Effect of temperature and texture on the reorientation of martensite variants in NiTi shape memory alloys

Accepted Manuscript Effect of temperature and texture on the reorientation of martensite variants in NiTi shape memory alloys

G. Laplanche, T. Birk, S. Schneider, J. Frenzel, G. Eggeler PII:

S1359-6454(17)30033-2

DOI:

10.1016/j.actamat.2017.01.023

Reference:

AM 13485

To appear in:

Acta Materialia

Received Date:

04 October 2016

Revised Date:

09 January 2017

Accepted Date:

10 January 2017

Please cite this article as: G. Laplanche, T. Birk, S. Schneider, J. Frenzel, G. Eggeler, Effect of temperature and texture on the reorientation of martensite variants in NiTi shape memory alloys, Acta Materialia (2017), doi: 10.1016/j.actamat.2017.01.023

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ACCEPTED MANUSCRIPT

Effect of temperature and texture on the reorientation of martensite variants in NiTi shape memory alloys G. Laplanche* a, T. Birk a, S. Schneider a, J. Frenzel a, G. Eggeler a

a

Institut für Werkstoffe, Ruhr-Universität Bochum, 44801 Bochum, Germany *corresponding author: [email protected]

Abstract Martensitic Ni50Ti50 wires and sheets with different textures were tensile tested in the temperature range between -100°C and 60°C. The effect of texture and temperature on reorientation of martensite variants was investigated. After deformation, all material states were heated into the austenite regime to study their shape memory behavior. During room temperature tensile testing, in-situ digital image correlation revealed that the reorientation of martensite variants is associated with the nucleation and propagation of a macroscopic Lüders band. A comparison between the mechanical data obtained for wire and sheet specimens revealed a strong effect of texture. The plateau stresses of sheets were found to be 25 – 33% larger and their recoverable strains were 30% lower than for wires. However, the product of plateau stress and recoverable strain, which represents the external work per unit volume required for martensite variants reorientation does not depend on texture. The tensile tests performed at different temperatures revealed that in the temperature range considered the recoverable strain does not depend significantly on temperature. In contrast, the plateau stress as well as the external work required to reorient martensite decrease with increasing deformation temperature. We use a

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thermodynamic approach involving the elastic strain energy associated with the growth of reoriented martensite variants to rationalize these temperature dependencies.

Keywords: Shape memory; digital image correlation; martensitic phase transformation; texture

1. Introduction and objectives NiTi alloys are the most successful shape memory alloys (SMAs) in the medical [1] and engineering [2] sectors due to their superior mechanical and functional properties compared with other systems such as Cu- [3, 4], Fe- [5, 6], Ru- [7, 8] and Ti-based [9-11] SMAs. Various shape memory properties show an orientation dependence [12-16]. Therefore the texture which develops during thermomechanical processing of SMAs has a strong influence on their mechanical and functional properties [16-22]. Textures of NiTi drawn wires and rolled sheets have received considerable attention in the literature. It is now well-documented that drawn NiTi wires after recrystallization exhibit a strong <111>B2 fiber texture along the wire axis [23, 24]. In contrast, NiTi rolled sheets are characterized by <011>B2 fiber texture parallel to the rolling direction whose amplitude depends on the rolling temperature [25-30]. For shape memory alloys, four specific temperatures T are important: the martensite start and finish temperatures (Ms and Mf) which describe the transformation from austenite to martensite when cooling from the high temperature regime and the austenite start and finish temperatures (As and Af) which characterize the reverse transformation from martensite to austenite when heating from the low temperature regime. The mechanical and functional behaviors of shape memory alloys have been shown to strongly depend on temperature and two temperature regimes can be distinguished.

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In the high temperature regime, not too far away from Af [31], the material is austenitic and exhibits a B2 (CsCl type) crystal structure. During tensile testing, a stress induced phase transformation is observed which accounts for up to 10% strain, i.e. the austenite B2 transforms into martensite (B19’ monoclinic crystal structure) [32]. As the stress induced martensite is metastable in the high temperature regime, it transforms back to austenite upon unloading and the material recovers its original shape. This mechanical memory is usually referred to as superelasticity or pseudoelasticity. In the high temperature regime, the critical stress for the formation of martensite strongly increases with increasing temperature and this phenomenon can be rationalized by a Clausius-Clapeyron type of relationship, e.g. [33]. In the low temperature regime below the martensite finish temperature Mf [32], the material is fully martensitic and exhibits a self-accommodated microstructure. During straining, favorably oriented martensite variants grow at the expense of less favorably oriented ones and this can provide up to 10% pseudoplastic strain [12]. After unloading, the martensitic alloy remains deformed but the material can recover its original shape on heating above Af (where martensite fully transforms to austenite). The fact that all martensite variants have to transform to austenite which cannot form variants, results in thermal memory which is often referred to as the one way effect (1WE) [34]. In the low temperature regime, the apparent yield stress is shown to increase with decreasing temperature [35, 36]. While the temperature dependence of the mechanical properties of pseudoelastic alloys has been extensively studied in the literature [35, 37-40], the reorientation of martensite variants and the associated temperature dependence of functional properties have received less interest [36]. To our knowledge, no attempt has been undertaken so far to model the effect of temperature on the reorientation of martensite variants.

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In this study, after a brief review of the elementary deformation mechanisms which govern the reorientation of martensite variants, an attempt is made to explain the effect of temperature and texture on the reorientation of martensite variants. To reach this goal, two Ni50Ti50 SMAs with different textures were subjected to tensile testing at temperatures between -100 °C and +60 °C. The obtained results are rationalized on the basis of a thermodynamic approach which accounts for the growth of reoriented martensite variants in a self-accommodated microstructure.

2. Experimental 2.1 Melting and Casting High purity nickel pellets (Ni, > 99.98 wt.%) of an average size of 9 mm and titanium slabs with a quadratic shape (Ti, > 99.995 wt.%), 8 mm thick and 50 mm large, were purchased from Ampere GmbH (Frankfurt, Germany) and Hauner Metallische Werkstoffe GmbH (Röttenbach, Germany), respectively. Ti rapidly oxidizes in air and therefore its oxide layer was removed by grinding (80 grid SiC paper) prior to melting. The raw materials were melted under a high purity argon atmosphere of 500 mbar (99.998 vol.%) by vacuum induction melting (VIM, type VSG 010 from PVA TePla AG) and subsequently poured into steel molds preheated to 500°C. Two molds were used in the present study, a cylindrical mold (40 mm diameter, 120 mm height) and a rectangular mold with dimensions of 20 × 76 × 90 mm. More details about the melting parameters used in the present work can be found in the literature [30, 41, 42]. Prior to inserting the molds into the VIM furnace, their walls were coated with an yttria slurry. Two 1 kg ingots of the equiatomic Ni50Ti50 alloy were produced. An effort was made to keep carbon and oxygen impurity levels low, because their presence significantly affects phase transformation temperatures [43] and mechanical properties [44]. The cast ingots were homogenized for 10 h at

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1000°C under an argon atmosphere, followed by water quenching to achieve a homogeneous microstructure.

2.2 Thermomechanical processing of Ni50Ti50 wires The homogenized cylindrical ingot was processed by means of rotary swaging using a four dies swaging machine of type HMP R6-4-120-21S (HMP Umformtechnik GmbH, Pforzheim, Germany). The cross-sectional reduction by swaging, φ, is given by

φ = 2 ln(di / di+1)

(1),

where di and di+1 are the rod diameters before and after swaging step i, respectively. Swaging was performed at 800°C and the diameter of the cylindrical NiTi ingot was reduced from 40 mm to 5.5 mm (total cross-sectional reduction of 2.0) in 14 successive steps with a cross-sectional reduction of about 0.3 in each step [45]. Before each swaging step, the NiTi rod was annealed at 800°C for 10 min. After the final diameter reduction, the cylindrical specimen was annealed at 800°C for 10 min, followed by water quenching. Then the 5.5 mm rod was cold drawn using a HMP machine of type ZPR 2000 6 down to a final diameter of 1.7 mm (total cross-sectional reduction of 1.2). The NiTi rod was cold drawn with a drawing speed of 2.4 mm/s in several successive steps with φ = 0.07. After three successive drawing steps, intermediate annealing at 800°C for 10 min followed by water quenching was performed. Finally the as-drawn wire was straight annealed under a tensile stress of 50-150 MPa at 800°C for 10 min. Recrystallization occurs during this heat treatment which is followed by water quenching to freeze the solutionized high temperature state.

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2.3 Thermomechanical processing of Ni50Ti50 sheets After homogenization, the rectangular ingot (initial thickness of 20 mm) was rolled at 800°C in 22 steps down to a thickness of 1.8 mm which corresponds to a cross-sectional reduction by rolling of φ = |ln (ti / tf)| ≈ 2.4 where ti and tf are the initial and final thicknesses, respectively. Before each hot rolling step, the material was annealed at 800°C for 10 min. The thickness of the hot rolled sheet was then reduced by rolling at room temperature from 1.8 to 1.2 mm (total crosssectional reduction of 0.4) in three steps. More details about each rolling step can be found in the literature [30]. Between each cold rolling step, the material was annealed at 800°C for 10 min followed by water quenching. After the final cold rolling step, the resulting NiTi sheet was recrystallized at 800°C for 10 min and water quenched.

2.4 Thermal and microstructural characterization The phase transformation temperatures Ms, Mf, As and Af of our alloys were measured by differential scanning calorimetry (DSC), for more details about DSC see [46]. The transformation temperatures of Ni50Ti50 wires and sheets are similar and are given in Table 1. In order to characterize the microstructure of the NiTi wires and sheets, longitudinal sections were etched at room temperature using a stirred solution containing 14 g K2S2O5, 1000 ml distilled water, 200 ml HCl (32 wt.%), and 24 g (NH4)HF2. The specimens were subsequently characterized by optical microscopy using polarized light. A representative optical micrograph for a NiTi wire is shown in Fig. 1 where the swaging/drawing direction is highlighted by a white arrow in the upper right corner of the micrograph. Etching of the martensitic alloy allows to reveal the previous grains of the parent austenitic phase. Indeed the inset in the lower left corner

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of the optical micrograph shows a prior austenitic grain which consists of several martensite variants after cooling. Sizes of the prior austenitic grains were measured with the Heyn linear intercept method outlined in ASTM E112-10 [47] using ten parallel and equidistant reference lines of identical length. Two sets of intercepts were used, one parallel and another perpendicular to the deformation direction (wire axis and rolling direction) in order to measure whether grains are equiaxed or elongated along the wire axis/rolling direction. One reference line i of length li intersecting a number ni of grain boundaries yields a grain size of di = li / ni. Typical grain size measurements involved about 500 intercepts per metallographic cross-section. Both recrystallized wires and sheets revealed equiaxed grains with a mean grain size of 15 ± 5 µm.

2.5 Mechanical characterization The NiTi wires and sheets were tensile tested in tension using a Zwick/Roell test rig of type Z100 equipped with a 100 kN load cell and a clip-on extensometer [48]. The NiTi sheets were tested along the rolling direction. Specimens with a gauge length of 25 mm were used. Displacement-controlled tensile tests were conducted in the temperature range -100 < T < +60°C using a cross-head speed of 0.5 mm/min. A climate chamber kept the temperature constant during tensile testing. Prior to mechanical testing, the NiTi specimens were cooled down to -100°C to establish a fully martensitic microstructure prior to mechanical loading. The wires and sheets were deformed up to 9% and 6% strain, respectively, i.e. about 1% strain beyond the end of their respective stress-plateau. After unloading, the shape memory behavior was investigated by heating the material in 10°C per minute up to 150°C (50°C above Af). This temperature was held for 30 min and the material was cooled down (10°C/min) to room temperature.

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Additional tensile tests were carried out at room temperature and strain fields were documented using digital image correlation (DIC). These tensile tests were performed using a miniature 10 kN tensile test rig from Kammrath & Weiss GmbH, Dortmund, at a displacement rate of 0.5 mm/min. A random speckle pattern was applied by spraying black and white paints onto the specimen surface. The system for DIC strain measurements consisted of one pair of 5 megapixel cameras that acquired 5 images per second. Strains were determined from the displacement fields using the software Vic3D (Limess, Pforzheim, Germany).

3. Results 3.1 Digital image correlation (DIC) A typical stress-strain curve for the Ni50Ti50 wire recorded during the DIC experiment at room temperature is shown in Fig. 2a where the circled numbers 1 to 12 indicate at which stage DICmeasurements (shown in Fig. 2b) were performed. The colors in the DIC-images represent the local strain distribution as shown in the color code on the right side of Fig. 2b. The initial linear elastic part of the stress-strain curve is followed by a pseudo-plastic regime where inelastic deformations are homogeneously distributed over the whole wire (compare DIC-images 1 and 2 in Fig. 2b). However, between stage 2 and 3, a load drop occurs which reflects the formation of a band of localized deformation near the bottom grip. This so-called Lüders band propagates towards the upper grip at a constant stress and a constant velocity, see DIC-images 3 to 9 in Fig. 2b. For each deformation state (points 3 to 9), the strain distribution can be divided into three regions. The first region is located at the bottom of the wire (where the Lüders band nucleated) and exhibits a homogeneous tensile strain of 8.5%. The second region is at the top of the wire and shows small deformation levels not exceeding 2%. These two regions are separated

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by a third transition region (or Lüders band) where the local deformation varies progressively from 2 to 8.5%. At each subsequent data point on the plateau stress (3 to 9), additional strain increments occur exclusively by the motion of the transition region which has a finite width of 2 mm, comparable to the diameter of the martensitic wire (1.7 mm). More detailed features regarding distributions of local strains and local strain rates are presented in Fig. 3 for a macroscopic strain of 4% (see 5 in Fig. 2b). These strain and strain rate data are representative for all macroscopic strains along the stress-plateau (points 3 to 9 in Fig. 2b). The distribution of local strains εx as a function of the position x along the wire axis exhibits a step-like shape (see Fig. 3b). In the region near the bottom grip (8.5% local strain, Figs. 3a and 3b) where martensite variants were reoriented, local strains correspond to the end of the macroscopic stress-plateau (Fig. 2a). In contrast, local strains (< 2%) inside the region at the top of the wire, where martensite variants still form a self-accommodated microstructure, correspond to the macroscopic strain at the onset of the stress-plateau. Now regarding local strain rates (see Fig. 3c), they are nearly equal to zero in the regions at the bottom and at the top of the wire while a strain rate peak is observed in the transition region. This distribution of local strain rates shows that all deformation/reorientation takes place in the narrow transition region. The end of the stress-plateau in Fig. 2a occurs when the Lüders band reaches the upper loading grip, DICimage 10. When continuing testing beyond the stress-plateau, stresses sharply increase in an elasto-plastic regime. These results are similar to those observed for austenitic NiTi alloys during the stress induced austenite to martensite phase transformation. Therefore, it is likely that the Lüders-type of deformation mechanisms for martensitic and austenitic alloys have a similar nature. The unloading part of the stress strain curve in Fig. 2a is approximately linear elastic up to point 11 which is followed by a nonlinear unloading part between points 11 and 12 in Fig. 2a.

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This indicates that inelastic recovery processes are occurring. During unloading, the DIC-images 11 and 12 suggest that the NiTi wire exhibits a homogeneous deformation.

3.2 Effect of temperature and texture on mechanical properties Representative stress-strain curves obtained at different temperatures for NiTi sheets tensile strained along their rolling direction (RD) are presented in Fig. 4. In addition to the stress-strain response, recoverable strain which occurs when the 1WE is triggered by heating to 150°C, is shown as a black dashed line. Fig. 5 shows representative stress-strain curves for the wire (blue) and the sheet (green) materials deformed at 20°C up to 9% and 6% strain, respectively (~1% strain beyond the end of their respective stress-plateau). The general shapes of these curves are similar. However, due to the different textures of both materials the magnitude of the stressplateau, its length as well as the recoverable strain are different. The temperature dependence of the stress-plateau and recoverable strain are shown in Figs. 6a and 6b, respectively. In the investigated temperature range, the plateau stresses monotonically decrease with increasing temperature (Fig. 6a) while the recoverable strains remain constant (Fig. 6b). The magnitude of the stress-plateau of the sheets is always larger than that of the wires (between 25 and 33% larger). In contrast, recoverable strains εrec of the NiTi wires and sheets are 5.6 ± 0.5% and 4 ± 0.5%, respectively, i.e. they are always larger for wires. Interestingly, when the product of the stress-plateau and the recoverable strain is plotted as a function of temperature in Fig. 6c, all data points fall onto one common master curve, regardless of the texture of wires and sheets.

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4. Discussion 4.1 Elementary deformation mechanisms associated with the reorientation of martensite variants In the present study, martensitic NiTi wires and sheets with similar grain sizes and different textures were tensile strained at temperatures between -100°C and +60°C. Stress-strain curves of both wires and sheets exhibit a similar shape with a characteristic stress-plateau which is associated to the propagation of a Lüders band, Fig. 2. Both wires and sheets were strained 1% beyond the end of their respective stress-plateau, unloaded and subsequently heated 50°C above Af to trigger the 1WE. The obtained data revealed a strong effect of texture on the magnitude of the stress-plateau, its length, and the recoverable strain obtained after the 1WE. However, the product of the stress-plateau and the recoverable strain, which represents an energy per unit volume required to reorient martensite variants, is found to be independent of texture, Fig. 6. This strongly suggests that this energy is characteristic of a unique elementary deformation mechanism which includes texture effects. In the following, after a brief review of the elementary deformation mechanisms which govern the reorientation of martensite variants, attempts will be made to explain the fact that the external work per unit volume required to reorient martensite variants is independent of texture and to rationalize its temperature dependence. Although the inhomogeneous nature of the stress induced austenite to martensite phase transformation has been well documented by several researchers [49-64], few efforts have been made so far to study the reorientation of martensite variants by DIC [65, 66]. In this section we briefly summarize the governing elementary deformation mechanisms, compiled from various literature sources, responsible for the appearance of tensile stress-strain curves and the propagation of the Lüders band observed by DIC. A typical stress-strain curve showing the 1WE

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is shown in Fig. 7. This stress-strain curve can be subdivided into four regimes. The corresponding elementary deformation mechanisms are illustrated in Figs. 8a-e. Prior to mechanical loading, the NiTi SMA consists of 24 self-accommodated martensite variants. The self-accommodated microstructure has a multiscale character. At the nanoscale, the martensite variants in NiTi SMAs are internally twinned (mostly [011]B19’ type II twinning) [67]. For the sake of simplicity, these internally twinned martensite variants are hereafter referred to as variants. At a larger scale, different neighboring variants in self-accommodated microstructures can share a coherent junction plane [68] also termed as macrotwin [69], midrib [70, 71], or conjugation boundary [72, 73]. These two different types of boundaries at the nano and macroscale may be mobile under stress. Their motion allows martensite variants, which are present in the initial self-accommodated microstructure, to reorient. For more details about macrotwins motion, see [74-76]. We now discuss a scenario which reflects the elementary processes governing re-orientation of martensite variants during mechanical loading. In Fig. 8a, grain boundaries of the parent austenitic phase are black, twin planes are blue and macrotwin boundaries between two neighboring variants are represented by red lines. Regime I in Fig. 7 defines the initial elastic stage of the material. The elastic stage is followed by a first pseudo-plastic regime II, lying between an apparent yield stress (~100 MPa) and an upper yield stress (~175 MPa), Fig. 7. At the apparent yield stress, favorably oriented twin boundaries are mobile which allow detwinning, see blue arrows in Fig. 8b. Moreover, macrotwin boundaries are also mobile and allow favorably oriented twinned domains to grow at the expense of unfavorably oriented ones, see red arrows in Fig. 8b. The motion of these boundaries is the first step of the reorientation of martensite variants. Twin boundaries favorably oriented with respect to the direction of loading (high

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resolved shear stresses) move first and less favorably oriented ones require higher stresses to become mobile. From a macroscopic point of view the deformation of the specimen is homogeneous within regime II as shown by digital image correlation, image 2 in Fig. 2b. All the variants cannot reorient by motion of pre-existing macro and nanotwin boundaries. As a consequence, the level of internal stresses increases and the material hardens until the upper yield point (end of regime II in Fig. 7). Here the critical stress to nucleate new favorably oriented martensite variants has been reached, see elliptical nuclei at the bottom of Fig. 8c. Between regimes II and III, a stress drop occurs followed by a stress-plateau. The stress drop corresponds to the formation of a macroscopic Lüders band close to the bottom grip [77, 78], see orange region in Fig. 8c. In Figs. 8c to e, the region where martensite has been reoriented is highlighted in red while the propagating transition layer is colored in orange. Within this Lüders band, new favorably oriented martensite variants nucleate and grow within unfavorably oriented martensite variants [77, 79]. The interfaces between the old and new martensite variants are coherent ((100)B19’/(010)B2 junction planes [79]). Then the deformation in regime III proceeds in a catalytic manner which was also termed as domino detwinning [80], i.e. the interaction of reoriented martensite variants with grain boundaries in the Lüders band generate stress concentrations due to deformation incompatibilities between grains. These internal stresses can be relieved by the nucleation of new variants in neighboring grains outside the Lüders band. This behavior provides the basis for localized reorientation of martensite variants in a Lüders-like manner. The stress drop in Fig. 7 is due to the fact that the nucleation of the Lüders band requires higher stresses than for its propagation [49]. Note that this localized deformation mechanism is very similar to the stress induced formation of martensite in austenitic NiTi SMAs [50]. The end of the stress-plateau in Fig. 7 is reached when the Lüders band has propagated through the whole

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specimen, see Fig. 8e. Due to incompatibilities between grains (former austenitic grains), the reorientation process is never complete, i.e. grains of the parent phase cannot transform into a single crystalline domain of martensite. Depending on the magnitude of the critical stress, dislocation slip can accompany the reorientation of martensite variants in stage III [33]. When the critical stress to reorient martensite variants is lower than the critical stress to induce dislocation plasticity (~ 150 MPa in solution-treated single crystals [81]), the inelastic strains in stage III can be fully recovered through phase transformation upon heating (1WE). However, when the critical stress which is required to reorient martensite variants is higher than that for slip, irreversible deformations hamper full strain reversibility during the 1WE. In this latter case, the deformation is partially reversible as shown in Fig. 7. Between stages III and IV, a short elastic stage (between 145 and 160 MPa) is likely followed by yield which indicates a transition to irreversible plastic deformation due to dislocation activities, reorientation of martensite variants and/or deformation twinning [82, 83].

4.2 Effect of texture on the reorientation of martensite variants To rationalize the strong effect of texture shown in Fig. 6 we use a Taylor-type model for textured polycrystals. While the orientation dependence of the Taylor factor has been reported in the literature for the stress induced transformation from austenite into martensite [84], to our knowledge nothing has been reported so far regarding the reorientation of martensite. In other words, the involved shear systems may differ between the reorientation of martensite and the austenite to martensite transformation. However, if we assume the Taylor factors to be the same, a textured polycrystal, in which the crystallite axes along the loading direction are preferentially

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aligned along the <111>B2 directions, has a Taylor factor of 2.4, 3.2 for a <101>B2 fiber texture, 4.9 for a <001>B2 fiber texture and 3.2 for a random texture [84]. Based on the expected texture of the wire and sheet, i.e. <111>B2 fiber texture along the wire axis and <011>B2 fiber texture along the rolling direction of the sheet, we expect the Taylor factor of the wire and sheet to be 2.4 and 3.2, respectively. The relationship between the resolved shear stress τ, the stress-plateau σplateau and the Taylor factor M is σplateau = τ × M and the relationship between the shear magnitude associated with the martensitic transformation γ13 = 0.13 (see section 4.3), the recoverable strain εrec and the Taylor factor M is εrec = γ13 / M [85]. According to these equations it is expected that the plateau stresses of wires are 30% lower and their recoverable strain 30% larger than for sheets. These results are in excellent agreement with those shown in Fig. 6. It is also worth mentioning that the product σplateau × εrec = τ × γ13, which is the energy per unit volume required to reorient martensite, is independent of the Taylor factor and therefore independent of texture, as shown in Fig. 6c.

4.3 Temperature dependence of the interaction energy for the reorientation of martensite variants In previous sections, we have shown that the Lüders band propagates at a constant stress σplateau corresponding to a constant energy per unit volume (σplateau × εrec). As the nucleation and growth of new martensite variants plays a key role in the propagation of the Lüders band, it is likely that it is responsible for the temperature dependence of the external work required to reorient martensite variants. To investigate this possibility, we treat an isolated reoriented martensite nucleus in a self-accommodated microstructure as an ellipsoidal inclusion embedded in a homogeneous effective medium invoking Eshelby’s theory [86]. The total Gibbs free energy

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change ΔG due to the formation of an ellipsoidal reoriented martensite variant with radius r and semi-thickness t is given by [87]





4 4 G   r 2 t g ch   r t 2 K  2  r 2  3 3

(2),

where g ch is the chemical free energy change per unit volume accompanying the reorientation of the martensite variant. The second term in Eq. (2) is the elastic strain energy and the last term represents the interfacial energy, which shape memory researchers usually neglect [88]. In Eq. (2), K is a strain energy parameter and γ is the specific interfacial energy of a fully coherent inclusion. In the case of the reorientation of martensite variants during uniaxial loading, the chemical free energy corresponds to the interaction energy for twinning into a preferential variant (external work) [89, 90]. It is expressed as

g ch  σ plateau ε rec

(3).

Right after nucleation, the nucleus grows radially faster than along the thickness direction. This continues until the nucleus growth is radially stopped at a grain boundary of the parent phase [91]. Subsequently, the reoriented martensite variant can only grow in the thickness direction in response to the driving force  G / t where t is the semi-thickness of the reoriented martensite variant. This thickening does not cease until the driving force decreases to zero ( G / t  0 ) [92], which yields

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σ plateau εrec 2

K

t r

(4).

Therefore, when a unit volume of self-accommodated martensite has been swept by the Lüders band, the thickening of a reoriented martensite variant stops when its elastic strain energy has reached half the magnitude of the external work [92]. Assuming isotropic elasticity, the parameter K in Eq. (4) can be defined as [87]

K

  2  ν  2 μ 2 γ13  ε33 8 1  ν  4 1  ν 

(5),

where μ is the elastic shear modulus, ν the Poisson’s ratio, γ13 denotes the shear and ε33 the volume change associated with the reorientation of martensite variants. As the reorientation of martensite variants does not involve any volume change, we have ε33 = 0. Now we develop an expression for γ13. It has been reported in the literature that the reorientation of martensite in NiTi shape memory alloys involves twinning on the (100)B19’ plane in the [001]B19’ direction [79]. An elementary reorientation process is schematically illustrated in Fig. 9 where a white single variant is being reoriented into a new grey variant under the action of a shear stress by twinning on the (100)B19’ plane in the [001]B19’ direction. From Fig. 9, it can be seen that the shear angle θ associated with the reorientation of martensite is related to the monoclinic angle β of the NiTi unit cell by

  θ 2β  2 

(6).

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The deformation gradient F associated with the simple shear shown in Fig. 9 is given by

1 0 θ    F  0 1 0 0 0 1  

(7),

and the strain tensor can be calculated using

 0 0 θ / 2 T   1    F  F  I   0 0 0  2  θ / 2 0 0   

(8).

T

In Eq. (8), F is the transpose of the matrix F and I is the identity matrix. Therefore from Eq. (8), it can be seen that

γ13 = θ / 2 = β – π / 2

(9).

When Eqs (5) and (9) are inserted in Eq. (4), we find

σ plateau εrec

  2  ν   t    μ  β     2  4 1  ν   r 

2

(10).

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Assuming that the ratio t / r of fully-grown reoriented domains is independent of temperature and has a value in the range [0.01 - 0.04] [91], Eq. (10) shows that the temperature dependence of the interaction energy per unit volume can only result from the evolution of the elastic moduli (µ and ν) and the evolution of the monoclinic angle β with temperature. Using ultrasonic measurements, Brill et al. [93] have measured the temperature dependence of the Poisson’s ratio and elastic shear modulus µ in Ni50.5Ti49.5 single crystal which transformed into a self-accommodated microstructure upon cooling. The Poisson’s ratio was shown to exhibit a step-like behavior from ν = 0.43 (austenite) in the high temperature regime to ν = 0.35 (self-accommodated martensite) in the low temperature regime [93]. The evolution with temperature of the elastic shear modulus µ obtained by Brill et al. [93] and the evolution of the monoclinic angle β determined using Xray diffraction by Prokoshkin et al. [94] are shown in Figs. 10a and 10b, respectively. The red trend lines represent best fits to the experimental data for interpolation purposes. Note that the exact chemical composition of the NiTi alloys which were investigated by Brill et al. [93] and Prokoshkin et al. [94] differs slightly from the equiatomic concentration (Ni50.5Ti49.5 and Ni50.26Ti49.74, respectively). To avoid any discrepancy related to alloy composition, we subtract Ms (martensite start temperature) from the temperature. Fig. 10 shows the evolution of µ and β with T - Ms, Fig. 10. As the shear angle β increases with decreasing temperature (see Fig. 10b), the shear magnitude γ13 associated with the reorientation of martensite variants increases by 5% when the temperature decreases from 0 to -100°C. The recoverable strains should also increase by the same amount in this temperature range. The recoverable strain of the wire and sheet are expected to increase from 5.6% to 5.88% and 4% to 4.2%, respectively, when the temperature decreases from 0 to -100°C. However, since the experimental error on recoverable strains is as large as 0.5%, i.e. the

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recoverable strains of the wire and sheet are 5.6 ± 0.5% and 4.0 ± 0.5%, respectively, it is therefore not possible to experimentally detect the slight increase of the recoverable strain with decreasing temperature as shown in Fig. 6b. To summarize, while an increase of recoverable strain with decreasing temperature may occur, the experimental error does not allow us to detect it. Inserting the experimental temperature dependence of µ and β as shown in Figs. 10a and 10b into the energy term presented in Eq. (10) we obtain the calculated black line shown in Fig. 6c with Ms = 60°C and t / r = 0.0112. It is a striking new finding that this calculated line which is only based on elastic moduli (µ) and martensite crystallography (β) rationalizes the experimental data. The excellent match between experimental and theoretical data in Fig. 6c clearly shows that the temperature dependence of the plateau stress (Fig. 6a) is related to a softening of the elastic shear modulus and a decrease of the monoclinic angle with increasing temperature.

5. Summary and conclusions In the present study, fully martensitic Ni50Ti50 wires and sheets with similar transformation temperatures, similar grain sizes but different textures (<111>B2 fiber texture along the wire axis, <011>B2 fiber texture along the rolling direction of the sheet) were produced. Their stress-strain curves exhibit the well-known plateau behavior, which is governed by the propagation of a Lüders band. The experiments were interrupted after a small additional strain interval (~1%) beyond the end of the stress-plateau. After unloading, the specimens were heated to trigger the one way shape memory effect and the associated recoverable strains were measured. From the results obtained in the present work the following conclusions can be drawn:

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1) A stress drop at the beginning of the stress-plateau indicates the nucleation of a Lüders band. Digital image correlation shows that the Lüders band nucleates at one grip of the test rig, from where it propagates through the specimen until it reaches the other grip. 2) Despite their similar transformation temperatures and grain sizes, NiTi wires and sheets exhibit strong differences in their tensile and shape memory behaviors due to their different textures. The stress-plateau of the sheets is 25 – 33% larger and their recoverable strains are 30% lower than those of the wires. However, the external work per unit volume required for the reorientation of martensite variants, defined in the present study as the product of the stressplateau and the recoverable strain, is independent of texture. All these results can be rationalized by a Taylor-type model for polycrystals. 3) The recoverable strain was found to be independent of temperature while the stress-plateau increases monotonically with decreasing temperature. As a results, the external work per unit volume required to reorient the martensite variants increases with decreasing temperature. This is due to the fact that the elastic strain energy per unit volume which hampers the reorientation of martensite variants is increasing with decreasing temperature. 4) The decrease of the elastic strain energy per unit volume with increasing temperature results from two factors. First, the elastic shear modulus softens when the temperature increases towards the martensite start temperature (this represents the major contribution 60 - 70%). Second, the magnitude of the monoclinic angle and therefore the magnitude of the shear associated with the reorientation of martensite variants are decreasing with increasing temperature (minor contribution 30 - 40%).

Acknowledgements

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G.L. acknowledges funding from the Alexander von Humboldt (AvH) Foundation. G.E., T.B. and J.F. acknowledge funding provided by the German Research Association (DFG: Deutsche Forschungsgemeinschaft) through project FR 2675/2-1.

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ACCEPTED MANUSCRIPT Figure captions

Fig. 1: Optical micrograph of an etched Ni50Ti50 wire where the grains of the parent phase were split in several variants forming a self-accommodated microstructure.

Fig. 2: Stress strain characteristics of martensitic Ni50Ti50 wire deformed at room temperature. (a) Stress-strain curve which indicates 12 positions where strain distributions were measured. (b) Color coded strain fields (DIC measurements) at positions highlighted in Fig. 2a.

Fig. 3: Strain and strain rate distributions for position 5 in Fig. 2a. (a) Color coded strain distribution. (b) Local strains ε x plotted as a function of wire position x. (c) Local strain rates εx as a function of the position x along the wire axis (0 marks a position at the middle of the

wire). For details see text.

Fig. 4: Loading / unloading tensile stress-strain curves together with strain recovery associated with the one way effect (black dashed line) obtained at different temperatures for Ni50Ti50 sheets strained along their rolling direction.

Fig. 5: Comparison of tensile stress-strain curves and strain recovery (dashed lines) obtained at 20 °C for NiTi wires and sheets.

Fig. 6: Temperature dependence of shape memory properties for NiTi sheets and wires. (a) plateau stress, (b) recovery strains, and (c) products of plateau stresses and corresponding recovery strains. The black line in (c) is calculated using the thermodynamic approach developed in section 4.3, for details see text.

ACCEPTED MANUSCRIPT Fig. 7: Typical tensile stress-strain curve, for a fully martensitic Ni50Ti50 wire, for details see text.

Fig. 8: Schematic representation of the elementary deformation mechanisms occurring during martensite variant reorientation. (a) Stress-free self-accommodated microstructure where grain boundaries are black, twin boundaries are blue and coherent junction planes between two neighbouring twinned martensite plates are represented by red lines. (b) Martensite variant reorientation by movement of junction planes. (c) Lüders band (orange region) formation where favourably oriented martensite variants nucleate (red ellipses between the old and new martensite variants). (d) Lüders band propagation (swept area colored in red). (e) The Lüders band has propagated through the whole sample. Due to incompatibilities between grains, the martensite variant reorientation process is not complete.

Fig. 9: Schematic drawings illustrating the reorientation of the monoclinic (B19’) unit cell (with lattice parameters a, b, c and β) by twinning on the (100)B19’ plane in the [001]B19’ direction under a shear stress. The angle θ shown in the right part of the figure represents the shear angle associated with the reorientation of martensite.

Fig. 10: Experimental literature data for martensitic NiTi alloys as a function of temperature plotted as T - Ms. (a) shear modulus µ [93] and (b) monoclinic angle β [94].

ACCEPTED MANUSCRIPT Table Captions

Table 1: Transformation temperatures (in °C) of Ni50Ti50 wires and sheets determined by DSC (precision: ± 5°C). wires

sheets

Af

100

105

As

70

80

Ms

60

70

Mf

30

50