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Abstract

Self-sensing is one of the key features of intelligent materials. Thermal Shape Memory Alloy (SMA) wires change their electrical resistance during phase transformation from martensite to austenite and vice versa. During cooling, the intermediate R-Phase may form, which has a significant impact on electrical resistance. SMA models incorporating all three phase fractions are required for the design and control of advanced SMA actuators.

This paper investigates the feasibility of a three-phase SMA model that accurately captures both strain and resistance behavior. Based on experimental data from a commercial SmartFlex^® wire, the transformation processes between the three material phases are identified. The proposed model uses phase transformation surfaces to accurately represent stress-dependent changes in transformation characteristics. It is shown how parallel transformation processes that compete for one phase fraction can be modeled. A parameterization process is suggested that uses the measured mechanical contraction and electrical resistance at different stress levels as well as the measured transformation enthalpies of the SMA wire. Model data are compared with experimental results. It is analyzed up to which stress level the R-phase has an impact on electrical resistance, and what role it plays in today’s self-sensing schemes.

Keywords

shape memory alloys constitutive model three-phase model R-phase phase transformation surfaces self-sensing

Introduction

Motivation and objectives

Over the past three decades, shape memory alloys (SMAs) based on nickel-titanium have proved their worth in various application fields. Actuators based on SMA wires provide benefits such as ease of miniaturization, noiseless actuation, and high energy density. In addition, SMA wires also change their electrical resistance with their phase transformation, enabling self-sensing actuator concepts. To date, there is an ongoing trend from simple ‘binary’ actuators moving between two distinct positions toward more complex position-controlled actuators.

However, the highly non-linear and hysteretic behavior of the SMA material poses major challenges. Models are required for the design of more complex actuators, as well as for advanced model-based control strategies. These models should reflect the complex behavior of SMA wires accurately, while being able to run on standard microcontrollers in real-time.

Current SMA models typically describe the phase transformation between the low-temperature martensite phase and the high-temperature austenite phase. Depending on alloy composition, thermal treatment, and aging of the SMA wire, a third rhombohedral (R)-phase may develop. This intermediate phase has little influence on SMA wire contraction and is therefore often neglected. However, the R-phase has a significant impact on the electrical resistance of the SMA material. To date, there are no SMA wire models which include all three phase fractions and calculate wire strain and wire resistance accurately.

This study investigates the feasibility of an SMA wire model that includes all three phase fractions and is capable of representing both the strain and resistance behavior accurately. In addition, the model aims to capture also the transformation enthalpy required for the phase transformations, as this is needed to realistically reflect the heating and cooling of the wire in air. The methodology of this paper is based on a description of material behavior in temperature-stress-strain space and temperature-stress-resistance space as (strain and resistance) surfaces. These contain all information regarding material parameters and transformation behavior (Kazi et al., 2023).

SMA models incorporating the R-phase

Modeling the complex behavior of SMA materials is an ongoing challenge. Numerous models have been developed to understand the complex material behavior and to develop active and passive SMA applications. In their review, Cisse et al. (2016) differentiate between microscopic thermodynamic models, micro-macro models and macroscopic phenomenological models. This section focuses on macroscopic phenomenological models that describe the input-output behavior of entire SMA elements and lend themselves to modeling of SMA actuators. Lagoudas (2008) reviews various macroscopic phenomenological models and distinguishes between free energy-based and phase diagram-based approaches.

Free energy-based SMA models (Ballew and Seelecke, 2019; Paiva et al., 2005; Zhu and Zhang, 2007) describe the thermodynamic states of the material using the Gibbs or Helmholtz free energy. The models commonly have a high degree of adaptability and are therefore able to reflect the measured behavior well. Furthermore, it is possible to calculate three-dimensional stress states, which makes this type of model very valuable for finite element analysis (FEA) of complex geometries. However, free energy-based models are mathematically complex and typically require iterative calculations to minimize the energy landscape at each simulated time step.

Sedlák et al. (2012) developed a model based on Helmholtz free energy and compared it to experimental results from a superelastic 56.0 wt% Ni wire in tensile tests at various temperatures and thermal cycling tests (Pilch et al., 2009). The work focused on the mechanical response associated with the transformation between austenite and R-phase. The simulation of a bulk component was successfully performed on finite element code using parallel processing of eight processor cores.

Rigamonti et al. (2017) extended the Helmholtz free energy model by (Zhu and Zhang, 2007) by including the R-phase. The work considered tensile tests at various temperatures. By adjusting the model parameters, experimental strain data from two commercial Smartflex^® SMA wires with different heat treatments could be represented.

Helbert et al. (2017) extended the model by Bouvet et al. (2004) to three phase fractions. Tensile tests at different temperatures and strain rates were performed with a superelastic 56.3 wt% Ni wire (Helbert et al., 2014). The model matched the general behavior of tensile tests at different strain rates.

While much research in recent years has focused on free energy SMA models, the first SMA models from a historical perspective were based on phase diagrams (Bekker and Brinson, 1997; Govindjee and Kasper, 1999; Liang and Rogers, 1990; Tanaka, 1986). Phase diagrams describe the dependence of the transition temperatures on the stress level. To calculate the phase fractions, different mathematical transformation functions have been used. The advantages of phase diagram-based SMA models are the simplicity of their mathematics and parameterization, and the intuitive presentation of the transformation behavior. However, due to the restricted possibility of adjustment, the accuracy of phase diagram models strongly depends on the transformation function chosen. Phase diagram-based models describe one-dimensional stress states, which is sufficient for modeling SMA wires, but not for SMA elements with complex geometries.

Brammajyosula et al. (2011) introduced a phase diagram-based model for a temperature induced phase transformation. The model used a cosine transformation function to calculate the phase fractions of martensite, austenite, and R-phase. Experimental data was collected with a commercial SMA wire of an unspecified type. Depending on the stress level, the model switched between a direct transformation from austenite to martensite and a transformation from austenite to full R-phase to martensite. The model was shown to be able to reproduce the time behavior of the experimental strain and resistance data qualitatively.

Wang et al. (2021a) used a compound transformation function that combined arctan and sinh to capture the stress-induced phase transformation during tensile tests at different temperatures. Experimental stress-strain data from a 49.5 at% Ni SMA wire provided by Šittner et al. (2006) was used for comparison. It was concluded that both the shape memory effect and the pseudo-elasticity could be represented by the model.

In conclusion, only a surprisingly small number of studies integrate the R-phase into a macroscopic phenomenological SMA model. In most cases, the mechanical stress-strain behavior of a superelastic SMA wire is investigated, in which the R-phase shows as a small characteristic plateau during stress-induced phase transformation. Only the work by Brammajyosula et al. (2011) addresses the behavior of both mechanical strain and electrical resistance at the same time. However, this model does not cover the general case where austenite is transformed into martensite with variable degrees of intermediate R-phase formation. Neither does this model comprise the phase transformation from R-phase to austenite, which is required for inner hysteresis loops. The cosine transformation function is barely able to capture the real transformation behavior of an SMA wire with adequate accuracy.

Conceptual approach

In this section, the concept of the proposed model is developed. For the commercial SmartFlex^® SMA wire used in this study, different types of experimental data are analyzed regarding thermally induced phase transformations and their impact on SMA material behavior (‘Analysis of phase transformations’ section). The ‘SMA model based on phase transformation surfaces’™ section outlines how these phase transformations can be described in the model. Finally, a suitable mathematical description is selected in the ‘Transformation function’™ section.

Analysis of phase transformations

In general, six different phase transformations can occur between the three phase fractions of martensite, austenite, and R-phase. The goal of this section is to determine which of these six phase transformations are relevant for the model, and how they manifest themselves in the SMA material behavior.

For this purpose, an SMA wire sample is put through a heating and cooling process with differential scanning calorimetry (DSC). This method measures the transformation enthalpy via the amount of energy introduced, enabling the identification of the phase transformation processes. On the other hand, heating and cooling of an SMA wire in a temperature controlled silicone oil bath (Kazi et al., 2023) shows the effects of the phase transformations on its mechanical contraction and electrical resistance. The challenge when trying to combine experimental data from both methods arises from different stress levels. While DSC measurements are performed at zero stress, it is essential to apply a minimum stress level to measure the mechanical contraction in the oil bath.

Even small changes in stress will lead to changes in transformation temperatures. In this study, the comparability of mechanical contraction and electrical resistance to the DSC data was achieved by extrapolating measured data from five different stress levels (50–10 MPa) down to 0 MPa. The ‘virtual’ mechanical contraction and electrical resistance at 0 MPa gained from this procedure can be compared to the DSC data.

This work uses a commercial SmartFlex^® NiTi wire of 76 µm diameter produced by SAES Getters, Italy (Fumagalli et al., 2009). This SMA wire is widely used in industrial actuator applications. The SMA wire sample was to be characterized in a somewhat used condition with a pronounced two-way effect rather than ‘fresh from the spool’. Thus, the wire sample was subjected to a run-in procedure of approximately 1500 full transformation cycles under a constant load of 200 MPa before characterization.

After the run-in procedure, the contraction and electrical resistance behavior of the SMA wire sample was measured in the temperature-controlled silicone oil bath. Directly afterward, the same test sample underwent the DSC measurements. Figure 1 shows the resulting data of applied power, strain and specific resistance in the temperature range between 20°C and 100°C. To visualize the introduced energy for the phase transformations, the DSC data are postprocessed with an offset correction according to Michael et al. (2018).

Figure 1.

Comparison of DSC heating power, strain, and specific resistance. The dashed line in the left subfigures marks a single observed phase transformation during heating. The two dashed lines on the right mark two distinct phase transformations during cooling.

During the heating process shown on the left side of Figure 1, a change in applied heating power, strain, and specific resistance is visible around 75°C. While the transformation enthalpy generates a positive peak, the strain and the specific resistance of the SMA wire drop in an almost stepwise fashion. All three changes occur simultaneously over the same temperature range. As the martensite phase fraction has a higher pseudoplastic strain and specific resistance than the austenite fraction, the observed phase transformation is most likely from martensite to austenite (‘M2A’). The data do not provide any evidence of an intermediate formation of the R-phase.

During the cooling process on the right side of Figure 1, the DSC data show two negative peaks around 60°C and 40°C. The first phase transformation at a temperature of 60°C causes a rise in the specific resistance without much impact on the strain. The second transformation around 40°C leads to an elongation of the SMA wire, while the specific resistance changes only gradually. Obviously, the cooling process of the SMA wire comprises two different phase transformations. The R-phase has a high specific resistance, which leads to the conclusion that the first transformation at 60°C is from austenite to R-phase (‘A2R’). The A2R transformation is followed by the transformation from R-phase to martensite (‘R2M’) around 40°C. The elongation of the SMA wire matches well with the high pseudoplastic strain of the martensitic phase.

The phase transformation in the SMA wire is also of interest at higher stress levels. DSC measurements can only be performed at zero stress. However, experiments in the temperature-controlled silicone oil bath can be performed at arbitrary stress levels. Figure 2 compares the strain and resistance data at 0 and 250 MPa for a heating and cooling cycle of the SMA wire.

Figure 2.

Comparison of 0 MPa (left) and 250 MPa (right) experimental runs. The trajectories during heating are colored red and the trajectories during cooling are colored blue. The dashed lines mark the observed changes in strain and specific resistance.

As in Figure 1, the M2A transformation at 0 MPa (red lines in the left subfigures in Figure 2) can be observed in the reduction of both strain and specific resistance around 75°C (A). The cooling process (blue lines) first sees the A2R transformation starting around 73.5°C (B), causing a significant increase in specific resistance. This is followed by the R2M transformation starting around 48°C (C) causing the elongation of the SMA wire. At 250 MPa (right subfigures in Figure 2), the changes in strain and specific resistance occur simultaneously both during heating (around 98°C, D) and cooling (around 80°C, E). This indicates that at high stress levels, no intermediate R-phase is formed, but austenite is transformed directly into martensite (‘A2M’).

It is concluded that the model for the SmartFlex^® wire should comprise five phase transformation processes between the three phase fractions of martensite (M), austenite (A), and R-phase (R). Figure 3 shows three phase transformation processes (A2M, A2R, R2M) during forward transformation (cooling/increase of stress level) in blue, while the two phase transformation processes (M2A, R2A) of the reverse transformation (heating/reduction of stress level) are shown in red. On the basis of the observations reported above, no need is seen to include a phase transformation from martensite to the R-phase.

Figure 3.

The three phase fractions martensite, austenite and R-phase with the phase transformations in-between.

SMA model based on phase transformation surfaces

SMA models intended for the use in model-based actuator control should require a low computational effort in order to be capable of running in real-time. The iterative calculations typically involved in free energy-based SMA models make them appear less attractive for this purpose. Phase diagram-based models are usually computationally less demanding. Their limitation to one-dimensional stress states is not a drawback as long as actuators driven by SMA wires are targeted. However, conventional phase diagram-based SMA models may not be able to represent the contraction and electrical resistance behavior of the SMA wires with a sufficient degree of accuracy.

This work seeks to extend the concept of phase diagram-based SMA models to enable them to reflect the measured mechanical and electrical behavior of SMA wires more accurately. If phase diagrams are added with a third axis to reflect the phase composition, three-dimensional phase transformation surfaces are obtained (Liang and Rogers, 1990). Figure 4 illustrates how phase transformation surfaces are able to represent a phase transformation more accurately: the phase transformation surfaces show an increasingly rounded and asymmetric shape of the transformation for higher stress levels. The shape of the transformation is not reflected in the transition temperatures that would be shown in a phase diagram.

Figure 4.

Phase transformation surfaces for the forward (blue) and reverse (red) transformation between austenite and martensite. The straight blue and red lines show the phase diagram-like transition temperatures.

Figure 4 depicts the two phase transformation surfaces that govern the martensite-to-austenite (M2A) and austenite-to-martensite (A2M) transformations in the temperature-stress-phase fraction domain. However, the proposed three-phase model requires a total of five phase transformation surfaces. In this work, these phase transformation surfaces are derived from the SMA material response viewed from two complementary perspectives: the measured surfaces in temperature-stress-strain space and in temperature-stress-resistance space for forward and reverse phase transformation.

These strain and resistance surfaces can be characterized using the experimental methodology outlined by Kazi et al. (2023). The experiments start from initial states of ‘full martensite’ or ‘full austenite’, followed by a suite of different thermomechanical testing trajectories. For a continuous forward or reverse transformation, each measured trajectory of the tested SmartFlex^® wire proceeded on the same surfaces. The SMA model must be able to capture these strain and resistance surfaces with sufficient accuracy. For this purpose, the selection of an appropriate transformation function is of key importance.

Transformation function

Transformation functions are defined either for temperature- or stress-induced phase transformation. In SMA models developed for actuator applications, transformation functions are usually temperature dependent. Several different transformation functions have been used. ‘Classic’ transformation functions like cosine (Liang and Rogers, 1990), hyperbolic tangent (Ikuta et al., 1991), or arc tangent (Nascimento et al., 2009) depend only on the transition start and finish temperatures. Their adaptability to experimental data is therefore limited.

Experimental data recorded from a SmartFlex^® SMA wire show a rounded shape at the beginning and end of the transformation, with a steep transition in between (see Figure 5). Clearly, the classic transformation functions are not able to represent the experimental data properly.

Figure 5.

Transformation functions compared to experimental data from a SmartFlex^® SMA wire.

A more modern approach to mathematically represent the transformation between two SMA material phases is the use of Bézier curves (Enemark and Santos, 2016). Bézier curves are constructed by recursively performing linear interpolation between support points defined in the T-z-plane. The Bézier curve in Figure 5 uses four support points (i.e. eight parameters) and is calculated with the help of a Bernstein polynomial. Compared to the classic transformation functions, the fit of the Bézier curve with the experimental data is much improved. The fit could be further optimized with an increased number of support points; their number would also need to be increased to cover a larger temperature range. For the user, the coordinates of the support points of a Bézier curve tell little about its shape and are not very intuitive. Another disadvantage is that each point on the curve is calculated from an internal variable. Thus, an iterative process is required to determine the phase fraction as a function of temperature.

A recent suggestion still largely unknown is the so-called ‘ihyp’ function (Kazi et al., 2021). The idea is to define the inverse transformation function T(z) based on the sum of two hyperbolic functions and a linear function (see Figure 6).

Figure 6.

The individual ihyp components are two hyperbolic and one linear function.

The curvatures at both ends of the transition are defined by the hyperbolic functions, while the linear function is used to adjust the gradient in the steep section of the transition. The resulting function can also display variable degrees of asymmetry. By reformulation, the function can be expressed in terms of the common transition start and finish temperatures. The asymmetry parameter z_T shifts the turning point between the two hyperbolic functions toward the initial phase or the target phase. The curvature parameter Γ controls the ‘radii’ adjacent to the steep section. Algebraic inversion of the function is possible using Cardano’s formula. More details on the ihyp function can be found in (Kazi et al., 2021).

The authors of this paper also studied the use of higher order hyperbolic functions to match experimental data with sharper ‘bends’ in the phase transformation. However, this makes numeric methods based on root-finding algorithms necessary for inversion. These algorithms result in significantly (e.g. factor 100) higher computation times, which makes higher order hyperbolic functions less attractive for the use in real time applications.

Already with standard hyperbolic functions, the ihyp transformation function achieves a high degree of adaptability with only four intuitive parameters. It can be tailored to match the experimental data well (see Figure 5). It can be expected that the ihyp function could also be adapted to different alloy compositions and thermomechanical histories. The phase fraction can be calculated for a given temperature and stress without the need for time-consuming iterations. Therefore, it is selected as the mathematical basis to describe phase transformation surfaces in this study.

Model structure

After the essential ingredients of the proposed model have been introduced in the previous section, the model structure is now outlined. While the basic layout is designed for two phase fractions, the innovation in the ‘Model layout’ section is the inclusion of the additional R-phase fraction. The complexity resulting from the interaction between the phase transformations is investigated in the ‘Transformation hysteresis’ section.

Model layout

In general terms, the SMA model links the input variables temperature T and stress σ to the output variables strain ε and specific resistance ρ. The martensite phase fraction z_M and the newly added R-phase fraction z_R can be considered as internal variables. Since z_M and z_R are defined between 0% and 100%, the austenite phase fraction z_A is determined by

z_{A} = 1 - z_{M} - z_{R}

Figure 7 shows the layout of the model.

Figure 7.

General layout of the SMA model with martensite phase fraction (blue) and R-phase fraction (green).

The model comprises three major building blocks: the Transformation Hysteresis block computes the phase composition of the SMA wire from its temperature and stress and implements the phase transformation surfaces. The Stress-Strain Relation block produces the mechanical behavior from the phase composition. The Electrical Resistance block derives the specific resistance ρ also from the phase composition, but may also have to take the stress σ into account. The behavior of all three blocks has to be derived from the measured strain and resistance surfaces, which can be regarded as the result from the combination of the Transformation Hysteresis with the Stress-Strain Relation or the Electrical Resistance blocks, respectively.

A fourth Energy Balance block, linking the specific electrical heating power to the SMA wire temperature, is not part of this study, as the SMA wire temperature in the characterization experiments is directly controlled by means of a silicon oil bath. However, the Energy Balance block also requires the phase fractions of martensite z_M and the R-phase z_R as inputs. Thus, the Energy Balance block is able to reflect the transformation enthalpies, which have a significant impact on the heating and cooling of the SMA wire in air.

Compared to a conventional two-phase model structure, the Transformation Hysteresis block has to calculate the R-phase fraction z_R as an additional output, while the Stress-Strain Relation and Electrical Resistance blocks have to process the R-phase fraction as an additional input. For the Stress-Strain Relation block, a serial or a parallel arrangement of the phase fractions can be assumed (Concilio et al., 2021).

Due to the high degree of flexibility of the ihyp function, the choice of structural arrangement can be expected to have a comparatively minor influence on the quality of the fit with the measurement data. In previous two-phase models developed by the authors, the serial configuration produced good results when modeling mechanical strain. Therefore, it was chosen as the starting point for the proposed Stress-Strain Relation. This leads to adding the strain of the phase fractions according to

ε (σ) = z_{M} \cdot ε_{M} (σ) + z_{R} \cdot ε_{R} (σ) + z_{A} \cdot ε_{A} (σ)

The Electrical Resistance block is more complex, since the specific resistance of a phase fraction depends on its temperature as well as on the stress level. The specific relationship will be analyzed during parameter identification in the ‘Model parameters for different stress levels’ section.

As a starting point, a serial arrangement of the phase fractions was also assumed for the Electrical Resistance block.

ρ (T, σ) = z_{M} \cdot ρ_{M} (T, σ) + z_{R} \cdot ρ_{R} (T, σ) + z_{A} \cdot ρ_{A} (T, σ)

By far the most complex part of the model is the Transformation Hysteresis block. It is discussed in the following section.

Transformation hysteresis

The Transformation Hysteresis distinguishes between two directions: the forward transformation toward martensite is associated with decreasing temperature and/or increasing stress, while the reverse transformation toward austenite is caused by rising temperatures and/or decreasing stress. For improved clarity, the following visualization is based on a temperature-induced phase transformation, even though stress-induced phase transformation is also covered by the model. Starting from the well-known switching process between two phase fractions in ‘Switching the transformation direction’, the sections ‘Heating process for three phase fractions’ and ‘Cooling process for three phase fractions’ describe the interaction of three phase fractions during the reverse and forward transformation.

Switching the transformation direction

SMA wires used in actuators do not only perform complete phase transformations, but also inner hysteresis loops. Thus, the transformation must be able to start from any phase composition. Liang and Rogers (1990) suggested to simply scale the transformation function directly to the current phase fraction value. An example trajectory is given in Figure 8.

Figure 8.

Scaling of the outer hysteresis curves for inner hysteresis loops.

Assuming an incomplete heating process along the outer M2A transformation up to the temperature T_rev, scaling of the cooling function (A2M) is performed such that

z_{A 2 M} (T_{rev}, σ_{rev}) = z_{M_rev}

z_{A 2 M} (T \to - \infty) = 1

More recent work suggests to adapt the transformation function when scaling (Nascimento et al., 2009; Wang et al., 2021a). For simplicity, this study used scaling without adaptation as a starting point. Figure 8 indicates that an exact representation of the outer hysteresis loops is crucial, as also the inner hysteresis loops are derived from them. However, in a three-phase SMA model the mathematical description becomes more complex, as several phase transformation processes may run in parallel, and not all of these processes are independent of each other. This has severe implications on the required scaling of the transformation functions.

Heating process for three phase fractions

The heating process simultaneously produces austenite from the existing martensite (M2A) and R-phase (R2A) fractions. Both phase transformation processes are not competing for a common resource (i.e. a phase fraction). Thus, they can be assumed to run independently of each other. Figure 9 shows a graphical representation of this situation.

Figure 9.

During heating, two independent phase transformation processes (M2A, R2A) are producing austenite.

In the diagram, the 100% range of the phase fractions is delineated by dashed lines, with the martensite phase fraction counting upward from the lower line and the R-phase fraction downward from the upper line. The different phase fractions are colored blue (martensite), green (R-phase), and red (austenite). At each simulation time step, the hypothetic phase fraction values of the outer hysteresis loops are calculated based on the current temperature T and stress σ. The transformation functions are then again scaled to the current martensite and R-phase fraction z_M and z_R such that

\begin{matrix} z_{M 2 A} (T, σ) = z_{M} \\ z_{R 2 A} (T, σ) = z_{R} \end{matrix}

For infinite temperatures ( $T \to + \infty$ ), the scaled transformation functions are made to strive toward

\begin{matrix} z_{M 2 A} (T \to + \infty) = 0 \\ z_{R 2 A} (T \to + \infty) = 0 \end{matrix}

Cooling process for three phase fractions

The cooling process is significantly more complex than the heating process. First, the transformation processes from austenite to martensite (A2M) and from austenite to R-phase (A2R) are not independent, but rather compete for the available austenite fraction. Second, the third transformation process from the R-phase to martensite (R2M) acts simultaneously on the martensite and R-phase fraction. Both aspects are visualized in Figure 10.

Figure 10.

During cooling, the A2M and A2R phase transformation processes are competing for the available austenite. The R2M transformation has to reflect the R-phase becoming available at each instant.

The example in Figure 10 shows the transformation starting at the temperature T. Like in Figure 9, the clearly foreseeable phase fraction segments are colored blue, red, and green. White segments cannot be directly assigned to the martensite or R-phase fraction, as either the A2M or the A2R transformation could possibly dominate the cooling process.

In the proposed model, this is implemented by calculating the phase fraction values of the outer hysteresis loops from the current temperature T and stress σ at each simulation time step and rescaling the transformation functions to the available phase fractions z_M and z_R such that

\begin{matrix} z_{A 2 M} (T, σ) = z_{M} \\ z_{A 2 R} (T, σ) = z_{R} \end{matrix}

The competitive nature of the A2M and A2R phase transformations is incorporated by scaling the corresponding transformation functions for ( $T \to - \infty$ ) to

\begin{matrix} z_{A 2 M} (T \to - \infty) = 1 - z_{R} \\ z_{A 2 R} (T \to - \infty) = 1 - z_{M} \end{matrix}

That is, each transformation function assumes that it can have all available austenite for itself. In particular, there is no distinct decision boundary that determines which transformation function is active; both transformations operate simultaneously across all states. Transition temperatures play a crucial role: the earlier a transformation starts, the more austenite it will transform to its target phase. After each simulation time step, the transformation functions are scaled to the remaining austenite.

The R2M transformation does not compete with the A2M or A2R transformation. Scaling will follow

\begin{matrix} z_{R 2 M} (T, σ) = 1 - z_{R} \\ z_{R 2 M} (T \to - \infty) = 1 \end{matrix}

The R2M transformation decreases the R-phase fraction, while simultaneously increasing the martensite fraction. Thus, the martensite and R-phase fractions cannot be derived only from z_A2M or z_A2R. At the same time, the contribution of the R2M transformation does not correspond to the absolute value of z_R2M. Instead, the incremental change of the R2M transformation $Δ z_{R 2 M}$ between simulation time steps needs to added to z_A2M and subtracted from z_A2R:

\begin{matrix} z_{M} = z_{A 2 M} (T, σ, z_{R}) + Δ z_{R 2 M} (T, σ) \\ z_{R} = z_{A 2 R} (T, σ, z_{M}) - Δ z_{R 2 M} (T, σ) \end{matrix}

Experiments

In Kazi et al. (2023), the authors introduced a novel approach for accurately scanning the measured surfaces of SMA wires for both strain and specific resistance by combining a sophisticated fluidic experimental setup with a specific experimental procedure. The ‘Experimental setup’ section gives a brief overview and addresses recent modifications. In the context of three phase fractions, the experimental procedure had to be adapted (‘Experimental procedure’ section). Finally, the outcome is discussed in the ‘Experimental results’ section.

Experimental setup

The experimental setup uses a silicon oil bath for heating and cooling of the SMA wire sample homogeneously over its full length. Temperature, force, and displacement of the SMA wire under test can be controlled individually or in combination, allowing for a precise control and measurement of temperature-induced and stress-induced phase transformation. Figure 11 shows the schematic layout and a picture of the experimental setup, illustrating its key components.

Figure 11.

Layout and picture of the experimental setup with a silicon oil bath for homogeneous heating and temperature control of the SMA wire sample under test (Kazi et al., 2023).

A commercial bath thermostat was selected as an oil bath. To enhance the cooling rate, a water-cooling system was added. A linear actuator adjusts the stress and strain of the SMA wire during experiments. The displacement of the moving upper end of the SMA wire is tracked by a laser triangulation sensor. A force sensor is placed between the actuator and the upper fixation point. To minimize thermal deformations, critical components were manufactured from materials that are largely insensitive to temperature variations. Efforts were also made to isolate the sensors and the attachment structure from thermal gradients, while preserving a high mechanical stiffness of the setup. An electric fan introduces an airflow across the surface of the oil bath, significantly decreasing the heating of the components located above it.

Recent improvements in the experimental setup include a commercial refrigerator unit to speed up cooling of the oil bath even more. Electrical resistance measurements were enhanced by incorporating a commercial digital multimeter with four-wire measurement. To allow measurements at low stress levels down to 10 MPa, where the development of the R-phase was expected to be most pronounced, the varying buoyancy forces resulting from the thermal expansion of silicon oil were compensated in the readings of the force sensor.

Experimental procedure

The measurement trajectories in Kazi et al. (2023) were composed of three experimental conditions: constant force (ConstF) maintains constant stress during heating or cooling to test temperature-induced phase transformations. Constant temperature (ConstT) varies the strain and stress at a fixed temperature to test stress-induced phase transformations. Constant length (ConstL) keeps the strain constant while varying temperature, combing both temperature- and stress-induced phase transformations. In this work, a fourth experimental condition was introduced: constant stiffness (ConstK) implements a force control that changes its reference value depending on the current length of the SMA wire, yielding a spring-like behavior. This condition enables the use of the material fully up to the limits specified by the manufacturer, leading to a higher stress-strain range in the experimental procedures.

The procedure described in the ‘Parameter identification’ section aims to fit the parameters of the ihyp transformation function individually at different stress levels. From this, the stress dependence of the parameters is derived in a second step. Thus, the experimental procedure uses ConstF data over the broadest possible range within the allowed limits of the material. The experimental procedure shown in Figure 12 describes the correlation of SMA wire temperature (T), force (F), and length (L).

Figure 12.

Experimental procedure with the combination of three test scenarios to scan the strain and resistance surfaces.

The heating trajectory between A and F starts with ConstT from a (for practical purposes) fully martensitic wire condition. After reaching a stress level of 100 MPa, at which the SMA wire manufacturer’s definition of the limitation of the material starts (point B), the temperature increases within a ConstF condition to the specified strain. The following ConstK condition follows the stress-strain boundary specified by the manufacturer until the target stress level at D is reached. The main phase transformation occurs during the following ConstF condition. A (for practical purposes) fully austenitic wire condition is reached by reducing the stress level to 0 MPa during the ConstT condition between points E and F.

The cooling process starts from the austenitic SMA wire in the reverse order of experimental conditions. The dotted lines in Figure 12 illustrate the paths for different stress levels during the ConstF condition. For higher stress levels, the cooling trajectories will converge with the heating trajectories at point B. Lower stress levels result in reduced strain due to the formation of twinned martensite. Since the SMA model only accounts for detwinned martensite so far, the formation of twinned martensite is mathematically compensated in the parameter identification procedure.

Experimental results

To avoid overheating, the temperature range of the SMA wire in the experimental runs was limited to 20°C…150°C. Stress levels were distributed over the stress range from 10 to 400 MPa, with an emphasis on stress levels between 10 and 60 MPa to obtain detailed information on the formation of the R-phase. In total, 15 experimental runs were performed. Figure 13 presents the measured surfaces for temperature, stress, and strain (left) and for temperature, stress, and specific resistance (right).

Figure 13.

Experimental results of heating and cooling in a 3D representation with temperature, stress, strain (left) and temperature, stress, specific resistance (right).

The strain and specific resistance trajectories of the different experimental runs form consistent surfaces for heating and cooling. The critical stress-strain combinations specified by the manufacturer (C, D in Figure 12) are satisfied by the ConstK condition (A, B). The strain values for full austenite at the end of each heating procedure match very well (C). As expected, strain paths corresponding to high stress levels above 100 MPa converge at a strain of approx. 4.5% when reaching the minimum temperature of 20°C (D, see B in Figure 12). Cooling paths at stress levels below 100 MPa exhibit a somewhat reduced elongation due to the formation of twinned martensite. The typical increase in specific resistance from the formation of the R-phase during cooling can be observed around (E). At low stress levels, the change in specific resistance (F) occurs at higher temperatures than the change in strain (G).

Parameter identification

After the strain and resistance surfaces have been measured, the challenge remains to extract the key parameters of the SMA model. These comprise the material parameters in the Stress-Strain Relation and Electrical Resistance blocks, but also the parameters defining the phase transformation surfaces in the Transformation Hysteresis block. The general methodology is outlined in the ‘Parameterization method’ section. The parameters are fitted in three sequential steps: after fitting the parameters for a stress-free phase transformation, the model parameters for different stress levels are examined. Finally, the stress dependence of model parameters is fitted.

Parameterization method

In two-phase SMA models based on phase diagrams, the parameters are often fitted using strain data measured at different stress levels. In this study, the parameters of the three phase fractions and the relevant phase transformation surfaces were obtained by simultaneously analyzing both strain and resistance data. In addition, data from DSC measurements was used.

The SMA model described in ‘Model structure’ section was implemented in MATLAB/Simulink (R). All five phase transformations were incorporated without restrictions. However, to simplify parameterization in this feasibility study, the phase transformations toward austenite (i.e. M2A and R2A) were assigned identical parameters. The same approach was applied to the phase transformations toward martensite (i.e. A2M and R2M). Consequently, the number of phase transformation surfaces that require parameterization was reduced from five to three.

The first step of the parametrization method determines the parameters of the Transformation Hysteresis block for a stress-free SMA wire. This step uses DSC measurement data, focusing on the rate of phase fraction evolution during the transformation process. The identified parameters serve as the starting point for parameterization of the model at higher stress levels: the strain and specific resistance outputs of the model are iteratively fitted to the measured data, progressing sequentially from lower to higher stress levels. This simultaneous parametrization method takes all phase transformation processes at each stress level into account. The final step derives the stress dependencies of the individual parameters.

Stress-free phase transformation

For fitting the model parameters of the transformation functions in a stress-free state, the DSC measurement data were compared to the model following an identical temperature profile. The modeled transformation rates were computed from the derivative of the transformation functions, and subsequently scaled to match the power levels recorded in the DSC data. Figure 14 shows the results of this step, with experimental heating and cooling data represented by red and blue curves. The black curve depicts the scaled phase fraction change rates derived from the model.

Figure 14.

Modeled phase transformation rate (black) fitted to the DSC experimental heating data (red) and experimental cooling data (blue).

The alignment between the experimental and simulated data in Figure 14 confirms the ability to replicate the stress-free transformation behavior of the SMA wire. The correlation validates the use of DSC measurements to identify not only transition temperatures, but also the shape-specific parameters (z_T, Γ) of the ihyp transformation function. The transformation enthalpies required for the Energy Balance block are identified by the scaling factors in this section.

Model parameters for different stress levels

In this section, the model parameters are derived from experimental strain and specific resistance data measured at different stress levels. The parameterization of the heating process can be performed similar to that of a two-phase model. However, it needs to be ensured that the model adequately captures both strain and specific resistance. Due to the competitive nature of the phase transformation processes during cooling, it is not possible to derive parameters directly from individual data segments. Instead, the modeled cooling response must be evaluated collectively with all phase transformations processes running in parallel. The experimental and model data are displayed as strain-temperature diagram and specific resistance-temperature diagram, as illustrated in the subfigures in Figure 15.

Figure 15.

Two-dimensional fit of the model (black) to strain and specific resistance data (green) at 10 MPa stress level.

The iterative parameterization procedure begins with fitting the material parameters in the Stress-Strain Relation and Electrical Resistance blocks. For a given stress level σ, the (pseudoelastic) strain values ε_x of the different phase fractions were assumed not to depend on temperature, that is,

ε_{x} (σ, T) = ε_{x} (σ)

In the left subfigure of Figure 15, the martensitic strain ε_M is shown in blue, the austenitic strain ε_A in red and the strain of the R-phase ε_R in green. ε_M and ε_A are the asymptotic values that the strain would approach for infinitely low or high temperatures, respectively. ε_R marks the strain the SMA wire will assume during cooling at low stress levels before the R2M transformation sets in.

The specific resistances of the different phase fractions ρ_x were assumed to linearly depend on temperature, following the relation

ρ_{x} (σ, T) = ρ_{x_T} (σ) \cdot (1 + α_{x} (σ) \cdot (T - T_{0}))

Here, $ρ_{x_T}$ is the specific resistance at the reference temperature T₀, while α_x represents the temperature dependency. The specific resistances of the different phase fractions are shown in the right subfigure of Figure 15 in their assigned colors. Again, ρ_M and ρ_A are fitted to reflect the behavior of the specific resistance at low or high temperatures. Interestingly, ρ_R corresponds only loosely to the measured resistance during cooling, as the model predicts that even at low stress levels a significant amount of martensite will be formed.

The next stage in the parameterization process involves fitting the transition temperatures for each phase transformation. This is carried out in temperature regions where the rates of change in strain and specific resistance are at their maximum. For the M2A/R2A phase transformation during heating, the transition temperatures are given by the intersections between the tangents and the strain/resistance of the corresponding material phases (see Figure 15). For cooling, the A2R transition temperatures are fitted using the changes in specific resistance, while those for the A2M/R2M transformation are fitted in the strain-temperature diagram.

The final stage of the parameterization process adjusts the shape parameters z_T and Γ of the ihyp function. Special attention should be paid to the temperature interval between 80°C and 100°C, where the specific resistance trajectories of the M2A/R2A and A2R transformations are in close proximity (see A). The values of z_T and Γ for these transformations must be chosen carefully to prevent a crossing-over of the modeled heating and cooling trajectories, which would be an unrealistic physical behavior.

Since adjusting a single parameter can influence the strain and specific resistance throughout the entire temperature range, the parameterization process for each stress level is iterative and needs to be repeated several times. Once a satisfactory parameter set has been identified, the values are used as the starting point for the parameterization at the next stress level, working the way up from low to high stress levels. The result of this procedure will be sets of model parameters tailored to each stress level.

Figure 15 shows the results of the parameterization at a stress level of 10 MPa, where the formation of the R-phase is the most prominent. For cooling, the model proved to be capable of capturing both strain and specific resistance with a high degree of accuracy. This is also true for all higher stress levels. For heating, on the other hand, the model did not achieve a satisfactory fit at the end of the steep section of the M2A/R2A transformation. In this region (B), the model tends to overestimate the strain while simultaneously underestimating the specific resistance. Within the current model, it is not possible to improve the fit with the experimental data, as adjustments of the modeled strain will directly lead to larger deviations of the modeled specific resistance and vice versa. This may indicate that strain and electrical resistance do not both combine serially from the strain and resistance values of the respective phase fractions, as assumed in the ‘Model layout’ section. Strain and specific resistance could be calculated assuming different structural arrangements. The topic is addressed by ongoing research.

Stress dependence of model parameters

This section describes the stress dependencies of the different model parameters. The parameter values obtained from the fitting procedures in the previous sections (‘Stress-free phase transformation' and ‘Model parameters for different stress levels') are used as support points for the correlation between each parameter and the applied stress. The ‘Stress dependence of the transformation parameters’™ section examines the parameters of the Transformation Hysteresis block, while the ‘Stress dependence of the material parameters’ section looks at the parameters of the Stress-Strain Relation and Electrical Resistance blocks.

Stress dependence of the transformation parameters

Figure 16 illustrates the transition temperatures found in the previous sections. Data points and functions associated with M2A/R2A are colored red, those associated to A2M/R2M are colored blue, and those associated to A2R are colored green.

Figure 16.

Phase diagram-like stress dependencies of the transition temperatures. Start and finish temperatures of A2R (green), A2M/R2M (blue) and M2A/R2A (red).

The illustration in Figure 16 corresponds to the widely used phase diagrams. The red data points, which represent the start and finish temperatures of the M2A/R2A phase transformation, show a linear dependence on the applied stress. The blue data points, corresponding to the A2M/R2M transformations, exhibit a more complex pattern. The data points above 100 MPa could be well approximated by straight lines. The same is true for the data points below 100 MPa—however with a different slope. This may be caused by the A2M transformation being dominant above 100 MPa, while the R2M transformation being more prominent below 100 MPa. To maintain model simplicity, an individual parameterization of the A2M and R2M phase transformations was not attempted. Instead, a second-order polynomial function was employed.

It should be noted that at stress levels above 100 MPa, the development of the R-phase can still be observed in the experimental data, but is not sufficiently pronounced such that model parameters could be determined. This limits a realistic fit for the A2R transformation at higher stress levels. Consequently, the green A2R start and finish temperatures were fitted based on stress levels below 100 MPa and modeled as stress-independent constants.

The variation of the shape-specific parameters of the ihyp function with the stress level are shown in Figure 17.

Figure 17.

Stress dependencies of the shape-specific Ihyp parameters. Curvature parameter Γ and asymmetry parameter z_T for the A2R (green), A2M/R2M (blue) and M2A/R2A (red) phase transformations.

The variation in parameter values with stress is particularly evident for the curvature parameter Γ. While the variation of z_T across different stress levels is less pronounced, it remains noticeable. Mostly, the parameters vary approximately linearly with stress. An exception to this is the A2R z_T parameter, which is treated as a constant in this analysis.

In summary, defining the stress dependencies for the shape-specific ihyp parameters proved to be essential to achieve a good fit between the experimental data and the model. The stress-dependent shape of the transformation functions had been the driver for using transformation surfaces rather than phase diagrams as the basis for this model already in the ‘Conceptual approach’ section. The model parameters obtained from the DSC measurements (‘Stress-free phase transformation' section) and from the measured strain and resistance surfaces (‘Model parameters for different stress levels' section) are generally in good agreement. The only exception are the shape parameters of the M2A/R2A transformation in which the deviation between strain and specific resistance could not be resolved (see ‘Model parameters for different stress levels' section). The two measurement methods appear to complement each other well. Their combination may pave the way to reflect the transformation enthalpies realistically in phase diagram-based or phase transformation surface-based SMA models.

Stress dependence of the material parameters

With the above assignments, the model parameters within the Transformation Hysteresis block are fully specified. Now the material parameters in the Stress-Strain Relation and the Electrical Resistance blocks are analyzed. As illustrated in Figure 18, data points and functions associated with the austenite fraction are marked in red, those associated with the martensite fraction in blue, and those associated with the R-phase fraction in green.

Figure 18.

Stress dependencies of the mechanical and electrical parameters of the three phase fractions martensite (blue), austenite (red) and R-phase (green).

The left subfigure illustrates the relationship between strain and stress, where the slope of the graph corresponds to the Young’s modulus of the different phase fractions. The strain proportion ε_x of each phase fraction x is mathematically expressed as:

ε_{x} (σ) = ε_{x_0} + \frac{σ}{E_{x}}

Herein, $ε_{x_0}$ represents the pseudoplastic strain, σ is the stress and E_x denotes the Young’s modulus. A comparison revealed that the determined austenitic Young’s modulus E_A is in good agreement with literature (Brammajyosula et al., 2011; Helbert et al., 2017; Scholtes et al., 2024; Sedlák et al., 2012; Wang et al., 2021b). The martensitic Young’s modulus E_M is approximately 1.2–2 times higher than values reported in prior studies. The discrepancy stems from the fact that in this model, E_M describes the hypothetical stress-strain relationship at infinitely low temperatures (see ‘Model parameters for different stress levels' section) which is different from the measured behavior at room temperature. The Young’s modulus of the R-phase E_R is hard to compare, as its value is highly sensitive to alloy composition. In any case, the value derived in this work is more than a factor of five smaller than the typical literature values. Again, the R-phase stress-strain relationship in this work is a hypothetical value at infinite temperatures that should not be directly compared to a measured Young’s modulus.

The right subfigure in Figure 18 presents the relationship between specific resistance and stress. It is evident from the data that the specific resistances of austenite and R-phase increase with increasing stress, whereas the specific resistance of martensite does not exhibit a noticeable stress dependence. The specific resistance $ρ_{x_T} (σ)$ for each phase is formulated as

ρ_{x_T} (σ) = ρ_{x_0} \cdot (1 + k_{x} \cdot σ)

Herein, $ρ_{x_0}$ is the specific resistance in a stress-free state, while k_x represents the stress dependency. For the temperature coefficients α_x the variations across the stress levels were not significant and not pointing into a specific direction. Therefore, the temperature coefficients α_x were assumed to be constant for all phase fractions.

The specific resistance of thin SMA wires has been reported only in a few studies (Brammajyosula et al., 2011; Scholtes et al., 2024). The parameters found in this study correspond well with the reported values, with a maximum deviation of approx. 20%.

Comparison of experimental data and model

With the parameters identified, the ‘Measured and modeled strain and resistance surfaces’ section assesses the model’s ability to accurately predict the measured data. The ‘Impact of R-phase on strain-resistance relationship’ section’ looks at the same experimental and simulation results from a self-sensing perspective: it is investigated up to which stress levels an impact of the R-phase can be observed, and how it manifests itself in the strain-resistance behavior.

Measured and modeled strain and resistance surfaces

The temperature/stress trajectories from the experimental runs in the ‘Experiments’ section were applied to the model. Figure 19 compares the measured data to simulation results with respect to strain (left subfigure) and specific resistance (right subfigure). For better clarity, only the trajectories spaced in 50 MPa intervals are displayed.

Figure 19.

Comparison of measured (colored) and simulated (black) strain and specific resistance.

Overall, the simulated data (black) and the experimental data (colored) are in good agreement. The stress dependent transformation characteristics, like steepness and curvature, are represented well by the model. Notably, during cooling the model successfully captures variation in transformation temperatures for both strain (A) and electrical resistance (B). The formation of twinned martensite in the experimental data is not represented by the model yet. This is observed by the differences in strain at low temperatures (C). Integration of a martensite detwinning kinetics could enhance the predictive accuracy of the model at this point. The most notable discrepancy occurs around the steep transition during heating (see D, E), as already described in the ‘Model parameters for different stress levels’™ section. This systematic deviation reduces with increasing stress, but is present across all stress levels.

The above comparison focuses on the measured and simulated trajectories of the outer hysteresis loops. On inner hysteresis loops, a distinct shift of transition temperatures was observed in experimental data. Future research will address model improvements to also reflect this behavior.

Impact of R-phase on strain-resistance relationship

Self-sensing in SMA actuators makes use of the relation between wire strain and electrical resistance, which has been shown to be fairly linear and contain little hysteresis already by (Ikuta et al., 1988). Figure 20 shows the strain-resistance curves at different stress levels measured in the experiments on the left, while the corresponding simulation results are shown on the right. The color scheme of the simulation results reflects the distribution of phase fractions at each simulation step in RGB colors (red: austenite, green: R-Phase, blue: martensite).

Figure 20.

Comparison of experimental data (left) and modeled data (right) between the heating and cooling process in strain-resistance-stress space.

At stress levels below 150 MPa, both the experimental and simulation data show a wide gap in the strain-resistance curves (A, B). This can be attributed to the formation of the R-phase (B). The ‘bulging’ of the resistance during cooling is not due to the high specific resistance of the R-phase, but rather to the early increase in the specific resistance (A2R) before the elongation of the SMA material sets in (R2M; see also Figure 1). At stress levels of 250 MPa and above, the data again display a pronounced hysteresis (C, D). It is assumed that at these high stress levels, very little R-phase is formed. At intermediate stress levels between 150 and 250 MPa, the hysteresis in the strain-resistance curves remains relatively small (E). At these stress levels, the bulging caused by the R-phase during cooling favorably combines with the natural hysteresis, so that both effects more or less cancel out.

The model in the right subfigure captures the behavior of the real SMA wire quite well. It even shows the compensation mechanism at intermediate stress levels somewhat clearer than the experimental data. At 10 MPa, the obvious ‘greening’ of the cooling trajectory indicates the dominance of the R-phase (approximately 70%, B). At 200 MPa, where the hysteresis between resistance and strain is at a minimum, the R-phase remains below 11% and is barely visible in the line color (E). However, this small amount of R-phase significantly affects the electrical resistance of the SMA wire. Even at a stress level of 400 MPa, the model predicts the formation of small amounts of R-phase (e.g. 3.8%). It is concluded that SMA models aiming to accurately reflect the electrical resistance also need to incorporate the R-phase, even if only small quantities are formed.

The modest hysteresis in the strain-resistance regime which forms the basis for current self-sensing schemes can be attributed to the emergence of the R-phase. However, this phenomenon is predominantly confined to intermediate stress levels. When operating SMA wires outside the stress window between 150 and 250 MPa, non-linear observers based on SMA models that incorporate the R-phase can prove to be highly useful.

SMA models to be incorporated in observers running on a microcontroller are also required to run in real-time. The presented model addresses this point by avoiding iterative calculations. For example, an actuation with a simulated duration of 47.3 s and a simulation time step of 0.0001 s will need to calculate 473,000 operating points of the SMA wire. The runtime of the model on a standard consumer CPU (Intel^® Core™ i7-10,510U) in MATLAB/Simulink is only 17.2 s (only one core being used, no optimization for computational efficiency). This indicates that the proposed model should also be able to cover future real-time requirements.

Summary and outlook

In SMA models, accurate prediction of both SMA wire contraction and electrical resistance requires incorporation of the R-phase alongside with the martensite and austenite phase fractions. In the commercial SmartFlex^® NiTi wire studied in this paper, the R-phase had a significant impact on electrical resistance up to stress levels of almost 250 MPa. In a stress window between 150 and 250 MPa, the intermediate formation of the R-phase during cooling created the relatively linear and hysteresis-free relation between strain and electrical resistance which is the basis of current self-sensing schemes. For model-based design and control of SMA actuators that operate outside of this stress window, accurate SMA models are needed that are also capable of running on a microcontroller in real time.

This paper shows the feasibility of SMA models that integrate all three phase fractions in order to capture both SMA wire strain and resistance behavior accurately. Comparison of Differential Scanning Calorimetry (DSC) with measurements of strain and specific resistance in a temperature-controlled silicone oil bath revealed five distinct phase transformation processes to be modeled (no indication for a transformation from martensite to R-phase during heating was found). The proposed model is based on the concept of transformation surfaces, making it possible to represent stress dependent changes in the shape of the transformation curves. To achieve an accurate fit with the experimental data, the novel ‘ihyp’ function was employed. Besides the common transition start and finish temperatures, this transformation function incorporates two intuitive parameters controlling curvature and asymmetry that allow to adapt the shape of the transformation.

For heating, two parallel transformation processes were assumed to independently produce austenite from martensite and R-phase. During cooling, however, two phase transformations are competing for the available austenite: the amount of austenite transformed into martensite is not available for transformation into R-phase and vice versa. At the same time, a third phase transformation process reduces the R-phase while increasing the martensite. In the proposed model, this is implemented by rescaling the transformation functions to the available phase fractions after each simulation time step. There is no decision mechanism how much austenite is transformed either into martensite or R-phase, but this results from the differences in the transition temperatures at the different stress levels.

For parameterization of the SMA model, a three-stage process is proposed. Initially, the shape of the transformation function is fitted to DSC data at zero stress. Subsequently, strain and specific resistance data recorded in a silicone oil bath are used to determine both the parameters for the transformation functions and the mechanical/electrical properties at stress levels between 10 and 400 MPa. Due to the competitive nature of the model, the experimental data has to be compared to simulations with all phase transformation processes running in parallel rather than deriving parameters directly from individual data segments. Finally, parameter values determined at individual stress levels serve as support points for establishing stress dependencies of the model parameters.

The parameters obtained by the above process were generally in good agreement with comparable values found in the literature. The parameters derived from the DSC measurements were also in line with the parameters obtained in the silicone oil bath. The use of DSC data paves the way for capturing the transformation enthalpies in the model, which have a significant impact on the heating and cooling of an SMA wire in air. The incorporation of stress dependencies proved to be essential to achieve a good fit between model and experimental data. While a good fit could be achieved between the model and the experimental data at all stress levels for the cooling processes, discrepancies emerged during heating that could not be resolved. Current research is addressing this limitation. In addition, the model will be also refined to capture shifts of transition temperatures during inner hysteresis loops observed in experimental data.

The SMA model presented in this paper addresses the computation time by avoiding iterative calculations. A simple experiment showed that at a simulation time step of 0.1 ms, the model will run significantly faster than real-time on a single core of a standard microprocessor already without optimizations for computational efficiency. This indicates that future incorporation in an observer running on a microcontroller should be feasible, thereby enhancing the potential for accurate self-sensing in SMA actuators over a wide range of operational conditions.

Still, a significant amount of research remains to be conducted. The remaining deviations of the model from the measured strain and resistance trajectories during heating may indicate a need for a different arrangement of the phase fractions (e.g. serial/parallel). Work addressing this topic is already under way. In the experiments, also differences between outer and inner hysteresis loops were observed. Model accuracy could be improved by adapting the scaling procedure of the transformation function. The current model aims to reflect the behavior of an SMA wire in a somewhat used condition. It could be extended by integrating a twinning/detwinning kinetics to capture the behavior of wires straight from the spool. This would also pave the way toward models describing the behavior of SMA wires over their entire lifetime.

Footnotes

ORCID iDs

Max Hauber

Arif Kazi

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors wish to thank the German Ministry of Education and Research (BMBF) for funding this work within the context of the project SMArtMed (grant no. 13FH080KX1).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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