Shape Memory Effect and Superelasticity of [001]-Oriented FeNiCoAlNb Single Crystals Aged under and without Stress

: The two-step ageing of Fe-28Ni-17Co-11.5Al-2.5Nb (at.%) single crystals under and without stress, leads to the precipitation of the γ (cid:48) - and β -phase particles. Research has shown that γ – α (cid:48) thermoelastic martensitic transformation (MT), with shape memory effect (SME) and superelasticity (SE), develops in the [001]-oriented crystals under tension. SE was observed within the range from the temperature of the start of MT upon cooling M s , to the temperature of the end of the reverse MT upon heating A f , and at temperatures from A f to 323–373 K. It was found that at γ – α (cid:48) MT in the [001]-oriented crystals, with γ (cid:48) - and β -phase particles, a high level of elastic energy, ∆ G el , is generated, which signiﬁcantly exceeds the energy dissipation, ∆ G dis . As a result, the temperature of the start of the reverse MT, while heating A s , became lower than the temperature M s . The development of γ – α (cid:48) MT under stress occurs with high values of the transformation hardening coefﬁcient, Θ = d σ /d ε from 2 to 8 GPa and low values of mechanical ∆ σ and thermal ∆ T h hysteresis. The reasons for an increase in ∆ G el during the development of γ – α (cid:48) MT under stress are discussed.


Introduction
FeNiCoAlX alloys (X = Ta, Nb, Ti), undergoing thermoelastic martensitic transformation (MT) from the fcc (γ)-phase to bct (α )-martensite upon cooling/heating and under stress, are characterized by high values of superelasticity (SE) from 5 to 13.5% and values of shape memory effect (SME) of up to 8%; these alloys can compete with TiNi-based alloys . The necessary conditions for the development of thermoelastic γ-α MT are achieved: (i) due to the precipitation of nanosized γ -phase particles, with L1 2 -type ordering of between 3 and 8 nm in size, which have coherent lattice conjugation with hightemperature γ-phase and retain coherence during MT in α -martensite [1][2][3][4]28]; (ii) particles of the γ -phase strengthen the high-temperature phase, reduce the transformation strain by comparison with the single-phase state [1,5] and contribute to an increase in the tetragonality of martensite c/a to 1.1-1.23 [1,28]. This, in turn, decreases the value of the shear strain, g, at γ-α MT and the Burgers vector, b, of twinning dislocations a/6<211>(111), arising during MT [1][2][3][4][5]18].
In the FeNiCoAlX polycrystals (X = Ta, Nb, Ti) during ageing at high temperatures, T~873-973 K, the precipitation of β-phase particles with B2-type ordering at the grain boundaries occurs simultaneously with the precipitation of γ -phase particles in the grain body, leading to a catastrophic decrease in plasticity [5,31,32]. Alloying with boron to 0.05 at.% suppresses the precipitation of β-phase particles along the grain boundaries, which contributes to the manifestation of SME and SE in the polycrystals of these alloys [5,31,32]. In addition, an increase in the grain size to 100-200 µm and the creation of a sharp <001>{035} texture in polycrystals, as well as the production of oligocrystals, leads to the appearance of SE from 5 to 13.5% in these alloys [5,6,31,32].

Materials and Methods
Ingots of the Fe-28Ni-17Co-11.5Al-2.5Nb (FeNiCoAlNb) (at.%) alloy were melted from pure elements in a resistance furnace (InterSELT, St. Petersburg, Russia) in a helium atmosphere. To achieve a homogeneous distribution of the elements in the bulk of the ingots, they were remelted three times. Single crystals of the FeNiCoAlNb alloy were grown by the Bridgman method in a helium atmosphere, on a Russian-made Redmet installation (Firm "Kristallooptika", Tomsk, Russia). To determine the orientation of the crystals, the diffractometric method was used by means of a DRON-3M X-ray diffractometer (Bourevestnik, St. Petersburg, Russia) with monochromatic Fe Kα radiation. Dog-bone-shaped tension samples had a gauge length of 12 mm and a cross section of 2 × 1.5 mm 2 . Samples were cut using wire electrical discharge machining ARTA-5.9 (DELTA-TEST, Fryazino, Moscow region, Russia). The damaged surface layer was ground off mechanically, and then electrically polished in 200 mL of an H 3 PO 4 + 50 g CrO 3 (phosphoric acid with chromium trioxide) electrolyte at room temperature. All samples were initially homogenized at 1573 K for 24 h in a quartz tube, in a helium atmosphere, followed by water quenching. Two-step ageing was used to precipitate γ -and β-phase particles. The first step of ageing was carried out at a temperature of 973 K for 5 h without stress, in an MRU-STF15180-01_301 resistance furnace (Carbolite Gero, Hope Valley, UK) within a quartz tube, in a helium atmosphere with water quenching. The second step of ageing was performed in a vacuum at 873 K, for 2 and 4 h in a special device (Firm "Kristallooptika", Tomsk, Russia) with a vacuum chamber, able to apply tensile stresses of 120 MPa during ageing, in the following way. Two samples were simultaneously used during each ageing treatment in the second step, with their axes in vertical and horizontal positions. The temperature was monitored by a thermocouple, attached to the vertical sample. The device was pre-heated to 473 K, subsequently, a constant tensile stress of 120 MPa was applied to the vertical sample, whereas the horizontal sample remained in a stress-free condition. Next, the device was heated to 873 K at a rate of 20 K/min and kept for 2 h and 4 h at 873 K. A controlled stress of 120 MPa, applied to the vertical sample, was maintained during all the ageing treatment. The simultaneous ageing of two samples (under and without applied stress) allows for a better comparison of these, to ascertain the effect of the applied external stress, as the samples adhere to exactly the same temperature evolutions during ageing. After completion of the ageing process under stress, the device was slowly cooled to 373 K over 1.5 h and unloaded. Start M s and finish M f temperatures of the forward γ-α MT during cooling and start A s and finish A f temperatures of the reverse γ-α MT during heating, were determined by the intersection of tangents on the temperature dependence of the electrical resistance ρ(T) on a Russian-manufactured installation (Firm "Kristallooptika", Tomsk, Russia). The MT temperatures under stress were determined in terms of deviation from the linear dependence in the "transformation strain-temperature" (ε tr (T)) curves. The value of thermal ∆T h and mechanical ∆σ hysteresis was determined in the middle of the ε tr (T) and "stress-strain" (σ(ε)) hysteresis loops, respectively. Mechanical properties and SE were investigated using an Instron 5969 universal testing machine (Instron, Norwood, MA, USA) at the deformation rate of 4·10 −4 s −1 . Tests at room temperature were carried out in air. Tests within the temperature range of 185-350 K were carried out in a special cooling/heating chamber, which is included in the equipment of the Instron 5969 universal testing machine (Instron, Nowood, MA, USA). The heating/cooling rate of the chamber was 2 K/min. The sample was inserted into grips, heated/cooled together with the chamber, kept at each temperature for 30 min before testing, and then deformed. SME at a constant tensile stress in the cycle, was studied using a Russian-made dilatometer (Firm "Kristallooptika", Tomsk, Russia) during cooling and heating within a temperature range of 77-400 K, with a heating/cooling rate of 10 K/min. Temperature variations at 10 K/min were achieved by liquid nitrogen, flowing in copper tubes, coiled around the grips and by induction heating of the grips. The sample temperature was measured by a thermocouple, fixed to the sample gauge section. The lowest sample temperature, attainable by the dilatometer, was~77 K. A miniature extensometer, attached to the sample, was used for the strain measurements. The crystal structure after ageing was investigated using a JEOL-2010 transmission electron microscope (TEM) (JEOL, Tokyo, Japan) at an accelerating voltage of 200 kV. The thin foils were prepared using double-jet electropolishing (TenuPol-5, "Struers", Ballerup, Denmark), with an electrolyte containing 20% sulfuric acid in methyl alcohol at room temperature, with 12.5 V applied voltage. In this paper, the following denotations are adopted for heat treatments: Crystal I-ageing without stress at 973 K for 5 h and then at 873 K for 2 h without stress and with slow cooling; Crystal II-ageing without stress at 973 K for 5 h and then ageing under stress 120 MPa at 873 K for 2 h with slow cooling; Crystal III-ageing at 973 K for 5 h, then at 873 K for 4 h without stress and with slow cooling; and Crystal IV-ageing without stress at 973 K for 5 h and then ageing under stress 120 MPa at 873 K for 4 h with slow cooling. For each ageing type, namely, Crystals I-IV, SE and SME were investigated on separate samples after determining the MT temperatures.

Effect of Ageing under and without Stress on Temperatures of γ-α MT and the Precipitation of γ -and β-Phase Particles
The temperatures of γ-α MT, depending on the ageing condition under and without stress, and determined from the temperature dependence of the electrical resistivity ρ(T), are presented in Figure 1 and Table 1. It may be observed that in Crystals III with regard to two-step ageing, an increase in the ageing time at the second step at 873 K from 2 to 4 h without stress leads to an increase in the M s temperature by 29 K compared to Crystals I. In Crystals II and IV, ageing under stress at the second step leads to an increase in the M s temperature by 15 and 11 K, respectively, relative to Crystals I and III.  TEM studies ( Figure 2) showed that in Crystals I-IV, the γ′-and β-phase particles are precipitated. The γ′-phase particles of the equiaxed shape, with L12-type ordering, which is confirmed by the presence of superstructural reflections (Figure 2c), have the chemical composition, (FeNiCo)3AlNb [5,13,19,31,32]. Equiaxial γ′-phase particles have a size d ~ 8-12 nm and coherent conjugation with the high-temperature γ-phase. These data are consistent with previously published results in [5,13,19,31,32]. TEM studies have shown that the particle size of the γ′-phase depends weakly on the ageing time at the second step and the external, applied stresses during ageing (Figure 2a   TEM studies ( Figure 2) showed that in Crystals I-IV, the γ -and β-phase particles are precipitated. The γ -phase particles of the equiaxed shape, with L1 2 -type ordering, which is confirmed by the presence of superstructural reflections (Figure 2c), have the chemical composition, (FeNiCo) 3 AlNb [5,13,19,31,32]. Equiaxial γ -phase particles have a size d~8-12 nm and coherent conjugation with the high-temperature γ-phase. These data are consistent with previously published results in [5,13,19,31,32]. TEM studies have shown that the particle size of the γ -phase depends weakly on the ageing time at the second step and the external, applied stresses during ageing (Figure 2a,b).
The β-phase particles of the non-equiaxial shape with B2-type ordering have the chemical composition, (FeNiCo)(AlNb) [31,32] and are non-coherent with the high-temperature γphase (Figure 2d,e). Due to the significant difference in the lattice parameters (a γ = 0.36 nm, a β = 0.28 nm), the β-phase particles lose coherence at small sizes, d < 5 nm [35]. Similar lattice parameters of the γand β-phases were obtained in the FeNiCoAlTaB polycrystals (a γ = 0.3604 nm, a β = 0.2880 nm) [5,31,32]. In the FeNiCoAlTaB polycrystals, β-phase particles are precipitated at the grain boundaries with the formation of regions free from γ -phase particles. In these polycrystals, the nucleation of β-phase particles was also found on the Ta particles, with a lattice parameter close to the β-phase. The Ta particles decrease the energy barrier, associated mainly with the generation of elastic energy, for the β-phase nucleation [31,38]. In the FeNiCoAlNb single crystals, β-phase particles are precipitated in the volume of the crystal, and zones free from γ -phase particles are not observed. During ageing without and under stress at the second step, four crystallographically equivalent variants of the β-phase particles, with an orientation relationship (111) γ ||(110) β , [29,38]. With an increase in the ageing time at the second step, the volume fraction and particle size of the β-phase increase from f = 0. Ageing under a tensile stress of 120 MPa does not lead to a notable difference in the morphology of the particles of the γ -and β-phases compared to ageing without stress. In crystals aged under stress, an increased dislocation density was observed at the "matrix-β-phase" interface (Figure 2e), which was less pronounced during ageing without stress. 225  155  210  260  35  70  50  3.7 TEM studies ( Figure 2) showed that in Crystals I-IV, the γ′-and β-phase particles are precipitated. The γ′-phase particles of the equiaxed shape, with L12-type ordering, which is confirmed by the presence of superstructural reflections (Figure 2c), have the chemical composition, (FeNiCo)3AlNb [5,13,19,31,32]. Equiaxial γ′-phase particles have a size d ~ 8-12 nm and coherent conjugation with the high-temperature γ-phase. These data are consistent with previously published results in [5,13,19,31,32]. TEM studies have shown that the particle size of the γ′-phase depends weakly on the ageing time at the second step and the external, applied stresses during ageing (Figure 2a  TEM studies facilitate an explanation of the dependence of the M s temperature on the ageing conditions [2]. Firstly, in Crystals I-IV, the volume fraction and particle size of the γ -phase change insignificantly (Figure 2a,b) and, therefore, the M s temperature change is not associated with this phase. Secondly, in Crystals I-IV, the volume fraction and particle size of the β-phase change in relation to the ageing time at the second step. The volume fraction and size of β-phase particles are larger in Crystals III and IV, than in Crystals I and II. This leads to a decrease in the nickel concentration in the matrix and explains the increase in the M s temperature in Crystals III and IV, in comparison with Crystals I and II [1][2][3][4][5][6]38]. Thirdly, the differences in the sizes and volume fractions of β-phase particles, as a consequence of the ageing conditions under and without stress, were not found in the study of the crystal structure. Consequently, an increase in the M s temperature in Crystals II and IV compared to Crystals I and III, respectively, cannot be explained from the standpoint of changes in the concentration of alloying elements in the matrix, and in the morphology of the particles of the γ -and β-phases. The increase in the M s temperature in Crystals II and IV in comparison with Crystals I and III is due to the formation of internal oriented stress fields <σ G > from the β-phase particles. A qualitative confirmation of the <σ G > appearance is the formation of dislocations at the "matrix-β-phase" boundaries (Figure 2e), as was previously observed in TiNi crystals, aged under stress and containing partially non-coherent Ti 3 Ni 4 precipitates [29,30,39,40].

Crystal IV
As can be seen in Table 1, γ-α MT during heating-cooling in a free state, is associated with thermoelastic MT of the second type, in which A s ≤ M s [29,30,41]. Another feature is the large value of overcooling ∆M = M s − M f , with a forward γ-α MT and overheating of ∆A = A f − A s with a reverse α -γ MT, varying from 50 to 74 K in combination with low values of thermal hysteresis, ∆T h = A f − M s = 18-40 K. According to the results of thermodynamic analysis for the thermoelastic MT of the second type [41][42][43][44][45], the ratio of the stored elastic energy, ∆G el , is equal to or exceeds the doubled value of the dissipated energy, 2∆G dis , ∆G el ≥ 2∆G dis . The values of ∆G el and ∆G dis are estimated from the relationships obtained in [33,[41][42][43][44][45]: M s , M f and A s , A f are, respectively, the temperatures of the start and finish of the forward γ-α MT during cooling and the reverse during the heating of the α -γ MT; ∆S ch is the change in entropy under MT. Estimates show that ∆G el /∆G dis varies from 3 to 6.6 ( Table 1). High values of ∆G el , as will be shown below, determine the temperature range of SE in Crystals I-IV.

SE and SME in the [001]-Oriented FeNiCoAlNb Crystals Aged under and without Stress
SE in Crystals I-IV was found within a broad temperature range ( Figure 3). A comparison of the γ-α MT temperatures, obtained from the ρ(T) dependence ( Figure 1) and the SE temperature range (Figure 3), shows that SE in Crystals I-IV takes place at M s temperature, within the temperature range M s < T < A f and above the A f temperature. According to well-known experimental and theoretical studies, the conditions for SE are achieved at T ≥ A f , when martensite becomes thermodynamically unstable, when the load is removed [29,30]. SE within the temperature range from M s to A f was observed in FeNiCoAlNb single crystals [33], FeNiCoAlTaB [31,32] and FeNiCoTi polycrystals [9]. Reversible deformation within the temperature range between 77 K and M s is not associated with the development of γ-α MT under stress, but is caused by the reversible motion of "austenite-martensite" interphase and twin boundaries of a cooling α -martensite in "load-unloading" cycles. It was previously found in FeNiCoAlNb single crystals [33] and FeNiCoAlTaB polycrystals [31,32]. This was defined as having rubber-like behavior [33]. In this paper, the reversible deformation at T < M s was not investigated.
It may be observed that σ cr Ms for the onset of stress-induced γ-α MT within the temperature range from M s to 373 K increases, with an increase in the test temperature, and the dependence, σ cr Ms (T), turns out to be characteristic of alloys, undergoing MT under stress at T > M s (Figures 3 and 4). This dependence is described by the Clapeyron-Clausius relationship [29,30,45]: dσ Ms ∆H ch and ∆S ch are the enthalpy and entropy changes, respectively, at MT; T 0 is the temperature of the chemical equilibrium of phases and ε tr is the transformation strain at MT. Figure 4 demonstrates that the value α = dσ cr Ms /dT varies from 3.2 to 3.8 MPa/K in Crystals I-IV. In the case of Crystals II and IV, α = dσ cr Ms /dT is greater than for Crystals I and III. A comparison of the α = dσ cr Ms /dT values for Crystals I-IV with the known literature data, obtained on the [001]-oriented FeNiCoAlX crystals (X = Ta, Nb, Ti) under tensile strain and containing only γ -phase particles, 3-8 nm in size and particles of the γand β-phases [25,27,33,34], showed that these values, α = dσ cr Ms /dT, are in close proximity to one other.
FeNiCoAlNb single crystals [33], FeNiCoAlTaB [31,32] and FeNiCoTi polycrystals [9]. Reversible deformation within the temperature range between 77 K and Ms is not associated with the development of γ-α′ MT under stress, but is caused by the reversible motion of "austenite-martensite" interphase and twin boundaries of a cooling α′-martensite in "load-unloading" cycles. It was previously found in FeNiCoAlNb single crystals [33] and FeNiCoAlTaB polycrystals [31,32]. This was defined as having rubber-like behavior [33]. In this paper, the reversible deformation at T < Ms was not investigated.   Determination of critical stresses ( σcr Ms , σcr Mf , σcr As and σcr Af ) and value of mechanical hysteresis Δσ is shown in (a) and (b).
It may be observed that σcr Ms for the onset of stress-induced γ-α′ MT within the temperature range from Ms to 373 K increases, with an increase in the test temperature, and the dependence, σcr Ms (T), turns out to be characteristic of alloys, undergoing MT under stress at T > Ms (Figures 3 and 4). This dependence is described by the Clapeyron-Clausius relationship [29,30,45]: ΔHch and ΔSch are the enthalpy and entropy changes, respectively, at MT; Т0 is the temperature of the chemical equilibrium of phases and εtr is the transformation strain at MT. Figure 4 demonstrates that the value α = dσcr Ms /dT varies from 3.2 to 3.8 MPa/K in Crystals I-IV. In the case of Crystals II and IV, α = dσcr Ms /dT is greater than for Crystals I and III. A comparison of the α = dσcr Ms /dT values for Crystals I-IV with the known literature data, obtained on the [001]-oriented FeNiCoAlX crystals (X = Ta, Nb, Ti) under tensile strain and containing only γ′-phase particles, 3-8 nm in size and particles of the γ′-and β-phases [25,27,33,34], showed that these values, α = dσcr Ms /dT, are in close proximity to one other.
Using relationship (3) In Crystals II and IV, the value of <σ G > calculated by relationship (4) is 35-57 MPa at the ∆M s = 11-15 K and α = (3.2-3.8) MPa/K. SE loops with a sequential increase in ε tr in the "load-unloading" cycle until fracture at M s and the temperature above A f of Crystals I-IV, are presented in Figure 5. The results obtained for Crystals I are close to those obtained earlier in [33]. In relation to Crystals I, at M s temperature, the maximum SE was 3.3%. With an increase in the transformation strain, ε tr , in the "load-unloading" cycle, an increase in the mechanical hysteresis, ∆σ, is observed, and rather high values of the transformation hardening coefficient, Θ = dσ/dε, equal to 2 GPa, are found. At 273 K, the SE was 3%, the mechanical hysteresis, ∆σ, increased three times relative to the M s temperature and became equal to 170 MPa and Θ = 4.3 MPa. Among Crystals II, at an M s temperature, the SE was 3.5%, ∆σ increased to 135 MPa and Θ = dσ/dε, up to 3.2 GPa, compared to Crystals I at this temperature. At 273 K, the SE was 4%, ∆σ = 190 MPa and Θ = 8 GPa. Thus, ageing under stress leads to a significant increase in Θ = dσ/dε and ∆σ, compared to ageing without stress. SE loops with a sequential increase in εtr in the "load-unloading" cycle until fracture at Ms and the temperature above Af of Crystals I-IV, are presented in Figure 5. The results obtained for Crystals I are close to those obtained earlier in [33]. In relation to Crystals I, at Ms temperature, the maximum SE was 3.3%. With an increase in the transformation strain, εtr, in the "load-unloading" cycle, an increase in the mechanical hysteresis, Δσ, is observed, and rather high values of the transformation hardening coefficient, Θ = dσ/dε, equal to 2 GPa, are found. At 273 K, the SE was 3%, the mechanical hysteresis, Δσ, increased three times relative to the Ms temperature and became equal to 170 MPa and Θ = 4.3 MPa. Among Crystals II, at an Ms temperature, the SE was 3.5%, Δσ increased to 135 MPa and Θ = dσ/dε, up to 3.2 GPa, compared to Crystals I at this temperature. At 273 K, the SE was 4%, Δσ = 190 MPa and Θ = 8 GPa. Thus, ageing under stress leads to a significant increase in Θ = dσ/dε and Δσ, compared to ageing without stress.
In Crystals III and IV, an increase in the ageing time at the second step to 4 h at 873 K, as shown above, increases the volume fraction of β-phase particles in comparison with Crystals I and II. As a result, the SE decreases, while Δσ and Θ, on the contrary, increase and an irreversible strain appears of up to 0.5% at temperatures above Af ( Figure 5). An analysis of the SE loops for Crystals I-IV in the temperature range from Ms to 373 K, shows that in all studied crystals, the stresses, σcr Ms , for the onset of stress-induced γ-α′ MT, are lower than the stresses, σcr As , of the start of the reverse α′-γ MT, when unloading (Figures 3 and 5). Using the results of the thermodynamic analysis of SE loops [33,[42][43][44][45], the contribution of ΔGel and ΔGdis can be determined from the values of σcr Ms , σcr Mf   In Crystals III and IV, an increase in the ageing time at the second step to 4 h at 873 K, as shown above, increases the volume fraction of β-phase particles in comparison with Crystals I and II. As a result, the SE decreases, while ∆σ and Θ, on the contrary, increase and an irreversible strain appears of up to 0.5% at temperatures above A f ( Figure 5).
An analysis of the SE loops for Crystals I-IV in the temperature range from M s to 373 K, shows that in all studied crystals, the stresses, σ cr Ms , for the onset of stress-induced γ-α MT, are lower than the stresses, σ cr As , of the start of the reverse α -γ MT, when unloading (Figures 3 and 5). Using the results of the thermodynamic analysis of SE loops [33,[42][43][44][45], the contribution of ∆G el and ∆G dis can be determined from the values of σ cr Ms , σ cr Mf and σ cr As , σ cr Af : ∆G el = (σ cr Mf (T) − σ cr Ms (T))ε tr (5) ∆G dis = 1 2 (σ cr Mf (T) − σ cr As (T))ε tr (6) σ cr Ms and σ cr Mf are the stresses at the start and end of the forward γ-α MT under load, and σ cr As and σ cr Af are the stresses at the start and end of the reverse α -γ MT, when the load is removed. The estimate of the ratio ∆G el /∆G dis will depend on ε tr , therefore, its calculations for Crystals I-IV for different temperatures, were tested in relation to the close values of ε tr = 2% (Figure 3). Estimates showed that ∆G el /∆G dis vary from 5.5 to 6.3, within the range between M s and A f, and at T > A f .
The condition for the appearance of a closed loop at T ≥ M s was obtained in [33] according to the assumption that σ cr Af (T) > 0, and is written as follows: ∆G ch (T) = (T − T 0 )∆S ch is the chemical energy change at MT. At T ≥ M s , condition (7) is satisfied since ∆G el = (5.5-6.3)∆G dis and the reverse α -γ MT occurs, due to the large stored elastic energy. A high level of ∆G el determines high values of the transformation hardening coefficient Θ = dσ/dε at T ≥ M s . This is due to the fact that β-phase particles have a non-coherent conjugation of the lattice with the high-temperature phase. These particles are stable and their atomic structure locks the MT inside the β-phase. Consequently, in the case of a reversible γ-α MT, the β-phase particles are only deformed elastically and can assist to the predominant nucleation of α -martensite near the "β-phase-high-temperature phase" interphase surfaces, as was previously observed in the TiNi-based composites, containing TiC particles [46,47]. High values of Θ = dσ/dε at a low level of strain ε ≤ 5%, were observed in fcc poly-and single crystals, containing non-coherent, plastically undeformed particles during slip deformation in Cu-SiO 2 , Cu-Al 2 O 3 [48,49] and upon twinning deformation in Cu-Al-Co with non-coherent CoAl particles [50]. In this case, an increase in ∆G el and, accordingly, <σ G >, depending on the level of plastic deformation ε pl by slip, twinning and γ-α MT in the absence of relaxation processes, is written as [48,49]: GPa is the shear modulus of martensite, f is a volume fraction of β-phase particles, which is estimated as being between 10-15%, ε pl is plastic deformation and m is the Schmid factor for twinning in the [001] orientation in martensite, which is equal to 0.41 at tensile strain [33]. The experimental Θ = dσ/dε values for Crystals I-IV, which ranged from 2 to 8 GPa, were close to the calculated Θ = dσ/dε values according to relationship (9). It is interesting that in the study of SE of the [001]-oriented FeNiCoAlX crystals (X = Ta, Nb, Ti), containing only the nanosized γ -phase particles, the value of Θ = dσ/dε was equal to 2-3 GPa, σ cr Ms > σ cr As , M s < A s . Therefore, MTs are transformations of the first type, according to the Tong-Wayman classification [11,12,18,21,27]. SME studies during cooling/heating, under the constant stresses of Crystals I-IV are shown in Figures 6 and 7. The values of SME ε SME , overcooling ∆M = M s − M f and overheating ∆A = A f − A s , as well as thermal hysteresis ∆T h = A f − M s , depend on the level of applied stresses, σ cr , in the "cooling-heating" cycle and heat treatment ( Figure 6).
In the studied Crystals I-IV, the M s σ temperature of the onset of the γ-α MT under stress increases with an increase in the level of applied stresses, σ cr , and coincides with the M s σ temperature obtained in experiments on the SE study (Figures 3 and 4). The values of ∆M, ∆A, ∆T h and ε tr also increase with increasing level of applied stresses, σ cr . Figure 4 displays the stresses of the finish of the reverse MT, σ cr Af , when the load is removed, which like the stresses for the onset of the stress-induced MT σ cr Ms , were obtained from the results of studying the SE temperature range (Figure 3) and the SME under stress ( Figure 6). It can be seen that similar values of σ cr Af and σ cr Ms were obtained for Crystals I-IV in both experiments. The σ cr Af curves for Crystals I and II are parallel to the σ cr Ms curves, which indicates a weak dependence of ∆σ on the test temperature and thermal hysteresis ∆T h on the level of external stresses. In Crystals III and IV, ∆σ and ∆T h are greater than in Crystals I and II and increase with growth of the test temperature and external stresses, respectively. In this case, the σ cr Ms and σ cr Af curves are not parallel to each other. Therefore, in Crystals III and IV, ∆σ and ∆T h depend, respectively, on the test temperature and external stresses. The physical reason for the stronger dependence of the ∆σ and ∆T h , respectively, on the test temperature and external stresses in Crystals III and IV as compared to Crystals I and II requires additional structural studies.
The physical reason for the stronger dependence of the Δσ and ΔTh, respectively, on the test temperature and external stresses in Crystals III and IV as compared to Crystals I and II requires additional structural studies.
A comparison of Crystals I and III with Crystals II and IV reveals a number of differences. Firstly, the values of εtr and dεtr/dσ in Crystals III and IV differ significantly ( Figures  6 and 7). At a stress level of 300 MPa, in case of Crystals IV, εtr reaches 3.3%, whereas εtr reaches 1.4% in Crystals III. The value of εtr in these crystals is limited by their destruction. In the [001]-oriented FeNiCoAlX crystals (X = Ta, Nb, Ti), containing only the γ′-phase particles, εtr = 5.5-8.7% in experiments relating to cooling/heating under constant stress [12,18] and up to 13.5% in experiments on the SE study [5,27]. Similar results were obtained for Crystals I and II (Figures 6 and 7). Therefore, as studies of the SE and SME in single crystals containing particles of the γ′-and β-phases have shown, the particles of the β-phase lead to a decrease in the plasticity associated with the stress induced γ-α′ MT.
Secondly, among Crystals I-IV at the level of constant stress of σcr = 300-350 MPa, high values of ΔM and ΔA were found, which reached 150 and 190 K; 110 and 140 K; 150 and 190 K; and 125 and 200 K, respectively, for Crystals I, II, III and IV. At the maximum level of constant stress of σcr = 300-350 MPa, the values of thermal hysteresis, ΔTh, in Crystals I, II, III and IV are equal to 150 K, 50 K, 50 K and 100 K, respectively. In all studied crystals on the εtr(Т) curves under constant stress, the As temperature is lower than the Ms temperature ( Figure 6). Therefore, such MTs under constant stress are related to the thermoelastic MT of the second type [41]. This indicates that in the [001]-oriented FeNiCoAlNb crystals, with particles of the γ′-and β-phases, a high level of ΔGel accumulates at the γ-α′ MT under stress. These results confirm the aforementioned conclusions regarding the second type of γ-α′ MT, obtained in the study of the temperature dependence of ρ(Т) ( Figure 1) and SE (Figures 3 and 5).    Thirdly, ageing under stress leads to the formation of internal oriented stress fields <σG>, which is confirmed by an increase in the Ms temperature in Crystals II and IV as compared to Crystals I and III with the same ageing time at the second step. The source of these internal stress fields <σG> = 35-57 MPa are dislocations arising at the "matrix-βphase" interface during ageing under stress (Figure 2e). Earlier, in the [001]-oriented crystals of another Fe-28Ni-17Co-11.5Al-2.5(TiNb) (at.%) (FeNiCoAlNbTi) alloys, after ageing under stress, as in Crystals II and also containing particles of γ′-and β-phases, similar values of ΔMs = 15 K and <σG> = 65 MPa were found [51].
Thus, the development of thermoelastic γ-α′ MT in the [001]-oriented FeNiCoAlX crystals (X = Ta, Nb, Ti) occurs in a disordered fcc high-temperature phase, in which the nanosized γ′-phase particles with a size d = 3-12 nm are precipitated. The fundamental role of the γ′-phase particles is that they act as "memory elements", which determine the restoration of the initial orientation of austenite and the realization of the fact that reverse α′-γ MT is "exactly backward" [1,2,4,5,28].
The γ′-phase particles have coherent conjugation with the high-temperature phase and retain coherence with the α′-martensite. With d > dcr at γ-α′ MT, there is a loss of coherence of the γ′-phase with α′-martensite and relaxation of the elastic energy, ΔGel, stored during the MT. As a result, the transformation becomes non-thermoelastic [1,2]. In relation to Crystals I-IV, containing particles of γ′-and β-phases, the γ-α′ MT is not suppressed by the presence of the β-phase particles. In contrast to the γ′-phase particles, the β-phase particles have non-coherent conjugation with the γ-phase and with α′-martensite. This leads to a decrease in plasticity and, accordingly, to a decrease in the value of SE and SME at MT under stress. Non-coherent β-phase particles act as preferential sites for the nucleation of α′-martensite, which grows in the regions between the β-phase. The high A comparison of Crystals I and III with Crystals II and IV reveals a number of differences. Firstly, the values of ε tr and dε tr /dσ in Crystals III and IV differ significantly (Figures 6 and 7). At a stress level of 300 MPa, in case of Crystals IV, ε tr reaches 3.3%, whereas ε tr reaches 1.4% in Crystals III. The value of ε tr in these crystals is limited by their destruction. In the [001]-oriented FeNiCoAlX crystals (X = Ta, Nb, Ti), containing only the γ -phase particles, ε tr = 5.5-8.7% in experiments relating to cooling/heating under constant stress [12,18] and up to 13.5% in experiments on the SE study [5,27]. Similar results were obtained for Crystals I and II (Figures 6 and 7). Therefore, as studies of the SE and SME in single crystals containing particles of the γ -and β-phases have shown, the particles of the β-phase lead to a decrease in the plasticity associated with the stress induced γ-α MT.
Secondly, among Crystals I-IV at the level of constant stress of σ cr = 300-350 MPa, high values of ∆M and ∆A were found, which reached 150 and 190 K; 110 and 140 K; 150 and 190 K; and 125 and 200 K, respectively, for Crystals I, II, III and IV. At the maximum level of constant stress of σ cr = 300-350 MPa, the values of thermal hysteresis, ∆T h , in Crystals I, II, III and IV are equal to 150 K, 50 K, 50 K and 100 K, respectively. In all studied crystals on the ε tr (T) curves under constant stress, the A s temperature is lower than the M s temperature ( Figure 6). Therefore, such MTs under constant stress are related to the thermoelastic MT of the second type [41]. This indicates that in the [001]-oriented FeNiCoAlNb crystals, with particles of the γ -and β-phases, a high level of ∆G el accumulates at the γ-α MT under stress. These results confirm the aforementioned conclusions regarding the second type of γ-α MT, obtained in the study of the temperature dependence of ρ(T) (Figure 1) and SE (Figures 3 and 5).
Thirdly, ageing under stress leads to the formation of internal oriented stress fields <σ G >, which is confirmed by an increase in the M s temperature in Crystals II and IV as compared to Crystals I and III with the same ageing time at the second step. The source of these internal stress fields <σ G > = 35-57 MPa are dislocations arising at the "matrix-β-phase" interface during ageing under stress (Figure 2e). Earlier, in the [001]-oriented crystals of another Fe-28Ni-17Co-11.5Al-2.5(TiNb) (at.%) (FeNiCoAlNbTi) alloys, after ageing under stress, as in Crystals II and also containing particles of γ -and β-phases, similar values of ∆M s = 15 K and <σ G > = 65 MPa were found [51].
Thus, the development of thermoelastic γ-α MT in the [001]-oriented FeNiCoAlX crystals (X = Ta, Nb, Ti) occurs in a disordered fcc high-temperature phase, in which the nanosized γ -phase particles with a size d = 3-12 nm are precipitated. The fundamental role of the γ -phase particles is that they act as "memory elements", which determine the restoration of the initial orientation of austenite and the realization of the fact that reverse α -γ MT is "exactly backward" [1,2,4,5,28].
The γ -phase particles have coherent conjugation with the high-temperature phase and retain coherence with the α -martensite. With d > d cr at γ-α MT, there is a loss of coherence of the γ -phase with α -martensite and relaxation of the elastic energy, ∆G el , stored during the MT. As a result, the transformation becomes non-thermoelastic [1,2]. In relation to Crystals I-IV, containing particles of γ -and β-phases, the γ-α MT is not suppressed by the presence of the β-phase particles. In contrast to the γ -phase particles, the β-phase particles have non-coherent conjugation with the γ-phase and with α -martensite. This leads to a decrease in plasticity and, accordingly, to a decrease in the value of SE and SME at MT under stress. Non-coherent β-phase particles act as preferential sites for the nucleation of α -martensite, which grows in the regions between the β-phase. The high values of Θ = dσ/dε, due to MT in these crystals, which are characteristic of the development of deformation by slip and twinning [48][49][50], indicate an increase in ∆G el with an increase in the volume fraction of α -martensite. The combination of a high level of ∆Gel and small values of ∆Gdis creates conditions for the development of SE within the temperature range from Ms to Af, which are not usually observed during the development of thermoelastic MTs. Ageing under stress at the second step leads to an increase in Ms temperature, compared to ageing without stress, which is due to the appearance of the internal oriented stress fields, <σG>. To elucidate the mechanism of <σG> formation, additional studies are required, in which it will be necessary to change the ageing conditions at the second step, in order to amend the size and volume fraction of the β-phase particles. This will facilitate the creation of unique composite materials, with a simultaneous precipitation of γ -and β-phase particles, high transformation strain values and the development interval of the γ-α MT within the high-temperature region due to a decrease in nickel in the fcc high-temperature phase, as a result of the precipitation of the β-phase particles.
In the present paper, it is shown that the maximum values of the transformation strain, ε tr , obtained in experiments when studying the SE and SME during cooling/heating under constant stress, are equal to 4 and 3.4%, respectively. Theoretical estimates of ε tr , carried out in [5,12], show that the value of ε tr for the [001]-oriented crystals under tension decreases from 8.3 to 5.5% with an increase in the tetragonality of martensite c/a from 1.1 to 1.15%. This estimate assumes that the γ-α MT occurs in the alloy without the γ -phase particles or when these particles undergo elastic deformation together with the fcc matrix. The experimental values of the SME and SE under tension along the [001] direction vary from 14.5 to 5.5% for the γ -phase particle sizes of 3 nm and 8-12 nm [5,12,18,26,27]. The noncoherent β-phase particles, the sizes of which are much larger than the γ -phase particles, cannot accommodate transformation strain in the matrix due to elastic deformation and contribute to the generation of multiple variants of martensite from the "matrix-β-phase" interface [46,47]. As a result, ε tr significantly decreases to between 3.4 and 4% and the determination of the total resource of ε tr is limited by the low plasticity of Crystals I-IV due to the effect of the β-phase on transformation plasticity.

Conclusions
The studies of development of the thermoelastic γ-α MT under stress using [001]oriented FeNiCoAlNb single crystals, containing non-coherent β-phase particles after ageing under and without stress, showed that the effect of the β-phase particles on SME and SE in single crystals and polycrystals of FeNiCoAlX alloys (X = Ta, Nb, Ti) is fundamentally different. Based on the results of the SME and SE studies, crystal structure after ageing under and without stresses and their analysis, the following conclusions can be summarized as the main findings:

1.
A new two-step ageing has been proposed for the FeNiCoAlNb single crystals, oriented along the [001]-direction for tensile strain, including ageing at the first step at 973 K for 5 h without stress and at the second step, at 873 K for 2 and 4 h without and under tensile stress at 120 MPa. At the first step of ageing at 973 K for 5 h, particles of the γ -phase are precipitated, and at the second low-temperature step at 873 K for 2 and 4 h, a structure, consisting of particles of the γ -and β-phases is formed.

2.
An increase in the ageing time at the second low-temperature step from 2 to 4 h leads to an increase in the M s temperature, which is associated with a decrease in the nickel concentration in the matrix due to an increase in the volume fraction of the nickel-rich β-phase particles with the composition, (FeNiCo)(AlNb).

3.
Ageing under a tensile stress of 120 MPa at the second low-temperature step for 2 and 4 h leads to an increase in the M s temperature, in comparison with ageing without stress by 11-15 K, which is associated with the generation of internal oriented tensile stresses during ageing under stress. 4.
In a structure containing particles of the γ -and β-phases, conditions are created for observing SE within the temperature range from M s to A f due to the formation of a high level of accumulated elastic energy, ∆G el , which significantly exceeds the value of the dissipated energy, ∆G dis .