Introduction

Shape Memory Alloys (SMAs) are a unique class of materials that have gained significant attention in the scientific community due to their ability to return to a pre-determined shape after deformation. The shape memory effect is achieved through a thermal treatment and is based on the thermo-elastic martensitic transition. This transition involves a change in the crystal structure of the material, resulting in the presence of an austenitic phase at high temperatures and a martensitic phase at low temperatures. The behavior of SMAs is captured in the phase transformation diagram, which displays the four transformation temperatures (Ms, Mf, As and Af) associated with the material [1]. These temperatures play a critical role in the functioning of SMAs and are essential to consider when designing applications that utilize the shape memory effect. Understanding the relationship between the transformation temperatures and the applied stress is crucial in the optimization and utilization of SMAs. SMAs have attracted significant attention, both in the scientific and commercial sector. In applications including automotive and aerospace [2,3,4,5,6], robotics [7,8,9] and constructions [10,11,12,13], as well as for biomedical applications [6, 14,15,16]. SMAs can be used as replacement of traditional linear actuators, by providing a joule heating activation using a suitable amount of electrical current. One of the recent challenges is using SMAs for sensing, i.e. to monitor their deformation through the measurement of their resistance at known conditions. In this context, self-sensing resistance techniques are particularly promising [17,18,19]. These techniques allow to evaluate the resistance of a SMA during use. For example, a feedback circuit can be used to maintain the deformation of the material at a predetermined value statically or in autonomous cyclic movement [20, 21]. The study presented in [22] explores the utilization of resistivity variation in actuation systems. Building upon previous research on a single shape memory alloy-spring system, this work introduces a second opposing shape memory alloy wire. The main objective is to establish a mapping between resistance and strain, crucial for self-sensing applications.

With these findings, the NiTi wire can be utilized as an actuator in applications where the measurement of wire length is not feasible but still required. By analyzing the data collected, the shape memory alloy wire can exhibit self-sensing capabilities, which is the ability of the transducer to monitor its own motion during the operational process. As stated by Abdullah et al. [23], even though direct control of the system’s position is more effective, it is also possible to exploit the material’s resistivity to control its actuation, thus eliminating the need for external sensors. This concept is further corroborated by Lambert et al. [24], who assert that, by assuming that the resistance remains constant throughout the SMA wire, the wire itself can be employed as a self-sensing element to determine its length in various control schemes through a dual measurement technique.

The literature contains numerous studies reporting on the measurement of resistivity in shape memory materials, with some focusing on self-sensing applications. In [25], the results of electrical resistivity and strain recovery measurements are presented, investigating the stability of the R-phase in NiTi shape memory alloy during thermal cycling under a constant tensile stress of 100 and 200 MPa. Similarly, in [26], the variation in electrical resistance of NiTi shape memory alloy wires is examined during thermo-mechanical loading, applying stresses of 50 and 400 MPa, respectively [27]. analyzes a single load level, conducting thermal cycling tests under a constant applied load of 200 MPa to investigate the deformation, electric behavior, and transformations associated with both the R and martensitic transformations in a near equiatomic NiTi alloy. Some studies have also explored multiple load levels for resistivity measurements, such as in [28], which investigates the evolution of electrical resistivity during thermal and mechanical tests of NiTi wires undergoing R-phase transformation. In [29], experimental characterization of the electrical resistance of a shape memory material is presented, examining its dependence on stress levels ranging from 0 to 150 MPa and correlating the resistance variation with the strain recovered by the shape memory material. Finally, in [30], the authors describe experimental methods used to investigate the electrical transport properties of shape memory alloys. They utilize a newly developed apparatus that enables precise control of the current waveform applied to the shape memory element while measuring electrical resistance and displacement. Two specific materials are tested under several constant stress states, ranging from 9 to 400 MPa. Additionally, [31] demonstrates that electrical resistance (ER) measurements can be simultaneously detected during stress-strain tensile tests performed on NiTi specimens, providing insights into macro and micro-structural changes in shape memory alloys for quality control and material characterization purposes.

In order to enable sensing, it is necessary to accurately determine the electrical resistivity of both the austenitic and martensitic phases under varying stress conditions. The literature reveals a deficiency in the development of easily reproducible automatic measurement systems in the laboratory, lacking the measurement scheme and associated measurement tools. Furthermore, there is no reported data on the measurement uncertainty, which cannot be evaluated without knowledge of the measurement device model. In addition, the literature does not describe a clear methodology for defining the requirements and regulations to be employed in electrical resistance measurements. The objective of this work is to provide a comprehensive set of engineering-scale characterization parameters for both the electrical and thermo-mechanical/mechanical aspects of a commercial shape memory alloy filament from SAES Getters [32]. These parameters are extracted from commonly used constitutive models for shape memory materials as expressed in [33] and [34]. Furthermore, the current study provides guidance on selecting the appropriate instrumentation based on the prevailing regulations. Specifically, a mechanical characterization was conducted through static tensile testing in order to estimate the elastic moduli of the different phases of the material. Additionally, a thermomechanical characterization was performed to estimate the phase diagrams, ensuring a comprehensive understanding of the transformation temperatures under varying applied stress. The present study explores the evolution of electrical resistivity under a relatively narrow range of stress and temperature conditions. The investigation ranges were selected according to the following criteria: (i) The stress levels (10 to 293 MPa) aim to cover the operating range typically used for SMA wires based on supplier specifications and prior literature [32, 35]; (ii) the temperature span of 25 to 160 °C was selected to fully capture the phase transformations between martensite and austenite. The measurement methodology and proposed model do not account for partial transformation cycles or incomplete phase transformations.

Expanding the characterization to larger stress and temperature ranges would certainly enable developing more complete models and comprehensive understanding of the coupled electro-thermo-mechanical phenomena. However, extreme stresses and temperatures level, that is beyond the critical transformation values, could cause material degradation with permanent effects like plastic deformation or microstructural changes.

Finally, a dedicated experimental setup was developed to accurately measure the variation of electrical resistivity as a function of temperature on the SMA filament under various levels of constant stress, utilizing electrical measurement equipment and a unique custom software developed in LabView.

Materials and Methods

Thermo-Mechanical Characterization

The material used in this investigation is a commercial Ti rich NiTi-based SMA wire Ti 51 at % (SmartFlex, SAES Memry, USA [35]) with diameter d = 480 μm. Figure 1(a) report significant thermo-mechanical properties of the SMA, as obtained from characterization tests, that is in terms of isothermal stress strain responses (Fig. 1(b)) and DSC thermogram (Fig. 1(a)).

The thermogram in Fig. 1(a) was obtained from a heating/cooling cycle (\(\dot{\text{T}}=1.67\text{*}{10}^{-2}{ ^\circ \text{C*s}}^{-1}\)) between − 29.5 and 149.5 °C. The measured values of the transformation temperature (TTs), namely martensite start (\({\text{M}}_{\text{s}}\)), martensite finish (\({\text{M}}_{\text{f}}\)), austenite start (\({\text{A}}_{\text{s}}\)) and austenite finish (\({\text{A}}_{\text{f}}\)), are also shown on the figure. The peak observed between 80 °C and 45 °C during the cooling phase (Fig. 1(a)) clearly indicates the presence of the R-phase. However, it is important to note that this study is solely focused on evaluating the properties of the austenitic and martensitic phases. Any other effects, including the R-phase transformation, are beyond the scope of this work. Figure 2(b) illustrates isothermal stress-strain response of the two material phases (A and M), as obtained from strain controlled quasi static loading tests (\(\dot{{\upepsilon }}={1\text{*}10}^{-4}{\text{s}}^{-1}\)) by using an electromechanical testing machine (MTS Criterion M42, 1kN N Load Cell). In particular, austenitic and martensitic phases were tested at T = 130 °C > > Af and T = 25 °C, respectively. Main mechanical parameters of the two phases are also shown in the figure, that are in terms of Young’s moduli of austenite (EA), twinned (\({\text{E}}_{\text{M}}^{-}\)) and detwinned martensite (\({\text{E}}_{\text{M}}^{+}\)), detwinning stresses (\({{\upsigma }}_{\text{dtw}}^{\text{s}}\), \({{\upsigma }}_{\text{dtw}}^{\text{f}}\)). Unloading curves can provide additional insights related to superelasticity and functional fatigue but they were not included since the focus was on parameters required for modeling the shape memory effect.

Fig. 1
figure 1

a DSC thermogram reporting transformation temperatures values, b Thermomechanical properties of the investigated SMA: isothermal stress strain curve for T = 130 °C > > Af and for T = 25 °C < Mf.

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Constant Stress Recovery Tests

In order to conduct recovery tests under constant stress conditions, a specially designed experimental testing setup was utilized (as depicted in Fig. 2). The test frame was assembled using modular aluminium profiles of appropriate stiffness. The top rod of the frame was fitted with a load cell (HBM Model U9C, 1 kN) to which the upper end of the SMA wire was connected. The lower end of the wire was connected to a support structure for staking dead weights that was directly linked to an HBM Model WA LVDT displacement transducer (+/- 5 mm) mounted on the bottom of the frame. Both ends of the SMA wire were connected using special clamping elements with threaded grains. To apply controlled electric current profiles, a programmable current power supply unit (PowerSupply AimTTi CPX400DP) was employed. The force and displacement signals from the sensors were captured by a data acquisition unit (HBM QuantumX DAQ). Non-contact temperature measurements were obtained using an infrared (IR) camera (FLIR A615). Due to the limited surface area of the shape-memory alloy (SMA) wire, direct temperature measurement using a IR Camera would yield inaccurate results. To overcome this limitation, a strip of Kapton tape with a known emissivity value of 0.97 was affixed to the SMA wire. This expanded the measurable surface area, making it more compatible with the thermal camera optics. This methodology is corroborated by the protocol outlined in the FLIR Tools software manual [36], which was employed for data processing. This experimental setup enabled accurate testing and analysis of the SMA wire’s recovery behavior under constant stress conditions.

Fig. 2
figure 2

Representation of the experimental setup designed for conducting recovery tests under constant stress conditions

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Quasi static tests were performed to analyses the effects of the thermo-mechanical coupling of the SMA wire, i.e. to measure the thermal recovery and the corresponding transformation temperatures (TTs) under different applied stresses. To this end, the samples were preliminary pre-strained (ɛL = 3.82%) at T < Mf and the activation was carried out by means of a current ramp (\(\dot{\text{I}}\)= 0.0375 A/s) to a maximum value of 1.5 A, to obtain the complete strain recovery by full austenitic transformation (T > Af). The electric current was then reduced to zero (\(\dot{\text{I}}\)= -0.075 A/s) after a holding time of 30 s (see Fig. 3(a)) to reset the original detwinned martensitic structure (T < Mf). Figure 3(a) shows the current profile together with the corresponding temperature evolution. Ten subsequent activation cycles were carried out with a 50% duty-cycle, to obtain a stable response of the SMA wire. The stabilized thermal hysteresis loops were used to identify the phase transition temperatures (TTs) and the strain recovery by means of the tangent method [1], as shown in Fig. 3(b).

Fig. 3
figure 3

a Electrical current profile and the corresponding temperature evolution during the constant stress recovery tests; b tanged method used to determine phase transition temperatures on the stabilized 10th cycle

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Electrical Resistance Measurements

Special tests were designed and executed to capture the evolution of material resistivity during thermally-induced phase transformations, taking into account the complex electro-thermal-mechanical coupling mechanisms in SMAs. The material resistivity is a key parameter to define the electric input in SMA-based actuators, that is to allow complete thermally induced phase transformation (T > Af), but its evolution can be also used in self-sensing applications that is the ability of the actuator to monitor its own motion during the operational process. To this aim electrical measurements were carried out during thermal activation under different applied stresses and measuring the temperature evolution. Testing equipment and measurements methods are described in next subsections.

Design Requirements and Specifications

The electrical characterization for resistivity measurement involves several parameters and strict standards for instrumentation, samples and the testing environment must be considered for accurate/reliable results. Unfortunately, neither specific standards nor guidelines for measuring electrical resistivity of shape memory alloys are available and therefore basic requirements were obtained from existing general standards. In particular, ASTM F2516 [37], ASTM F2082 [38] and ASTM F2004 [39] standards and some literature works [40, 41] were considered. ASTM F2516 provides guidelines for conducting tensile tests for super elastic NiTi and both ASTM F2082 and F2004 provide specifications for the determination of transformation temperature, whereas no specific standard is available for the electrical characterization of SMA wires. The most significant requirements come from ASTM F2004, which suggests controlling the heating and cooling procedure throughout the entire test duration. Table 1 summarizes all the specifications considered during the experimental setup design phase.

Table 1 Specifications used during the construction of the experimental setup for characterizing the electrical resistivity of SMA wires and for building the acquisition system
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Considering the lack of standard methodologies for measuring electrical properties of SMA materials, various experimental setups have been proposed in the literature. Abdullah et al. [23] proposed a setup to measure the electric resistance of a SMA wire subjected to stress-induced phase transformations (see Fig. 4(a)). To this aim a horizontally positioned SMA wire is loaded with weights via pulleys and its length variation is measured. The voltage measurements are performed using a cDAQ 9174 Data Acquisition System, incorporating a NI 9201 module. The module, with an ADC resolution of 12 bits, allows for an input voltage range of ± 10 V with a typical accuracy of ± (0.04% of reading + 0.07% of range) under ambient temperature conditions. Song et al. [10] developed a setup (see Fig. 4(b)) to capture the electrical resistance of the SMA wire during thermally-induced phase transformations under constant mechanical loads. The wire is connected to a Wheatstone bridge as an unknown resistance and powered by a 12 V DC power supply. The voltage across the wire is measured and collected using a Quanser MultiQ PCI data acquisition interface. The temperature of the wire is measured by a thermocouple. However, no details on the accuracy of the instruments are provided. Antonucci et al. [43] proposed a thermostatic chamber-based setup (see Fig. 4(c)), in which the wire is cooled and heated at constant rate. The electrical resistance is measured using a custom-made four-probe setup. A constant electric current of 478 µA + 0.1% is provided via a self-built generator, and the resistance is retrieved using a low noise signal amplifier with a gain factor of 1025 ± 2% and a 25 kHz bandwidth. The data was acquired using a NI DAQ-PAD 6052E acquisition system and processed using LabVIEW. However, no information was provided on the accuracy of the electrical measurements, or the acquisition system used.

Fig. 4
figure 4

Comparison of three experimental setups for measuring the resistance behavior of shape memory alloys (SMA) wires: a Abdullah et al. proposed a metallic frame-based setup; b Song et al. proposed a horizontal setup, and Antonucci et al. and c proposed a thermostatic chamber-based setup

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Proposed Measurement Setup

As a result of a comprehensive analysis of current methods and systems, there is a significant need to develop a system with a systematic and well-defined methodology that is specifically designed to evaluate low-value resistances and enable the accurate determination of the resistivity for SMA wire material. The proposed measurement system aims to perform dynamic measurement of the SMA wire temperature and electrical resistivity during thermally-induced phase transformations caused by complete heating/cooling cycles. The heating of a SMA wire is achieved through the phenomenon of Joule heating, which occurs because of the flow of an electrical current through the wire.

To minimize effects of variable ambient conditions, all electrical characterization tests were conducted inside an enclosed thermal chamber maintained at a constant temperature of 22 ± 1 °C [26]. This stabilized test temperature was verified at the beginning of each experimental trial using a thermocouple sensor positioned adjacent to the SMA wire sample. The stabilized ambient temperature was then entered into the thermal camera software as a reference TEnv value for accurately imaging the SMA wire surface temperature throughout testing. By controlling airflow and maintaining a steady ambient temperature surrounding the wire, potential measurement variations caused by unstable environmental conditions were minimized.

The system is composed of two multimeters, a power supply, and an infrared (IR) camera. These instruments are arranged to form an automatic measurement system that can be remotely configured and controlled through a dedicated software platform developed using the LabVIEW environment. The configuration of this system eliminates the need for operator intervention during the measurement process, thus reducing the possibility of introducing random errors. The system measures the temperature of the wire using an IR camera, while the electrical resistivity is calculated based on the measured electrical resistance value. The system performs these measurements dynamically, allowing for real-time analysis of the wire’s temperature and resistivity during both heating and cooling phases. The ability to monitor the wire’s properties in real-time provides valuable insights into the behavior of the material and its response to external stimuli. Additionally, the system’s remote-control capability allows for convenient and efficient operation, enabling researchers to focus on data analysis and interpretation.

  • Keithley 2002 multimeter;

  • Agilent E3631A power supply;

  • GW Instek GDM9061 multimeter;

  • FLIR A615 thermal camera;

  • Keysight E5810B LAN/GPIB/USB gateway.

The Keithley 2002 Multimeter is utilized for measuring the voltage across the Shape Memory Alloy (SMA) wire, while the Instek Multimeter is used to measure the current flowing through the SMA wire. The temperature of the wire is measured contact-free using the IR camera, and the control of the heating current is achieved through the Agilent Programmable Power Supply. The Keysight Gateway is necessary to facilitate communication between the two GPIB instruments, the Keithley multimeter and Agilent power supply, to connect them on the system network. The configuration of the system is schematically shown in Fig. 5.

Fig. 5
figure 5

Configuration of the experimental system for measuring the electrical resistivity of shape memory alloy wires: illustrating the utilization of Keithley 2002 Multimeter, Instek Multimeter, Thermal Camera, Agilent Programmable Power Supply and Keysight Gateway

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Measurement Method and Data Acquisition System Working Principle

Due to the low electrical resistance value of the SMA wire (< 1 Ω) and the strict requirements for temperature control by the electrical current, the most appropriate measurement configuration is the Voltmeter-Ammeter method, as depicted by the equivalent circuit in Fig. 6(a). The Voltmeter is positioned downstream of the Ammeter, which results in the optimal configuration for measuring low resistance values.

Fig. 6
figure 6

Schematization of the equivalent circuit for the optimal Voltmeter-Ammeter method for measuring low resistance values in a temperature regulated wire

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The resistance evaluation using the Voltmeter-Ammeter Method is based on the measurement of the voltage (VRx) across the unknown resistance (Rx) and the current (IRx) flowing through the wire. The resistance is determined through the application of Ohm’s law:

$${\text{R}}_{\text{x}}=\frac{{\text{V}}_{\text{Rx}}}{{\text{I}}_{\text{Rx}}}$$
(1)

Upon the determination of the resistance (Rx), the resistivity can be obtained by Equation (2):

$${\rho}={\mathrm{R}_{\mathrm{x}}}\frac{\mathrm{S}}{\mathrm{l}}$$
(2)

where l is the sample length (see Fig. 6(b)), as measured between the two measurement points, and S is the cross-sectional area of the sample. The resistivity values were obtained using a probe distance of 38 mm and assuming a constant wire diameter of 0.181 mm2. The cross-sectional area S of the wire is constant as its variations are deemed negligible. The length l is set constant as the distance between two sliding probes. This setup allows to consider the effects of wire length variation, possibly due to shape memory recovery, on the electrical resistance of the wire. This effect is negligible, with respect to resistivity changes linked to phase transformations between martensite and austenite, when dealing with small transformation strain, but it could play a more important role when increasing the wire length variations occurring under high applied stresses.

The entire system, which encompasses multiple measurement cycles including the management of heating and cooling of the wire, is controlled by a custom-built software platform in the LabView development environment. The implemented virtual instrument (VI), as depicted in Fig. 7, consists of a first section for instrument setup, followed by activation of the power supply and initial temperature acquisition to determine the starting temperature of the wire.

A second section of the VI is devoted to managing the sample heating phase, where temperature, voltage, and current measurements are taken, and the resistance is evaluated. This is followed by a section like the second but dedicated to the cooling phase of the sample.

Fig. 7
figure 7

Logical schematic representation of the Virtual Instrument (VI) for resistance evaluation in sample heating and cooling phases

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The maximum current value was selected after conducting multiple experiments, as shown in Fig. 8(a). This figure shows the maximum temperature reached for five different levels of current supply. A current value of 1.8 A was found to raise the wire temperature to approximately 150 °C and enable complete phase transformation even under high applied stresses.

During each experimental trial, 10 measurement cycles were conducted (see Fig. 8(b)) to verify the stability/repeatability of the measurement procedure. Each cycle was partitioned into 180 points and the current input is provided by increments according to this discretization. The initial flat temperature segments observed in Fig. 8(a) are a result of a deliberate delay in the onset of the heating cycle, during which the current is set to zero amps to allow the measurement system to stabilize. This delay varies in duration across tests, but it does not influence subsequent measurements.

Fig. 8
figure 8

a The temperature progression observed during heating stages conducted at varying levels of electrical current; b An illustration of 10 consecutive heating and cooling cycles during electrical resistance measurement

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Results and Discussion

The following subsections report main results on electro-thermal-mechanical coupling in SMA wire, that is in terms of the evolution of thermal recovery properties and electric resistivity under different applied mechanical stresses.

Thermal Recovery Properties: Effects of Applied Stress

Figure 9 shows stabilized strain-temperature hysteresis cycles, as obtained from 10 subsequent thermal cycles, under different values of applied stress. It is shown that both strain recovery and TTs are affected by the applied stress. Furthermore, a complete functional stabilization is observed at the 10th cycle in terms of both strain recovery and TTs.

Figure 10(b) shows the evolution of recovery strain at the 1st and 10th thermal cycles as function of the applied stress. Strain recovery at the first cycle is almost stress independent, that is linked to the training process carried out by material manufacturer. On the contrary, a marked effect of the stress is observed on the stabilized strain recovery (10th cycle), with a rapid increase in the range 0-100 MPa and a subsequent stable response at higher stress levels. It is attributed to incomplete martensite reorientation on cooling below martensite detwinning plateau.

Figure 9 also shows an increase in transformation temperatures (TTs) with increasing applied stress, according to Clausius-Clapeyron relation. Measured TTs under the different test conditions are summarized in Fig. 10(a). The Clausius-Clapeyron constants for the four TTs (CMs, CMf, CAs, and CAf) were obtained from linear fits of the measured points. Different slopes were observed for the four TTs. Additionally, an intersection between the Ms and As curves was observed when increasing the applied stress (σ ~ 170 MPa), as previously reported in [1]. This convergence of the curves results in a narrowing of the hysteresis loop with increasing applied stress. It’s worth noting that more intricated phenomena occurs in the near TT0 regions of the phase diagram, leading to possible negative values of Clausius-Clapeyron constants [44]. These effects could be captured by more precise DSC experiments that are out of the scope of this investigations, where only the global response is captured.

Furthermore, data scattering present for some transformation temperature points in the phase diagram of Fig. 10(a) can be attributed to both measurement uncertainties and inherent material behavior. The simplified linear trends in the stress-temperature phase diagram may fail to capture some complex features, as noted in [44], that are not considered in this investigation. While the overall increasing transformation temperature dependence on applied stress agrees with expectations, localized deviations can arise from limitations in precisely identifying phase transformation onset/finish as well as subtleties in the thermo-mechanical response.

A summary of all thermo-mechanical parameters of the SMA wire resulting from the thermo-mechanical characterization is reported in Table 2.

Fig. 9
figure 9

Stabilized hysteresis cycles for different values of applied constant stress condition (σ = 10 up to 265 MPa)

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Fig. 10
figure 10

a Stress-Temperature phase diagram of the shape memory alloy (SMA) wire obtained from constant stress recovery experiments; b Recovery strain vs. stress, illustrating that the recovery effect in the first cycle is very close to the nominal value imposed by the supplier (3.8%), while the 10th cycle shows a sharp decrease in recoverable strain when the load levels are low

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Figure 10 shows a mismatch between the zero-stress transformation temperatures and those measured by DSC in Fig. 1(a). This effect is primarily due to thermomechanical cycling. Specifically, 10 consecutive actuation cycles were performed prior to determining the phase transformation temperatures. In contrast, the initial DSC results provide an indication of behavior before any training. As reported in [1], cycling induces dislocations and non-uniform internal stresses at the microscale. This localizes the transformations and tends to flatten the DSC curves with lower evidence of the transformation peaks of the DSC thermograms. Therefore, following stabilization, alternative methods like the constant stress recovery testing used here become necessary. This allows accurately determining the stabilized zero-stress transformation temperatures relevant for modeling actuation applications after training has occurred.

Table 2 Thermomechanical parameters extracted from experimental characterization tests and commonly required by phenomenological models describing the behavior of shape memory alloys
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Electrical Characterization Results

Electrical characterization tests of SMA wires were carried out under complete heating/cooling cycles at different levels of applied stress, in the range between 10 and 293 MPa. Figure 11 reports the hysteresis cycles (10th cycle) resistance vs. temperature (graphs on the left) and resistivity vs. temperature (graphs on the right) as obtained by Equation (2). The resistivity change depends mainly on the volumetric fraction of austenite (ρA) and martensite (ρM). These latter values can be directly compared with those reported by the manufacturer in [32].

Fig. 11
figure 11

Summary of stabilized cycles (10th ) showing electrical resistance measurements and resistivity values at different levels of constant stress applied to the SMA wire, ranging from σ = 10 to 293 MPa

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A summary of the maximum and minimum resistance and resistivity values, as obtained from the graphs shown in Fig. 11, is reported in Table 3, here the uncertainty on the resistance values are of ± 0.00001 Ω, based on the used instrumentations datasheet. An increase of both resistance and resistivity is observed with increasing of the applied stress. The graphs in Fig. 12 report, on a double y-axis scale, the evolution of recovery strain (εL) and resistivity (ρ) as a function of the temperature for the stress values of 10, 85, and 216 MPa.

Table 3 Maximum and minimum resistance and resistivity values obtained from the conducted experimental tests. The results demonstrate a consistent increase in the electrical parameters of resistance and resistivity with the applied stress on the SMA wire
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Electrical characterization data are of very critical concern in both numerical and analytical modelling of SMA actuation systems because electrical input required for SMA activation is directly linked to the heat generated by Joule effect. In addition, the real-time evolution of electrical resistance can be used in self-sensing applications, that is for monitoring the state of the SMA actuator [45]. To this aim, the trend of recovery deformation with respect to electrical resistance can be obtained by combining the recovery strain curves (Fig. 9) with the electrical resistivity curves (Fig. 11). These results are shown in Fig. 13 for the three values of the applied stress of Fig. 12. Regression models for predicting the electrical resistivity of SMA as a function of applied stress and temperature were developed. Figure 14(a) shows the evolution of austenite and martensite resistivity (ρA and ρM) as a function of the applied stress. An increase of the electrical resistivity for both phases is observed, and a linear fit was made within the stress range 10–293 MPa. The graph in Fig. 14(b) shows a fitting surface of the electrical resistivity as a function of stress and temperature.

Fig. 12
figure 12

Recovery strain (εtr) and resistivity values measured at stress values of 10, 85, and 216 MPa, during the SMA wire activation phase

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Fig. 13
figure 13

Combined plot of Recovery Strain and Electrical Resistivity curves, which describes the direct variation of recovered deformation with respect to the electrical resistance of the material

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A distinct resistivity peak is observed prior to the onset of phase transformation, as seen in Fig. 13c. In the temperature range below As, the resistivity follows a linear relationship as a function of temperature and applied stress [45]:

$$\rho\left(\sigma, \mathrm{T}\right)=\rho_0\left(\mathrm{T}_0\right)+\alpha\left(\mathrm{T}-\mathrm{T}_0\right)+\beta \alpha$$
(3)

where ρ0 is the resistivity at reference temperature, α is the temperature coefficient of resistivity (TCR), and β is the stress coefficient of resistivity (SCR). As shown both in Figs. 12 and 13, this peak resistivity value becomes increasingly larger at higher stresses, primarily due to the additive contribution of the stress-dependent β term [45]. Also the temperature have a great contribution, this is due the increasing ΔT needed to complete the transformation at higher stress.

Additionally, phase transformation velocity accelerate at higher stresses, as evidenced by steeper slopes between As and Af during constant stress recovery tests, Fig. 9. The increasing recovery strain causes elongation and “stretching” of the hysteresis loops. This manifests as more rapid resistivity decrease following the peak, since the transformation occurs more readily under increased applied stress.

Fig. 14
figure 14

a Linear interpolation of electrical resistivity values for the martensitic and austenitic phases at different levels of constant applied stress; b Surface representation of the relationship between electrical resistivity, temperature, and applied stress

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A non-linear surface fitting was made, and the expression (4) represents the regression model of the material resistivity during the heating phase within the stress and temperature domains 10\(-\)293 MPa and 25-160 °C, respectively.

$${\uprho }\left(\stackrel{-}{\text{T}},\stackrel{-}{{\upsigma }}\right)={{\uprho }}_{0}+{\upalpha }\text{cos}\left(\frac{\text{T}}{{\text{w}}_{1}}\right)+{\upbeta }\text{sin}\left(\frac{\text{T}}{{\text{w}}_{1}}\right)+ {\upgamma }\text{cos}\left(\frac{{\upsigma }}{{\text{w}}_{2}}\right)+{\updelta }\text{sin}\left(\frac{{\upsigma }}{{\text{w}}_{2}}\right)$$
(4)

where \({{\uprho }}_{0}, {\upalpha }, {\upbeta }, {\upgamma }\) and \({\updelta }\), whose values are given in Table 4 and are the coefficients of the interpolating function that best approximate the surface represented in Fig. 14(b). The regression model was validated by comparing the heating curves obtained from the model with the experimental one as reported in Fig. 15, that shows a good agreement between the two results. The adopted empirical fitting function was found to provide the best fit to the experimental data after evaluating other regression functions. The proposed expression is valid within the specified temperature and stress ranges and has been validated only under the condition of constant stress recovery during the heating phase. It does not, however, take into account more complex loading conditions, involving partial transformations, such as constrained recovery, partial load/unload with internal loop, and biaxial load/unload conditions. More complex physically-based models are available in [45, 46].

Table 4 Parameters extracted from the 3D nonlinear fit for the function given by expression (4)
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Fig. 15
figure 15

Comparison between experimental and interpolated resistivity values of the material under heating conditions

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Limitations and Future Work

The proposed methodology and models have some key limitations that the reader wanting to use the data should take into account:

  • The measurement methodology and proposed model do not account for partial transformation cycles or incomplete phase transformations. The model has been validated only for complete heating/cooling cycles that result in full phase transformations between martensite and austenite;

  • The stress and temperature ranges considered for model development were 10–293 MPa and 25–160 °C.

  • The simplified linear trends in the stress-temperature phase diagram may fail to capture some complex features that are not considered here;

  • The empirical fitting function used for the resistivity model provides good agreement with the experimental data but has limitations, indeed it can be used only inside the stress-temperature ranges analyzed.

  • The wire length variation effects can be assumed negligible in this work. These could become more prominent at larger recovery strains induced by high stress levels.

Future work could focus on:

  • Expanding the stress-temperature range for resistivity measurements and model development;

  • Considering more complex loading conditions like partial phase transformations, constrained recovery, etc.;

  • Incorporating physics-based modeling approaches for the resistivity;

  • Accounting for wire length change effects under high recovery strains.

Conclusion

The effect of applied stress on the functional response of SMA-based wire actuators is analyzed, that is in terms of electric resistivity, activation temperatures and shape recovery properties. This information is of major concern when designing SMA actuator systems due to the complex electro-thermal-mechanical coupling mechanisms in SMAs. In fact, thermal activation in SMA wires is normally provided by an electric current through the Joule’s effect. Both activation temperature and electric resistivity are mainly affected by the applied stress, and this represents a major issue in designing the electric actuation system. Main results of this investigation can be summarized as follows:

  • A thorough methodology for evaluating low-value electric resistances in SMA wires during thermally induced phase transformations under applied stresses is proposed;

  • Precise determination of the electric resistivity of a commercial SMA wire was carried out as a function of temperature and stress;

  • A non-linear regression model was developed the represents a comprehensive relationship between electrical resistivity, temperature, and applied mechanical stress;

  • The variation in electrical resistance can be utilized as a sensor to track the SMA actuator’s movement condition in different analytical models for SMA actuators that make use of the electrical characterization data.

  • Mechanical and functional characterization was carried out to measure the evolution of both transformation temperature and shape recovery properties of the SMA wire under complex thermos-mechanical loading conditions.