The results of dynamic tests of the TiNi and CuAlNi shape memory alloys are given. Compressive and tensile tests of the TiNi alloy were carried out in the temperature range of 20–300 °C. A significant change was revealed in the elastic modulus before the dislocation plastic flow and the dislocation yield stress with a change in the test temperature in the range of the reverse martensitic transformation.

For the CuAlNi alloy, the effect of the strain rate on the phase yield stress and on the phase and dislocation moduli of elasticity was insignificant. In this case the value of the dislocation yield stress increases markedly with increase in the strain rate. Using the CuAlNi alloy as an example, the method for determining the duration of the reverse martensitic transformation was implemented and the shape recovery diagram during this transformation was constructed.

1 Introduction

Thanks to unique properties of shape memory alloys (SMA), namely high corrosion resistance and strength, good strain recovery parameters, high recovery stresses, excellent biocompatibility, high damping ability, they are successfully used in many areas of engineering [1, 2], medicine [3,4,5], space technology [2, 6, 7]. Nonlinear elastic properties of these alloys are used to create energy-absorbing devices for seismic protection of buildings and structures [8,9,10,11] as well as for various drives and converters [12, 13].

Shape memory alloys belong to a group of materials that, due to phase transformations, can recover their original shape when heated. Examples of such alloys are Ag-Cd, Au–Cd, Cu–Al–Ni, Cu–Sn, Cu–Zn–(X), In-Ti, Ni–Al, Ni–Ti, Fe–Pt, Mn–Cu, Fe–Mn–Si and others. The shape memory effect is due to their temperature-dependent crystal structure. When the SMA is below phase transition temperatures it has a crystallography with low yield stress. In this state the material may be deformed into other shapes by relatively small force. A new form is maintained, provided that the material remains below the temperature of reverse transformation. When heated above this temperature, the material returns to its parent structure and recovers its original shape.

Thus, in the typical operating temperature range SMAs have two phases with the different crystal structure each, and therefore have different properties. One of them is the high-temperature phase called austenite (A), and the other is the low-temperature phase called martensite (M). Austenite (usually cubic) has a crystal structure different from martensite (tetragonal, rhombic or monoclinic). Transformation from one structure to another is not a diffusion of atoms, but a distortion of the crystal lattice. Such a transformation is known as martensitic transformation. To initiate such transformations in the material either the mechanical stress or the temperature change can be used.

A set of martensitic variants can exist in two forms. The first is the twinned martensite (Mt) which is formed by a combination of “self-sufficient” martensitic variants. The second is the deterministic or reoriented martensite, in which the characteristic variant (Md) prevails. The reversible phase transformation from austenite (parent phase) to martensite and vice versa is the basis of the unique behavior of SMA. This behavior is found at the temperature range of Mf–Af which are the temperatures of the ends of the direct and reverse martensitic transformations respectively.

The transformation temperatures increase with increasing of the load magnitude, regardless of loading type (tension or compression). When an uniaxial tensile load is applied with the corresponding stress σ, the new transformation temperatures are represented as Mσf, Mσs, Aσs and Aσf for the martensitic finish, martensitic start, austenitic start and austenitic finish temperatures, respectively.

If the material temperature is slightly higher than Af then, after loading the material, accompanied by direct martensitic transformation, the complete recovery of the shape occurs due to reverse martensitic transformation during unloading. This material behavior is called the superelastic effect. The associated change in macroscopic shape due to the applied load is reflected in the resulting stress-strain curve as shown schematically in Fig. 1. The stresses, at which the initiation and the completion of the direct martensitic transformation during loading occur, are marked as σMs and σMf respectively. Similarly, when the SMA is unloading, the stresses at which the reverse transformation into austenite is initiated and completed are marked, respectively, as σAs and σAf. If the material in the austenitic phase is tested above the Ms temperature but below the Af temperature, only a partial recovery of the shape is observed.

Fig. 1
figure 1

Schematic superelastic σ–ε diagram

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Now there are the papers on study of structural features, martensitic transformations [14,15,16], mechanical properties [17,18,19,20,21,22] and behavior modeling of SMA [23,24,25]. However, as noted in [20], the description of the mechanical behavior of SMA in a wide range of strain rates requires the knowledge of empirical dependencies of the deformation characteristics on loading parameters.

One of the most common alloys with shape memory is titanium nickelide (TiNi) which has wide application in various fields from medicine to space technology. However many applications require the characteristics titanium nickelide does not hase. For example, the narrow hysteresis which allows deformation under almost constant stress, or a perfect shape memory at high temperatures are required. In this regard, it is of interest to consider copper-based alloys which both have the above functional properties and also are less expensive.

An important feature of SMA is the possibility to generate the recovery stresses (the ability to develop stresses under constrained strain conditions, due to the shape memory effect during heating of predeformed specimens). This process is widely used in various mechanical couplings and control devices, so the study of its response time to high-strain-rate loading is very important.

In this paper the results of our study on the behavior of TiNi alloy during high-strain-rate deformation in the temperature range of the reversible martensitic transformation and the deformation diagrams of CuAlNi alloy at the strain rates of 2000 and 6000 s−1 are represented. In addition, for the CuAlNi alloy the time of the reverse martensitic transformation is estimated and the shape recovery diagram is constructed.

2 Specimens for Testing

For compression tests of the TiNi alloy the specimens in the form of short cylinders with 8 mm in diameter and 4 mm in height were used. For high-strain-rate tension the cylindrical specimens were used with threaded heads with the diameter and the length of the working part of 5 mm and 10 mm respectively. All specimens were made of hot-rolled TiNi-alloy bars. To reduce residual stresses and create an equilibrium structure the specimens were annealed at 500 °C for 1 h and then cooled in a furnace.

The temperatures of the direct and reverse martensitic transformations of the alloy were determined in the Mettler Toledo 822e differential scanning calorimeter (DSC). The rate of cooling and heating of the samples in the calorimeter was 10 °/min. The results of determining the temperatures of martensitic transformations by the DSC method are presented in Fig. 2. As can be seen, after aging at 500 °C for 1 h and subsequent cooling in the furnace the alloy had the following temperatures of reversible martensitic transformation Ms = 74 °C, Mf = 32 °C, As = 74 °C, Af = 98 °C.

Fig. 2
figure 2

Temperature dependence of heat release and heat absorption of TiNi alloy after aging at 500 °C for 1 h and cooling in a furnace

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These features of the structural-phase state of the alloy after aging should have affected its mechanical behavior under quasistatic and high-strain-rate tension. To separate the contributions of the structure and high strain rate into mechanical behavior of the alloy we conducted quasistatic tension of the specimens at the strain rate of 10−3 s−1, which was most often used to study the mechanical behavior of quenched single-phase TiNi alloys under quasistatic tension.

Figure 3a shows the typical diagrams of quasistatic tension of the TiNi alloy specimens at the temperatures of 20 and 130 °C. The choice of these diagrams is due to the fact that, according to the characteristic temperatures of martensitic transformations, the specimens at these test temperatures were in the martensitic and austenitic states, respectively. A distinctive feature of these diagrams from the well-known TiNi binary alloys diagrams is the absence of the stress peak usually observed at the beginning of the second stage of quasistatic tension and associated with the formation, growth and fusion of localized deformation zones. If we turn to the deformation diagrams under high-strain-rate loading at the same temperatures, which are shown in Fig. 3b, in this case the stress peak is clearly visible. The diagrams highlight three stages of deformation (I, II, and III), the choice of which is also based on the different nature of the stress change.

Fig. 3
figure 3

Typical diagrams of quasistatic a and dynamic b TiNi alloy loading in martensitic (at 20 °C) and in austenitic (at 130 °C) states

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In addition, the specimens of the CuAlNi alloy in the martensitic state (ρ = 7.2 g/cm3, E = 65 GPa, σt = 50 MPa, Tpr = 1500 °C) in the form of tablets with 6 mm in diameter and 3 mm in height were made to study them under compression.

3 Methods of Testing

The dynamic tests of TiNi and CuAlNi alloys at the strain rate of (1–6) × 103 s−1 were carried out on the installations that implement the split Hopkinson pressure bar method [26] with measuring bars of 10 mm in diameter (compression) and 20 mm (compression and tension). The tensile tests were carried out according to the Nicholas scheme [27, 28].

To provide the required test temperature, the coaxial electric heater was used which was placed at the ends of the measuring bars and the specimen between them. The specimen temperature was measured by a miniature thermocouple. When testing the TiNi alloy, the test temperatures near the temperatures of martensitic transformations were achieved by two ways. In the first, the specimens were heated to the selected temperature from room temperature. In the second, the specimens were preheated to 180 °C to convert them into the austenitic state, and then they were cooled to the required test temperature. This made it possible to change the phase composition at the same temperature within the interval of reversible martensitic transformations and to compare the mechanical response of the alloy to high-strain-rate and quasistatic tension, depending on its phase state.

The quasistatic tension of TiNi specimens at the strain rate of 10−3 s−1 under the same temperatures was carried out in the Lloyd 30 K Plus universal mechanical testing machine equipped with a heat chamber. The residual strain was 12–25%.

To determine the time of duration of the reverse martensitic transformation and to plot the σ-ε curve in the process of shape superelastic recovering, a preformed specimen of CuAlNi alloy was placed between the measuring bars and heated in the temperature range of the reverse martensitic transformation. During the heat-induced transformation, the specimen recovered its shape rapidly. This process was accompanied by pulsed loading of the measuring bars ends in contact with the specimen. As a result, the elastic deformation waves in the bars were recorded by strain gauges. The scheme of the specimen layout as well as pulses registration are shown in Fig. 4a. Figure 4b shows the recorded strain pulses in the measuring bars.

Fig. 4
figure 4

The scheme of the experiment to determine the time of the reverse martensitic transformation (a) and pulses recorded in the measuring bars (b)

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The proposed method allowed us to estimate the duration of the process of recovering the original (parent) shape during heating.

4 Results of TiNi Alloy Testing

Before testing the alloy was in the martensitic state.

Tests at elevated temperatures were carried out as follows. The specimen was placed between the end faces of measuring bars, then heated to a temperature of +180 °C and then cooled to the required test temperatures (+137, +117, +87 °C). After that the specimen was subjected to high-strain-rate deformation. In addition, the tests were performed when the specimen was heated to +300 °C. The selected test temperatures correspond to the dilatogram points of this alloy (Fig. 5).

Fig. 5
figure 5

TiNi alloy dilatogram

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Figure 6 shows the deformation curves of the TiNi alloy under compression at a strain rate of 2000 s−1, obtained under various temperature conditions. The effect of the test temperature and of the way of the test temperature achievement on the phase yield stress, phase modulus and phase hardening modulus is insignificant. An increase in the test temperature leads to significant decreases in the dislocation yield stress and in dislocation modulus of elasticity. At the temperature of +300 °C there is no phase transition.

Fig. 6
figure 6

Dynamic deformation diagrams of TiNi alloy under compression

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Figure 7 shows the deformation curves of TiNi alloy under compression and tension at the strain rates of 2500 and 1000 s−1 respectively.

Fig. 7
figure 7

TiNi alloy deformation diagrams under compression and tension

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To analyze the effect of the phase composition on the mechanical behavior of TiNi quasistatic and high-strain-rate tension of the alloy were carried out at the temperatures of 60, 77, 87 and 100 °C. To reach these test temperature two ways were used. In one case the specimens were heated from the room temperature (martensitic state) to the temperature at which the test deformation was performed. In the other case the material was heated to 180 °C to transform it completely into the austenitic state, and then the specimen was cooled to the specified temperatures at which the deformation was performed. This approach is due to the fact that at the same temperature the alloy could be in the martensitic, austenitic or in the mixed state with a different content of the martensitic and austenitic phases depending on how the test temperature was reached. The phase composition of the alloy at these temperatures can be estimated as follows. At 60 °C in the first case (when the specimen was heated from room temperature) the alloy was in a stable martensitic state, in the second case (preheating to 180 °C and then cooling) the alloy contained the austenitic and martensitic phases in approximately equal proportions. At 77 °C in the first case the alloy was in martensitic state with small amount of austenite, while in the second case it was in austenitic premartensitic state probably with small amount of martensite. At 87 °C in the first case the alloy contained martensitic and austenitic phases in approximately equal proportions, while in the second case it was in the austenitic premartensitic state. At 100 °C in the first case the alloy was in austenitic state with small proportion of martensite, while in the second case it was in austenitic premartensitic state.

Figure 8 shows the quasistatic and dynamic stress-strain diagrams of the alloy with different ways to achieve the temperature at which the loading was performed, namely the heating from the room temperature to the test temperature or the cooling from 180 °C to the test temperature. These diagrams show that the lengths of first stages on the curves of quasi-static tension of martensite were determined by the transformations that took place under stress.

Fig. 8
figure 8

Quasistatic (a) and dynamic (b) deformation diagrams of the TiNi alloy under tension when two modes to achieve the required temperature were used

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It is revealed that the phase and dislocation yield stresses behave differently when the test temperature increases, but they have approximately the same values at the room temperature. The phase yield stress continuously increases in the dynamic case, and in the quasi-static case it firstly decreases to a certain minimum and only then begins to increase. This means that the processes of twinning, detwinning and reorientation of the martensitic phase are very sensitive to the loading rate. And the value of the phase yield stress at high-strain-rate loading is always higher than that at quasistatic loading.

The high tensile rate has influence on the dependence of the phase yield stress on the test temperature, both for martensite and for austenite, and specifies at what temperature the dislocation mechanism of deformation begins to act.

5 Results of CuAlNi Alloy Testing

The tests of the CuAlNi alloy in the martensitic state under compression at room temperature were carried out using an installation with measuring bars of 10 mm in diameter. Because the phase transformation temperatures for this alloy were about +150 °C, the specimens during loading received the residual strain of 8% (at strain rate of 5 × 103 s−1) to 10% (at strain rate of 9 × 103 s−1). The phase transformation in this case occurred under the applied stress. The above-mentioned increased residual strain was associated with plastic deformation of the material after phase transformation. Repeated loading of such specimen did not reveal a noticeable residual strain after loading. Annealing at a temperature of +170 °C caused complete recovering of the original specimen shape. Double reloading of the annealed specimen showed a similar character of deformation. Figure 9 shows typical σ-ε curves (solid lines) and the corresponding dependences of the change in strain rate (dashed lines in the lower part of the figure), where 1 is the 1st cycle, initial state of the specimen; 2 is the 2nd cycle, initial state; 3 is the 1st cycle, the specimen after annealing; 4 is the 2nd cycle, after annealing.

Fig. 9
figure 9

Dynamic diagrams of CuAlNi alloy before and after annealing at a strain rate of ~6000 s−1

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The residual strain of the specimens after testing was 8.4% for the specimens in the initial state and after annealing, as well as 0.4% for the specimens after their preliminary elastoplastic deformation in the initial state and after annealing.

On the basis of the obtained stress-strain curves the elastic moduli in the parts of active loading and unloading and the hardening modulus, as well as the conventional yield stress were determined (see Table 1).

Table 1 Mechanical characteristics of the CuAlNi alloy
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Because there is practically no a section of ideal yield of 8–9% on the repeated loading diagrams (curves 2 and 4), unlike the first cycles (curves 1 and 3), the maximum achieved stress in the specimen is accepted as the conventional yield stress for curves 2 and 4.

Under other loading modes (according to the strain rate) the similar results were obtained. The obtained σ-ε curves and corresponding dependences of the change in the strain rate of CuAlNi alloy in the initial (martensitic) state and after annealing of deformed specimens are shown in Fig. 10.

Fig. 10
figure 10

Mechanical behavior of CuAlNi alloy during compression in the initial state (a) and after annealing (b)

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To determine the duration of the reverse martensitic transformation and to plot the diagram σ–ε in the process of unloading and shape recovering, a predeformed specimen of the CuAlNi alloy was placed between the measuring bars and then heated in the temperature range of reverse martensitic transformation (Fig. 4a). When constructing the recovery diagram (the lower curve in Fig. 11), displacements and forces on the two ends of the specimen were determined on the basis of the strain pulses εi(t) (i = 1, 2) in the measuring bars (Fig. 4b).

$$ u_{i} (t) = c\int\limits_{0}^{t} {\varepsilon_{i} } dt, $$
$$ P_{i} (t) = EA\,\varepsilon_{i} (t), $$

where c, E and A are the sound speed, the modulus of elasticity and the cross-sectional area of the measuring bars, respectively. Then, using the Kolsky formulas [27], the engineering stress and strain of the specimen were calculated as shown below

$$ \sigma_{n} (t) = \frac{P}{{A_{S}^{0} }} = \frac{{P_{1} + P_{2} }}{{2A_{S}^{0} }} $$
$$ \varepsilon_{n} (t) = \frac{{u_{2} (t) - u_{1} (t)}}{{L_{0} }}, $$

where \( A_{S}^{0} \) and L0 are the initial area and the length of the specimen, respectively.

Fig. 11
figure 11

Parameters of the deformation and shape recovery

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Figure 11 shows the change in stress and in strain of the specimen in time (Fig. 11a), and the diagram of the shape recovery when the specimen was heated (lower curve) after active loading (upper curve) (Fig. 11b). It is clearly seen that the deformation in martensitic state under applied dynamic stress takes about 20 μs in a time, whereas the reverse martensitic transformation (martensite → austenite) (recovery of the shape) with slow heating of the specimen occurs quickly taking about 220 μs.

6 Conclusion

Using the Kolsky technique, we obtained the diagrams of deformation of TiNi alloys with the shape memory effect in the temperature range of the reverse martensitic transformation, as well as that of CuAlNi alloy at room temperature. Using the CuAlNi alloy as an example, we developed a method for determining the duration of the reverse martensitic transformation and constructed a diagram of the alloy shape recovery.