Orientation Dependence of B19’-Martensite Reorientation Stress and Yield Stress in TiNi Single Crystals

Abstract

This paper deals with the effect of crystal orientation on the B19’-martensite reorientation stress and yield stress in compression in TiNi single crystals with different Ni contents varying from 50.4 to 51.2 at.%. It was experimentally shown that the martensite yield stress appears to be higher for the [111]_B2-oriented single crystals than for the [001]_B2-oriented single crystals regardless of Ni content. The difference between martensite yield stress for the two investigated orientations increases with the growth of Ni content. The maximum difference between martensite yield stress σ_cr^M for two investigated orientations is 996 MPa at Ni content of 51.2 at.% (σ_cr^M = 1023 MPa for the [001]_B2-orientation and σ_cr^M = 2019 MPa for the [111]_B2-orientation). As a result of comparison with the B2-austenite yield stress, it was found that this is not an ordinary case. The [001]_B2 orientation is a high-strength in B2-austenite and a low-strength in B19’-martensite. It was experimentally shown that the B19’-martensite reorientation stresses weakly depend on the orientation and chemical composition compared with the martensite yield stress. The reasons for the orientation dependence of the martensite yield stress in compression and the deformation mechanisms of B19’-martensite are discussed.

1. Introduction

The TiNi alloy is the headliner among shape memory alloys (SMAs) in terms of knowledge. This determines the use of this alloy in medical practice, space, mining industry, robotics, the production of thermal mechanical devices, and the development of special technologies. Binary quenched TiNi alloys with a nickel content of C_Ni < 51.2 at.% undergoes thermoelastic martensitic transformation (MT) from the cubic B2 austenite lattice to the monoclinic B19’ martensite lattice and back in stress-free cooling/heating [1,2]. In quenched high-nickel alloys with C_Ni ≥ 51.2 at.% there is a strain-glass transition during stress-free cooling/heating, while B2-B19’ MT occurs only under applied stress [3,4].

Most of the research works on the TiNi alloy are related to the study of the critical stress levels for reorientation and detwinning of multivariant the B19’-martensite structure σ_reor (T < M_s), the critical stress levels for the B19’-martensite formation σ_cr in the stress-induced MT temperature range (M_d > T > A_f) and the levels of B2-austenite yield stress σ_cr^A (T > M_d) (Figure 1). This is understandable because these stress levels σ_reor, σ_cr, and σ_cr^A are responsible for the temperature interval and characteristics of the stress-induced B2-B19’ MT and related phenomena, such as shape memory effect (SME) and superelasticity (SE) [1,2,5,6,7,8,9].

It is well known that at M_d temperature, the critical stress for the stress-induced martensite formation and B2-austenite yield stress are equal to each other [2,5,8,9,10]. Therefore, MT cannot be stress-induced above M_d temperature, and this temperature determines the criterion and temperature range for SE (Figure 1). The M_d temperature strongly depends on the microstructure, chemical composition, and orientation of single crystals or texture of polycrystals. It varies from 323 to 423 K [11,12]. The higher level of B2-austenite yield stresses and the greater the ratio of σ_cr(M_d)/σ_cr(M_s) > 5÷7 (the Wayman–Otsuka condition [8]), the more probable the manifestation of SE in the SMAs. One of the ways to obtain a wide temperature range of SE with a high level of σ_cr^A is to minimize the critical stress for the stress-induced martensite formation σ_cr and, accordingly, to minimize the coefficient α = dσ/dT (Figure 1) [9]. The orientation dependence of the B2-phase yield stress is well studied in both single- and hetero-phase TiNi single crystals [11,12,13,14]. The Schmid factors analysis for the a <001> _B2{110}_B2 slip system in the B2-phase, experimental data of the temperature dependence of the yield stress and deformation mechanisms of the B2-phase for the single crystals of TiNi alloys show that the <001>_B2 orientation is a high-strength and is characterized by high yield stress ~G/100 (G is the shear modulus of austenite). The main mechanism of the B2-phase plastic deformation for this orientation is twinning [12,13,14,15,16,17,18]. For example, the B2-phase yield stress of σ_cr^A = 1300–1800 MPa for the [001]_B2 orientation is 1.6 times higher and the SE interval of ~200–250 K is wider than for the [111]_B2 orientation (σ_cr^A = 800–1100 MPa and SE interval of <150 K) in quenched Ti-(50.7–51.8 at.%)Ni single crystals in compression [9,11,13].

The SMAs must have the high-strength properties of both the austenite and martensite to show a wide temperature range of SE, a high reversible strain of the MT, as well as good cyclic stability of the SE. The martensite yield stress σ_cr^M characterizes the ability of the alloy to resist plastic deformation of martensite. The mechanism of B19’-martensite plastic deformation has been studied in the TiNi alloy. Numerous experimental studies mainly on polycrystalline equiatomic TiNi and Ti-(50.5–50.9 at.%)Ni bars and wires report the activity of the and <001>{100}_B19’ dislocation slip in B19’-martensite in combination with the twinning [2,10,19,20,21,22,23,24,25]. It should be noted that the thermo-induced martensite, existing without stress below M_f temperature, consists of 24 different crystallographically equivalent internally twinned variants. The two twin-related variants constitute the correspondent variants pair (CVP). The CVPs form a self-accommodating structure of martensite and define an invariant habit plane {0.89 0.40 0.22} [7,26,27,28,29,30,31]. Type II <011>_B19’ and the type I {$\overline{1}\overline{1}$1}_B19’ are the preferred twinning modes required to obtain the lattice invariant strain and the habit plane at the B2-B19’ MT in the quenched TiNi alloys [7]. Type II <011>_B19’ twinning mode is described in terms of the twinning elements as follows K₁ = (0.7205 1$\overline{1}$), η₁ = [011], K₂ = (011), η₂ = [1.5727 1$\overline{1}$]. Type I {$\overline{1}\overline{1}$1}_B19’ is described as follows K₁ = ($\overline{1}$11), η₁ = [0.5404 0.49597 1], K₂ = (0.2469 0.561 1), η₂ = [$\overline{2}\overline{1}$1]. Another twinning mode is the compound {001}_B19’ twins (K₁ = (001), η₁ = [100], K₂= (100), η₂ = [001]), which are not a solution for the phenomenological crystallographic theory of the MT and can exist both in the single- and hetero-phase TiNi alloys in the stress concentration places [7,27]. Such twins grow during thermo-induced MT, do not lead to macrodeformation of the sample, and are not inherited to austenite when heated above A_f during the reverse MT. If the B19’-martensite yield stress is reached, then the plastic deformation, in addition to the dislocation slip, can proceed via the twinning of types {100}_B19’, {20$\overline{1}$}_B19’, or {113}_B19’ in the martensite B19’-phase. After unloading and heating above A_f, these twins can be inherited to the austenite phase. The {114}_B2 twins of B2-austenite originates from the {20$\overline{1}$}_B19’ twins of the B19’-martensite, the {211}_B2 twins of B2-austenite originates from the {113}_B19’ twins of the B19’-martensite [2,7,9,10,23].

The B19’-martensite yield stress σ_cr^M, as well as the stresses of σ_cr, σ_reor, and σ_cr^A, depends on the deformation way, chemical composition, and single crystal orientation or polycrystal texture. On Ti-(50.5 at.%)Ni and Ti-(50.8–50.9 at.%)Ni wires in tension, it is shown [19,32,33] that the martensite yield stress σ_cr^M at T < M_s depends on the Ni concentration (σ_cr^M = 1150 MPa for Ti₄₉.₅Ni₅₀.₅ wire, σ_cr^M = 1420 MPa for Ti₄₉.₂Ni₅₀.₈ wire, σ_cr^M = 1000 MPa for Ti₄₉.₁Ni₅₀.₉ wire) and exceeds the B2-phase yield stress at T~M_d (σ_cr^A = 800 MPa for Ti₄₉.₅Ni₅₀.₅ wire, σ_cr^A = 1230 MPa for Ti₄₉.₂Ni₅₀.₈ wire, σ_cr^A = 830 MPa for Ti₄₉.₁Ni₅₀.₉ wire). With an increase in the test temperature to T = M_d, the B19’-martensite yield stress σ_cr^M decreases and becomes close to the B2-austenite yield stress at T~M_d [19,32,33]. Šittner et al. have found that the oriented martensite plastic deformation can be either homogeneous or localized, depending on the yield stress and strain hardening rate [10,19,33]. The martensite plastic deformation proceeds via a peculiar deformation mechanism which combines {100}_B19’ deformation twinning with <100>_B19 dislocation slip based on kinking in nanocrystalline Ti₄₉.₁Ni₅₀.₉ and Ti₄₉.₅Ni₅₀.₅ wires. The reverse MT after the martensite plastic deformation during unloading and heating leaves behind a large irreversible strain and a high density of lattice defects in austenite. The martensite transforms into the B2-austenite and the {20$\overline{1}$}_B19’ deformation twins transform to {114}_B2 austenite twins [10,19,33].

It was shown that the B19’-martensite yield stress in the Ti₄₉.₂Ni₅₀.₈ wire, Ti₄₈.₅Ni₅₁.₅ and Ti₄₉.₂Ni₅₀.₈ single crystals with <112>_B2, <148>_B2, <111>_B2, and <100>_B2 orientations increases due to the precipitation hardening after aging at T = 573–723 K compared with aging at T = 773–823 K [12]. For example, for single crystals oriented along the <112>_B2, <148>_B2, <111>_B2, and <100>_B2-directions after aging at 723 K the martensite yield stress in compression at T = T_room > A_f is σ_cr^M = 2000–2100 MPa, and after aging at 823 K σ_cr^M decreases and becomes equal to 1500–1700 MPa. At the same time, the yield stress σ_cr^M does not depend on the orientation in the aged at 723–823 K Ti₄₈.₅Ni₅₁.₅ and Ti₄₉.₂Ni₅₀.₈ single crystals with dispersed Ti₃Ni₄ particles.

Thus, currently, there are disorganized data on the study of the B19’-martensite yield stress on TiNi polycrystals and single crystals for the separate orientations/texture, composition, test temperatures and thermomechanical treatments. The effect of test temperature and deformation way on the B19’-martensite yield stress is considered only for polycrystalline alloys in tension, such as equiatomic TiNi, Ti₄₉.₅Ni₅₀.₅ and Ti-(50.8–50.9 at.%)Ni bars and wires [19,21,22,32,33]. The effect of orientation on the B19’-martensite yield stress in compression has been mainly considered only on heterophase Ti₄₉.₅Ni₅₁.₅ and Ti₄₉.₂Ni₅₀.₈ single-crystals and only at room temperature in SE conditions at T > A_f [12,13]. The scientific and practical interest is the systematic study of the orientation dependence of the B19’-martensite yield stress in the single-phase state without the influence of the secondary phases in TiNi single crystals with various nickel contents and the refinement of the plastic deformation mechanisms of B19’-martensite. The purpose of this work is to investigate the orientation dependence of the critical stress for the stress-induced B19’-martensite reorientation and the B19’-martensite yield stress in the quenched TiNi single crystals with various Ni content.

2. Materials and Methods

The single crystals of Ti₄₉.₆Ni₅₀.₄ (at.%) (marked as 50.4Ni), Ti₄₉.₃Ni₅₀.₇ (at.%) (marked as 50.7Ni), Ti₄₈.₈Ni₅₁.₂ (at.%) (marked as 51.2Ni) alloys were grown by the Bridgman technique. The compressive test specimens had [001]_B2 ([001]_B2-oriented crystals) and [111]_B2 ([111]_B2-oriented crystals) deformation axes. Samples had a cuboid shape with a cross-sectional area of 9–25 mm² and a length of 6–10 mm. The orientation of the samples was determined by X-ray diffraction analysis on a Dron-3 diffractometer (Burevesnik, St. Petersburg, Russia). The cut samples were annealed at 1253 K for 1.0 h in a helium atmosphere and quenched in room-temperature water.

The microstructure of single crystals was investigated on a transmission electron microscope (TEM) HT-7700 (Hitachi, Tokyo, Japan). The foils for TEM were disks with a diameter of 3 mm and a thickness of 100 μm. They were finally thinned out using double-jet electrochemical polishing by a Tenupol-5 (Struers, Ballerup, Denmark).

The MT temperatures were determined by differential scanning calorimetry (DSC) using a DSC 404F1 Pegasus (NETZSCH, Bayern, Germany). The cooling and heating rates of DSC analysis were 10 K/min. The loading/unloading cycles were performed using a VHS 5969 universal testing machine (Instron, High Wycombe, UK). Errors in strain and stress measurements during mechanical tests were 0.2% and 2 MPa, respectively. The strain rate during the study of SE response was 2.0·10⁻³ s⁻¹. The strain-glass transformation was investigated by a DMA/SDTA861 dynamic mechanical analyzer (DMA) (Mettler Toledo, Columbus, OH, USA).

3. Results

DSC curves are shown in Figure 2, and B2-B19’ MT temperatures are summarized in

Table 1. B2-B19’ MT temperature during cooling/heating for quenched TiNi single crystals.

Ni, at.%	Ms(±2), K	Mf(±2), K	As(±2), K	Af(±2), K	Δ1(±2), K	Δ2(±2), K
50.4	286	272	299	316	14	17
50.7	240	190	218	266	50	48
51.2	Undefined

for the quenched single crystals with different Ni content. An increase in Ni concentration from 50.4 to 51.2 at.% leads to a decrease in the MT temperatures and an extension of the forward and reverse MT intervals Δ₁ = M_s − M_f and Δ₂ = A_f − A_s. The MT intervals are Δ₁ = 14 K and Δ₂ = 17 K at a Ni concentration of 50.4 at.%. With an increase in Ni concentration by 0.3 at.% the MT intervals expand by 36 K for Δ₁ and by 31 K for Δ₂, and shift towards low temperature. With a further increase in Ni concentration, the endothermic and exothermic peaks are not observed in the DSC curves, which indicates the absence of the B2-B19’ MT during cooling/heating in 51.2Ni single crystals. Instead, a strain-glass transition occurs, as the DMA study shows (Figure 3).

The frequency-temperature dependencies of the elastic modulus E and defining the internal friction tan δ at cooling for [001]_B2-oriented quenched 51.2Ni single crystals are shown in Figure 3. The dip in the temperature dependence of the elastic modulus E(T) in the region of the internal friction peak (tan δ) indicates a change in the microstructure of the material or a phase transition (Figure 3). As the frequency increases, the internal friction peak reduces, and the temperature of the internal friction peak T_g shifts towards high temperatures. The temperature T_g depends on the frequency according to the Vogel–Fulcher relation [3]:

$$\mathsf{\omega }={\mathsf{\omega }}_0\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{{\mathrm{E}}_{\mathrm{a}}}{{\mathrm{k}}_{\mathrm{B}}}\left({\mathrm{T}}_{\mathrm{g}}-{\mathrm{T}}_0\right)\right]$$

(1)

where T₀ = 180 K is the temperature T_g at a frequency of ω→0 Hz, which is consistent with [4], E_a is the activation energy, k_B is the Boltzman constant. This frequency dependence of internal friction is the main characteristic of the strain-glass transition, which is observed in 51.2Ni single crystals (Figure 3). The DMA curves show no shift of the internal friction peak on frequency during the MT [3].

The stress-strain σ(ε) curves under uniaxial compression for the quenched TiNi single crystals are demonstrated in Figure 4 and Figure 5. The σ(ε) curves for the 50.4Ni and 50.7Ni single crystals were obtained at temperatures T = 258 K and T = 203 K, respectively, i.e., the samples without stress were below temperature M_s in the martensite phase. The lowest possible temperature T = 203 K was chosen for the stress-induced MT in the high-nickel single crystals of 51.2Ni alloys not undergoing the MT upon stress-free cooling. Two yield stress σ_cr(σ_reor) < σ_cr^M and four deformation stages can be distinguished on the σ(ε) curves. Stage I is an elastic deformation of the initial structure under stress. The initial structure of the 50.4Ni and 50.7Ni single crystals is the self-accommodating structure of the thermal-induced B19’-martensite. The initial structure of 51.2Ni single crystals is the austenite B2-phase. The σ_reor stress corresponds to the critical stresses for the B19’-martensite reorientation in 50.4Ni and 50.7Ni single crystals and is weakly dependent on the orientation and Ni content (Figure 4). The critical stress is σ_reor = 169–192 MPa for 50.4Ni single crystals and σ_reor = 138–153 MPa for 50.7Ni single crystals. In the high-nickel 51.2Ni single crystals undergoing a strain-glass transition upon cooling, the B2-B19’ MT can be induced under the action of uniaxial stresses [3,4]. In the 51.2Ni crystals, the value of σ_cr characterizes the beginning of the stress-induced B19’-martensite formation and significantly exceeds stress for the martensite variants reorientation in the single crystals with C_Ni = 50.4–50.7 at.%. In addition, a strong orientation dependence of σ_cr is observed in the 51.2Ni single crystals: σ_cr = 876 MPa for the [111]_B2 orientation is 2.1 times higher than σ_cr = 412 MPa for the [001]_B2 orientation (Figure 5).

Stage II length depends on the composition and orientation of single crystals. In the case of the [001]_B2 orientation, stage II is observed up to a given strain of 6.0–7.0% in 50.4Ni and 50.7Ni single crystals and up to 9.5% in 51.2Ni single crystals. Stage II ends at a given strain of 6.0% in 50.4Ni single crystals, 3.5% in 50.7Ni single crystals, and 7.5% in 51.2Ni single crystals with the [111]_B2 orientation. The plastic deformation occurs in stage II with a low strain hardening coefficient θ = dσ/dε because of the rearrangement of B19’-martensite twin structure by twin boundaries movement. As a result, the preferred oriented martensite variants by the action of the external load grow, and their detwinning occurs in the 50.4Ni and 50.7Ni single crystals. In the high-nickel 51.2Ni single crystals only the oriented B19’-martensite is formed at stage II due to the deformation process (martensite reorientation and/or detwinning) responsible for the stress-induced MT. The single crystal orientation defines the stage length and strain hardening coefficient θ. The [001]_B2-oriented crystals are characterized by low coefficient θ and high length of stage II compared with the [111]_B2-oriented single crystals independently of Ni content (Figure 4 and Figure 5).

In the case of the [001]_B2 orientation, stage III is up to a given strain of 10.5% in 50.4Ni and 50.7Ni single crystals and 13.0% in 51.2Ni single crystals. Stage III for the [111]_B2 orientation is observed up to a given strain of 12.0% in 50.4Ni single crystals, 9.0% in 50.7Ni single crystals, and 11.5% in 51.2Ni single crystals. Stage III is accompanied by a higher coefficient θ and is usually associated with the elastic deformation of the formed oriented B19’-martensite. In addition, the further oriented B19’-martensite reorientation and the detwinning, MT undergoing, as well as the formation of new twin modes, are possible at this stage. Such a combination of various deformation processes leads to the different values of θ = dσ/dε at stage III depending on the crystal orientation and the Ni content and does not explicitly characterize the elastic modulus of oriented B19’-martensite (Figure 4 and Figure 5).

The samples were unloaded at the given strain of ε_g < 10% when the stress was lower than σ_cr^M. The loading/unloading curves are shown in the insets of Figure 4 and Figure 5. The SME occurs in the low-nickel 50.4Ni and 50.7Ni single crystals. The complete shape and size recovery of these samples is observed after unloading and heating to T > A_f due to the reverse MT. In the high-nickel 51.2Ni single crystals, the given strain is completely reversible without heating due to SE effect. The [001]_B2-oriented crystals with long stages II are characterized by higher reversible strain ε_rev compared with the [111]_B2-oriented crystals regardless of the Ni content.

The B19’-martensite plastic deformation occurs above σ_cr^M at stage IV. The σ_cr^M stress is the yield stress of the oriented B19’-martensite. It has been experimentally shown that there is a strong dependence of the martensite yield stress σ_cr^M on the Ni content and the single crystal orientation in the studied TiNi single crystals (Figure 4 and Figure 5). The martensite yield stress σ_cr^M for the [111]_B2-oriented crystals is higher than for the [001]_B2-oriented crystals. The difference Δσ_cr^M = σ_cr^M[111] − σ_cr^M{001] at one Ni concentration shows how much the martensite yield stress for the [111]_B2-oriented crystals is higher than in the [001]_B2-oriented crystals. The value of Δσ_cr^M increases from 272 MPa to 996 MPa with rising the Ni concentration from 50.4 to 51.2 at.%. The martensite yield stress σ_cr^M = 2019 MPa for 51.2Ni single crystals with the [111]_B2 orientation is two times higher than the one for single crystals with the [001]_B2 orientation, where it is σ_cr^M = 1023 MPa.

The microstructure after plastic deformation of B19’-martensite up to the given strain ε_g = 12.0–15.5% was studied in detail in the [001]_B2- and [111]_B2-oriented single crystals of 50.4Ni and 50.7Ni alloys by the TEM. It should be noted that the study of the microstructure after deformation in the martensite was carried out after the unloading, heating to room temperature, and subsequent preparation of thin foils from deformed samples. In the process of heating and preparation of thin foils, the reverse MT was in the deformed structure of the samples. Then, the thermo-induced martensite may appear with a decrease in temperature of the foil. Therefore, the microstructure observed by TEM differs from the one after plastic deformation of oriented B19’-martensite up to ε_g = 12.0–15.5%. It is very difficult to identify the stages at which one or another defective structure of austenite or martensite was formed, and what processes preceded this.

It has been experimentally shown that the B2-phase areas with a high dislocation density are observed after deformation above the B19’-martensite yield stress in compression and subsequent heating, regardless of the crystal orientations (Figure 6). As the samples were deformed plastically in B19’-martensite, the dislocation slip occurred in B19’-martensite, and this dislocation structure was inherited by the B2-phase from B19’-martensite during reverse MT. The elongated dislocations were observed along the (110)_B2, (101)_B2, and (01$\overline{1}$)_B2 (Figure 6b). No large difference could be distinguished in the dislocation density between [001]_B2 and [111]_B2 crystal orientations (Figure 6).

The multiple deformation twinning of B2-phase and lamellas of the residual martensitic phase are detected by TEM in [001]_B2-oriented 50.4Ni single crystals (Figure 7). The azimuthal smearing of reflections in the B2-matrix and B19’-martensite in [001]_B2-oriented single crystals indicates the presence of a misoriented substructure caused by the intense action of dislocation deformation mechanisms. Misorientation of the structure is formed as a result of the forward and reverse stress-induced B2-B19’ MT with the retention of the boundaries of the martensite twin structure. The twin boundaries are retained in the B2-austenite structure after the reverse MT because they already interacted with dislocation slip in the matrix. The material of the former martensite twins has become slightly misoriented [34].

A more detailed analysis of the microstructure of the [001]_B2-oriented single crystals showed that the {411}_B2 twins of the B2-phase are observed after plastic deformation of B19’-martensite and subsequent heating to room temperature (Figure 8). Figure 8 shows dark field images in the matrix and twin reflections of the {411}_B2 twins up to ~100 nm wide.

In [111]_B2-oriented single crystals the morphology of deformation structure differs from the morphology of the [001]_B2-oriented single crystals after the plastic deformation above the B19’-martensite yield stress to ε_g = 12.0–15.5% with followed by heating to room temperature. A V-shaped microstructure is formed (Figure 9), which was repeatedly observed in the deformed thin TiNi wire with the [111]_B2 texture [28]. The microstructure of plastically deformed TiNi wire consists of a high density of the {114}_B2 austenite twins that often form wedges-like twins inside strongly dislocated austenite phase [28].

The analysis of structure shows that there is a mix of B2- and B19’-phases in the [111]_B2-oriented 50.4Ni single crystals deformed in martensite and heated to room temperature. The bands of B2-austenite (bright bands in the bright field image in Figure 10a) and of B19’-martensite (dark bands in the bright field image in Figure 10a) with widths of up to ~100 nm are detected by TEM. Moreover, the SADPs from martensite variants c and d are rotated by 90° around their common [100]_B19’ zone axis.

A similar V-shaped microstructure is observed in [111]_B2-oriented 50.7Ni single crystals (Figure 11). Only B2-phase reflections are present in the joint SADP from dark and light bands (Figure 11b). The two zones are simultaneously in the reflective position: the [011]_m matrix zone and the [111]_tw twin zone. The misorientation angle between these types of zones in a cubic lattice is 54.8°. The reorientation vector of the crystal lattice at twinning is the <110>_B2-direction. The orientation relation {011}_m||{111}_tw definitely characterizes the {114}_B2 twinning in the B2-phase [35]. It can be seen from the joint SADP that the structure shown in Figure 11a,b is fragmented by the twinning in the two systems with {114}_B2 planes in the B2 phase. The two systems of twins are misoriented by 60° around the {011}_m||{111}_tw zone axis. The width of {114}_B2 twins is ~100–200 nm.

Thus, the main defect structure elements after plastic deformation above the B19’-martensite yield stress up to ε_g = 12.0–15.5% in compression and subsequent heating are the dislocation structure and {114}_B2 twins of the B2-phase regardless of the TiNi single crystal orientation.

4. Discussion

It has been experimentally shown that the critical stresses for the stress-induced B19’-martensite reorientation σ_reor during the martensite deformation in compression (T < M_f) are practically independent of the chemical composition and orientation as opposed to the martensite yield stress (Figure 4). The critical stress for the stress-induced martensite reorientation is σ_reor = 169–192 MPa in 50.4Ni single crystals and σ_reor = 138–153 MPa in 50.7Ni single crystals (Figure 4). This is primarily determined by the self-accommodating structure formation, which consists of 24 CVPs (

Table 2. Lattice correspondences between B2-austenite and B19’-martensite. Data from [28,29,30].

Martensite Domain	[100]B19’	[010]B19’	[001]B19’
1	[100]B2	[011]B2	[0$\overline{1}$1]B2
1’	[$\overline{1}$00]B2	[0$\overline{1}\overline{1}$]B2	[0$\overline{1}$1]B2
2	[100]B2	[0$\overline{1}$1]B2	[0$\overline{1}\overline{1}$]B2
2’	[$\overline{1}$00]B2	[01$\overline{1}$]B2	[0$\overline{1}\overline{1}$]B2
3	[010]B2	[101]B2	[10$\overline{1}$]B2
3’	[0$\overline{1}$0]B2	[$\overline{1}$0$\overline{1}$]B2	[10$\overline{1}$]B2
4	[010]B2	[10$\overline{1}$]B2	[$\overline{1}$0$\overline{1}$]B2
4’	[0$\overline{1}$0]B2	[$\overline{1}$01]B2	[$\overline{1}$0$\overline{1}$]B2
5	[001]B2	[110]B2	[$\overline{1}$10]B2
5’	[00$\overline{1}$]B2	[$\overline{1}\overline{1}$0]B2	[$\overline{1}$10]B2
6	[001]B2	[$\overline{1}$10]B2	[$\overline{1}\overline{1}$0]B2
6’	[00$\overline{1}$]B2	[1$\overline{1}$0]B2	[$\overline{1}\overline{1}$0]B2

). Moreover, these CVPs are not randomly distributed in the matrix. They form the combination of a triangular or hexagonal shape that minimizes the elastic strain energy during transformations [7,26,27,28,29,30,31]. There are 12 lattice correspondences between the B2- and B19’-phase (Table 2). The lattice invariant strain and the habit plane formation are achieved by the CVP formation, which consist of two different martensite domains in single-phase TiNi crystals. The relationship between the martensite domain pairs exists as type II-1 <011>_B19’ twin (Table 2 and

Table 3. The 24 CVPs and system for the habit plane indices relative to the B2-phase. Data from [28,29,30].

CVP	Martensite Domain Pairs	Habit Plane
1(+)	1–2	(−0.89 0.22 −0.40)
1(−)	1–2’	(−0.89 −0.40 0.22)
1’(+)	1’–2	(0.89 −0.40 0.22)
1’(−)	1’–2’	(0.89 −0.22 0.40)
2(+)	2–1	(0.89 0.22 0.40)
2(−)	2–1’	(0.89 0.40 0.22)
2’(+)	2’–1	(−0.89 0.40 0.22)
2’(−)	2’–1’	(−0.89 0.22 0.40)
3(+)	3–4	(−0.40 0.89 0.22)
3(−)	3–4’	(−0.22 0.89 0.40)
3’(+)	3’–4	(−0.22 −0.89 0.40)
3’(−)	3’–4’	(−0.40 −0.89 0.22)
4(+)	4–3	(0.40 0.89 0.22)
4(−)	4–3’	(0.22 0.89 0.40)
4’(+)	4’–3	(0.22 −0.89 0.40)
4’(−)	4’–3’	(0.40 −0.89 0.22)
5(+)	5–6	(0.22 −0.40 0.89)
5(−)	5–6’	(0.40 −0.22 0.89)
5’(+)	5’–6	(−0.40 0.22 0.89)
5’(−)	5’–6’	(−0.22 0.40 0.89)
6(+)	6–5	(0.22 0.40 0.89)
6(−)	6–5’	(0.40 0.22 0.89)
6’(+)	6’–5	(−0.40 −0.22 0.89)
6’(−)	6’–5’	(−0.22 −0.40 0.89)

) [7], while the relationship between different CVPs appears as type I {$\overline{1}\overline{1}$1}_B19’ twin [31]. Such a twinned structure is isotropic and is characterized by low critical stresses for the twin boundaries motion.

The σ_cr value for high-nickel 51.2Ni single crystals significantly exceeds the σ_cr values for 50.4Ni and 50.7Ni single crystals. The σ_cr value for high-nickel 51.2Ni single crystals characterizes the stress for the start of stress-induced martensite formation and strongly depends on the crystal orientation. The σ_cr = 876 MPa for [111]_B2-oriented crystals is two times higher than σ_cr = 412 MPa for [001]_B2-oriented crystals (Figure 5). The strain-glass transition is observed in the cooling/heating cycles, and the self-accommodating structure of thermo-induced martensite is not formed in TiNi single crystals with a nickel content C_Ni ≥ 51.2 at.%. As shown in Ref. [3], there are numerous point defects in the matrix at a high nickel content that suppress the MT. The defects form local distortions that frustrate the long-range order of the crystal and, therefore, prevent the appearance of macroscopic martensite upon cooling. Instead, nano-sized martensite domains are formed with randomly distributed domains, within which there is a local order. It is known [11,13,36] that there is stress-induced MT in the strain glass crystals. This MT requires high critical stress σ_cr > 400–800 MPa depending on the orientation. According to the Clausius–Clapeyron relation [8,37], the theoretical critical driving stress for the MT σ_cr can be expressed by the following equation:

$${\mathsf{\sigma }}_{\mathrm{c}\mathrm{r}}={\mathsf{\sigma }}_0+\frac{∆\mathrm{S}}{{\mathrm{V}}_{\mathrm{m}}{\mathsf{\epsilon }}_{\mathrm{t}\mathrm{r}}}(\mathrm{T}-{\mathrm{M}}_{\mathrm{s}}^{{\mathsf{\sigma }}_0})$$

(2)

where T is the test temperature, V_m is the molar volume, ε_tr is the theoretical transformation strain at MT, M_s^σ0 is the temperature of the MT start under the action of the minimum uniaxial stress σ₀ able to induce thermoelastic MT in these crystals. The theoretical transformation strain at B2-B19’ MT is ε_tr = 4.38% for the [001]_B2 orientation and ε_tr = 3.58% for the [111]_B2 orientation in the quenched TiNi single crystals in compression [11,12,14,15]. This primarily determines the orientation dependence of the critical stress for the B19’-martensite formation at the same test temperature σ_cr [111] > σ_cr [001] in quenched high-nickel 51.2Ni single crystals. Thus, in accordance with Equation (2), the lower transformation strain ε_tr = 3.58% corresponds to higher critical stress for martensite formation σ_cr = 876 MPa in [111]_B2-oriented crystals compared with [001]_B2-oriented crystals.

It has been experimentally found that there is an orientation dependence of the strain hardening coefficient θ at stages II and III, stress hysteresis Δσ at the SE, and the reversible SME/SE strain in all the studied TiNi single crystals (Figure 4 and Figure 5). This is due to the different transformation strain resources and the presence of the martensite detwinning contribution ε_detw for the different oriented crystals.

The theoretical transformation strain at B2-B19’ MT, taking into account the martensite CVP formation and its subsequent martensite detwinning ε_CVP + _detw in the single crystals of TiNi alloy, was calculated in the Refs. [11,12,14,15]. The [001]_B2 orientation is characterized by a larger transformation strain ε_CVP + _detw = 4.38% and the absence of the detwinning contribution ε_detw→0 in the general strain of the MT. In contrast to the [111]_B2 orientation, where the transformation strain is ε_CVP + _detw = 3.58% and the detwinning contribution is ε_detw = 0.6%. The CVP detwinning under stress leads to a deviation of the habit plane from the invariant position. This is accompanied by an increase in the friction force for the interfacial boundary motion and energy dissipation during the MT, which is characterized by stress hysteresis Δσ. The experimental reversible strain at the SME and SE for the [001]_B2 orientation is higher than for the [111]_B2 orientation. This corresponds to the theoretical calculations. The maximum reversible strain in 51.2Ni crystals is not achieved due to the selected test temperature and the low given strain in the loading/unloading cycle.

Thus, the [111]_B2-oriented crystals are characterized by a higher strain hardening coefficient θ at stage II, a wide stress hysteresis Δσ at the SE in 51.2Ni single crystals, and a small reversible SME/SE strain, in contrast to the [001]_B2-oriented crystals with the detwinning contributions of ε_detw→0 (Figure 4 and Figure 5). A wide stress hysteresis Δσ also observed in ferromagnetic single crystals with B2-L1₀ MT, such as NiFeGa(Co), NiMnGa, and CoNi (Al, Ga) [38,39,40].

The strain hardening coefficient at stage III increases compared with stage II, but the orientation dependence of θ is retained θ[111] > θ[001]. The elastic deformation of oriented B19’-martensite proceeds at stage III in all studied single crystals. Moreover, the B19’-martensite reorientation and detwinning processes typical for stage II continue at stage III in 50.4Ni and 50.7Ni single crystals. Further MT undergoing and the B19’-martensite detwinning typical for stage II continue at stage III in 51.2Ni single crystals. This is evidenced by the reversible strain value, which does not reach its theoretical resource of transformation strain after unloading at stage II (in the insets of Figure 4 and Figure 5).

As a result of the completion of the oriented martensite formation, a different number of active martensite CVPs is formed at stage III, depending on the crystal orientation. These active CVPs have a maximum Schmid factor under the action of compressive stress. It is known [12] that there are eight active CVPs in the [001]_B2-oriented crystals and six active CVPs in [111]_B2-oriented crystals (

Table 4. Active CVPs, calculated Schmid factors of martensite, and experimental yield stress of martensite (this study) depending on crystal orientation in TiNi single crystal.

Compression Axis in B2-Phase	Compression Axis in B19’-Martensite	Active CVPs [12,15]	Schmid Factors of Martensite		σcrM, MPa
			<001>B19’ {100}B19’ Dislocation Slip	<$\overline{1}$0$\overline{2}$>B19’ $\{20\overline{1}$}B19’ Twinning	50.4 Ni	50.7 Ni	51.2 Ni
[001]B2	[011]B19’ (1)	(1,3) (3,1) (1,4) (4,1) (2,3) (3,2) (2,4) (4,2)	0.49	0.37–0.39	692	1124	1023
	[0$\overline{1}$1]B19’ (1’)
	[01$\overline{1}$]B19’ (2, 3, 4’)
	[0$\overline{1}\overline{1}$]B19’ (2’, 3’, 4)
	[100]B19’ (5, 6)
	[$\overline{1}$00]B19’ (5’, 6’)
[111]B2	[120]B19’ (1,3, 5)	(3,1) (4,1) (7,5) (8,5) (11,9) (12,9)	0.27–0.46	0.39–0.5	964	1714	2019
	[$\overline{1}\overline{2}$0]B19’ (1’, 3’, 5’)
	[$\overline{1}$0$\overline{2}$]B19’ (2’, 4’, 6’)
	[10$\overline{2}$]B19’ (2, 4, 6)

). Thus, the combination of different deformation processes and different morphology of martensite formed under stress lead to different values of θ = dσ/dε at stage III depending on the crystal orientation and the Ni content and do not characterize the elastic modulus of B19’-martensite (Figure 4 and Figure 5).

The deformation behavior of single crystals from σ_cr^M and above is associated with the plastic deformation of B19’-martensite. The deformation above σ_cr^M does not recover even after heating to T > A_f. As previously noted, there is a strong orientation dependence of the B2-phase yield stress σ_cr^A in TiNi single crystals: [001]_B2-oriented crystals are “hard” and have a higher B2-phase yield stress compared with the “soft” [111]_B2-oriented crystals (σ_cr^A[111] < σ_cr^A[001]) in the quenched and aged states [9,12,13]. For example, in the quenched TiNi single crystals with nickel concentration of C_Ni = 50.0–51.8 at.% the austenite yield stress is σ_cr^A = 1300–1800 MPa for the [001]_B2 orientation. The austenite yield stress σ_cr^A is lower by 500–700 MPa and is equal to 800–1100 MPa for the [111]_B2 orientation [9,11,12,13]. In this work, it is experimentally shown for the first time that the quenched TiNi single crystals have an inverse orientation dependence of the B19’-martensite yield stress σ_cr^M in contrast to the conventional orientation dependence of the B2-phase yield stress. The [111]_B2-oriented crystals are characterized by a higher level of σ_cr^M compared with the [001]_B2-oriented crystals σ_cr^M[111] > σ_cr^M[001]. The difference of the B19’-martensite yield stress between two orientations Δσ_cr^M increases from 272 MPa to 996 MPa with an increase in Ni concentration from 50.4 to 51.2 at.% (Figure 4 and Figure 5). Moreover, an increase in the difference Δσ_cr^M occurs as a result of a large growth of σ_cr^M in [111]_B2-oriented crystals.

In studies by Sehitoglu et al. [12,13], the orientation dependence of the martensite yield stress in compression at temperature T = T_room > A_f was not found in the aged Ti-51.5 at.% Ni and Ti-50.8 at.% Ni single crystals. These studies are concerned only with single crystals aged at 723–823 K containing dispersed Ti₃Ni₄ particles. In the precipitation-hardened single crystals after aging at 723 K, the martensite yield stress is σ_cr^M = 1700–2100 MPa, regardless of the <112>_B2, <148>_B2, <111>_B2, and <100>_B2 orientations. Aged TiNi single crystals are characterized by a weakening of the orientation dependence of the critical stress for the martensite formation and the SME strain compared with quenched ones. This is determined, firstly, by the change in the morphology and twin structure of B19’-martensite. The dispersed Ti₃Ni₄ particles change the type of B19’-martensite twinning from type II <011>_B19’ twin to the compound <001>{100}_B19’ twin, and the density of the compound twins increases as the interparticle distance decreases [7,41]. Secondly, this is the multiple nucleation of martensite, generated both by elastic fields from several crystallographic variants of particles and by external applied stress. Such a multivariate martensite structure determines the difficulty of its reorientation and detwinning under the action of applied stress. As a result, the martensite morphology becomes independent of the crystal orientation and is controlled by microstructure parameters (particle size and interparticle distance).

On the basis of the TEM studies of the microstructure of quenched 50.4Ni and 50.7Ni single crystals after plastic deformation of B19’-martensite at T < M_f and subsequent heating, it was concluded that the crystals deformed in martensite contain a mix of B2- and B19’-phases (Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11). Firstly, there is a high dislocation density in the B2-phase inherited from B19’-martensite (Figure 6). Secondly, there is a complex twinned structure consisting of B2- and B19’-phases. The main twinning system is {411}_B2 twins in the B2-phase, regardless of the crystal orientation (Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11).

Numerous previous studies on the TiNi polycrystals and thin wires have shown that the typical defects in the B2-phase after plastic deformation of martensite and subsequent heating are as follows [2,7,9,19,20,23]. The <100>{011}_B2 dislocation slip acts as the <001>{100}_B19′ slip in the corresponding B19’-martensite. The {411}_B2 twins of B2-phase correspond to the {201}_B19′ twins in the B19’-martensite. As shown in [42], the <001> {100}_B19′ dislocation slip, the {100}_B19′ and {001}_B19′ twins are energetically favorable over the {201}_B19′ twinning. Nevertheless, {201}_B19’ twinning is one of the most common mechanical twins observed during plastic deformation of B19’-martensite. Kadeřávek et al. proposed a plastic straining mechanism of B19’-martensite, which explains the formation of {201}_B19’ deformation-twin bands. This mechanism of plastic deformation is called kwinking and involves a combination of deformation twinning and dislocation-based kinking [19,33,42,43].

Since plastic deformation was realized in B19’-martensite in the studied crystals, the Schmid factors analysis m was carried out for dislocation slip systems and twinning in B19’-martensite. The Schmid factors are calculated as follows m = cosλcosχ, where λ is the angle between the normal to the habit plane and the compression axis, χ is the angle between the shear direction and the compression axis. The crystallographic indices of the deformation axis in B19’-martensite can be determined considering the lattice correspondences between the B2- and B19’-phases (Table 3) and active CVPs with the maximum Schmid factor under compressive stress along the [111]_B2- and [001]_B2-deformation axis [7,19,20]. The [001]_B2 crystal orientation in the austenite will correspond to one of the four possible {011}_B19’-directions in the B19’-martensite, while the [111]_B2 orientation will correspond to one of the four {120}_B19’ directions in the B19’-martensite (Table 4). Calculations of the Schmid factors for the <001>_‘{100}_B19’ slip systems in B19’-martensite and the {20$\overline{1}$}_B19’ twinning in B19’-martensite are presented in Table 4.

Calculations showed that the Schmid factor for the {20$\overline{1}$}_B19’ twinning in B19’-martensite is close for the two studied orientations. However, for the [111]_B2 orientation m = 0.39–0.5 turns out to be slightly higher than for the [001]_B2 orientation m = 0.37–0.39 (Table 4). The Schmid factor for the <001>{100}_B19’ slip systems in the [111]_B2 orientation strongly depends on the martensite orientation, while for the [001]_B2 orientation, the m value is the same for all martensite orientations (Table 4). In contrast, the Schmid factor for a slip in one domain is higher than in another for one CVP in the [111]_B2-oriented crystals (Figure 12). For example, the Schmid factors for slip are m = 0.27 in domain 2’, and m = 0.46 in domain 1 in active CVP 2’(+). As a result, the plastic deformation of B19’-martensite is difficult. The various Schmid factors of CVP domains can determine the high B19’-martensite yield stress in the [111]_B2-oriented crystals and the structure morphology after plastic deformation in martensite.

The main difference between the microstructure of the [111]_B2 and [001]_B2-oriented single crystals after plastic deformation in B19’-martensite is the morphology of the twinned structure of B2- and B19’-phases: step microstructure in [001]_B2-oriented crystals and V-shaped microstructure in [111]_B2-oriented crystals (Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11). The difference in morphology is primarily determined by the number of active CVPs, which have the maximum Schmid factor for B2-B19’ MT and determine the oriented B19’-martensite formation under compressive stress along the corresponding directions. In the case of [001]_B2 orientation, the number of CVPs that are activated is 8. The number of active CVPs of [111]_B2-oriented crystals is 6. It is assumed that there is a high dislocation density and/or residual martensite pinned by dislocations in one twin domain of CVP with the maximum Schmid factor m = 0.46 compared with another domain with a low Schmid factor m = 0.27 for the <001>{100}_B19’ slip system in V-shaped microstructure of the [111]_B2-oriented crystals after plastic deformation in martensite.

Thus, in the present work, it is shown that the orientation dependence of the B19’-martensite yield stress σ_cr^M can have inverse regularities, in contrast to the orientation dependence of the B2-austenite yield stress in TiNi single crystals and textured polycrystals. The “hard” [001]_B2-oriented single crystals in B2-austenite, which are characterized by a high yield stress level σ_cr^A, on the contrary, are the “soft” in B19’-martensite σ_cr^A[001] > σ_cr^M[001]. For example, in the quenched [001]_B2-oriented crystals of TiNi with C_Ni = 51.0–51.8 at.% the B2-austenite yield stress at T = M_d is σ_cr^A ≈ 1300–1800 MPa [9,11,13]. Whereas the B19’-martensite yield stress is below and equal to σ_cr^M = 1023–1124 MPa for the [001]_B2-oriented crystals of TiNi with C_Ni = 50.7–51.2 at.% (Table 4). As a result, the plastic deformation of martensite can be achieved faster than that of austenite in the [001]_B2-oriented single crystals. When σ_cr^M is reached, the plasticity sets in, and the reversible stress-induced MT is no longer possible because irreversible dislocation slip begins in B19’-martensite. Recoverable strain decreases after reaching σ_cr^M.

In the case of the [111]_B2-oriented crystals, an inverse dependence of σ_cr^A[111] < σ_cr^M[111] takes place. The B2-austenite yield stress is equal to σ_cr^A≈800–1100 MPa at C_Ni = 51.0–51.8 at.% [9,11,13], and the B19’-martensite yield stress is equal to σ_cr^M = 1714–2019 MPa at C_Ni = 50.7–51.2 at.% (Table 4). The degradation of reversible strain at stress-induced MT in this orientation will primarily occur due to the austenite plastic deformation, which is characterized by a lower yield stress compared with the B19’-martensite.

In addition, the strong orientation dependence of the B19’-martensite yield stress can be the cause of a nonhomogeneous deformation of TiNi polycrystals in the martensite. The plastic deformation of martensite begins earlier in the grains with the <001>_B2 orientation than in the grains with the <111>_B2 orientation with a high B19’-martensite yield stress. This is especially pronounced in high-nickel TiNi single crystals with C_Ni ≥ 51.2 at.% where the B19’-martensite yield stress is σ_cr^M = 2019 MPa for the [111]_B2 orientation, and it is almost 1000 MPa lower and equal to σ_cr^M = 1023 MPa for [001]_B2 orientation.

Therefore, as the dependence of the B19’-martensite yield stress σ_cr^M on the crystal orientation established in this work, the orientation dependence of the B2-austenite yield stress σ_cr^A must be carefully taken into account when estimating the SE temperature range, the cyclic stability of the SME and SE.

5. Conclusions

The orientation dependence of the plastic deformation of B19’-martensite in compression has been investigated on quenched TiNi single crystals with various Ni content from 50.4 to 51.2 at.%: Stress-strain curves up to plastic deformation of martensite were obtained, Schmid factors for the existing dislocation slip and twinning systems in martensite were calculated, and the mechanism of plastic deformation of martensite was studied. Based on the obtained results, it can be concluded that:

• The weak orientation dependence of the critical stresses for the reorientation of the thermo-induced B19’-martensite at T < M_f is observed in 50.4Ni and 50.7Ni single crystals (σ_reor = 169–192 MPa for 50.4Ni and σ_reor = 138–153 MPa for 50.7Ni) due to the same initial self-accommodating structure consisting of 24 B19’-martensite variants.
• In the high-nickel strain-glass 51.2Ni single crystals not undergoing thermoelastic MT during cooling/heating, the critical stress for the stress-induced B19’-martensite formation σ_cr depends on the crystal orientations. The level σ_cr is much higher than σ_reor for the reorientation of martensitic variants in single crystals with Ni content 50.4 and 50.7 at.%. In accordance with the Clausius–Clapeyron relation, the [111]_B2-oriented 51.2Ni single crystals are characterized by higher values of the critical stress for the stress-induced martensite formation σ_cr = 876 MPa because of a smaller transformation strain ε_tr = 3.58% compared with the [001]_B2-oriented single crystals (ε_tr = 4.38% and σ_cr = 412 MPa).
• It is shown that an inverse orientation dependence of the B19’-martensite yield stress σ_cr^M[111] > σ_cr^M[001] is observed in comparison with the B2-phase yield stress σ_cr^A[111] < σ_cr^A[001] in TiNi single crystals with different Ni content C_Ni = 50.4–51.2 at.% in compression. With the Ni content, the difference between the stresses σ_cr^M[111] and σ_cr^M[001] increases from 272 MPa at C_Ni = 50.4 at.% to 996 MPa at C_Ni = 51.2 at.%. The main mechanisms of plastic deformation of B19’-martensite in compression are <001>{100}_B19’ dislocation slip and {20$\overline{1}$ }_B19’ mechanical twinning.
• It is necessary to take into account the orientation dependence of martensite yield stress to predict the SE temperature range and cyclic stability of SME and SE in TiNi alloys. A significant effect of stress level σ_cr^M on the functional properties should be expected in high-nickel [001]_B2-oriented single crystals of TiNi alloys with C_Ni ≥ 51.2 at.%. In this case, martensite yield stress σ_cr^M can be lower than austenite yield stress σ_cr^A. This will contribute to the relaxation of elastic energy due to the plastic deformation of B19’-martensite during the stress-induced MT and degradation of functional properties.