after pixel 20). This decrease is due to the emissivity variation along the wire's surface, due to the experimental procedure of painting one "side" of the wire with black paint (for IR acquisition purpose) and the other "side" with the white (which has lower emissivity) and black speckle painting (to perform DIC analysis). To overcome this problem, it was simply decided to use data from the wire's zone with a uniform temperature, avoiding the non-uniform emissivity zone.

After analyzing the circumferential temperature profiles, longitudinal temperature profiles were plotted. For each image, the longitudinal profile was obtained by averaging the circumferential profiles. In other words, each point of the longitudinal profile (at y direction) was obtained by the mean value of the circumferential profile at the y pixel position. 2.4 Full field measurements Figure 2.19a presents a line traced along the longitudinal direction of the wire and its respective temperature variation in Fig. 2.19b. As it can be observed, the obtained temperature was uniform along the y direction. Because of that, the temperature was treated as independent of the y position and it was averaged along the wire longitudinal y direction.

Finally, the wire temperature variation during the test was calculated by the mean temperature variation (θ(t)) of the longitudinal profile at each image. Fig. 2.20 shows the temporal mean temperature evolution of a tested wire.

2.4.2.5.c Lost images during data acquisition

Infrared images acquisitions are made through a software provided by the camera manufacturer. As the camera is capable of a high acquisition rate, the software has two acquisition modes. The first one consists in writing every image directly on the computer's hard disk. The advantage of this mode is unlimited data acquisition (limited by the hard disk capacity). However, as the hard disk shows a low writing speed, for high rate acquisition, several images may be lost on the process. Because of that, a second acquisition mode is suggested for high rate acquisition. It consists in writing every image on the computer RAM memory, which presents a much faster writing speed. However, this mode is restricted to a limited number of data acquisition per test (10000 images). Even though, at high acquisition rates, some images may be lost also in this acquisition mode. The second acquisition mode was used in this work to minimize the number of lost images. Figure 2.21a shows a schematic representation of images lost during a test. This phenomenon is random and is present along the whole test duration. For example, in a test performed for this work, in a total of 4300 acquired images, 230 have been lost. To overcome this problem, a MATLAB code was developed to replace the lost data by linear interpolation of the data before and after the lost images.

The developed method consisted on identifying the gap position (on time) and adding the interpolated data in it. As the data were acquired in a constant frequency of 30Hz, the normal time between images was ∆t = 1/30s which is approximately ∆t = 0.0333s. Analyzing the values of ∆t along the data acquisition, it was observed that some values were two times bigger ∆t = 0.0667s. In these cases, it was confirmed that some images have been lost. Figure 2.21b shows an example of the time difference between images during a test. The bottom line shows the values of ∆t = 0.0333s, meaning that time between images were as expected, however, the top line shows a value of ∆t = 0.0667s, which means that the time between these images were the twice expected. From that, as one can observe on the detail of Fig. 2.21b, it was possible to determine in which position, these images were lost, in this case, at the positions: 622, 629, 637 and 645.

After identifying the position of the lost data, the mean temperature was interpolated and placed in the vacant position. Figure 2.22 shows the difference between the raw data (black curve) and the corrected data (red curve). The cumulative nature of the problem can be noticed by the increase of the gap between the first and second temperature peak. The correction of this problem was fundamental for the perfect temporal synchronization, further discussed in section 2.4.3.2. Chapter 2. Experimental Aspects

• PC 1: Thermal images were acquired by the infrared camera at a frequency of f IR = 30Hz for all tests. Image acquisition started at a time t 0(IR) .

• PC 2: Visible light images were acquired at a frequency of f V is = 10Hz in all tests.

Those images were then used by a DIC software to obtain the displacement and strain fields. Force was also acquired and automatically synchronized with strain fields by the same software. Image and force acquisition started at a time t 0(kin) , where "kin" stands for "kinematic" field.

In order to analyze the thermomechanical coupling, a synchronization of both data was necessary. As the acquisition was made separately and with different frequency rates, the temporal synchronization was made manually as presented in the following section 2.4.3.2. Due to the uniform nature of the acquired thermal field, spatial synchronization was not necessary [START_REF] Schlosser | Influence of thermal and mechanical aspects on deformation behaviour of NiTi alloys[END_REF].

Temporal synchronization

As the acquisitions were made independently, without a hardware synchronization, the initial time (t 0 ) was different for thermal and kinematic data. The synchronization was made manually and consisted in finding a characteristic event to determine the test beginning. For that, the beginning of test was set for the kinematic field, as the first point of the increasing deformation curve and for the thermal field as the first point of temperature variation.

Due to the constant acquisition rate of both systems (after fixing the issue of lost images during thermal acquisition, discussed in section 2.4.2.5.c) time synchronization was easily obtained after determining the test beginning. t 0(kin) t 0(IR)

7.5s

 (K)  (K) Figure 2
.24: a) Example of experimental data with both axial strain (ε) and temperature variation (θ) before synchronization. b) and after synchronization.

Repeatability of thermomechanical measurements

To investigate the repeatability of the experimental results, three tensile tests were performed under the same conditions (with the four steps described in section 2.3.2.1). These tests were performed in wires submitted to the heat treatment of 598K for 30 min.

The mechanical responses of the three samples (named HT 523 -1, HT 523 -2 and HT 523 -3) are presented in Fig. 2.25. One can notice qualitatively the same behavior for the three tested samples. Despite the lightly difference, a weak scattering with a standard deviation reaching 3% of the stress value is observed. Analogously, the temperature variations reported in Fig. 2.26 show qualitatively the same behavior between the samples. Even if the absolute temperature reached at the peak is lower for the HT 523 -3.

Chapter review

Chapter review

This chapter presented all the aspects of the experimental procedures used in this work. The used material, the performed heat treatments as well as the thermal and mechanical tests.

It was shown the superelastic response of a initially cold worked wire and the constant evolution of its behavior with different heat treatments. The details of the uniaxial tensile tests were also presented.

Full field measurement techniques such as digital image correlation (DIC) and infrared thermography have been described. Details of the experimental set-up and difficulties were presented for both techniques.

Concerning the DIC method, the issue of measuring the strain field on a tiny surface, such as 0.5mm diameter wire, was take into account by the difficulties of creating a speckle painting on its surface.

For the infrared measurement technique, several aspects have been detailed such as: the infrared camera operation, its calibration, the sample preparation and the experimental precautions. It was also described the post processing of the data in order to obtain the temperature variation of the measured sample as well as problematics of the signal synchronization.

The results of a brief study on the repeatability of the thermomechanical measurements have also been shown.

The experimental results of the simultaneous temperature and strain fields measurements will be presented and analyzed in the further chapters. Chapter 3 will show in more detail the procedure used to estimate the heat sources of phase transformation and the martensitic fraction in the wire submitted to the heat treatment of 598K for 30min. Chapter 4 performs the same analysis for the other heat treatments.

CHAPTER

THREE

SPECIFIC LATENT HEAT AND MARTENSITE FRACTION DURING UNIFORM CYCLIC SUPERELASTIC TENSILE DEFORMATION OF NANOSTRUCTURED NITI WIRES

This Chapter is written as a paper to be published soon.

ABSTRACT

Superelastic NiTi wires are manufactured by a sequence of hot and cold drawing which induces on the material a mixture of amorphous phase and highly strained nanocrystals. The present work analyses the thermomechanical tensile behavior of a cold rolled T i -50.9 at.% N i wire submitted to a heat treatment at 598 K for 30 min, which is below the recrystallization temperature (623 K). Such low temperature heat treatment promotes a fully superelastic loop without stress plateau and no Luders-like deformation. The specific latent heat in such nanostructured materials cannot be measured using Differential Scanning Calorimetry method due to the absence of peaks in the heat flow vs. temperature curve. In this work, a method using Digital Image Correlation (DIC) and Thermal Field Measurement (TFM) was developed and used to determine during superelastic tensile tests thermal power and thermal energy through a heat balance. From these results and using a thermodynamic approach, forward and reverse specific latent heats and martensite fraction in function of strain and stress were estimated.

Keywords: Kinematic and thermal full field approach, heat balance, enthalpy of transformation, superelastic NiTi, non-localized strain, nanostructured NiTi wires.

Introduction

NiTi shape memory wires are manufactured through a process of hot [START_REF] Ahadi | Effect of hot rolling on microstructure and transformation cycling behaviour of equiatomic NiTi shape memory alloy[END_REF] and cold working [START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF][START_REF] Mitwally | Effect of cold work and annealing on the structure and characteristics of NiTi alloy[END_REF]. Usual industrial wires are submitted to a further high temperature heat treatment (annealing), above 623K. Such a heat treatment promotes a fully recrystallized microstructure [START_REF] Peterlechner | Study of relaxation and crystallization kinetics of NiTi made amorphous by repeated cold rolling[END_REF][START_REF] Shi | Grain size effect on the R-phase transformation of nanocrystalline NiTi shape memory alloys[END_REF][START_REF] Sergueeva | Structure and properties of amorphous and nanocrystalline NiTi prepared by severe plastic deformation and annealing[END_REF][START_REF] Yu | Synchrotron high energy X-ray diffraction study of microstructure evolution of severely cold drawn NiTi wire during annealing[END_REF]. During heating and cooling of these annealed NiTi wires, an austenite to martensite phase transformation occurs and a large amount of latent heat is released or absorbed. These forward and reverse latent heats are commonly quantified through a Differential Scanning Calorimetry (DSC) test. These annealed wires stretched in tension are capable of achieving large amounts of pseudoelastic deformation associated to a stress-induced phase transformation (SIPT).

The pseudoelasticity, also called superelasticity, is the ability of the material to undergo a phase transformation when an external stress is applied. Superelastic tensile deformation of annealed NiTi wire is accompanied by a forward stress plateau during loading and a reverse stress plateau during unloading. The deformation along the wire during these two plateaux is non uniform and exhibits a large mechanical hysteresis.

Cold worked wires can be heat treated at lower temperatures (473K to 598K). When submitted to DSC tests to quantify the transformation latent heat, these aged materials show inconclusive results since a weak or no thermal peak is observed [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF][START_REF] Li | Cold Rolling-Induced Multistage Transformation in Ni-Rich NiTi Shape Memory Alloys[END_REF][START_REF] Chang | Transformation sequence in severely cold-rolled and annealed Ti50Ni50 alloy[END_REF][START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF][START_REF] Miller | Influence of cold work and heat treatment on the shape memory effect and plastic strain development of NiTi[END_REF]. Many authors showed that such wires exhibit the superelastic effect, however, without stress plateau and with a smaller mechanical hysteresis when compared to annealed materials [START_REF] Kan | Experimental observations on rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy[END_REF][START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF][START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF]. Moreover, no deformation localisation phenomenon is observed for these low temperature aged wires.

The knowledge of the amount of latent heat of phase transformation is mandatory in many applications, such as those which use the elastocaloric effect of NiTi alloys in cooling devices (Tušek et al., 2016b;[START_REF] Schmidt | Elastocaloric Cooling: from Fundamental Thermodynamics to Solid State Air Conditioning[END_REF]. These devices make use of the latent heat of stress induced phase transformation (SIPT) to perform a temperature change in a designed system. Working as heat pumps, a cyclic behaviour of phase transformation is needed to maintain the cooling effect, which in turn, implies in a huge fatigue life. Aged materials present a suitable mechanical behaviour to obtain a longer fatigue life since less strain deformation is obtained in the presence of phase transformation (Qian et al., 2016b). However inconclusive DSC tests are a barrier to determine the transformation latent heat and its further use on the design of elastocaloric applications.

To overcome this problematic, the analysis of the temperature variation during a SIPT can be performed through a thermal approach to obtain the involved heat sources. A method to estimate heat sources from temperature fields obtained with an infrared thermal camera has been presented by [START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF]. The same method was used to investigate the heat source of phase transformation [START_REF] Schlosser | Image Processing to Estimate the Heat Sources Related to Phase Transformations during Tensile Tests of NiTi Tubes[END_REF] and the deformation mechanism [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] of NiTi tubes.

Material and experimental procedure

In this context, this paper aims to quantify the heat sources generated during a SIPT of a cold worked wire aged at 598K for 30 min. For this purpose, temperature field was acquired using an infrared thermal camera. Strain field was also acquired using a visible camera and analyzed via digital image correlation (DIC). Using a thermodynamic framework based on the Gibbs free energy, the specific latent heat on loading and unloading were determined in function of strain and stress. Also, the martensite fraction was estimated and presented in function of strain and stress. Finally the obtained specific latent heat values were compared with the ones founded in literature for fully annealed NiTi alloys when tested using purely thermal methods (i.e. DSC).

Material and experimental procedure 3.2.1 Material

The material used in the experimental campaign was a commercial cold worked NiTi wire of d=0.5 mm diameter with a composition of T i -50.9 at.% N i provided by Fort Wayne Metals. The wire was heat treated (HT) at 598K for 30 min. Figure 3.1a shows the uniaxial tensile test at room temperature (T 0 = 297 K) performed to observe the superelastic behavior until rupture. Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires

Superelastic tensile tests

The NiTi wire sample was mounted on a standard gripping system on a tensile testing machine (Gabo Qualimeter) equipped with a 500N load cell. The gauge length of the sample was L 0 = 20 mm. Before being clamped, the wire was heated in boiling water and then cooled down to room temperature. This procedure was conducted to transform as much as possible the material to the austenite phase, allowing tests to start in the superelastic state.

The tensile test was performed by loading the NiTi wire at a pre-set global strain of 6% at a strain rate of 10 -2 s -1 . During the complete load-unload cycle, two hold-time were performed, one at the end of loading and the other at the end of unloading, during 50s each. Subsequently, the wire was submitted to a tensile test until rupture using the same strain rate.

Thermal and kinematics full field measurements during tensile tests

Temperature field was measured with a fast multi detector infrared camera (FLIR SC6000 series) with a resolution of 640x512 pixels equipped with a medium wavelength infrared (MWIR) macro 1 : 3.0 objective lens. Spatial resolution (pixel size) was 16 µm × 16 µm.

The surface of the wire observed by the IR camera was coated with a highly emissive black paint in order to obtain black body properties compatible with the infrared camera calibration. The thermal field was measured in section of length l 0 = 8.2mm, shorter than L 0 (see Fig. The displacement field was obtained using a visible-light digital camera (ALLIED, 3.2 Material and experimental procedure 2048x2048 pixels) and Digital Image Correlation "VIC 2D" processing software. The visible camera recorded the opposite side of the IR zone. The wire surface was covered with a random pattern of white paint speckles over a black paint coat to allow displacement and strain field measurements through image correlation. The resolution of the visible light camera ZOI was 2048x72 pixels. However, the ZOI used to perform the DIC analysis was 1181x72 pixels, which corresponded to the same length l 0 = 8.2mm filmed by the thermal camera. At this ZOI a subset of 25 pixels with a step of 17 pixels led to a spatial resolution of ∼ 0.17 mm in the axial direction and ∼ 0.12 mm at the circumferential direction. From that, the strain profiles obtained were calculated taking into account only the axial component ε yy of the true strain (logarithmic) tensor.

In the following, the true strain tensor ε yy will be named ε. The uniform temperature variation computed as the difference between the averaged absolute temperature in the IR ZOI T and the room temperature T 0 will be named θ , i.e. θ = T -T 0 .

From the full field measurements it was possible to obtain the Fig. 3.3. The stress vs. strain behavior, on loading, of the superelastic NiTi wire is plotted in Fig. 3.3a. Each square marker represents a strain level in which the strain profiles were plotted in Fig. 3.3b and the temperature variation profiles were plotted in Fig. 3.3c. The curves shown in Fig. 3.3b indicates an uniform strain behavior of the NiTi wire with no Luders-like localization mechanism [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF][START_REF] Leo | Transient heat transfer effects on the pseudoelastic behavior of shape-memory wires[END_REF]. This behavior is ratified by the Fig. 3.3c where the temperature variation profiles also present a uniform response over the wire's length. As temperature variation θ and strain ε were considered uniform over the wire's length, their values were averaged in this zone in order to construct their average quantities θ and ε along the axial length l 0 . Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires

Heat sources and thermal energy estimation

In the present work, due to the uniform temperature variation along the tested wire, a time dependent thermal approach was chosen. The thermal framework used to estimate heat sources during a superelastic tensile test is presented hereafter.

For a simple geometry (wire) and uniformly distributed heat sources (uniform temperature distribution), if the specimen is not too far from thermal equilibrium with the ambient medium T 0 , the thermal model reads [START_REF] Chrysochoos | Energy balance of thermoelastic martensite transformation under stress[END_REF][START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF][START_REF] Schlosser | Influence of thermal and mechanical aspects on deformation behaviour of NiTi alloys[END_REF]:

c ∂ θ ∂t + θ τ eq = q (3.1)
where q represents the uniform heat sources or thermal power per unit of mass.

As shown by [START_REF] Chrysochoos | Energy balance of thermoelastic martensite transformation under stress[END_REF][START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF], the heat source in the right member of Eq.3.1 can be calculated from experimental data. It is obtained from estimations of the two terms of the left member after some filtering and numerical derivative operations.

In the following and in order to simplify the notation, the uniform heat sources will be noted q. The specific heat c is assumed to be uniform and constant. The parameter τ eq represents a characteristic time of the heat losses by both convection and radiation through the specimen surface and by conduction towards the grip's zone.

Thermal energy E th can also be estimated. It is calculated through time integration of the heat sources per unit of mass q; as shown in Eq. 3.2.

E th (t) = t 0 qdt (3.2)

Determination of mechanical power and energy

Mechanical power P mech as well as the mechanical energy E mech were calculated. Both values are determined per unit of mass. The first one is defined as:

P mech = σ ε ρ (3.3)
where σ is the uniaxial tensile true stress, ε the uniaxial strain rate and ρ the mass density. True stress σ is calculated from the formula σ = σ 0 (1 + ε n ) where σ 0 is the engineering stress and ε n is the nominal strain [START_REF] Quan | A characterization for the dynamic recrystallization kinetics of as-extruded 7075 aluminum alloy based on true stress-strain curves[END_REF].

Results

The mechanical energy per unit of mass E mech was calculated by time integrating the mechanical power, as described in the following: • Stage I: Temperature raises as the strain goes up to 4.8%. Due to the fast release of the latent heat of phase transformation the temperature increase was θ = 11.8K.

E mech (t) = t 0 P mech dt (3.4)

Results

Strain and temperature field evolution

• Stage II: The hold time allows the temperature to decrease to T 0 while ε = 0 and ε remained almost constant.

Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires

• Stage III: The recovering of strain promoted the absorption of latent heat due to reverse transformation, leading to a negative temperature variation of θ = -8.7K. It can be noticed that at the end of the stage III (end of unloading) the temperature started to rise again just before the strain reaches zero.

• Stage IV: Second hold time allowing once again the temperature to increase to T 0 while ε = 0 and ε remained constant.

Estimation of thermal power

The averaged thermal power q was calculated using the averaged temperature θ shown in Fig. 3.4b and a processing of Eq. 3.1. The material properties used were ρ = 6500 Kgm -3 and c = 500 JKg -1 K -1 .  eq = 7.9 s Figure 3.5: a) Uniform temperature variation θ and Thermal power q , presented on left and right axis, respectively, in function of time. b) Mean and standard deviation of q, calculated in the time interval ∆t shown in a), obtained for different values of τ eq . Figure 3.5a shows the temperature variation θ and an example of the temporal evolution of the thermal power ( q) computed with a value of τ eq = 5s.

The best value of τ eq was calculated on the hold-time of stage IV. During this stage no heat source was present. From Eq. 3.1 and for each imposed τ eq value, the q quantity was computed on the period ∆t, with its statistical characteristic (mean and standard deviation). In Figure 3.5b the blue thick line represents the mean value of Thermal Power in function of τ eq , while the black lines show the standard deviation. With this procedure, it was possible to determine a value of τ eq = 7.9 s, considered as the optimal value leading to a thermal power close to zero. Similarly to Fig. 3.4b, it is possible to divide Fig. 3.6 in four distinct stages. In stage I, one can observe the sudden increase of thermal power and its continuous growth until it reaches a peak at approximately q = 2.1 W/g. The same qualitative behavior is shown by the mechanical power, which reaches at the same time a maximum at P mech = 1.0 W/g. After the thermal peak in the stage I, the decrease on thermal power highlights the decrease of the phase transformation "activity" before reaching the end of loading. Stage II, as well as the stage IV , which corresponds to the hold-times of Fig. 3.4b, are characterized by zero thermal power. Finally, in stage III (unloading) one can see the reversible thermomechanical behavior of the martensite transformation. The thermal power shows the same qualitative response compared to the stage I (loading) but with a peak at q = -1.7 W/g which, despite its lower amplitude, is around twice as large as the mechanical power, which reaches P mech = -0.8 W/g. Figure 3.7 shows the thermal E th and mechanical E mech energies per unit of mass (J/g) as a function of time. These results were calculated through Eqs. 3.2 and 3.4 using data shown in Fig. 3.6. On loading (stage I), one can observe that the monotonic increase of thermal and mechanical energies reaching a maximum of E th = 8.3 J/g and E mech = 4.3 J/g respectively. During the first hold-time (stage II) a little drop on energy level can be observed at the beginning for both thermal and mechanical energies. This behavior is associated with a stress decrease due to the specimen cooling and can be explained by the Clausius-Clapeyron relation. This relation states that the transformation stress variation is directly proportional to the temperature variation. After that, the energies show a constant trend once no power is generated in this stage. During unloading, a decrease of thermal and mechanical energies can be observed. This response is due to the endothermic reverse transformation and to the total strain recovery. Regarding the mechanical energy, a value of E mech = 1.4J/g is observed at the end of stage III. This residual value represents the hysteresis loop area on the stressstrain curve. In the same way, the thermal energy shows a close value (E th = 1.8J/g) at the end of the cycle. This behavior leads to the conclusion that one has the condition of a thermodynamic cycle where P mech = q.

Results

Thermal and mechanical power and energy in function of time

Thermal and mechanical power and energy as function of

strain and stress 3.3.4.1 Cyclic test

Powers

Figure 3.8 shows, as a function of true strain ( ε ), the thermal ( q) and mechanical (P mech ) power per unit of mass (on the left ordinate axis) and strain rate ( ε) (on the right ordinate axis). The dots represent the loading and the empty circles represent the unloading. Since the beginning of loading, one can notice the increasing behavior of thermal power. Also, a non-zero value of thermal power at such low strain level confirms the presence of an early beginning of phase transformation. The increasing behavior stopped at approximately 3% strain, where it reached its maximum value. Beyond this point, a slight decreasing on thermal power is observed until the end of unloading. This response suggests a decrease of the thermal activity, possibly due to the decrease of phase transformation rate. The same trend of evolution is observed on the mechanical power. In the same way, since the beginning of unloading, thermal power started to decrease. It reached its minimum value at about 1.7% strain. Mechanical power, however, showed a monotonically decreasing response from its minimum, starting at 4.5%, until reach zero. The recoverable nature of this phenomenon is qualitative revealed as an almost symmetric load-unload path shown by the dots and empty circles.

Results

Power

Due to the non-constant strain rate during the test, the thermal and mechanical power were normalized by the strain rate. With that, the misleading interpretations due to the strain rate effect can be excluded.

The normalized powers were plotted in Figs. 3.9a and 3.9b in function of strain and stress, respectively. Observing the normalized thermal power in Fig. 3.9a, one can notice the same qualitative response presented in Fig. 3.8. From that, one can still claim the strong presence of phase transformation from 0 to 3% strain and its decrease from 3% strain until the end of loading.

Analogously, in Fig. 3.9b the thermal power peak takes place around 700 MPa on loading, which is approximately the end of superelastic plateau observed in Fig. 3.4a. During unloading, the thermal peak appears at approximately 350 MPa, which is the end of the unloading plateau at the stress-strain curve shown in Fig. 3.4a. Such response highlights the thermal coupling present on SIPT. Power/ (J/g) ε .

Power/ (J/g) ε .

Energies

Figure 3.10 shows thermal and mechanical energies per unit of mass (J/g) as a function of (a) strain (%) and (b) stress (MPa). In Fig. 3.10a, mechanical energy increased monotonically until 4.3 J/g at the end of loading and decreased continuously to 1.4 J/g at the end of unloading. Thermal energy also showed the same qualitative response, although it reached 8.3 J/g at the end of loading, about twice the magnitude of the mechanical energy. During unloading, thermal energy decreased, reaching a minimum of 1.8 J/g, around the same mechanical energy level.

In Fig. 3.10b at loading, thermal energy increased to 2 J/g in the range of 0 to 600 MPa. With further loading, from 600 MPa to 800 MPa, it increased from 2 J/g to 8.3 J/g. The same qualitative behavior was observed for the mechanical energy where the most part of energy was released during the transformation plateau. the decreasing response of the normalized thermal power followed the same behavior presented at Fig. 3.9a. In Fig. 3.11a, the decreasing behavior can be observed until approximately 10% strain. After that, a slight increase on the normalized thermal power is observed before the rupture. This behavior may be explained by a decrease of the thermal power generated by phase transformation and its increase due to the presence of plastic deformation. In this case, one may conclude that the mechanism of plastic deformation takes place around 10% strain.

Figure 3.11b also shows the same qualitative behavior of the thermal and mechanical energies shown at Fig. 3.10a. A maximum of 19.3 J/g is reached by thermal energy while the mechanical one showed 15.3 J/g at the end of test. However, due to the presence of plastic transformation the amount of energies obtained cannot be associated only with the phase transformation.

Discussion

Deformation mechanisms in SMA

It is well known that deformation mechanisms for NiTi include elastic strain of the atomic lattice associated with martensitic transformation ( Šittner et al., 2006). Also, plastic deformation may be involved but is effective only for large strains [START_REF] Brinson | Stress-induced transformation behavior of a polycrystalline NiTi shape memory alloy: Micro and macromechanical investigations via in situ optical microscopy[END_REF]. In this way, the classical additive decomposition of the total strain rate ( ε) is assumed as:

ε = εel + εin = εel + εtr + εpl (3.5)
where the εel = σ/E and εin are the elastic and inelastic strain rates, respectively, and E is the Young modulus. The inelastic strain rate ( εin ) includes the strain rate due to phase transformation ( εtr ) and the one due to plasticity ( εpl ).

In this work, the value of the elastic modulus E=70 GPa is used. This value accounts for the austenite phase and it was determined through three different experimental methods (tensile tests, in situ synchrotron x-ray diffraction, and dynamic mechanical analysis) and estimated via theoretical calculation from elastic constants by Šittner et al., 2014.

Discussion

Analogously, the thermal power is also decomposed as shown in Eq. 3.6: q = qthel + qin = qthel + qtr + qpl (3.6) where the thermoelastic heat source qthel = -(αT /ρ) σ is calculated with α as the coefficient of thermal expansion (α = 1.1 × 10 -5 K -1 ), T the test temperature, ρ the mass density and the true stress rate σ = dσ/dt; qtr is the heat source associated with the phase transformation and qpl is the part of the intrinsic dissipated power due to plasticity.

The mechanical power per unit of mass can also be decomposed as shown in Eq. 3.7:

P mech = P mech-el + P mech-in (3.7)
Where the elastic and inelastic mechanical power per unit of mass are: P mech-el = 1 ρ σ εel and P mech-in = 1 ρ σ εin , respectively.

Furthermore, thermal and mechanical energies are also decomposed in elastic and inelastic portions, as described before. They are calculated through time integration of each portion of correspondent powers.

Impact of thermoelasticity on thermal Power and Energy

Figure 3.12a shows, in function of strain, on the left ordinate axis, thermal power normalized by the strain rate ( q/ ε) and the inelastic thermal power ( qin / ε) also normalized by strain rate. The evolution of the ratio between inelastic thermal power and mechanical power ( qin /P mech-in ) in function of strain is also plotted in Fig. 3.12a. The same values are plotted in Fig. 3.12b in function of stress.

One can observe that the normalized inelastic thermal power ( qin / ε) is slightly higher than the thermal one ( q/ ε). This difference is more evident at the beginning of loading between 0 -2%, at Fig. 3.12a, and 0-600 MPa, at Fig. 3.12b. These ranges represent the elastic portion on the superelastic loop. In this range, it is possible to observe that thermal power attains negative values at the beginning of loading. Such a response puts in evidence the endothermic nature of thermoelasticity during loading. Thereafter, once the phase transformation shows a stronger effect, between 2 -4.8% (Fig. 3.12a) and 600-800MPa (Fig. 3.12b) the thermoelasticity effect is less pronounced. Also, observing the Fig. 3.12, the ratio between inelastic thermal and mechanical powers ( qin /P mech-in ) immediately increases to a value close to 6 but shows a decreasing trend until reach a value around 2. During unloading, this ratio increased monotonically from a value around 2 and reaches a ratio of 7 at the end of unloading. Such a high ratio value implies a significant thermal effect in relation to the mechanical power. This ratio commonly reported on literature as the Taylor and Quinney coefficient, is usually less than 1 for plastic deformation of metals or polymers [START_REF] Zaera | On the Taylor-Quinney coefficient in dynamically phase transforming materials. Application to 304 stainless steel[END_REF][START_REF] Zaera | A constitutive model for analyzing martensite formation in austenitic steels deforming at high strain rates[END_REF] Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires . ε

Thermal Power/ (J/g) .

Loading Unloading

Ratio: q in /P mech_in .

. q/ε .

. q in /ε .

Figure 3.12: a) Normalized inelastic thermal power ( qin / ε) and normalized thermal power ( q/ ε) on the left ordinate axis. Ratio between inelastic thermal power ( qin ) and inelastic mechanical power on right ordinate axis ( qin /P mech-in ) in function of true strain. b) Same values in function of nominal stress. [START_REF] Rittel | On the conversion of plastic work to heat during high strain rate deformation of glassy polymers[END_REF]. A thermal and mechanical power ratio greater than 1 can be explained by a deformation mechanism different than plasticity, here due to phase transformation. Figure 3.13a shows, on the left ordinate axis, the thermal energy (E th ) and the inelastic thermal energy (E in ). The thermal energy was calculated using Eq. 3.2 and data shown in Fig. 3.6. The evolution of the ratio between inelastic thermal energy and mechanical energy (E in /E mech-in ) as function of strain is also plotted in Fig. 3.13a. The same values are plotted in Fig. 3.13b as function of stress.

In Fig. 3.13 it is possible to observe a slight difference between thermal energy (E th ) and the inelastic thermal energy (E in ). It is more evident at the end of loading. This difference is explained by the thermoelasticity occurring on the beginning of the loading. However, no difference is observed at the end of unloading. Such behavior highlights the reversibility of the thermoelastic effect on the tensile test.

It is also possible to observe the ratio between inelastic thermal energy and inelastic mechanical energy in Fig. 3.13. During loading, a sudden rise from 0 to a value around 6 is observed; after that, a decreasing tendency to a ratio of 2.5 is observed. Similarly, Peyroux et al., 1998b andBalandraud et al., 2005, studying the ratio between thermal and mechanical energy, reported a value of 10 but for a CuZnAl SMA. Thermal Energy (J/g) Ratio Ratio

Discussion

Thermal Energy (J/g)

E in E th

Ratio: E in /E mech_in Loading Unloading

Thermodynamics of martensitic transformation

Thermodynamic framework

Several thermodynamic framework have been proposed to describe the thermomechanical behavior of shape memory alloys. Among them, the one used in this work is based on the work developed by Ortin and Planes and Wollants et al. [START_REF] Ortín | Thermodynamic analysis of thermal measurements in thermoelastic martensitic transformations[END_REF][START_REF] Ortín | Thermodynamics of thermoelastic martensitic transformations[END_REF][START_REF] Ortín | THERMODYNAMICS AND HYSTERESIS BEHAVIOUR OF THERMOELASTIC MARTENSITIC TRANSFORMATIONS[END_REF][START_REF] Wollants | Thermally-and stress-induced thermoelastic martensitic transformations in the reference frame of equilibrium thermodynamics[END_REF]. It expresses the martensitic transformation as a diffusionless first order reversible transformation at the scale of an elementary volume. Following this approach, the occurrence of a thermoelastic martensitic transformation is determined by the Gibbs free energy (G) balance at equilibrium for the transformation, which is given by the condition dG = 0. For a thermoelastic martensitic transformation in a polycrystalline matrix, the free energy per unit mass changes over an infinitesimal step of martensite fraction df m of transformation and is expressed as shown in [START_REF] Ortín | Thermodynamics of thermoelastic martensitic transformations[END_REF][START_REF] Wollants | Thermally-and stress-induced thermoelastic martensitic transformations in the reference frame of equilibrium thermodynamics[END_REF]:

dG = (∆H c -T ∆S c )df m + ∂E st + ∂E f r -W tr mech = 0 (3.8)
where:

Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires

• dG = (∂G/∂f m )df m = G df m ;
• ∆H c is the difference between the specific enthalpies of the martensite and austenite phase;

• ∆S c is the difference between the specific entropies of the martensite and austenite phase;

• f m is the mass fraction of martensite defined in the elementary volume; depending only of time. Its time derivate dfm dt is noted ḟm .

• T is the temperature;

• ∂E st = (∂E st /∂f m )df m = E st (f m )df m
is the stored energy in the elementary volume transformed by an infinitesimal step df m ;

• ∂E f r = (∂E f r /∂f m )df m = E f r (f m )df m
is the friction energy dissipated by the infinitesimal step df m ;

• ∂W tr mech = (∂W mech /∂f m )df m = W mech (f m )df m is the external mechanical work per unit mass of the elementary volume during the infinitesimal step df m .

All terms in Eq. 3.8 are algebraic values and they are expressed as energy per unit of mass in Jg -1 , where, at full austenite phase f m = 0 and at full martensite phase f m = 1. The infinitesimal martensite mass fraction step (df m ) is positive during the forward transformation and negative during the reverse transformation. The term ∂E f r is the sum of all irreversible energies dissipated, such as the energy consumed in moving dislocations and interfaces. The term ∂E f r is always dissipated, thus is always positive. Rewriting then Eq. 3.8, one has:

∆H c -T ∆S c + E st + E f r -W mech = 0
(3.9)

Transformation heat source

The transformation heat source qtr is defined as:

qtr = qtr-chem + qtr-diss = ∆H tr (f m ) ḟm (3.10)
where ∆H tr is the specific latent heat and qtr is the specific transformation heat sources including both chemical qtr-chem and dissipative qtr-diss components.

From Eqs. 3.8 and 3.10, ∆H tr can be written as:

∆H tr = -T ∆S c + E f r (3.11) 3.4 Discussion or ∆H tr = -∆H c -E st + W mech (3.12)
3.4.3.3 Forward and reverse specific latent heat Observing the Eqs. 3.11 and 3.12 in a cyclic loading analysis one has:

• On loading (A -M ), with the change of f m = 0 to f m ≤ 1:

∆H A-M tr = -T A-M ∆S c + E A-M f r (3.13) and ∆H A-M tr = -∆H c -E A-M st + W A-M mech (3.14) • On unloading (M -A), with the change of f m ≤ 1 to f m = 0: ∆H M -A tr = -T M -A ∆S c + E M -A f r (3.15) and ∆H M -A tr = -∆H c -E M -A st + W M -A mech (3.16) with • ∆S c < 0; • ∆H c < 0; • E A-M f r > 0 and E M -A f r < 0; • E A-M st > 0 and E M -A st > 0; • W A-M mech > 0 and W M -A mech > 0.
Combining then, the Eqs. 3.13 and 3.15, one has:

∆H A-M tr -∆H M -A tr = -(T A-M -T M -A )∆S c + (E A-M f r -E M -A f r ) (3.17)
Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires

With the terms E

A-M f r > 0 and E M -A f r
< 0, the second term on the right hand side of the Eq. 3.17 will be always positive. Also, as the component ∆S c is a negative constant the first term of the right hand side will also be positive. However, its magnitude is strongly dependent on the temperature variation.

In the other hand, combining the Eqs. 3.14 and 3.16, one has:

∆H A-M tr -∆H M -A tr = -(E A-M st -E M -A st ) + (W A-M mech -W M -A mech ) (3.18)
The difference of the stored energy between loading and unloading (E

A-M st -E M -A st
) is very often assumed as zero since the stored energy is considered reversible. From that, the difference of specific latent heat between loading (∆H A-M tr ) and unloading (∆H M -A tr ) can be written as a difference of the mechanical work. Then, Eq. 3.18 is written as:

∆H A-M tr -∆H M -A tr = W A-M mech -W M -A mech (3.19)
3.5 Characterisation of the martensitic transformation 3.5.1 Determination of transformation strain (∆ε tr )

To determine the transformation strain ∆ε tr several superelastic tensile tests were carried out at different strain levels between 7% and 11% of superelastic deformation. The tests were performed on a Gabo Qualimeter testing machine with displacement control and crosshead speed of 10 -5 s -1 . The tests consisted in loading the wire (at room temperature) until a preset deformation level ε a and unloading until zero stress, then heating the wire at 373K. Figure 3.14a shows an example of the experimental superelastic loop in which it is possible to observe the decomposition of the strain. At the end of loading, the total strain ε a can be described as being composed by an elastic strain ε el a , a plastic strain ε pl a and a transformation strain ε tr a . After unloading, a final strain ε b is achieved. Heating above the critical phase transformation temperature allows the material to recover a part of this deformation and reach the final strain level ε c . The strain ε a is written as:

ε a = ε el a + ε pl a + ε tr a (3.20) with ε el a = σ a /E.
Assuming ε pl a = ε c , one can rewrite Eq.3.20 as:

ε tr a = ε a -ε c - σ a E (3.21)
All the values in the right hand side of the Eq. 3.21 can be measured experimentally.

Technically, the transformation strain ε tr a increases with increasing ε a until a maximum noted ∆ε tr , when the phase transformation is completed, i.e. the martensite fraction is f m = 1. The evolution of ε tr a with ε in is assumed to follow a curve similar to Fig. 3.15a. Figure 3.15b shows the experimental results of the tests carried out using the before mentioned method. In the present study, the tested wires reached rupture before the theoretical asymptotic point. Therefore, the maximum value shown by the data presented in Fig. 3.15b was assumed as being lower than the true transformation strain ∆ε tr . With this, the value of transformation strain was assumed as being ∆ε tr > 8%. [START_REF] Miyazaki | The habit plane and transformation strains associated with the martensitic transformation in Ti-Ni single crystals[END_REF]. One can observe that in the direction (011), same preferential direction of the used cold worked wire (see Chapter 2), the ∆ε tr is between 8% and 9%. Also, in a work published by [START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF], they observed that the ∆ε tr of a A -M transformation was around ε tr = 8%. Finally, based on the experimental determination and the literature support, the value ∆ε tr = 8% is used in the following of this study.

Strain Rate analysis

In the following, it is assumed that the deformation mechanism is composed only by elasticity and phase transformation. The effects of plasticity are neglected. In this way, the Eq. 3.5 is rewritten as follows:

ε = εel + εin = εel + εtr (3.22)
The transformation strain rate εtr will be assumed proportional to the martensitic transformation fraction. This is assumed for any phase transformation under tensile loading [START_REF] Mccormick | Intrinsic thermal-mechanical behaviour associated with the stress-induced martensitic transformation in NiTi[END_REF]:

εtr = ḟm ∆ε tr (3.23)
where ∆ε tr stands for the transformation strain for a complete transformation (f m = 1).

Thermal power assumptions

The total thermal power is assumed to be composed only by the presence of thermoelasticity and phase transformation. The dissipation power induced by plasticity ( qpl ) is neglected. With that, Eq. 3.6 is rewritten as follows: q = qthel + qin = qthel + qtr (3.24)

with qtr = ḟ ∆H tr .

Determination of ratio ∆Htr(fm) ∆εtr Using Eqs. 3.10 and 3.23, one obtain the following expression:

qtr εtr = qin εin = q + αT 0 σ ρ ε - σ E = ∆H tr (f m ) ∆ε tr (3.25)
Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires All terms of the right hand side of the Eq. 3.25 were obtained experimentally. The ratio qin / εin is plotted in function of strain and nominal stress as shown in Figs. 3.17a and 3.17b, respectively. q in /ε in (J/g) . . q in /ε in (J/g) . .

Figure 3.17: Ratio qin / εin , computed using experimental data and Eq. 3.25, as a function of a) strain and b) nominal stress for the cyclic test.

Observing Fig. 3.17a, it is possible to consider the ratio qin / εin constant for strains between ε = 1% and ε = 5%. Then, assuming the specific latent heat (∆H tr (f m )) as a constant in this strain interval it can be rewritten as: ∆H tr = ( qtr / εtr )∆ε tr (3.26)

Determination of ∆H tr

From Eq. 3.26, it is possible to experimentally obtain the value of the specific latent heat (∆H tr ). Calculating on the strain range between ε = 1% and ε = 4.5%, one obtains: ∆H A-M tr = 19.6 ± 0.4 J/g for the forward phase transformation and ∆H M -A tr = 14.9 ± 0.2 J/g for the reverse transformation. The standard deviation is also calculated to take into account the variation in this range.

These values can be compared to the ones obtained by [START_REF] Frenzel | On the effect of alloy composition on martensite start temperatures and latent heats in Ni-Ti-based shape memory alloys[END_REF] from DSC measurements. They showed that for fully annealed binary NiTi alloys, the values of specific latent heat decrease from about 30 J/g to 10 J/g with the increasing of at.%N i. For alloys with a composition of T i-50.9 at.% N i the values obtained were around 17J/g [START_REF] Khalil-Allafi | The effect of chemical composition on enthalpy and entropy changes of martensitic transformations in binary NiTi shape memory alloys[END_REF][START_REF] Frenzel | On the effect of alloy composition on martensite start temperatures and latent heats in Ni-Ti-based shape memory alloys[END_REF].

Analyzing the relation ∆H

A-M tr -∆H M -A tr = W A-M mech -W M -A mech
As presented in Eq. 3.12, ∆H tr is composed by a chemical enthalpy change ∆H c , an elastic energy E st stored or released during the process of transformation and a mechanical work W mech = 1 ρ σ∆ε tr . As shown in Eq. 3.19 the difference between forward and reverse transformation enthalpies (where ∆H A-M tr > ∆H M -A tr ) is further associated only with the mechanical energy (W mech ). Rewriting the Eq. 3.19, one have:

Characterisation of the martensitic transformation ∆H

A-M tr -∆H M -A tr = 1 ρ σ A-M ∆ε tr - 1 ρ σ M -A ∆ε tr (3.27)
From the thermal analysis, the difference between the specific latent heat on loading and unloading is ∆H A-M tr -∆H M -A tr = 4.7 ± 0.6 J/g. From the mechanical point of view, assuming σ A-M as the mean stress of the forward transformation plateau and σ M -A as the mean stress for the reverse transformation plateau, one obtains the difference of the mechanical energy between loading and unloading W A-M mech -W M -A mech = 3.5±0.7 J/g. The mean stress values as well as the standard deviation were calculated between 1% and 1.5% of transformation strain ε tr . Comparing then the values of ∆H A-M tr -∆H M -A tr = 4.7±0.6 J/g and W A-M mech -W M -A mech = 3.5 ± 0.7 J/g, a difference of 1.2 ± 1.3 J/g is observed. Because of the important standard deviation obtained from both mechanical and thermal approach, it can be assumed that the statement presented in Eq. 3.19 is true.

Analyzing the terms T ∆S c and E f r According to Eq. 3.11, the specific latent heat (∆H tr ) is composed by the entropy variation (T ∆S c ) and by the energy dissipated as friction (E f r ). With that, it is possible to calculate the term E f r for loading and unloading. Taking into account the test temperature variation during loading (θ A-M ) and unloading (θ M -A ), Eq. 3.11 can be rewritten as follows:

• During loading, A-M:

∆H A-M tr = -(T + θ A-M )∆S c + E A-M f r (3.28)
and,

• During unloading, M-A:

∆H M -A tr = -(T + θ M -A )∆S c + E M -A f r (3.29)
where T stands for the temperature test.

Figure 3.18 shows the average temperature variation θ during the test in function of transformation strain ε tr . The test temperature variation during loading (θ A-M ) and unloading (θ M -A ) are highlighted by the dashed red and blue lines, respectively. From that, their respective difference (θ A-M -θ M -A ) was calculated and is represented by the black thick line.

<> (K) θ A -M ( εt r ) θ M -A ( εtr ) θ A-M -θ M-A Figure 3
.18: Average temperature variation θ in function of the average transformation strain ε tr . The dashed red line stands for the temperature variation during loading (θ A-M (ε tr )) and the blue dashed line during unloading (θ M -A (ε tr )). The black curve stands for the difference between temperature variation during loading and unloading (θ A-M -θ M -A ).

Then, using the average value of the θ A-M -θ M -A = 11.8 ± 1.2 K and considering an entropy variation ∆S c = -63 ± 1 mJ/gK (Khalil-Allafi and Amin-Ahmadi, 2009), the terms of energy due to the entropy variation (T ∆S c ) and dissipated as friction (E f r ) were calculated using Eqs. 3.28 and 3.29.

Then, for a loading-unloading cycle, the energy value due to temperature and entropy variation is:

-(θ A-M -θ M -A )∆S c = 0.7 ± 0.09 J/g.
And the value due to the friction energy is:

E A-M f r + E M -A f r = 4.0 ± 0.7 J/g.
The comparison of these values shows that for a SIPT cycle the energy dissipation is predominantly due to the friction energy. However, due to the non isothermal aspect of the test, the thermomechanical coupling term (T + θ)∆S c is non-negligible.

3.5 Characterisation of the martensitic transformation 3.5.2.2 Rupture test Figure 3.19 presents the ratio qin / εin in function of true strain (Fig. 3.19a) and in function of nominal stress (Fig. 3.19b) for the rupture tensile test. Similarly to the curve observed in Fig. 3.17a, the ratio qin / εin in function of strain increases linearly until 1% and presents a rather constant value from ∼ 2% to ∼ 7.5% strain. Then, after 7.5%, the ratio shows a nearly linear decrease until the rupture point, around 11% strain. In Fig. 3.19b, the correspondent region is observed between ∼ 600 MPa and ∼ 1200 MPa. One can notice that after 7.5% (Fig. 3.19a) and ∼ 1200 MPa (Fig. 3.19b), the ratio is not constant anymore. In this way, one can assume that the plastic effect cannot be neglected after these values.

<ε>(%) Stress(MPa) a) b) q in /ε in (J/g) . . q
in /ε in (J/g) . . Figure 3.20a shows the nominal stress vs. true strain. This curve can be separated in three zones: I-from 0% to ∼ 2%, II-from ∼ 2% to ∼ 7.5% and III-from ∼ 7.5% until rupture. Zone I may be assumed as having predominantly elastic deformation while zone II shows mostly phase transformation. Finally zone III presents the beginning of a plastic deformation around ∼ 7.5%. Fig. 3.20b presents the ratio dσ/dε in function of true strain. This ratio is used to analyze the evolution of the apparent modulus with deformation. One can observe that the ratio varies with true strain. Such result indicates that no pure elastic zone is present during the tensile test, otherwise, a constant value of dσ/dε would be observed. It is known that at the beginning of SIPT, with the material in austenite phase, the elastic modulus decreases during loading, reaching a minimum at about the middle of the stress vs. strain plateau, and restarts to increase again until reaches the martensite phase. With further increase of strain, the ratio dσ/dε tends to decrease due to the presence of plastic deformation. Because of that, in the Fig. 3.20b, the inflection point around ∼ 7.5% strain indicates the begin of plastic deformation. These results support the assumption that the zone between ∼ 2% and ∼ 7.5% strain contains only (or at least predominantly) phase transformation. superelastic tensile deformation of nanostructured NiTi wires Then, the specific latent heat (∆H tr ) was treated constant on the interval between ε = 2% and ε = 7.5% strain. It was calculated using the experimental data shown in Fig. 3.19 and Eq. 3.25. The value ∆H A-M tr = 20.2±1.3J/g was obtained for a transformation strain ∆ε tr = 0.08. This value agrees with the one obtained for the cyclic test.

Estimation of martensite fraction (f m )

The martensite fraction was obtained only for the cyclic test, due to the absence of plastic effects. It was calculated by time integration of the martensite fraction rate deduced from Eqs. 3.23 and 3.26, where ḟm = qtr /∆H tr . For that, the imposed values of ∆H A-M tr = 19.6 J/g for the forward transformation (loading) and ∆H M -A tr = 14.9 J/g for the reverse transformation (unloading) were used. 3.5 Characterisation of the martensitic transformation Figure 3.21 shows the evolution of the martensite fraction in function of strain (a) and in function of nominal stress (b). Loading is described by the solid line between 0% and point "a" noted in Fig. 3.21. The red-line delimited by points "a" and "b" represents the hold time at the end of loading. Finally, the dashed lines represent the martensite fraction during unloading.

In Fig. 3.21a, the solid curve shows that since the beginning of strain the A -M phase transformation initiates, even before the superelastic plateau. Observing the Fig. 3.21b, during loading, around 5% of martensite is formed between 0 and 400 MPa and more than 10% at 600 MPa. This amount of martensite is present in the strain range of the tensile curve which is commonly considered to be purely elastic. This result ratifies the early increasing amount of thermal power observed at the Figs. 3.8, 3.9 and 3.12.

Between the points (a) and (b), a small decrease of the martensite fraction due to reverse transformation is observed. Such a weak decrease can be explained by the small decrease of the stress during this hold time due to relaxation associated with temperature decrease. In Fig. 3.21b at the beginning of the unloading, between the point (b) and 500 MPa , around 10 % of martensite transforms into austenite.

Despite the apparent full superelastic loop, a martensite fraction maximum of f m = 0.45 is achieved at the end of loading. This behavior does not follow what is commonly assumed in literature that, at the end of superelastic plateau, all the parent phase microstructure is transformed into martensitic phase [START_REF] Lagoudas | Shape Memory Alloys[END_REF].

New studies [START_REF] Stebner | In Situ Neutron Diffraction Studies of Large Monotonic Deformations of Superelastic Nitinol[END_REF][START_REF] Pelton | In Situ Neutron Diffraction Studies of Increasing Tension Strains of Superelastic Nitinol[END_REF] using neutron diffraction to measure the crystal phase structure show results of the martensitic fraction during a stress induced phase transformation. The path described by the martensite fraction in Fig. 3.21, where its value is higher during unloading than loading, is shown in the work of [START_REF] Pelton | In Situ Neutron Diffraction Studies of Increasing Tension Strains of Superelastic Nitinol[END_REF]. Stebner et al., 2015 obtained a maximum of f m = 0.95, at ε = 11% strain, in a stress relieved N iT i rod (T i -50.8 at% N i). Such high value of martensite fraction is expected in stress relieved materials since the grain size tends to increase with the heat treatment and so the phase transformation.

However, numerous studies [START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF][START_REF] Waitz | Phase Transformations of Nanocrystalline Martensitic Materials[END_REF][START_REF] Glezer | Martensite Transformation in Nanoparticles and Nanomaterials[END_REF] reveal that for nanostructured materials, the reduced grain size tends to partially suppress the martensite phase transformation. It has been shown that the grain size reduction decreases exponentially the potency of grains as nucleation sites [START_REF] Chen | Statistics of martensitic nucleation[END_REF]. In the present paper, the material used is assumed to have a nanoscale crystalline structure, since it was submitted to a severe cold work fabrication process and suffered a low magnitude heat treatment [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF][START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF] at temperatures below recrystallization. The manufacture history associated with the absence of transformation peak in DSC curve [START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF] ratifies then the low level of martensite fraction achieved in Fig. 3.21.

Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires

Conclusion

A thermomechanical analysis was performed on a nanostructured cold-worked T i -50.9 at% N i wire submitted to a heat treatment of 598K for 30min. Using a thermal field measurement (TFM) method it was possible to estimate thermal power and energy generated during a stress induced phase transformation. The same superelastic NiTi wire showed an inconclusive thermal response when tested via DSC technique.

Digital image correlation (DIC) allowed the estimation of mechanical power and energy from the kinematic field measurements. Moreover, it was possible to determine a thermal and mechanical power ratio ( qin /P mech-in ) higher than 2, for both loading and unloading, which is evidence of the phase transformation presence.

With the use of a thermomechanical framework it was possible to determine the specific latent heat of phase transformation (∆H tr ). The calculated values of specific latent heat were ∆H A-M tr = 19.6 ± 0.4 J/g for the forward and ∆H M -A tr = 14.9 ± 0.2 J/g for the reverse transformation. Values in the same range were found in literature for a fully annealed NiTi alloys tested via the DSC technique.

Finally, an estimation of the martensite fraction in function of strain and nominal stress was performed with the hypothesis of negligible plastic deformation. Under this condition, limiting the global strain to about 5%, a maximum of 45% martensite fraction was observed. It was also observed an increase from 0 to 5% of martensite fraction between 0 and 400 MPa on loading in a commonly said elastic region. Around 10% of martensite fraction recovered in the very beginning of unloading, between 680 MPa and 500 MPa, also in a supposedly elastic region. Such results put into evidence the premature phase transformation phenomenon on the stress induced phase transformation; and its presence along all the superelastic loop.

CHAPTER

FOUR

INFLUENCE OF HEAT TREATMENT ON THE SPECIFIC LATENT HEAT AND MARTENSITE FRACTION OF NANOSTRUCTURED NITI WIRES

This Chapter is written as a paper to be published soon.

ABSTRACT

Shape memory alloys are capable of inducing a considerable temperature change when they are mechanically charged. This phenomenon takes place due to an important thermal and mechanical coupling present during the phase transformation (P T ). The latent heat along the P T mechanism corresponds to the energy responsible for the temperature increase. The determination of the specific latent heat (∆H) of the phase transformation involved in the process, are generally performed by differential scanning calorimetry (DSC). However, depending on the fabrication history of the material, the obtained DSC response is useless on the determination of the ∆H. In this work, a method using digital image correlation (DIC) and thermal field measurements (T F M ) is used to obtain this property. A cold worked T i -50.9 at.% N i wire was submitted to four different heat treatments namely 523K, 548K, 573K and 598K for 30 min. Kinematic and thermal full fields were acquired during superelastic tensile tests. From the experimental results and a thermal approach associated with a thermodynamic framework, it was possible to determine the specific latent heat ∆H for all the heat treated wire during loading and unloading. The ∆H calculated values during loading showed a rather constant behavior, however, the values calculated during unloading showed a decreasing trend with the increase of the heat treatment temperature. The martensite fractions (f m ) were also calculated and showed an evolution with the heat treatments. At the same stress level, 800 MPa for example, values from f m = 0.07 to f m = 0.43 were obtained among the different heat treatments.

Chapter 4. Influence of heat treatment on the specific latent heat and martensite fraction of nanostructured NiTi wires Keywords: Full-field kinematic and thermal approach, nanostructured NiTi wires, heat balance, enthalpy of transformation, martensite fraction.

Introduction

Introduction

Thermomechanical effects present in martensitic phase transformation in shape memory alloys (SMA) have attracted considerable attention of researchers for the application in elastocaloric cooling devices. Recently, many studies have been conducted and devices designed based on this phenomenon [START_REF] Wu | Elastocaloric cooling capacity of shape memory alloys -Role of deformation temperatures, mechanical cycling, stress hysteresis and inhomogeneity of transformation[END_REF][START_REF] Álvarez-Alonso | Conventional and inverse elastocaloric effect in Ni-Fe-Ga and Ni-Mn-Sn ribbons[END_REF][START_REF] Pataky | Elastocaloric cooling potential of NiTi, Ni2FeGa, and CoNiAl[END_REF][START_REF] Schmidt | Elastocaloric Cooling: from Fundamental Thermodynamics to Solid State Air Conditioning[END_REF]. The elastocaloric effect takes place due to entropy change during the phase transformation. An endothermic response happens during the reverse phase transformation and an exothermic response on the direct phase transformation. Under adiabatic conditions the endothermic effect absorbs heat from the surroundings, dropping the material temperature considerably, reaching cooling peaks of 18.2K for NiTi alloys [START_REF] Wu | Elastocaloric cooling capacity of shape memory alloys -Role of deformation temperatures, mechanical cycling, stress hysteresis and inhomogeneity of transformation[END_REF][START_REF] Pataky | Elastocaloric cooling potential of NiTi, Ni2FeGa, and CoNiAl[END_REF]Tušek et al., 2016a;[START_REF] Tušek | Elastocaloric effect of Ni-Ti wire for application in a cooling device[END_REF][START_REF] Ossmer | Evolution of temperature profiles in TiNi films for elastocaloric cooling[END_REF]Schmidt et al., 2015b). As the elastocaloric effect is associated to the martensitic phase transformation, the amount of energy involved in this phenomenon depends directly on the volume of transformed phase.

Depending on the fabrication history and heat treatment submitted to SMA, the mechanical and thermal responses can be affected [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF]. NiTi SMA wires are commonly manufactured through a sequence of hot and cold working [START_REF] Ahadi | Effect of hot rolling on microstructure and transformation cycling behaviour of equiatomic NiTi shape memory alloy[END_REF][START_REF] Delville | Microstructure changes during non-conventional heat treatment of thin Ni-Ti wires by pulsed electric current studied by transmission electron microscopy[END_REF][START_REF] Mitwally | Effect of cold work and annealing on the structure and characteristics of NiTi alloy[END_REF] followed by a final heat treatment to perform a recrystallization [START_REF] Peterlechner | Study of relaxation and crystallization kinetics of NiTi made amorphous by repeated cold rolling[END_REF][START_REF] Shi | Grain size effect on the R-phase transformation of nanocrystalline NiTi shape memory alloys[END_REF][START_REF] Sergueeva | Structure and properties of amorphous and nanocrystalline NiTi prepared by severe plastic deformation and annealing[END_REF]. The severe plastic deformation during cold working induces a grain size reduction [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF] that may lead to a partial amorphization of NiTi SMA [START_REF] Yu | Synchrotron high energy X-ray diffraction study of microstructure evolution of severely cold drawn NiTi wire during annealing[END_REF][START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF]. This effect leads to a partial or total suppression of the DSC curve peaks [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF][START_REF] Li | Cold Rolling-Induced Multistage Transformation in Ni-Rich NiTi Shape Memory Alloys[END_REF][START_REF] Chang | Transformation sequence in severely cold-rolled and annealed Ti50Ni50 alloy[END_REF][START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF][START_REF] Miller | Influence of cold work and heat treatment on the shape memory effect and plastic strain development of NiTi[END_REF]. In this way, the use of DSC tests to determine the specific latent heat of phase transformation is useless. However, a superelastic loop is still observed and so a stress induced phase transformation (SIPT) is present [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF].

In this case, the specific latent heat may be obtained through the analysis of the thermomechanical coupling during a SIPT. This complex interaction between thermal and mechanical phenomena has been studied by many authors [START_REF] Ossmer | Evolution of temperature profiles in TiNi films for elastocaloric cooling[END_REF][START_REF] Zhang | Experimental study on rate dependence of macroscopic domain and stress hysteresis in NiTi shape memory alloy strips[END_REF][START_REF] Shaw | Thermomechanical aspects of NiTi[END_REF][START_REF] Yin | Temperature Variation in NiTi Shape Memory Alloy During Cyclic Phase Transition[END_REF]He and Sun, 2010). Heat sources were experimentally obtained through a thermal analysis in CuZnAl SMA by Peyroux et al., 1998b and[START_REF] Balandraud | Monotone strain-stress models for shape memory alloys hysteresis loop and pseudoelastic behaviorx[END_REF]. A thermal approach on the determination of the power generation on a NiTi tube was presented by [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]. Further, Chrysochoos and Louche, 2000and Schlosser et al., 2007 presented a study with a method to estimate heat sources from temperature fields observed by an infrared thermal camera.

In this context, this work aims to present an experimental and analytical study on the determination of the specific latent heat during a SIPT. Kinematic and thermal full fields measurement methods described in Chapter 3 were used. A thermomechanical analysis performed through the Gibbs free energy was used in the discussion. The tensile tests were performed on a NiTi cold worked wire submitted to heat treatments (HT) of 30 minutes at four different temperatures: 523K, 548K, 573K and 598K. Different levels of Chapter 4. Influence of heat treatment on the specific latent heat and martensite fraction of nanostructured NiTi wires heat sources and thermal and mechanical energies were estimated for each heat treated wire. The main results are discussed in the last part of this Chapter.

Material and experimental procedures 4.2.1 Material

The material used in the experimental campaign was a commercial NiTi wire of 0.5 mm diameter with a composition of T i -50.9 at.% N i provided by Fort Wayne Metals. The wire presents a hardened mechanical behavior due to its cold work manufacturing process. Wire specimens were submitted to four heat treatments at the following temperatures: 523K, 548K, 573K and 598K for 30 min in all cases. In the following, the wires are mentioned as HT 523K, HT 548K, HT 573K and HT 598K, in reference to their respective HT temperature.

Differential scanning calorimetry analysis

Prior to the tensile tests, a DSC analysis was performed on the heat treated wires. The tests were carried out between 223K and 363K with a heating rate of 10K/min in a Q200 (TA Instruments) DSC testing machine.

Experimental setup

Superelastic tensile tests were performed by loading the NiTi wires at a pre-set global deformation of 6% at a strain rate of 10 -2 s -1 with a further unloading at the same strain rate. During the complete load-unload cycle, two hold-time steps were performed (of 50 s each) at the end of loading and at the end of unloading, respectively. During all tests, thermal and kinematic full field data were acquired through IR and visible light cameras. The details of the experimental set-up and the data post-processing are explained in Chapter 2. For the wires HT 523K, HT 548K and HT 573K, one can observe that heat flow does not show any peak characterizing the presence of phase transformation. This absence is expected once the heat treatments were performed under the complete recrystallization temperature (623K) [START_REF] Peterlechner | Study of relaxation and crystallization kinetics of NiTi made amorphous by repeated cold rolling[END_REF][START_REF] Shi | Grain size effect on the R-phase transformation of nanocrystalline NiTi shape memory alloys[END_REF][START_REF] Sergueeva | Structure and properties of amorphous and nanocrystalline NiTi prepared by severe plastic deformation and annealing[END_REF][START_REF] Yu | Synchrotron high energy X-ray diffraction study of microstructure evolution of severely cold drawn NiTi wire during annealing[END_REF]. The wire HT 598K, however, shows a minor peak in the heat flow vs temperature curve. Despite its presence, the peak is not well pronounced and cannot be used on the determination of the specific latent heat nor phase transformations critical temperatures. Its use might lead to false conclusions. Such inconclusive DSC responses have already been reported in literature for cold worked materials showing an amorphous or nanosized polycrystalline structure [START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF][START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF].

Results

DSC responses

Superelastic mechanical responses

Figure 4.2 shows the superelastic true stress -strain curves of the four wires submitted to different heat treatment temperatures. One can observe the stress level decreasing with the increasing heat treatment temperature. Also, a minor growth on the hysteresis loop can be noticed. Despite the pre-set strain value of 6% imposed to all tests, each sample achieved a different strain level. This is due to the not controllable sliding of the sample on the clamps surface. However, even though showing different strain levels, the further analysis was not compromised.

The curves referent to the results of HT 523K and HT 548K wires show a steep loop without any signal of plateau. A kind of "plateau" starts to appear for the HT 573K wire and is almost fully established for the HT 598K. Finally, at the end of unloading it is Chapter 4. Influence of heat treatment on the specific latent heat and martensite fraction of nanostructured NiTi wires possible to observe the complete strain recovery of the material. Such a behavior ratifies the predominant presence of reversible elastic and phase transformation deformation mechanisms. Irreversible deformation mechanism such as plasticity can be neglected on the analysis. Firstly, one can observe an uniformity on strain and temperature variation profiles. Such stable curves, without any peak nor gradient, point out the absence of localized deformation. From the wire HT 523K, at 5% strain, the temperature increased 6K during loading. For almost the same level of strain (around 5%), the temperature increased 9K for the wire HT 548K and about 12K for the wires HT 573K and HT 598K.

Strain and temperature fields

As the observed strain (Fig. 4.3b) and temperature (Fig. 4.3c) profiles were uniform, they had been averaged and treated as time dependent only. In this way, each strain point at the σ -ε curve (Fig. 4.3a) represents the mean value ε of each strain profile ε of curves in Fig. 4.3b. The same analysis can be done for the temperature variation profiles θ. They were computed as the difference between the absolute temperature T in the gauge length l 0 and the room temperature T 0 , i.e. θ = T -T 0 . y/l 0 (%) y/l 0 (%) <ε> (%) y/l 0 (%) y/l 0 (%) <ε> (%) y/l 0 (%) y/l 0 (%) <ε> (%)

Results

y/l 0 (%) y/l 0 (%) <ε> (%) a) b) c)  (%)  (%)  (%)  (%)  (K)  (K)  (K)  (K)

Thermomechanical analysis

In this Section the results of the thermal and mechanical powers and energies are presented and discussed. These results were obtained experimentally through the experimental data shown in previous section (4.3.3) and the thermal approach extensively discussed in Sections 3.2.4 and 3.2.5 of Chapter 3.

Thermal and mechanical powers

Figure 4.5 shows the mechanical response as well as the thermal ( q) and mechanical power (P mech ) responses for all heat treated wires: HT 523K, HT 548K, HT 573K and HT 598K, in function of mean strain ε . The green curves represent the stress vs. strain behavior (without data from hold time). The red and blue curves are the thermal and mechanical power, respectively. Finally, the solid dots represent the response during loading, while the hollow dots represent the response during unloading.

Observing the loading portion of thermal power for the different heat treatments, one can notice that the maximum thermal power increases as the heat treatment temperature increases. Also, the thermal power maximum is always present in a strain range of 2% to 4.5%, which implies that the most phase transformation occurs in this strain range. Such strain range covers the middle section of the stress strain curve, away from the initial linear deformation regions where the elastic deformation is supposedly predominant. Between 0 and 1% of strain, a linear increase on thermal power is observed. Despite the predominance of elasticity in this region, this behavior ratifies the presence of phase transformation even for the wire submitted to the lowest heat treatment (HT 523K).

Due to cold work, the grain size reduction drastically decreases the nucleation of phase transformation [START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF]. With the grain growth promoted by the heat treatment, the nucleation barrier to phase transformation decreases. In this way, the thermal power released shows a considerable increase since more material transforms as the grains grow. On loading, the thermal power peaks values increased from 1.1W/g for the wire HT 523K to 2.1W/g for the one HT 598K, which represents a 91% increase. During unloading thermal power also showed an important decrease with the increasing heat treatment temperature. The difference observed reached 70% in the absolute value, from -1.0W/g for the wire HT 523K to -1.7W/g for the one HT 598K.

During loading, the maximum values of mechanical power showed a slight decrease with the increasing heat treatment, from 1.5W/g for the wire HT 523K to 1.0W/g for the HT 598K. During unloading the same variation observed was of -1.3W/g for the wire HT 523K to -0.8W/g for the HT 598K. Power vs. true strain on the right ordinate axis where the red dots represent thermal power ( q) and blue dots the mechanical power (P mech ). Dots represent the loading while the circles represent the unloading.

In order to take into account the variation of the strain rate during test (because of the sliding of sample), thermal and mechanical powers were normalized by the strain rate.

Figure 4.6 shows the thermal q/ ε and mechanical P mech / ε powers normalized by the strain rate in function of true strain. After normalization, one can observe the same qualitative response of thermal power. This result emphasizes the stronger presence of phase transformation after 2% strain observed for all heat treatments. The Quinney-Taylor coefficient establishes the ratio between the inelastic heat source and the inelastic work invested in deforming a material. Rittel, 1999 measured this fraction and found for copper specimens a value in the order of 0.9. For SMA this coefficient tends to be bigger once the heat is induced by the phase transformation.

Thermomechanical analysis

To investigate the relation between the thermal and mechanical powers, Fig. 4.6 shows in the right ordinate axis their ratio. One can observe the increasing behavior of the normalized thermal power over of the normalized mechanical one with the increase of heat treatment temperature. At 3% strain, the ratio increases from q/P mech ∼ = 0.8 to q/P mech ∼ = 2.1, between the HT 523K and HT 598K wires, respectively. These values quantify the increase of the phase transformation with the heat treatments. Previous results in the literature have shown a ratio of 10 for a polycrystalline CuZnAl SMA (Peyroux et al., 1998b).

Ratio

Power/ (J/g) ε .

Ratio Power/ (J/g) ε .

Ratio Power/ (J/g) ε .

Ratio Power/ (J/g) ε . Thermal and mechanical powers normalized by the strain rate ( q/ ε and P mech / ε, respectively) on the left ordinate axis and Ratio between thermal and mechanical powers q/P mech , on the right ordinate axis. For all the heat treatments, the curves of the thermal energies increase regularly during all the loading path. This increasing behavior is even observed at the beginning of loading, in the strain range of 0 to 1%, where is often assumed to present only elastic effect. As noted before while observing the power responses, such positive values of energy can be explained by the presence of phase transformation very early in the deformation process. After 1%-1.5% the evolution of thermal energy is more important, showing an increase of the exothermic heat sources of phase transformation. The same analysis can be done during unloading, the thermal energy decrease is more pronounced up to 1% strain.

Thermal and mechanical energies

Thermomechanical analysis

One can note also that thermal energy values increase with increasing temperature of heat treatment. As discussed in the previous section, as the grains grow due to the heat treatment magnitude, the amount of released/absorbed energy increases. The wire HT 523K showed a thermal energy at 4% strain of 4.3 J/g while the HT 598K showed 6.8 J/g for the same strain level, reaching 8.2 J/g at the end of loading.

The mechanical energy shows an opposite behavior. It shows a considerable decrease with the increase of heat treatment temperature, from 6 J/g for the HT 523K wire to 4.4 J/g for the HT 598K wire. However, the hysteresis area of the stress strain curve, increased from 1.0 J/g for the HT 523K wire to 1.4 J/g for the HT 598K wire. Both results agree with the fact that the increasing heat treatment temperature increases the grainsize which contribute to the SIPT. A similar result was obtained by Ahadi and Sun, 2014 when testing a Ni-rich NiTi sheet submitted to different heat treatments. Chapter 4. Influence of heat treatment on the specific latent heat and martensite fraction of nanostructured NiTi wires

The thermal energy at the end of unloading, was 0.9 J/g for the HT 523K wire and 1.8 J/g for the HT 598K wire. These values close to the values of mechanical energies at the end of the cycle (hysteresis area) lead to the same result given in Chapter 3, i.e. the mechanical cycle was a thermodynamic cycle.

Figure 4.8 shows the thermal and mechanical energies per unit of mass in function of stress, represented by red and blue dots respectively. Green dots stands for the ratio between them. Thermal and mechanical energies show a nonlinear response with stress, differently from the observed in Fig. 4.7. This behaviour becomes more evident with the increase of heat treatment temperature. At the same stress level, 800 MPa for example, the HT 523K wire shows a thermal energy of 1.1 J/g while the wire HT 598K shows a value of 7.2 J/g. The same qualitative behaviour is observed for the mechanical energy but in a reduced magnitude.

Analogously to the Fig. 4.6, one can observe the increasing behavior of the ratio between thermal and mechanical energies with the increase of heat treatment temperature. An increase from E therm /E mech ∼ = 0.8 to E therm /E mech ∼ = 2, is observed between the HT 523K and HT 598K wires, respectively. The specific latent heat was obtained from the results presented in the previous section computed through the thermodynamic framework based on the Gibbs free energy, presented more in detail in Chapter 3. However a succinct description is presented in the following.

In our analysis it is assumed that the deformation mechanism is composed only by elasticity and phase transformation. The effects of plasticity are neglected. Thus, the strain rate is written as: ε = εel + εin = εel + εtr (4.1)

with εel standing for the elastic strain rate, εin standing for the inelastic strain rate and εtr the transformation strain rate. The elastic strain rate is calculated from the relation εel = σ/E. Where the stress rate ( σ) is experimentally obtained and the elastic modulus used is E=70 GPa ( Šittner et al., 2014).

For any phase transformation, under tensile loading, the transformation strain rate 4.5 Influence of heat treatment temperature on latent heat and martensite fraction εtr is assumed to be proportional to the martensitic transformation fraction rate ( ḟm ) [START_REF] Mccormick | Intrinsic thermal-mechanical behaviour associated with the stress-induced martensitic transformation in NiTi[END_REF]:

εtr = ḟm ∆ε tr (4.2)
where ∆ε tr stands for the transformation strain for a complete transformation where the martensite fraction is f m = 1.

The heat source or thermal power released/absorbed during the tensile test reflects the manifestation of two mechanisms: thermoelasticity and phase transformation. Heat sources due to plasticity are neglected. Thus, thermal power is written in two components, a thermoelastic ( qthel ) and a phase transformation ( qtr ) one. Then, the thermal power shown in Subsection 4.4.1 can be written as:

q = qthel + qtr = - αT 0 σ ρ + qtr (4.3)
where the α is the coefficient of thermal expansion (α = 1.1 × 10 -5 K -1 ), T 0 the test temperature, ρ the mass density and σ = dσ/dt.

Finally, the specific latent heat (∆H tr (f m )) is calculated through the ratio between the transformation thermal power ( qtr ) and the transformation strain rate ( εtr ), as described by Eq. 4. The ratio between the two experimental data, namely the transformation thermal power and the transformation strain rate ( qtr / εtr ) is plotted in Fig. 4.9 in function of the true strain and stress, during loading (Fig. The ratio qtr / εtr in function of true strain during loading and unloading shows a rather constant trend between 1% and 5% strain. This behavior implies that the specific latent heat (∆H tr ) is constant and not a function of the martensite fraction (∆H tr (f m )) in this strain interval. Thus, the specific latent heat can be computed thanks to this ratio and the knowledge of the transformation strain (∆ε tr ) on the strain interval between ε = 1% and ε = 5%.

∆H tr = ( qtr / εtr )∆ε tr (4.5)

A value of ∆ε tr = 0.08, experimentally obtained in Section 3.5.1 of Chapter 3, was taken into account in the following.

Determination of ∆H tr for all heat treatments

Table 4.1 shows the mean values of the specific latent heat calculated between ε = 1% and ε = 5%, referent to the loading and unloading for all the heat treated wires. These values are in a good agreement with the ones obtained by Frenzel et al., 2015 from DSC 4.5 Influence of heat treatment temperature on latent heat and martensite fraction measurements. They showed that for fully annealed binary NiTi alloys, the values of specific latent heat decrease from about 30J/g to 10J/g with the increasing of at.% Ni. For alloys with a composition of T i-50.9 at.% N i the values obtained were around 17J/g [START_REF] Frenzel | On the effect of alloy composition on martensite start temperatures and latent heats in Ni-Ti-based shape memory alloys[END_REF]. Table 4.1: Mean values of the sçpecific latent heat during loading and unloading for all the heat treated wires.

∆H A-M tr ∆H M -A tr HT 523 K 19.6 ± 0.3 J/g 18.0 ± 0.7 J/g HT 548 K 20.6 ± 0.4 J/g 17.9 ± 0.8 J/g HT 573 K 19.6 ± 0.7 J/g 16.9 ± 0.4 J/g HT 598 K 19.6 ± 0.4 J/g 14.9 ± 0.2 J/g Figure 4.10a shows the obtained mean values and statistical data presented in Table 4.1. One can notice that the obtained values of specific latent heat (∆H A-M tr ) during loading (red curve) do not show any important variation with the increasing heat treatment temperature. Such a behavior is rather expected since it is known that the specific latent heat depends mostly on the alloy's composition. However, a decreasing behavior with the increasing HT temperature can be observed during unloading. So, the difference between loading (∆H A-M tr ) and unloading (∆H M -A tr ) specific latent heats increases with the increasing of HT temperature (see Fig. 4.10b). Analyzing then for a cycle, the chemical enthalpy change (∆H c ) and the elastic energy (E st ) are assumed to be equal for loading and unloading, so the difference between forward and reverse transformation specific latent heat may be written only in function of the mechanical work (W mech ). In this way, specific latent heat variation between loading and unloading (∆H A-M tr -∆H M -A tr ) can also be calculated from the mechanical energy difference between loading and unloading (W A-M mech -W M -A mech ). Then, rewriting Eq. 4.6, one has: HT 523 K 1.6 ± 1.0 J/g 2.2 ± 0.9 J/g HT 548 K 2.7 ± 1.2 J/g 2.4 ± 0.6 J/g HT 573 K 2.7 ± 1.1 J/g 2.8 ± 0.5 J/g HT 598 K 4.7 ± 0.6 J/g 3.5 ± 0.7 J/g

The values of ∆H A-M tr -∆H M -A tr were computed from Table 4.1 and are shown in Table 4.2 for all heat treated wires. The values of W A-M mech -W M -A mech were computed using the right member side of Eq. 4.7. The mean values of σ A-M and σ M -A and their respective standard deviations were calculated in a strain range between 1% and 1.5% of the transformation strain ε tr . This range was used in order to perform an equivalent analysis between all the tested wires. The obtained results are presented in Table 4.2. For a better analysis, the values presented in Table 4.2 were plotted in Fig. 4.11. Despite the slight difference between the thermal and mechanical values, one can observe the agreement between the difference of the specific latent heat variation between loading and unloading (∆H A-M tr -∆H M -A tr ) and the mechanical work between loading and unloading (W A-M mech -W M -A mech ). Such a result ratifies Eq. 4.7 and confirms the assumptions made to develop it.

Analyzing the friction energy variation E

A-M f r -E M -A f r
From the Gibbs free energy (presented and discussed in chapter 3), the specific latent heat ∆H tr can be written in terms of the entropy variation -T ∆S c (with T standing for the test temperature) and a friction energy E f r , see Eq. 3.11. From this statement, analyzing then the variation of the friction energy in a loading-unloading cycle, the Eqs. 3.28 and 3.29, can be written as follows:

∆H A-M tr -∆H M -A tr = -(θ A-M -θ M -A )∆S c + (E A-M f r -E M -A f r ) (4.8)
Figure 4.12 shows the difference of the temperature variation during loading and unloading (θ A-M -θ M -A ) in function of the average transformation strain (ε tr ). The methodology used to determine these curves has been presented in Chapter 3 by the .12: Difference between the average temperature variation during loading and unloading θ A-M -θ M -A in function of the average transformation strain ε tr for all heat treatment temperatures.

Calculating the mean values and their respective standard deviation from the curves showed in Fig. 4.12, one obtain for each test: ) between loading and unloading for all the heat treated wires.

-(θ A-M -θ M -A )∆S c E A-M f r + E M -A f r
HT 523 K 0.5 ± 0.04 J/g 1.1 ± 1.0 J/g HT 548 K 0.7 ± 0.05 J/g 2.0 ± 1.2 J/g HT 573 K 0.8 ± 0.06 J/g 1.9 ± 1.2 J/g HT 598 K 0.7 ± 0.09 J/g 4.0 ± 0.6 J/g

The obtained values show that the dissipated energy present in the SIPT cycle are composed by a non-negligible parcel from the temperature and enthalpy variation (-T ∆S c ) and a parcel from the friction energy (E f r ). One can observe that the friction energy term increases with the increasing of the HT temperature while the energy due to the entropy contribution remains rather constant. It is observed that independently of the heat treatment temperature, the dissipated energy is mostly due to the friction energy rather then the temperature and entropy term. The increase of the energy friction term is justified by the increase of phase transformation with the heat treatment temperature. This phenomenon takes place due to the increase of the moving interfaces during phase transformation [START_REF] Wollants | Thermally-and stress-induced thermoelastic martensitic transformations in the reference frame of equilibrium thermodynamics[END_REF].

Martensite fraction (f m )

The martensite fraction was calculated by time integration of the martensite fraction rate ( ḟm ) deduced from Eq. 4. On Fig. 4.13a, the solid curves show that since the beginning of strain (around 0.1% strain) the f m starts to increase. From 0 to 1% strain, an increase of almost 5% in f m is observed. A steeper increase is observed from 1% until the end of loading. Such behavior is observed in all heat treated wires.

It is known that martensitic phase transformation occurs by nucleation and growth of crystals in the parent phase matrix. It has been shown that the grain size reduction decreases exponentially the potency of grains as nucleation sites [START_REF] Chen | Statistics of martensitic nucleation[END_REF]. Numerous studies [START_REF] Tsuchiya | Martensitic transformation in nanostructured TiNi shape memory alloy formed via severe plastic deformation[END_REF][START_REF] Waitz | Phase Transformations of Nanocrystalline Martensitic Materials[END_REF][START_REF] Glezer | Martensite Transformation in Nanoparticles and Nanomaterials[END_REF] reveal that nanostructured materials tend to partially suppress the martensite phase transformation. The heat treatment acts backwards to this process increasing the grainsize and Chapter 4. Influence of heat treatment on the specific latent heat and martensite fraction of nanostructured NiTi wires promoting the phase transformation. The aforementioned behavior is then observed in Fig. 4.13 where at the same stress level, 800M P a for example, the amount of martensite transformed in the wire HT 523K was f HT 523K

Conclusion

A thermomechanical investigation was performed on cold worked wires submitted to a heat treatment at four different temperatures: 523K, 548K, 573K and 598K. Inconclusive DSC results were obtained for all heat treated wires since no thermal peaks were observed. Using an infrared thermal camera, uniform temperature profiles were obtained during tensile tests for all heat treated wires. With the increase of heat treatment temperature, the temperature variation of the wire during the tensile test increased. Uniform strain fields were observed with digital image correlation (DIC) on all tensile tests. The uniformity on the temperature and strain profiles allowed a 0D approach for experimental data computing.

Through a heat balance applied on the experimental data, thermal power and thermal energy generated during the SIPT were obtained. A substantial increase on thermal power peak was observed rising from 1.1 W/g for the wire HT 523K to 2.1 W/g for the one HT 598K. Also, the ratio between thermal power and mechanical power ( q/P mech ) increased from q/P mech = 0.8 to q/P mech = 2.1, for the wire HT 523K and HT 598K, respectively.

Conclusion

Thermal energy also showed an important increase with the heat treatment temperatures. The wire HT 523K showed a thermal energy at the end of loading of 4.9 J/g while the HT 598K presented a value of 8.2 J/g. Furthermore the hysteresis area of the stress strain curve increased from 1.0 J/g for the wire HT 523K to 1.4 J/g for the wire HT 598K.

With the support of a thermodynamic framework, it was possible to estimate the specific latent heat of the T i50.8%at.N i wire during a stress induced martensitic transformation. Despite the differences on thermal energies shown for each heat treated wire, the specific latent heat obtained in the tensile tests showed a mean value with a rather small dispersion on loading. The obtained value were ∆H A-M HT 523K = 19.6±0.3 J/g, ∆H A-M HT 548K = 20.6±0.4 J/g, ∆H A-M HT 573K = 19.6±0.7 J/g and ∆H A-M HT 598K = 19.6±0.4 J/g. During the unloading, the value obtained presented a decreasing behavior with the increase of HT temperature, they were ∆H M -A HT 523K = 18.0 ± 0.7 J/g, ∆H M -A HT 548K = 17.9 ± 0.8 J/g,∆H M -A HT 573K = 16.9 ± 0.4 J/g and ∆H M -A HT 598K = 14.9 ± 0.2 J/g. Values in the same range were found in literature for a fully annealed NiTi alloys tested via the DSC technique. Due to decreasing behavior during unloading the dissipated heat described as the difference between the specific latent heat on loading and unloading showed an increasing tendency. Such behavior was followed by the mechanical hysteresis showing rather the same values.

Moreover, it was possible to determine the martensite fraction for all the heat treated wires in function of strain and stress. It showed an important dependence with the heat treatment temperature. With the increase of the temperature heat treatment, more martensite was transformed for the same stress level. At approximately 800M P a, the fraction of martensite transformed were: f HT 523K

GENERAL CONCLUSION

This PhD thesis investigated the thermomechanical couplings associated with the tensile superelastic effect of a T i -50.9 at.% N i Shape Memory Alloy (SMA) nanocrystalline thin wire (diameter 0.5 mm), in a Cold Worked (CW) state. The wire was submitted to different heat treatment temperatures (HTT), ranging from 523 K to 598 K during 30 min. These heat treatments at low temperature promoted a fully superelastic loop without stress plateau and no Luders-like deformation. The main objectives were to measure and analyse the specific latent heat of phase transformation (∆H tr ) and the martensite fraction (f m ) during a stress induced phase transformation (SIPT), for different HTT. In such a material, classic DSC tests show inconclusive results for the determination of ∆H tr . To overcome this limitation, a method using digital image correlation (DIC) and thermal field measurement (TFM) associated with a thermodynamic analysis was proposed. These tasks were accomplished in several steps summarized in the following paragraphs.

The first chapter of this report presents a brief bibliography about the thermomechanical properties of the SMAs. Some potential engineering applications, using the effect of the thermomechanical coupling in superelasticity, are shown, as well as some experimental studies found in literature. A thermodynamic framework analysing the martensite transformation phenomenon is then given. Finally, this chapter discuss the influence: (1) of the microstructure on the superelasticity thermodynamic properties, (2) of the manufacturing process and heat treatment on thermomechanical responses.

The experimental aspects concerning the material, the applied heat treatments, the test procedure and the experimental issues are presented in chapter two. This chapter describes the methodology proposed to obtain and analyse the thermal and kinematic full field data.

Chapter three presents the results of the thermomechanical study of a NiTi wire, heat treated during 30 min at a single temperature (523 K), and submitted to a tensile loading. Thermal and mechanical powers are estimated from temperature variations and local strain fields measured with an infrared camera and a DIC system, respectively. The analysis of the thermal power shows that the phase transformation started since the beginning of deformation, rather than after reaching a critical stress, as it is usually described in literature.

General Conclusion

Analysing these results, and using a thermodynamic framework based on the Gibbs free energy, the specific latent heat of phase transformation is determined during the direct (∆H A-M ) and reverse (∆H A-M ) phase transformations. The latent heat estimated during a SIPT is in good agreement with the ones given in the literature for a fully annealed T i -50.9 at.% N i alloy tested via DSC technique.

The martensite fraction f m is also calculated in function of strain and stress. Variations of f m up to 10% are estimated during the initial loading stage (before the upper plateau) and during the unloading (before the lower plateau) stage. This result ratifies the presence of phase transformation in these classical "elastic deformation stages". At the end of loading, a maximum of 45% of martensite fraction is obtained, for an imposed strain of 4.5%.

Chapter four studies the influence of HTT on the thermomechanical couplings observed during a SIPT on the same NiTi wire but submitted to four HTT (523 K, 548 K, 573 K and 598 K) during 30 min.

An increase of the magnitude of the thermal power and a decrease of the mechanical power were observed with an increase in HTT. Despite the differences between the obtained thermal powers, the specific latent heat values during loading (∆H A-M ) show a rather small dispersion, around a mean value of 19.8 J/g. However, the specific latent heat values obtained during unloading (∆H M -A ), show a decreasing trend with the increasing of HT temperature. Also f m values, estimated for all HT wires at the same the same stress level, increase with the increase of HT temperature.

In general this work provided a method to determine the specific latent heat in SMAs which do not show any conclusive result when analyzed through DSC tests. It also contributed to give insights in the understanding of the thermomechanical couplings during a SIPT through the relation of the thermal and mechanical energies involved in the phenomenon.

PERSPECTIVES

A new field of investigation has been opened to study the thermomechanical behavior of superelastic Shape Memory Alloys. The used method allows the determination of the specific latent heat, as well as the amount of martensite transformation, using non contact measurements of temperature and displacement fields during a stress induced phase transformation.

As discussed in the present work, the specific latent heat is strongly dependent on the alloy composition. For binary NiTi Alloys the specific latent heat may vary from 10 J/g to 30 J/g. This variation is also strongly dependent on the composition for other ternary NiTi alloys (as NiTiV, NiTiCr, NiTiCu, etc.) or even for other SMA's compositions such as Cu based alloys (as CuAlMn, CuAlBe, CuAlNi). In this context the method used in this work could be applied to determine the specific latent heat in other compositions of superelastic SMA's.

Concerning the experimental analysis, thermal and mechanical results (i.e. powers and energies) are greatly sensitive to the geometry and size of the sample. From the thermal point of view, the determination of the τ eq may be affected by the specimen size and geometry. This effect is due to the small thermal inertia of small specimens. In this context, several sizes of specimens could be used to take into account the geometry influence. From the mechanical point of view, the determination of the strain rate ε is also affected by small specimens when using the method of digital image correlation. In order to take into account this possible variation, larger specimens could be used. Also, a physical strain gauge attached to the specimen could be used.

Lastly, the determination of the specific latent heat during a stress induced phase transformation is very useful for elastocaloric cooling applications. With the use of this approach, more efficient devices can be designed with the determination of the amount of martensite phase transformation and thermal power and energies.

  Memory Alloys (SMA) . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Mechanism of phase transformation . . . . . . . . . . . . . . . . . 3 1.1.2 Thermally-induced phase transformation . . . . . . . . . . . . . . 4 1.1.3 Stress-induced phase transformation . . . . . . . . . . . . . . . . 5 1.1.4 Shape Memory Effect . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.5 Superelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Thermomechanical coupling in superelasticity . . . . . . . . . . . . . . . 8 1.2.1 Engineering applications based on thermomechanical coupling in NiTi superelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.2 Thermodynamic framework of thermoelastic martensitic transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 Experimental studies on thermomechanical couplings . . . . . . . 20 1.3 Nanostructured NiTi wires . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.1 Specificities of the NiTi alloys . . . . . . . . . . . . . . . . . . . . 25 1.3.2 Influence of wire manufacturing process on crystallographic structure and mechanical behaviour . . . . . . . . . . . . . . . . . . . 27 1.3.3 Influence of heat treatment on microstructure and mechanical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 Chapter review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 Experimental Aspects 33 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 NiTi wire sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 CONTENTS 4.3.2 Superelastic mechanical responses . . . . . . . . . . . . . . . . . . 97 4.3.3 Strain and temperature fields . . . . . . . . . . . . . . . . . . . . 98 4.4 Thermomechanical analysis . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4.1 Thermal and mechanical powers . . . . . . . . . . . . . . . . . . . 101 4.4.2 Thermal and mechanical energies . . . . . . . . . . . . . . . . . . 103 4.5 Influence of heat treatment temperature on latent heat and martensite fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.5.1 The specific latent heat . . . . . . . . . . . . . . . . . . . . . . fraction (f m ) . . . . . . . . . . . . . . . . . . . . . . . 113 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. 1

 1 Schematic illustration of the Superelasticity or Superelastic Effect (SE) (a), and Shape Memory Effect (SME) (b) (Adapted from Schlosser, 2008). 3 1.2 Schematic illustration of thermal and mechanical induced phase transformation showing the (a) austenite, (b) twinned martensite and (c) detwinned martensite phases (Adapted from Cismasiu, 2010). . . . . . . . . 4 1.3 Schematics of the thermal-induced phase transformation (adapted from Lagoudas, 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Schematics of the stress-induced phase transformation (adapted from Lagoudas, 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Schematics representation of the shape memory effect on a NiTi SMA (adapted from Lagoudas, 2008). . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 A typical SMA superelastic loop (adapted from Lagoudas, 2008). . . . . . 8 1.7 Schematic showing the latent heat released/absorbed on a stress induced phase transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.8 (a) Schematic stress-strain response in isothermal tension (very low strain

  . . . . . . 22 List of Figures 1.15 The crystal structures of NiTi alloys. (a) Cubic austenite parent B2, (b) Monoclinic B19 martensite and (c) R-phase. (Adapted from Schlosser, 2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 a) Strain field obtained trough DIC and b) temperature field obtained trough IR camera of a dog-bone shaped NiTi specimen showing localization pattern. (Adapted from Xie et al., 2016) . . . . . . . . . . . . . . . . . . 1.17 Evolution of microstructure and corresponding superelastic functional properties in the Ni-Ti wires heat treated by electric pulses with pulse times t= 0, 6, 8, 10, 12, 14, 16 and 18 ms (a to h, respectively)(Adapted from Delville et al., 2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Effects of grainsize on the isothermal superelastic response (σ-ε) (Adapted from Ahadi and Sun, 2014) . . . . . . . . . . . . . . . . . . . . . . . . . 1.19 (a) Full-field thermographic images and (b) volume -averaged temperature

  2.1 a) Cold worked NiTi wire with a diameter of d=0.5mm. b) Tensile test of the CW NiTi wire showing in detail the loading path and their respective loading maximum (L) and rupture (R) points. . . . . . . . . . . . . . . . 2.2 Transmission electron microscopy showing a) a bright field picture b) and a diffraction pattern of the NiTi cold worked wire. . . . . . . . . . . . . . 2.3 Superelastic tensile behavior of the T i -50.9 at% N i wire CW and submitted to different heat treatments. . . . . . . . . . . . . . . . . . . . . . . . 2.4 a) Graphical representation of a classic DSC curve and methodology to determine the specific latent heat and the critical temperatures. b) Actual DSC results of heat treated NiTi wires. . . . . . . . . . . . . . . . . . . . 2.5 a) Eplexor Gabo tensile testing machine. b) Schematic representation of tensile test with both infrared and visible light camera. c) Schematic representation of the loading path imposed at each test. . . . . . . . . . . . . 2.6 a) "V-clamp" and "Flat clamp" models showing in detail the different contact points over the hold sample. b) sample mounted on the used clamps. 2.7 Example of a) a speckled image; b) an applied subset and grid. . . . . . . 2.8 Example of an image correlation grid a) prior and b) after shear deformation. 2.9 Example of natural and artificial textures. a) Biphasic polymer. b) Rock. c-e) Speckle paintings. (Adapted from Schlosser, 2008) . . . . . . . . . . 2.10 a) Big painting spot on the speckle pattern. b) Low contrast pattern. c) Ideal speckle painting with uniform pattern and good contrast. . . . . . . . 2.11 Electromagnetic radiation spectrum. . . . . . . . . . . . . . . . . . . . . . 2.12 Schematic view of the different forms of radiations. (Source: Schlosser, 2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Common emissivity values; Expanded scale between 0.8 and 1. (Adapted from: Schlosser, 2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 a) Used IR camera filming a scene with a standard objective lens. b) Thermal image obtained with the standard objective lens. c) Used IR camera filming a scene with the zoom objective lens. d) Thermal image obtained by the zoom objective lens. . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures 2.15 a)Schematic representation of an IR camera detector matrix b) and its respective digital signal output. . . . . . . . . . . . . . . . . . . . . . . . . 2.16 a) Schematic representation of the digital output and an example of a radiance image before NUC b) and after NUC. . . . . . . . . . . . . . . . 2.17 a) Example of thermal image of a tested wire. b) Temperature profile of the blue line shown in a). c) Spatial (x direction) temperature derivate curve used to determine the wire's position on thermal image. . . . . . . 2.18 a) Example of thermal image of a tested wire. b) Detail showing the 30 pixels in the wire's diameter and the 20 pixels used to avoid the image's non uniform emissivity zone. c) Temperature profile of the 30 pixels line shown in b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 a) Example of thermal image of the tested wire showing a longitudinal profile (blue line), b) and its respective longitudinal temperature profile. . 2.20 Wire's average temperature variation at each image. . . . . . . . . . . . . 2.21 a) Schematic representation of lost images during thermal data acquisition. b) Example of data showing time difference between images. . . . . 2.22 Example of the experimental data before (black curve) and after (red curve) fixing lost image problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.23 Schematic representation of the two acquisition systems. . . . . . . . . . 2.24 a) Example of experimental data with both axial strain (ε) and temperature variation (θ) before synchronization. b) and after synchronization. . . . . 2.25 a) Tensile test performed under same conditions on three wire samples. . 2.26 Temperature variation over time of the three tested wires with detail over the a)upper peak and b) lower peak. . . . . . . . . . . . . . . . . . . . . . 3.1 a) Tensile response up to rupture (with global deformation) at room temperature (297 K). b) DSC of the wire heat treated at 598 K (with temperature rate of 10K/min). . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Experimental set-up showing the NiTi wire in the center with extracted thermal and visible images. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 a) Single tensile Stress-Strain curve during loading and unloading of the NiTi wire, where ε is the averaged logarithmic strain over the length l 0 . b) DIC average logarithmic strain profiles over the wire's gauge length at different strain levels, as indicated by the colored marks in (a). c) Temperature variation profiles at different strain levels. . . . . . . . . . . 3.4 a) Stress-strain curve for a single cycle tensile test at ε = 10 -2 s -1 . b) Average temperature variation θ , average strain and average strain rate in function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 a) Uniform temperature variation θ and Thermal power q , presented on left and right axis, respectively, in function of time. b) Mean and standard deviation of q, calculated in the time interval ∆t shown in a), obtained for different values of τ eq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Thermal and mechanical powers per unit of mass in function of time for a single loop tensile test at ε = 10 -2 s -1 . At right, close-up for stages I and III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Thermal and mechanical energies in function of time for a single loop tensile test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures 3.8 Evolution with the strain of the thermal and mechanical power per unit of mass (left ordinate axis) and strain rate (right ordinate axis), on loading (dots) and unloading (empty circles). . . . . . . . . . . . . . . . . . . . . 3.9 a) Thermal and Mechanical Power normalized by strain rate in function of true strain. b) Same values in function of nominal stress. . . . . . . . 3.10 a) Thermal and Mechanical Energies in function of true strain. b) Same values in function of nominal stress. . . . . . . . . . . . . . . . . . . . . . 3.11 a) Thermal and Mechanical Power normalized by strain rate in function of true strain. b) Thermal and Mechanical Energy in function of true strain. 3.12 a) Normalized inelastic thermal power ( qin / ε) and normalized thermal power ( q/ ε) on the left ordinate axis. Ratio between inelastic thermal power ( qin ) and inelastic mechanical power on right ordinate axis ( qin /P mech-in ) in function of true strain. b) Same values in function of nominal stress. . 3.13 a) Thermal energy (E th ) and the inelastic thermal energy (E in ) on left ordinate axis and Energy ratio (E in /E mech-in ) on the right ordinate axis in function of true strain. b) Same values in function of nominal stress. . 3.14 a) Classic superelastic loop of a SMA with the decomposition of the strain components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 a) Theoretical evolution of ε tr a with the total inelastic strain ε in . b) Experimental data used to determine the value of transformation strain ∆ε tr . . 3.16 Inverse pole figure showing the orientation dependence of the transforma-

  3.20 a) Nominal stress vs. true strain. b) And ratio dσ/dε in function of true strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 a) Martensite fraction in function of strain. b) Martensite fraction in function of nominal stress. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 DSC results for the wires with the different heat treatments (The curves are plotted with an offset of 0.0125W/g). . . . . . . . . . . . . . . . . . . 4.2 Non-isothermal superelastic σ -ε tensile test responses for different heat treatment temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 a) Non-isothermal σ -ε superelastic tensile test responses, b) strain profiles and c) temperature variation profiles of the tensile test performed on the wires heat treated at 523K, 548K, 573K and 598K. The strain and temperature profiles correspond to the loading points highlighted in figures of column a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures 4.4 a) Non-isothermal σ -ε superelastic tensile test responses, b) strain profiles and c) temperature variation profiles of the tensile test performed on the wires heat treated at 523K, 548K, 573K and 598K. The strain and temperature profiles correspond to the unloading points highlighted in figures of column a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.5 Stress vs. true strain on the left ordinate axis represented by the green dots.

  . 4.4, as a function of strain and stress during a) loading and b) unloading for all heat treated wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.10 a) Specific latent heat, with their respective standard deviation, determined for each wire during loading (red data) and unloading (blue data). Dashed line representing the total average values of specific latent heat (∆H tr ) for loading and unloading. b) Evolution of the difference between the specific latent heat of loading and unloading (∆H A-M tr -∆H M -A tr ) in function of heat treatment temperature. . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.11 Comparison between specific latent heat variation between loading and unloading calculated with thermal data ∆H A-M tr -∆H M -A tr (black curve) and mechanical hysteresis W A-M mech -W M -A mech (blue curve). . . . . . . . . . . . 111 4.12 Difference between the average temperature variation during loading and unloading θ A-M -θ M -A in function of the average transformation strain ε tr for all heat treatment temperatures. . . . . . . . . . . . . . . . . . . . 112 4.13 Martensite fraction in function of strain (a) and stress (b) for all heat treated wires. Solid curves are for loading and dashed ones for unloading. of the properties used in this thermodynamic framework . . . . . . 16 1.2 Main characteristics of Cold and Hot working . . . . . . . . . . . . . . . 28 4.1 Mean values of the sçpecific latent heat during loading and unloading for all the heat treated wires. . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2 Mean values of the mechanical work (W A-M mech -W M -A mech ) and the specific latent heat (∆H A-M tr -∆H M -A tr ) between loading and unloading for all the heat treated wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3 Averaged temperature variation between loading and unloading θ A-Mθ M -A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.4 Values of the energies due to the temperature and entropy variations

FigFigure 1 . 1 :

 11 Figure 1.1: Schematic illustration of the Superelasticity or Superelastic Effect (SE) (a), and Shape Memory Effect (SME) (b) (Adapted from Schlosser, 2008).

Figure 1 . 2 :

 12 Figure 1.2: Schematic illustration of thermal and mechanical induced phase transformation showing the (a) austenite, (b) twinned martensite and (c) detwinned martensite phases (Adapted from Cismasiu, 2010).

  Figure 1.3: Schematics of the thermal-induced phase transformation (adapted from Lagoudas, 2008).

  Figure 1.4: Schematics of the stress-induced phase transformation (adapted from Lagoudas, 2008).

  Figure 1.5: Schematics representation of the shape memory effect on a NiTi SMA (adapted from Lagoudas, 2008).

Figure 1

 1 Figure 1.6: A typical SMA superelastic loop (adapted from Lagoudas, 2008).

  Figure 1.7: Schematic showing the latent heat released/absorbed on a stress induced phase transformation.

Figure 1

 1 Figure1.8: (a) Schematic stress-strain response in isothermal tension (very low strain rate and/or strong heat transfer); (b) and in non-isothermal tension (high strain rate and/or weak heat transfer)(adapted from[START_REF] He | On non-monotonic rate dependence of stress hysteresis of superelastic shape memory alloy bars[END_REF].

Figure 1

 1 Figure 1.10: Energy dissipation (hysteresis area) in function of strain rate at three different ambient conditions (a) still air, (b) airflow at v = 2m/s and (c) airflow at v = 17m/s (dapted from He et al., 2010).

  Figure 1.12: (a)Temperature variation in function of time for different cross-head speeds (in mm/min); (b) and variation of the critical stress for martensite nucleation (σ c ) as a function of temperature (adapted from[START_REF] Rodriguez | The thermal effect due to stress-induced martensite formation in -CuAlNi single crystals[END_REF].

Figure 1 . 13 :

 113 Figure 1.13: Thermal energy (W ch ), mechanical energy (W m ) and temperature variations (θ) at different strain rates (adapted from Peyroux et al., 1998b).

Figure 1 . 14 :

 114 Figure 1.14: Temperature measurement and heat sources estimation during a tensile test of a NiTi tube sample (adapted from Schlosser et al., 2007).

  Figure 1.15: The crystal structures of NiTi alloys. (a) Cubic austenite parent B2, (b) Monoclinic B19 martensite and (c) R-phase. (Adapted from[START_REF] Schlosser | Influence of thermal and mechanical aspects on deformation behaviour of NiTi alloys[END_REF]

Figure 1

 1 Figure 1.16: a) Strain field obtained trough DIC and b) temperature field obtained trough IR camera of a dog-bone shaped NiTi specimen showing localization pattern.(Adapted from[START_REF] Xie | Observation on rate-dependent cyclic transformation domain of super-elastic NiTi shape memory alloy[END_REF]

Figure 1

 1 Figure1.18: Effects of grainsize on the isothermal superelastic response (σ -ε) (Adapted from[START_REF] Ahadi | Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi[END_REF]

Figure 2

 2 Figure 2.1: a) Cold worked NiTi wire with a diameter of d=0.5mm. b) Tensile test of the CW NiTi wire showing in detail the loading path and their respective loading maximum (L) and rupture (R) points.

Figure 2 .Figure 2 . 2 :

 222 Figure 2.2b shows the TEM diffraction pattern (DP) of the tested wire. In this picture one can observe the presence of a polycrystalline austenite phase (B2) with crystal lattice preferentially oriented in the direction (110). This result will be further discussed in Chapter 3.

Figure 2

 2 Figure 2.3: Superelastic tensile behavior of the T i-50.9 at% N i wire CW and submitted to different heat treatments.

Figure 2 .Figure 2

 22 Figure 2.4b shows the results of DSC tests for the heat treated wires. The absence of peaks makes impossible to determine the specific latent heat using the before mentioned method.

Figure 2 .

 2 Figure 2.5: a) Eplexor Gabo tensile testing machine. b) Schematic representation of tensile test with both infrared and visible light camera. c) Schematic representation of the loading path imposed at each test.

Figure 2

 2 Figure 2.6: a) "V-clamp" and "Flat clamp" models showing in detail the different contact points over the hold sample. b) sample mounted on the used clamps.

Figure 2 . 7 :

 27 Figure 2.7: Example of a) a speckled image; b) an applied subset and grid.

Figure 2 . 8 :Figure 2 . 9 :

 2829 Figure 2.8: Example of an image correlation grid a) prior and b) after shear deformation.

Figure 2 .

 2 Figure 2.10: a) Big painting spot on the speckle pattern. b) Low contrast pattern. c) Ideal speckle painting with uniform pattern and good contrast.

Figure 2 .

 2 Figure 2.10 shows three different speckle paint patterns of the tested 0.5mm diameter

Figure 2 .

 2 Figure 2.11: Electromagnetic radiation spectrum.

Figure 2 .

 2 Figure 2.12: Schematic view of the different forms of radiations. (Source: Schlosser, 2008)

Figure 2 .

 2 Figure 2.14: a) Used IR camera filming a scene with a standard objective lens. b) Thermal image obtained with the standard objective lens. c) Used IR camera filming a scene with the zoom objective lens. d) Thermal image obtained by the zoom objective lens.

Figure 2 .

 2 Figure 2.14 shows the difference between a scene observed with a standard objective

Figure 2 .

 2 Figure 2.15: a)Schematic representation of an IR camera detector matrix b) and its respective digital signal output.

Figure 2 .Figure 2 .

 22 Figure2.15a shows a schematic representation of an IR camera operation. The incident IR radiation on each detector is transformed in a digital output signal read by the circuit's camera. Each detector acts as a temperature probe and responds to the absorbed IR radiation independently. Fig.2.15b shows for three detectors (A, B and C) a schematic representation of the digital level in function of the radiance (IR radiation). One can observe the random behavior shown by each detector. Because of that, a so called Non-Uniformity Correction (NUC) has to be performed in order to align each detector output signal.

  2.4.2.5.a Identifying the wire on thermal image

Figure 2 .

 2 Figure 2.17a presents an example of the processed infrared image showing the tested wire in the center. As one can notice, the wire is distinguished on the image because of the difference between its temperature and the background. When tracing the circumferential temperature profile, one can observe (see Fig.2.17b) that the image's background is around 300K (at this specific instant) while the wire is at 307K. So, by calculating the spatial derivative of the temperature profile along x direction 2 peaks are obtained as it can be seen in Fig.2.17c. From that, the peaks named p2 and p3 represent the initial and final positions of the wire when sweeping the thermal image from left to right.

Figure 2 .

 2 Figure 2.17: a) Example of thermal image of a tested wire. b) Temperature profile of the blue line shown in a). c) Spatial (x direction) temperature derivate curve used to determine the wire's position on thermal image.

Figure 2 .Figure 2 .

 22 Figure 2.18: a) Example of thermal image of a tested wire. b) Detail showing the 30 pixels in the wire's diameter and the 20 pixels used to avoid the image's non uniform emissivity zone. c) Temperature profile of the 30 pixels line shown in b).

Figure 2 .

 2 Figure 2.20: Wire's average temperature variation at each image.

Figure 2 .

 2 Figure 2.21: a) Schematic representation of lost images during thermal data acquisition. b) Example of data showing time difference between images.

Figure 2 .

 2 Figure 2.22: Example of the experimental data before (black curve) and after (red curve) fixing lost image problem.

Figure 2 .

 2 Figure 2.23: Schematic representation of the two acquisition systems.

Figure 2 .

 2 Figure 2.24 shows an example of the experimental data before (see Fig. 2.24a) and after (see Fig. 2.24b) synchronization. One can observe on the detail of Fig. 2.24a that the time gap between the two signals is 7.5s. Then, after determination of the test start, both signals presented the same initial time t 0(kin) = t 0(IR) (see detail in Fig.2.24b).

Figure 2 .

 2 Figure 2.25: a) Tensile test performed under same conditions on three wire samples.

Figure 2 .

 2 Figure 2.26: Temperature variation over time of the three tested wires with detail over the a)upper peak and b) lower peak.

Fig. 3 .

 3 Fig. 3.1b shows the result of a Differential Scanning Calorimetry (DSC) performed on the heat treated wire. A Q200 DSC testing machine from TA Instruments was used and the tests were carried out with a temperature rate of 10K/min between 223K and 363K.

Figure 3

 3 Figure 3.1: a) Tensile response up to rupture (with global deformation) at room temperature (297 K). b) DSC of the wire heat treated at 598 K (with temperature rate of 10K/min).

Figure 3 . 2 :

 32 Figure 3.2: Experimental set-up showing the NiTi wire in the center with extracted thermal and visible images.

Figure 3

 3 Figure 3.3: a) Single tensile Stress-Strain curve during loading and unloading of the NiTi wire, where ε is the averaged logarithmic strain over the length l 0 . b) DIC average logarithmic strain profiles over the wire's gauge length at different strain levels, as indicated by the colored marks in (a). c) Temperature variation profiles at different strain levels.

Figure 3 .Figure 3

 33 Figure3.4 shows nominal stress vs. true strain σ 0 -ε curve of the uniaxial tensile test (Fig.3.4a) and the temporal evolution of the uniform temperature variation θ and strain ε (Fig.3.4b). The uniform strain rate temporal evolution ε is also plotted in Fig.3.4b. It was calculated by numerical derivation of the strain temporal evolution.

Figure 3 .Figure 3 . 6 :

 336 Figure 3.6 shows the Thermal power and the Mechanical power per unit of mass (W/g) as function of time.

Chapter 3 .Figure 3 . 7 :

 337 Figure 3.7: Thermal and mechanical energies in function of time for a single loop tensile test.

Figure 3 . 8 :

 38 Figure3.8: Evolution with the strain of the thermal and mechanical power per unit of mass (left ordinate axis) and strain rate (right ordinate axis), on loading (dots) and unloading (empty circles).

Chapter 3 .

 3 Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires

Figure 3

 3 Figure 3.9: a) Thermal and Mechanical Power normalized by strain rate in function of true strain. b) Same values in function of nominal stress.

Figure 3 Figure 3

 33 Figure3.11 shows the normalized powers (a) and energies (b) in function of true strain for the tensile test until rupture. In Fig.3.11a one can observe the same qualitative response shown by the cyclic tensile test at Fig.3.9a. The increasing monotonic thermal response reaching a maximum at approximately 3% strain is observed. Also, after 3% strain,

Chapter 3 .Figure 3

 33 Figure 3.11: a) Thermal and Mechanical Power normalized by strain rate in function of true strain. b) Thermal and Mechanical Energy in function of true strain.

Figure 3 .

 3 Figure 3.13: a) Thermal energy (E th ) and the inelastic thermal energy (E in ) on left ordinate axis and Energy ratio (E in /E mech-in ) on the right ordinate axis in function of true strain. b) Same values in function of nominal stress.

Figure 3

 3 Figure 3.14: a) Classic superelastic loop of a SMA with the decomposition of the strain components.

Chapter 3 .

 3 Figure 3.15: a) Theoretical evolution of ε tr a with the total inelastic strain ε in . b) Experimental data used to determine the value of transformation strain ∆ε tr .

Figure 3 .

 3 Figure 3.16 shows an inverse pole figure describing the orientation dependence of the transformation strain ∆ε tr in a NiTi single crystal[START_REF] Miyazaki | The habit plane and transformation strains associated with the martensitic transformation in Ti-Ni single crystals[END_REF]. One can observe that in the direction (011), same preferential direction of the used cold worked wire (see Chapter 2), the ∆ε tr is between 8% and 9%. Also, in a work published by[START_REF] Orgéas | Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression[END_REF], they observed that the ∆ε tr of a A -M transformation was around ε tr = 8%. Finally, based on the experimental determination and the literature support, the value ∆ε tr = 8% is used in the following of this study.

Figure 3 .

 3 Figure 3.16: Inverse pole figure showing the orientation dependence of the transformation strain ∆ε tr of solution-treated NiTi single crystals (parenthesis) and calculated values (contour lines)(Adapted from[START_REF] Miyazaki | The habit plane and transformation strains associated with the martensitic transformation in Ti-Ni single crystals[END_REF].

3. 5

 5 Characterisation of the martensitic transformation 3.5.2 Experimental determination of forward and reverse specific latent heat 3.5.2.1 Cyclic test The equations developed in Sections 3.4.1 and 3.4.3 are used to further analyze thermal behavior and properties.

Chapter 3 .

 3 Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires

  Figure 3.19: Ratio qin / εin computed with Eq. 3.25 and experimental data, as a function of strain a) and as a function of nominal stress for the rupture test b).

Figure 3

 3 Figure 3.20: a) Nominal stress vs. true strain. b) And ratio dσ/dε in function of true strain.

Figure 3

 3 Figure 3.21: a) Martensite fraction in function of strain. b) Martensite fraction in function of nominal stress.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Material and experimental procedures . . . . . . . . . . . . . 4.2.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Differential scanning calorimetry analysis . . . . . . . . . . . . 4.2.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 DSC responses . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Superelastic mechanical responses . . . . . . . . . . . . . . . . 4.3.3 Strain and temperature fields . . . . . . . . . . . . . . . . . . . 4.4 Thermomechanical analysis . . . . . . . . . . . . . . . . . . . 4.4.1 Thermal and mechanical powers . . . . . . . . . . . . . . . . . 4.4.2 Thermal and mechanical energies . . . . . . . . . . . . . . . . . 4.5 Influence of heat treatment temperature on latent heat and martensite fraction . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 The specific latent heat . . . . . . . . . . . . . . . . . . . . . . 4.5.1.1 Method to determine ∆H tr . . . . . . . . . . . . of ∆H tr for all heat treatments . . . . 4.5.2 Analyzing the relation ∆H A-M tr -∆H M -A tr = W A-M mech -W M -A mech 4.5.3 Analyzing the friction energy variation E Afraction (f m ) . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4 .Figure 4

 44 Figure 4.1 shows the results of the DSC tests performed in all HT wires. The curves are plotted with an offset of 0.0125W/g in the ordinate scale in order to enable comparison, otherwise, the curves would be superimposed.

Figure 4 . 2 :

 42 Figure 4.2: Non-isothermal superelastic σ -ε tensile test responses for different heat treatment temperatures.

Figure 4 .

 4 Figure 4.3 shows the non-isothermal superelastic σ -ε tensile responses (a) with their respective strain (b) and temperature variation profiles (c) for each heat treatment temperature, i.e. HT 523K, HT 548K, HT 573K and HT 598K.

Figure 4

 4 Figure 4.3: a) Non-isothermal σ -ε superelastic tensile test responses, b) strain profiles and c) temperature variation profiles of the tensile test performed on the wires heat treated at 523K, 548K, 573K and 598K. The strain and temperature profiles correspond to the loading points highlighted in figures of column a.

Figure 4

 4 Figure 4.4: a) Non-isothermal σ -ε superelastic tensile test responses, b) strain profiles and c) temperature variation profiles of the tensile test performed on the wires heat treated at 523K, 548K, 573K and 598K. The strain and temperature profiles correspond to the unloading points highlighted in figures of column a.

Chapter 4 .Figure 4 . 5 :

 445 Figure 4.5: Stress vs. true strain on the left ordinate axis represented by the green dots.Power vs. true strain on the right ordinate axis where the red dots represent thermal power ( q) and blue dots the mechanical power (P mech ). Dots represent the loading while the circles represent the unloading.

  Figure4.6: Thermal and mechanical powers normalized by the strain rate ( q/ ε and P mech / ε, respectively) on the left ordinate axis and Ratio between thermal and mechanical powers q/P mech , on the right ordinate axis.

Figure 4 .Figure 4 . 7 :

 447 Figure 4.7 presents stress, thermal energy and mechanical energy in function of strain for all heat treated wires: HT 523K, HT 548K, HT 573K and HT 598K. The stress vs.

Figure 4

 4 Figure 4.8: Thermal (red curve) and mechanical (blue curve) energies in the left ordinate axis with their ratio (green curve) in right ordinate axis in function of stress.

4. 5

 5 Influence of heat treatment temperature on latent heat and martensite fraction 4.5.1 The specific latent heat 4.5.1.1 Method to determine ∆H tr

  4.9a) and unloading(Fig. 4.9b).Chapter 4. Influence of heat treatment on the specific latent heat and martensite fraction of nanostructured NiTi wires

Figure 4 . 9 :

 49 Figure 4.9: Ratio qtr / εtr , computed with experimental data and Eq. 4.4, as a function of strain and stress during a) loading and b) unloading for all heat treated wires.

Figure 4

 4 Figure 4.10: a) Specific latent heat, with their respective standard deviation, determined for each wire during loading (red data) and unloading (blue data). Dashed line representing the total average values of specific latent heat (∆H tr ) for loading and unloading. b) Evolution of the difference between the specific latent heat of loading and unloading (∆H A-M tr

Table 4 . 2 :

 42 σ A-M and σ M -A stand for the stress on forward (A -M ) and reverse (M -A) transformations. Mean values of the mechanical work (W A-M mech -W M -A mech ) and the specific latent heat (∆H A-M tr -∆H M -A tr ) between loading and unloading for all the heat treated wires. ∆H A-M tr -∆H M -A tr W A-M mech -W M -A mech

4. 5 Figure 4

 54 Figure 4.11: Comparison between specific latent heat variation between loading and unloading calculated with thermal data ∆H A-M tr

  Figure4.12 shows the difference of the temperature variation during loading and unloading (θ A-M -θ M -A ) in function of the average transformation strain (ε tr ). The methodology used to determine these curves has been presented in Chapter 3 by the Fig.3.18.

Table 4. 3 :

 3 Averaged temperature variation between loading and unloading θA-Mθ M -A . θ A-M -θ M -A HT 523 K 7.4 ± 0.6 K HT 548 K 10.5 ± 0.8 K HT 573 K 12.2 ± 0.9 K HT 598 K 11.8 ± 1.2 K Using the values of θ A-M -θ M -A from Table 4.3, the specific latent heat variation (∆H A-M tr -∆H M -A tr ) from Table 4.2 and considering an entropy variation ∆S c = -63 ± 0.2 mJ/gK (Khalil-Allafi and Amin-Ahmadi, 2009), the values of the energies involved in a SIPT cycle were calculated for all heat treatment temperatures. The energies due to the temperature and entropy variations (-(θ A-M -θ M -A )∆S c ) and the friction energy variations (E A-M f r -E M -A f r ) are shown in Table 4.4. 4.5 Influence of heat treatment temperature on latent heat and martensite fraction Table 4.4: Values of the energies due to the temperature and entropy variations (-(θ A-M -θ M -A )∆S c ) and friction energies (E A

  5. The values of the specific latent heat corresponding to loading (∆H A-M tr ) and unloading (∆H M -A tr ) used to calculate the fraction rate ( ḟm ) are the ones shown at Table 4.1.

Figure 4 .

 4 Figure 4.13 shows the evolution of the martensite fraction in function of strain (a) and in function of stress (b). The solid lines represent the f m evolution during loading, while the dashed lines represents the f m evolution during unloading.

Figure 4

 4 Figure 4.13: Martensite fraction in function of strain (a) and stress (b) for all heat treated wires. Solid curves are for loading and dashed ones for unloading.

  13 for the HT 548K, f HT 573K m = 0.27 for the HT 573K and f HT 598K m = 0.38 for the HT 598K. Finally, despite the full superelastic loop, the HT 598K wire showed a martensite fraction maximum of f HT 598K m = 0.43.

  38. Finally it was observed that for the HT 598K wire the maximum martensite fraction amount achieved was f HT 598K m = 0.43, when it is usually assumed to be f m = 1.

  . . . . . . . . 103 4.7 Stress (on the left ordinate axis) and energy (on the right ordinate axis) vs. mean true strain. The green dots represent the stress-strain curve, while the red and blue dots represent thermal and mechanical energies, respectively. Dots are for loading and circles represent unloading. . . . . . 104 4.8 Thermal (red curve) and mechanical (blue curve) energies in the left ordi-

nate axis with their ratio (green curve) in right ordinate axis in function of stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.9 Ratio qtr / εtr , computed with experimental data and Eq

  Heat sources estimation . . . . . . . . . . . . . . . . . 1.2.3.3.a General heat balance equation . . . . . . . . 1.2.3.3.b Simplified thermal model . . . . . . . . . . . 1.3 Nanostructured NiTi wires . . . . . . . . . . . . . . . . . . . . 1.3.1 Specificities of the NiTi alloys . . . . . . . . . . . . . . . . . . . Influence of wire manufacturing process on crystallographic structure and mechanical behaviour . . . . . . . . . . . . . . . 1.3.3 Influence of heat treatment on microstructure and mechanical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	General Introduction Chapter 1. Bibliography
	1.2.3.3 1.3.1.1 General Characteristics . . . . . . . . . . . . . . . . .
	1.3.1.2 Crystallographic microstructure of NiTi alloys . . . .
	1.3.1.3 Strain localization phenomenon in NiTi alloys . . . .
	1.3.2
	1.2.2.3 Sources of thermodynamic irreversibility . . . . . . .
	1.2.2.3.a Frictional work E f r . . . . . . . . . . . . . .
	1.2.2.3.b Stored elastic energy E st . . . . . . . . . . .
	1.2.2.4 Thermodynamic potential with irreversibility contri-
	butions . . . . . . . . . . . . . . . . . . . . . . . . . .
	Henrique Martinni RAMOS DE OLIVEIRA	xxiv

CHAPTER ONE BIBLIOGRAPHY Contents 1.1 Shape Memory Alloys (SMA) . . . . . . . . . . . . . . . . . . 1.1.1 Mechanism of phase transformation . . . . . . . . . . . . . . . 1.1.2 Thermally-induced phase transformation . . . . . . . . . . . . . 1.1.3 Stress-induced phase transformation . . . . . . . . . . . . . . . 1.1.4 Shape Memory Effect . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Superelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thermomechanical coupling in superelasticity . . . . . . . . 1.2.1 Engineering applications based on thermomechanical coupling in NiTi superelasticity . . . . . . . . . . . . . . . . . . . . . . . 1.2.1.1 Damping Applications . . . . . . . . . . . . . . . . . . 1.2.1.2 Elastocaloric Cooling Devices . . . . . . . . . . . . . . 1.2.2 Thermodynamic framework of thermoelastic martensitic transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2.1 The Gibbs free energy equation . . . . . . . . . . . . 1.2.2.2 Uniaxially stress-induced transformation . . . . . . . 1.2.3 Experimental studies on thermomechanical couplings . . . . . . 1.2.3.1 Local measurements . . . . . . . . . . . . . . . . . . . 1.2.3.2 Full field measurements . . . . . . . . . . . . . . . . . 1.4 Chapter review . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Shape Memory Alloys (SMA)

Table 1 . 1 :

 11 Units of the properties used in this thermodynamic framework

	Symbol	Property	Unit
	G	Specific Gibbs free energy	J/kg
	H	Specific enthalpy	J/kg
	P	Pressure	Pa
	T	Temperature	K
	S	Specific entropy	J/kg.K
	U	Specific internal energy	J/kg
	V	Specific volume	m 3 /kg
	W	Specific work	J/kg
	σ	Stress	Pa
	ε	Strain	-
	f m	Martensite fraction	-
	ρ	Mass density	kg/m 3
	E f r	Specific frictional work	J/kg
	E st	Specific stored elastic energy	J/kg
	1.2.2.1 The Gibbs free energy equation

:

  Table 1.2: Main characteristics of Cold and Hot working

	Cold Working	Hot working
	It is done at a temperature below the	It is done at a temperature above the
	recrystalization temperature	recrystalization temperature
	It is accomplished by strain hardening	Hardening due to plastic deformation is eliminated
	Crystalization does not take place	Crystalization takes place

Internal and residual stress are produced Internal and residual stress are not produced

No influence of oxydation

Strongly affected by oxydation

  Chapter 3. Specific latent heat and martensite fraction during uniform cyclic superelastic tensile deformation of nanostructured NiTi wires Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal and kinematics full field measurements during tensile tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Heat sources and thermal energy estimation . . . . . . . . . . 3.2.5 Determination of mechanical power and energy . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Chapter 3. Specific latent heat and martensite fraction during uniform cyclic
	superelastic tensile deformation of nanostructured NiTi wires
	NOTATION
	Contents
	L 0 (mm) l 0 (mm) E (GPa) ρ (kg/m 3 ) Mass density Sample gauge length Observed length 3.1 3.2 Material and experimental procedure . . . . . . . . . . . . . ∆ε tr (-) Transformation strain f m (-) Martensite fraction Elastic Modulus ḟ m (s -1 ) 3.2.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Martensite fraction rate 3.2.2 Superelastic tensile tests . . . . . . . . . . . . . . . . . . . . . . τ eq (s) Characteristic time T 0 (K) Initial/room temperature ∂ θ ∂t (K/s) Mean temperature derivative on time T (K) Measured temperature q (W/g) Heat source / total thermal power θ (K) Temperature Variation (T -T 0 ) qthel (W/g) Thermoelastic power θ (K) Mean temperature from temperature profiles qtr (W/g) Transformation thermal power σ (MPa) Nominal stress E th (J/g) Thermal energy σ F (MPa) Stress at middle of plateau for the forward transformation P mech (W/g) Mechanical power σ R (MPa) Stress at middle of plateau for the reverse transformation E mech (J/g) Mechanical Energy σ (MPa/s) Stress rate E st (J/g) Elastic energy released or stored ε (-) True/Logarithmic strain W mech (J/g) Mechanical work ε (-) Mean strain from strain profiles ∆H tr (J/g) Specific latent heat ε (s -1 ) Strain rate ε = d ε /dt ∆H A-M tr (J/g) Specific latent heat from austenite to martensite εel (s -1 ) Elastic strain rate εel = σ/E ∆H M -A tr (J/g) Specific latent heat from martensite to austenite εin (s -1 ) Inelastic strain rate ∆H F -R (J/g) ∆H A-M tr -∆H M -A tr εtr (s -1 ) Transformation strain rate εtr = ḟ m ∆ε tr ∆H m F -R (J/g) 1 ρ σ F ∆ε tr -1 ρ σ R ∆ε tr 3.2.3 3.5.3 Estimation of martensite fraction (f m ) . . . . . . . . . . . . . .
	3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.1 Strain and temperature field evolution . . . . . . . . . . . . . . 3.3.2 Estimation of thermal power . . . . . . . . . . . . . . . . . . . 3.3.3 Thermal and mechanical power and energy in function of time 3.3.4 Thermal and mechanical power and energy as function of strain and stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.1 Cyclic test . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2 Rupture test . . . . . . . . . . . . . . . . . . . . . . . 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Deformation mechanisms in SMA . . . . . . . . . . . . . . . . . 3.4.2 Impact of thermoelasticity on thermal Power and Energy . . . 3.4.3 Thermodynamics of martensitic transformation . . . . . . . . . 3.4.3.1 Thermodynamic framework . . . . . . . . . . . . . . . 3.4.3.2 Transformation heat source . . . . . . . . . . . . . . . 3.4.3.3 Forward and reverse specific latent heat . . . . . . . . 3.5 Characterisation of the martensitic transformation . . . . . 3.5.1 Determination of transformation strain (∆ε tr ) . . . . . . . . . 3.5.2 Experimental determination of forward and reverse specific latent heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.1 Cyclic test . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2.2 Rupture test . . . . . . . . . . . . . . . . . . . . . . .

2.4 Full field measurements

ACKNOWLEDGEMENTS

I would like to address my deep gratitude to my supervisors Prof. Denis Favier and Prof. Herve Louche for always being available to discuss and explain things clearly. With their