A neural network model for shape memory alloy actuation response with physical constraints for partial phase transformation

Abstract

In this work, we demonstrate a data-based Machine Learning (ML) framework that can capture the strain evolution during Shape Memory Alloy (SMA) actuation with a minimum of four state variable inputs.These inputs describe the thermomechanical stress-strain state of the SMA and changes in the stress state during thermal actuation. Furthermore, we identify the physics-based constraints to incorporate partial phase transformation and make predictions beyond the training regime. The ML framework uses a recurrent neural network (RNN) model to capture the nonlinear strain rate variation during phase transformation. Physics-based constraints are introduced in the RNN model to include the activation of thermoelasticity, while unloading away from transformation surfaces within the thermoelastic domain. The framework is trained using experimental thermal cycle responses of a NiTiHf SMA at different stress levels. The framework can then accurately predict actuation responses undergoing complete and partial phase transformations, and also extrapolate responses for new thermal cycling that involve complex thermomechanical loading paths. An uncertainty quantification analysis using ensemble training of the NN model is also presented to show the variability of the framework.

1. Introduction

Shape Memory Alloys (SMAs) are unique active materials with the ability to recover from large deformations through solid-solid phase transformation between austenite and martensite phases. The austenite phase is stable at high temperatures, and the martensite phase is stable at lower temperatures. Depending on the stress state, the martensite phase can be present either in a twinned or detwinned state. At zero stress, the martensite is in a twinned state, and at non-zero stress, it is in the detwinned state. Phase transformation can be induced by applying mechanical, thermal or combined thermo-mechanical loads (Lagoudas, 2008). The transformation between austenite and detwinned martensite results in the generation/recovery of inelastic strain, called transformation strain. When the SMA is constrained, the recovery of transformation strain can generate high stress and produce a higher work output compared to other active materials. This makes SMAs ideal for designing compact and lightweight solid-state actuators (Benafan et al., 2014; Jani et al., 2014; Simiriotis et al., 2021; Stroud and Hartl, 2020). SMAs are used in various applications in fields such as biomedical, aerospace, industrial, and wind energy (Antonucci et al., 2021; Balasubramanian et al., 2021; Bansiddhi et al., 2008; Calkins and Mabe, 2010; Costanza and Tata, 2020; Elahinia et al., 2012; Hartl and Lagoudas, 2007; Hernandez et al., 2018; Karakalas et al., 2020; Morgan, 2004; Rajput et al., 2022; Stroud and Hartl, 2020).

SMAs exhibit variety of hysteresis behaviors based on the thermo-mechanical loading path they undergo (Lagoudas, 2008). Shape Memory Effect (SME) is observed when SMA is deformed while in the twinned martensitic phase, which when unloaded and heated above the austenite finish temperature, recovers the original shape with the phase transformation back to the parent austenitic phase. Pseudoelastic behavior in SMAs is observed in the recovery of the strain state with stress-induced phase transformation at temperatures where austenite is stable. An actuation response is produced when SMAs at constant stress are taken through a cooling-heating cycle, resulting in the recovery of transformation strain. Depending on the loading conditions, SMAs can show partial phase transformation behavior when the thermal range is inadequate for a full transformation (Karakalas et al., 2020; Wang et al., 2021a, 2021b; You and Guo, 2022). A complete transformation cycle is obtained when an SMA is cooled below the martensitic finish temperature and heated above the austenitic finish temperature. Such cycles with complete phase transformations are referred to as “major cycle responses” and the responses with partial phase transformations due to insufficient cooling or heating are referred to as “minor cycle responses.”

Predicting the actuation response of SMAs efficiently and accurately is key for designing SMA based actuator and sensors. Especially, the minor cycle responses need a special modeling treatment as they involve a complex history of microstructural state involving mixed phase state. For example, initiating the minor cycle during the cooling branch of the major cycle has a different initial microstructure state with martensite phases are being nucleated in the austenite phase and whereas initiation of the minor loop on the heating branch has opposite microstructure state. The minor cyclic responses are of considerable relevance in the case of aerospace, wind turbines, and control applications where the SMA actuator can undergo partial transformation due to partial reversal of the loading. There have been many works to modeling the partial phase transformation, such as, phenomenological approaches (Alsawalhi and Landis, 2022; Brocca et al., 2002; Chemisky et al., 2011; Karakalas et al., 2019a, 2019b; Karakalas and Lagoudas, 2020; Lagoudas et al., 2006; Müller and Bruhns, 2006; Paiva and Savi, 2006; Savi and Paiva, 2005; Scalet et al., 2021; Wang et al., 2021a), empirical modeling efforts (Adeodato et al., 2022; Gorbet et al., 1998; Khan and Lagoudas, 2002) and those for control schemes (Jayender et al., 2008; Williams and Elahinia, 2008). Advanced phenomenological models have significantly improved the modeling of 1D SMA hysteretic thermomechanical behavior (Auricchio et al., 2007; Brinson, 1993; DeCastro et al., 2007; Shaw and Churchill, 2009; Shirani et al., 2017; Shu et al., 1997; Sittner et al., 2000; Song, 2020), particularly through the use of internal state variables that accurately capture the stress-strain-temperature responses. These models offer strong physical interpretability and have been instrumental in enhancing our understanding of SMA behavior. However, they may still require specific modeling assumptions and difficulty with parameter calibration. In contrast, data-driven Machine Learning (ML) models provide an alternative approach by leveraging experimental data to identify governing relationships without predefined constitutive assumptions. While ML models may not inherently offer the same physical interpretability as phenomenological models, they present advantages in flexibility and adaptability to complex material responses.

Data-driven ML approaches have become popular due to their ability to model complex behaviors with good accuracy and to also provide a computationally cost-effective alternative to the traditional physical models. Methods such as neural networks that can be trained directly using experimental responses can predict the specific material sample behavior efficiently. There has been considerable interest in using neural network-based models for modeling SMA behavior for actuation control applications, where a feed-back neural network model is used in the time-series modeling of SMA actuation in the control algorithm (Asua et al., 2010; Damle et al., 1995; Hmede et al., 2022; Narayanan and Elahinia, 2016; Nikdel and Badamchizadeh, 2015; Song et al., 2003; Tai and Ahn, 2012; Wang and Song, 2014). In these works, the neural network model is continuously trained in-situ from the displacement measurements and the predictions are limited to the SMA actuation in the immediate time step. Neural network based approaches has been proposed for the modeling of hysteresis behavior (Kilicarslan et al., 2011; Li et al., 2017; Mohammadi Nia and Moradi, 2022), actuation (Asua et al., 2010; Narayanan and Elahinia, 2014; Song et al., 2003) and cyclic actuation evolution (Owusu-Danquah et al., 2022) for phase transforming materials. These past studies do not introduce additional physics-based constraints/formulation of phase transforming materials, and as a result are limited to the modeling of hysteresis behavior only in the training data.

The focus in this work is to model actuation responses to predict the extent of actuation displacement which is important for the designing of SMA-based actuators. As a material system, NiTiHf based High Temperature SMAs that are ideal for many actuation applications with high transformation temperatures (Karaca et al., 2014) is chosen. A recurrent static neural network model is proposed to capture the hysteresis response of these materials with a minimal number of state variables. In a static neural network, the learning is only during the training phase, and does not evolve dynamically with a feedback. The training of the model is performed using the major cycle responses, with physics-based constraints at the initiation of minor cycle branches from the major cycle to extend its capabilities to predicting the minor cycle responses. A comparison of results from neural networks with and without such constraints showed a visible improvement in the prediction of minor cycle behavior. Therefore to correctly predict all possible states, using limited data for training, physics laws need to be imposed. Building on the work of Liu et al. (2023), which demonstrated that few as three internal variables are sufficient to describe the evolution of the elasto-viscoplastic behavior using a recurrent neural network, this study validates the finding by simulating the complex hysteretic response of SMAs. Unlike Liu et al. (2023), where internal variables were identified from microscopic responses but lacked physical interpretability, the present work shows that SMA actuation can be accurately learned using the physically measurable state variables—temperature, stress, and macroscopic strain—without explicitly predicting internal variables. Therefore it can be used for real-time predictions using sensor measurements, enhancing practical applicability. The authors have successfully applied the same algorithm to capture magnetic shape memory alloy actuation (Tian et al., 2024) and, in a NiTi SMA torque tube actuation application to evaluate internal stresses in real time.

The paper is structured as follows. In Section 2, a brief theory on SMA actuation behavior explaining the major and minor cycle responses is given. In Section 3, the experimental responses of the chosen NiTiHf SMA are discussed. In Section 4, details of the data based model used in the study are presented. In Section 5, the capability and predictions using the resulting model are discussed. Section 6 presents the key conclusions from the works.

2. SMA actuation behavior

2.1. Major cycle actuation response in SMA

A typical major cycle actuation behavior of SMA is as shown in Figure 1(a). The material is in the pure austenite phase at high temperature, represented by point A in the diagram, and in the pure martensite phase at low temperature, represented by point B in the diagram. The heating and cooling of the SMA between states A and B at a constant stress produces a major cycle actuation response of the SMA. The major cycle actuation responses of the SMA are characterized by transformation temperatures ( ${\mathrm{M}}_s^{\sigma }$ , ${\mathrm{M}}_f^{\sigma }$ , ${\mathrm{A}}_s^{\sigma }$ , and ${\mathrm{A}}_f^{\sigma }$ ) that can be calculated using four tangent lines as represented in Figure 1(a). The four transformation temperatures mark the start and finish of transformation in cooling ( ${\mathrm{M}}_s^{\sigma }$ and ${\mathrm{M}}_f^{\sigma }$ ) and heating ( ${\mathrm{A}}_s^{\sigma }$ and ${\mathrm{A}}_f^{\sigma }$ ) respectively. The subscript .“ $s$ ” denotes transformation start, and the subscript “ $f$ ” denotes transformation finish. The stress dependency of the transformation temperatures is captured in a phase diagram shown in Figure 1(b), where the lines plot the variation of four transformation temperatures with the external stress. The major cycle of cooling and heating at constant stress $\sigma$ is shown in the phase diagram with lines connecting A and B. The end states A and B are chosen after the transformation finish temperatures are reached, which indicates complete phase transformation in both directions.

Figure 1.

Actuation response and the corresponding loading path in a typical phase diagram: (a) representative strain-temperature response and calculation of transformation temperatures using the tangent lines approach and (b) loading path shown on a typical stress-temperature phase diagram for SMAs.

2.2. Minor cycle actuation response in SMA

Minor cycle actuation responses are created due to incomplete heating or cooling, resulting in partial phase transformation in either direction. Referring to the loading path in Figure 1(b), partial phase transformation is when the end states (A), (B) or both lie between the start and finish transformation temperatures in the phase diagram. Figure 2 schematically shows minor cycle responses with respect to a major cycle response. In Figure 2(a), the upper minor cycle response occurs when only the end of heating (“ ${\mathrm{A}}_U$ ”) results in partial phase transformation. The lower minor response occurs when only the end of cooling (“ ${\mathrm{B}}_L$ ”) results in partial transformation. A case where both “A” and “B” result in partial phase transformation is shown in Figure 2(b). In such cases, the partial cycle is contained within the major cycle response and is referred to as the “inner minor cycle.”

Figure 2.

Different types of minor cycle responses on corresponding major cycle responses: (a) upper and lower minor cycles and (b) inner minor cycle.

3. Experimental investigation

To train and validate the machine learning model, a complete material characterization test series along with additional minor cycle cases were carried out on the selected SMA. Dog-bone shaped specimens were cut from a Ni_50.3Ti_29.7Hf₂₀ sheet. Thermal cycles were applied to the material to induce complete phase transformation under various constant stress levels. The experiments were conducted using an MTS Insight testing machine equipped with an MTS load cell with a 30 kN load capacity. The strain measurements were carried out using a high-temperature Epsilon extensometer of 1-inch gage length attached to the specimen. The temperature measurements were made using a K-type thermocouple attached to the specimen. The specimen along with the grips were enclosed inside an insulated Thermcraft thermal chamber. The specimen was heated by raising the temperature of the air inside the chamber, and cooling was performed by passing cold liquid nitrogen through the chamber. The major thermal cycling responses were obtained at six stress levels (43, 93, 144, 192, 294, and 393 MPa) with heating and cooling between 80°C and 200°C.

Following the experimental procedure, the actuation responses in the selected SMA were obtained, which represent typical behavior for all SMAs. Figure 3(a) shows the experimental responses of major cycles at different stress levels. The noises in the experimental responses were removed using a moving average filter. In Figure 3(b) and (c), the upper and lower minor cyclic responses at stress level 395 MPa is shown, where each minor cycle response has five repetitions. Figure 3(b) shows the upper minor cycle responses obtained with temperature variation of 75°C from the reference phase of Martensite at 100°C, and Figure 3(c) shows the lower minor cycle responses obtained with temperature variation of 46°C from the reference phase of Austenite at 200°C.

Figure 3.

Experimental actuation responses in Ni_50.3Ti_29.7Hf₂₀ SMA: (a) shows major actuation cycles at different stresses, (b) shows upper minor cycles, and (c) shows lower minor cycle responses at 395 MPa on the corresponding major cycle. The Cycle 1–5 represent the five repetitions of the minor cycle in order.

An increase in strain is observed in the subsequent upper minor cycles compared to the first minor cycle. However, in the case of the lower minor cycles, a similar shifting of the strain is minimal. This major difference between upper minor cycles and lower minor cycles could be due to the difference in the microstructures at these two locations. In the upper minor cycle, harder austenite phases are nucleated in a softer martensite matrix phase, which can induce larger strains in the matrix from stress concentrations. In the lower minor cycles, softer martensite phases are nucleated in a harder austenite matrix, resulting in smaller strains. Because of this difference, the upper minor cycles are sensitive to localized high straining compared to the lower minor cycles. As a result, the response in subsequent upper minor cycles are significantly shifted compared to the initial response.

4. Machine learning framework with physical constraints

Machine learning methods can be classified as supervised learning or unsupervised learning. In supervised learning, the task is to construct a function that maps the input and target pairs provided from a training. In unsupervised learning, the targets are not identified a priori, and the model should self-discover the naturally occurring patterns in the training data. A common example would be the clustering of data, where the algorithm groups training data into categories based on some similarity. For the modeling of SMA actuation, the target is a model that fits the available experimental behavior and can also extend predictions for new loading conditions. In this scenario, supervised learning methods are used with a set of training data. The neural networks allow control of the complexity of a model by varying the number of layers and nodes within it, which allows the tuning of a model that works best for SMA responses.

4.1. Description of the machine learning model for thermal actuation

The thermal actuation response of SMA at a fixed stress $\sigma$ is a function of temperature $T$ , heating rate, $\stackrel{·}{T}$ and its loading history. This can be expressed as a differential equation of the strain evolution with time, that is,

$$\frac{d\epsilon }{\mathrm{dt}}=\hat{F}(\epsilon ,T,\stackrel{·}{T};\sigma ).$$

(1)

Assuming a steady-state loading condition, the effects of the loading rate can be neglected and the change in strain $\Delta \epsilon$ can be simplified in terms of temperature $T$ using the chain rule.

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d\epsilon }{\mathrm{dt}}=\frac{d\epsilon }{\mathrm{dT}}\frac{\mathrm{dT}}{\mathrm{dt}}=\frac{d\epsilon }{\mathrm{dT}}\stackrel{·}{T}\Rightarrow \mathrm{\Delta }\epsilon =\frac{d\epsilon }{\mathrm{dT}}\stackrel{·}{T}\mathrm{\Delta }t}\\ \multicolumn{1}{c}{=\left(\frac{d\epsilon }{\mathrm{dT}}\right)\mathrm{\Delta }T}\end{array}$$

(2)

An ML model is formulated to fit the slope ( $\frac{d\epsilon }{\mathrm{dT}}$ ) in the experimental curves with respect to the heating or cooling loading path. By targeting the slope, the same model is applicable for both major cycle and minor cycle thermal paths. Figure 4 summarizes the model, where the slope of strain ( $\frac{d\epsilon }{\mathrm{dT}}$ ) is trained with respect to the applied stress $\sigma$ , the value of strain $\epsilon$ , the temperature $T$ , and direction of heating $\mathrm{Sign}\left(\mathrm{\Delta }T\right)$ in the next step. Changes in $\sigma$ capture the difference in actuation response with the applied load. Values of $T$ and $\epsilon$ represent the state of the material in the temperature-strain domain. The input $\mathrm{Sign}\left(\mathrm{\Delta }T\right)$ is for differentiating the different response in heating and cooling. Table 1 summarizes the detailed description of input parameters and target parameter in the ML model. With the slopes predicted using the ML model at each heating or cooling step, the change in strain ( $\mathrm{\Delta }\epsilon$ ) for the next step is calculated. The proposed ML model can be summarized by equation (3), where the function ${\mathbb{F}}_{\mathrm{ML}}$ relates the $\frac{d\epsilon (T,\sigma )}{\mathrm{dT}}$ to $\sigma$ , $\epsilon$ , $T$ , and $\mathrm{Sign}\left(\mathrm{\Delta }T\right)$ . Note that these inputs describe the average stress-strain state, and the thermal boundary changes in the material.

$$\frac{d\epsilon (T,\sigma )}{\mathrm{dT}}={\mathbb{F}}_{\mathrm{ML}}[\sigma ,\epsilon ,T,\mathrm{Sign}\left(\mathrm{\Delta }T\right)].$$

(3)

Figure 4.

Machine learning framework for predicting the SMA response showing input, target, training, and prediction.

Table 1.

Input and output in the ML model for capturing the actuation response.

Parameter	Description
Input
$\sigma$	Current stress value
$\epsilon$	Strain value at current time step
$T$	Temperature at current time step
$\mathrm{Sign}\left(\mathrm{\Delta }T\right)$	Direction of temperature step
Target
$\frac{d\epsilon (T,\sigma )}{\mathrm{dT}}$	Derivative of strain variation

At any given point along the $\epsilon$  −  $T$ cycle, the next value of the strain is calculated incrementally with temperature using $\frac{d\epsilon (T,\sigma )}{\mathrm{dT}}$ from equation (3), that is,

$$\epsilon (T+\mathrm{\Delta }T,\sigma )=\epsilon (T,\sigma )+\frac{d\epsilon (T,\sigma )}{\mathrm{dT}}\mathrm{\Delta }T.$$

(4)

4.2. Physical constraints from transformation surfaces

To model the partial transformation responses, additional physics based constraints are needed to include the physics of transformation surfaces. In the SMA actuation, the partial transformation branches initiate at the reversal points of thermal loading in the major cycles where complete phase transformation is not achieved. At these reversal points, the SMA is unloading away from the transformation surface, and during that step only thermoelasticity is operating. Additional constraints are needed in the ${\mathbb{F}}_{\mathrm{ML}}$ to impose this physical understanding of activation of thermoelasticity. Note that the assumption of thermoelasticity is only true when there is no overlap of transformation surfaces ( ${\mathrm{M}}_s^{\sigma }>{\mathrm{A}}_s^{\sigma }$ ). If the minor loop initiation is in the overlapping region, some phase transformation is expected to occur immediately at the reversal points.

The hysteresis response in the thermal cycling of the SMA occur due to the different strain rates $\left(\frac{d\epsilon (T,\sigma )}{\mathrm{dT}}\right)$ while on the transformation surface and unloading away from the transformation surface. With the major cycle data, only the strain rates while on the transformations surfaces are specified. As discussed in the introduction section about the mixed phase states at the initiation of the minor loop, we hypothesize that the material thermal expansion is playing the key role at the reversal point before the transformation starts reversing. To incorporate this physical insight, additional data for strain rate are specified in the ML model along the major cycle path for the reversal of thermal direction. For each point in the cooling branch of the major cycle, the strain rates for heating are specified. Similarly, for the points in the heating branch, the strain rates for cooling are specified. The strain rate at the reversing of thermal load is chosen to be equal to a fixed thermal expansion coefficient of the material (i.e. $\frac{d\epsilon }{\mathrm{dT}}\approx {\alpha }_T=0.{00024}^{°}{\mathrm{C}}^{-1}$ ). This concept of additional constraints is summarized in Figure 5. Note that here the constraints are imposed in weak sense by providing additional training data compared to the approach in Physics-Informed Neural Networks (PINNs; Linka et al., 2021; Masi et al., 2021; Raissi et al., 2019), where the constitutive equations are strictly imposed. The resulting constrained ML model can now interpolate the strain rate in the $\epsilon$  −  $T$ space for points within the major cycle, and hence, the minor cycles are going to be predicted well and constrained within the major cycle, as will be seen in the Results and Discussion.

Figure 5.

Figure describing extra constraints to improve minor cycle response predictions.

4.3. Implementation of $\mathbb{F}$ _ML using Neural Network

The implementation of the presented ML framework involves: (1) processing experimental data and normalization of the data, (2) training the NN model ${\mathbb{F}}_{\mathrm{ML}}$ , and (3) predicting responses using ${\mathbb{F}}_{\mathrm{ML}}$ . The neural network architecture used here is shown in Figure 6. Four hidden layers with six nodes are chosen. The hyperbolic tangent sigmoid transfer function is used in the hidden layers. In the output layer, the linear transfer function is used. More details on the equations and matrix representation of the layers are presented in Appendix 1.

Figure 6.

Neural network architecture used for modeling actuation responses.

4.3.1. Noise removal and normalization of data

The noises in the experimental response may reduce the accuracy of the neural network model. To remove noise from the experimental data, the actuation responses are processed using the “smoothdata” function in MATLAB (2019) software with a Gaussian weighted moving average filter. The filtering is done separately for heating and cooling portions of the response to get accurate average values at the end points of cooling. In addition, the data is sampled at temperature steps of 1°C to remove additional irregularities. With the refined experimental response, the training data is generated.

Normalization of the input and target data is used in machine learning models to make the data on a similar scale. In the current approach, in addition to the conventional linear normalization, a nonlinear transformation with an exponential function is carried out. First, the strain rate $(d\epsilon (T,\sigma )/\mathrm{dT})$ data is normalized by taking a natural exponent form: that is, $\exp (ad\epsilon (T,\sigma )/\mathrm{dT})$ with $a=10$ so that the data of high and low slope regimes in the SMA response are captured with significance. Further, this transformed inputs and the outputs for the neural network model are normalized by linear mapping minimum and maximum values to [−1 1]. The linear mapping function is given by the following equation:

$$y=\frac{(y_{\max }-y_{\min })(x-x_{\min })}{(x_{\max }-x_{\min })}+y_{\min },$$

(5)

where, $x$ is the input variable and $y$ is the normalized value, $x_{\max }$ and $x_{\min }$ are the maximum and minimum values of the data used in the training, and $y_{\max }$ and $y_{\min }$ are the target maximum and minimum values, which in this case are taken to be 1 and −1 respectively. Table 2 summarizes the minimum and maximum value for the input and target.

Table 2.

The normalization parameters for the ML model.

Parameter	$x_{\min }$	$x_{\max }$
Input
$\sigma$	43.0000	393.0000
$\epsilon$	−0.1043	3.8275
$T$	80.0000	195.0000
$\mathrm{Sign}\left(\mathrm{\Delta }T\right)$	−1.0000	1.0000
Target
$\exp (10*\frac{d\epsilon (T,\sigma )}{\mathrm{dT}})$	0.0413	1.0650

The neural network model ${\mathbb{F}}_{\mathrm{ML}}$ is trained with the normalized inputs and targets. While making predictions using the ${\mathbb{F}}_{\mathrm{ML}}$ , an inverse transformation is performed to retain the actual values from the normalized values. The inverse transformation is summarized in equation (5).

$$x=\frac{(y-y_{\min })(x_{\max }-x_{\min })}{(y_{\max }-y_{\min })}+x_{\min }.$$

(6)

In the case of $\frac{d\epsilon (T,\sigma )}{\mathrm{dT}}$ , an additional logarithmic operation is performed to inverse the exponential transformation.

The algorithm for generating training data is summarized in Algorithm 1, where the input and target data are generated. Within each major cycle, for the stress, temperature, strain, and temperature change, the target derivative is estimated and stored. In addition, the data corresponding to the constraints in Section 4.2 are also stored.

Algorithm 1 Algorithm for making training data.

1: $m\leftarrow 1$   ⊳ index variable 2: for $\sigma ={\sigma }_1,{\sigma }_2,..$ . do  ▹ Constant $\sigma$ corresponds major cycle response 3:      for $i=1,2,3$ … ( $L_{\mathrm{steps}}-1$ ) do   $▹L_{\mathrm{steps}}$ is total steps in the $\sigma$ data 4:               $T_c\leftarrow T_i$   ▹Temperature at current step 5:               ${\epsilon }_c\leftarrow {ϵ}_i$   ▹Strain at current step 6:               $\mathrm{\Delta }T\leftarrow T_{i+1}-T_i$   ▹Temperature change 7:              $\frac{d\epsilon }{\mathrm{dT}}\leftarrow \frac{({\epsilon }_{i+1}-{\epsilon }_i)}{\Delta T}$   ▹Forward derivative 8:              $\mathrm{Inpu}{t}_m=\left[\begin{array}{c}\sigma {ϵ}_c{T}_c\mathrm{\Delta }T\hfill \end{array}\right]$   ▹Input to the ML model 9:              $\mathrm{Targe}{t}_m=\left[\begin{array}{c}\exp (10*\frac{d\epsilon }{\mathrm{dT}})\hfill \end{array}\right]$   ▹Target for the ML model 10:              $m\leftarrow m+1$    ▹Add m 11:              $Inpu{t}_m=\left[\sigma {ϵ}_c{T}_c-\Delta T\right]$   ▹Input corresponds to reversing thermal loading 12:            $\mathrm{Targe}{t}_m=\left[\begin{array}{c}\exp (10*{\alpha }_T)\hfill \end{array}\right]$  ▹Target for reversing thermal loading 13:            $m\leftarrow m+1$   ▹Add m 14:      end for 15: end for

4.3.2. Training and prediction

The ${\mathbb{F}}_{\mathrm{ML}}$ training is carried out on the major cycle responses of Ni_50.3Ti_29.7Hf₂₀ SMA in Figure 3(a). The major cycle experimental data of NiTiHf at stress levels 43, 93, 144, 192, 294, and 393 MPa with heating and cooling between 80°C and 200°C are used for training the model. With the input and target data generated using Algorithm 1, the weights and biases in the neural network model are trained in MATLAB software using “trainlm” which uses a Levenberg-Marquardt backpropagation algorithm (Hagan and Menhaj, 1994). Details of the weights and biases obtained through the training are summarized in Appendix 1.

For a given thermal loading path, the responses are predicted using the trained ${\mathbb{F}}_{\mathrm{ML}}$ , which follow the process summarized in Algorithm 2. First, the trained ${\mathbb{F}}_{\mathrm{ML}}$ is used to reproduce the experimental major cycle responses. For making prediction at any arbitrary stress levels, initial conditions are used. The initial condition for the SMA is taken to be the loading state at 195°C, where the SMA is in the austenite phase (point A in Figure 1(a)). The strain values at “A” as a function of applied stress are fitted assuming linear elasticity. For any arbitrary stress value, the initial strain value is calculated using the fit summarized in Figure 7. Since in the formulation the strain is estimated recursively from the previous value, the response at any arbitrary stress level can be predicted from this initial strain.

Algorithm 2 Algorithm for predicting response.

1: Specify $\sigma$ 2: $T_c\leftarrow T_0$   ▹Initial condition [Figure 7] 3: ${\epsilon }_c\leftarrow {\epsilon }_0$   ▹Initial condition [Figure 7] 4: for $i=1,2,3$ … ( $L_{\mathrm{steps}}-1$ ) do   $▹L_{\mathrm{data}}$ is total increments in the path 5:          $\mathrm{\Delta }T=T_{i+1}-T_c$    ▹Temperature change 6:         Evaluate ML model: $Y_c=\mathrm{ML}\left(\left[\begin{array}{c}\sigma {ϵ}_c{T}_c\mathrm{\Delta }T\hfill \end{array}\right]\right)$ 7:          $\frac{d\epsilon }{\mathrm{dT}}\leftarrow \frac{\log \left(\mathrm{Yc}\right)}{10}$   ▹Forward derivative 8:          ${\epsilon }_c\leftarrow {\epsilon }_c+\frac{d\epsilon }{\mathrm{dT}}\mathrm{\Delta }T$   ▹Prediction of strain for $T_c+\mathrm{\Delta }T$ 9:          $T_c\leftarrow T_c+\mathrm{\Delta }T$    ▹Update current temperature 10:            Store ${\epsilon }_c$ and $T_c$ 11: end for

Figure 7.

The strain at 195°C versus applied stress for the Ni_50.3Ti_29.7Hf₂₀ SMA. A linear fit captures the variation of the strain versus the applied stress at 195°C.

A comparison of the responses from neural network model at different stress levels is shown in Figure 8. The predictions follows the experiments very closely for all the six stress levels used for training. Although the presented neural network model ${\mathbb{F}}_{\mathrm{ML}}$ does not take any of the material parameters as input, it learns the governing rules of the material behavior from the data. Without being introduced to the temperature and stress dependencies of phase transformation in an SMA, the ${\mathbb{F}}_{\mathrm{ML}}$ can learn them during training with the weights and biases in it. Remarkably, the model can capture the evolution in a complex material response such as the SMA actuation using a minimal number of state variables. In the following Results and Discussion section, the learning capability of ${\mathbb{F}}_{\mathrm{ML}}$ using the constraints in Section 4.2 toward predicting the minor cycle responses is also evaluated.

Figure 8.

Experimental major cycle responses from the Ni_50.3Ti_29.7Hf₂₀ SMA reproduced using ${\mathbb{F}}_{\mathrm{ML}}$ at different stress levels. The neural network model captures the evolution in a complex material response such as the SMA actuation with few input variables describing the average stress-strain state and the changes in the boundary conditions.

5. Results and discussion

Partial phase transformation responses in the Ni_50.3Ti_29.7Hf₂₀ SMA are simulated using ${\mathbb{F}}_{\mathrm{ML}}$ . For this, the heating and cooling ranges are chosen so that they result in partial phase transformation. The predictions of partial transformation are made both with and without the extra constraints from Section 4.2 to demonstrate their significance.

5.1. Prediction of minor cycles without applying constraints

First the ${\mathbb{F}}_{\mathrm{ML}}$ is implemented without physical constraints and trained using the major cycle data from the experiments. This trained model captured the major cycle responses well, similar to the comparison in Figure 8. However, the trained model is failing to predict the minor loops. The minor loop predictions from the unconstrained ${\mathbb{F}}_{\mathrm{ML}}$ at stress level 395 MPa are shown in Figure 9 with the temperature history on top. The prediction of the upper minor cycle in Figure 9(a) diverges significantly from the experimental minor cycle behavior. For the prediction of the lower minor cycle shown in Figure 9(b), the responses cross through the major cycle behavior, and generate an unrealistic behavior. It can be seen that the model follows the major cycle well, but begins to diverge at the reversal points of the minor loop.

Figure 9.

Predictions of partial cycles in the Ni_50.3Ti_29.7Hf₂₀ SMA from ${\mathbb{F}}_{\mathrm{ML}}$ without constraints: (a) and (b) respectively show lower minor cycle and upper minor cycle predictions with respect to the major cycle response. The temperature history of the responses is plotted above. Without constraints, the presented ML model cannot predict the minor cycle responses accurately.

The particular predictions of partial transformation in Figure 9 can be improved by including also the experimental minor cycle responses from Section 3. However, the challenge is that the partial transformation has different responses while varying the combinations of endpoints of heating and cooling as described in Section 2. Although adding the particular experiments in Figure 9 can improve the prediction in the same path, it is not sufficient to map the whole combinations of partial transformation loops. To completely map the partial transformation region, one will have to do many experiments involving all combinations of partial transformations which could be expensive. Instead, in this work, the improvements in predicting partial phase transformation with the use of simple constraints are explored.

5.2. Prediction of partial cycles applying constraints

The predictions of partial phase transformation responses with ${\mathbb{F}}_{\mathrm{ML}}$ are improved significantly with the addition of constraints from Section 4.2. Figure 10 shows a comparison of partial transformation responses at stress level 395 MPa from ${\mathbb{F}}_{\mathrm{ML}}$ with constraints. In Figure 10(a), the $1^{\mathrm{st}}$ upper minor cycle response in Figure 3(b) is predicted using the model and compared with the experimental response. It is evident that the predictions improved compared to that in Figure 9(a) and match well with the experiments. Figure 10(b) shows a prediction of lower minor cycle response compared with experiments. The predicted response is similar to the experimental response, with a slightly higher strain at the lower region. This is because the major cycle responses in (a) and (b) had a different history than the cases used in the training.

Figure 10.

Predictions of minor cycle responses using the ${\mathbb{F}}_{\mathrm{ML}}$ for Ni_50.3Ti_29.7Hf₂₀ SMA at 395 MPa compared to the experiments: (a) upper minor cycle and (b) lower minor cycle. With physics-based constraints, the ML model predicts the minor cycle responses accurately.

5.3. Ensemble response and uncertainty quantification

Each training of the neural network model ${\mathbb{F}}_{\mathrm{ML}}$ can give different weights and biases because of using stochastic-based optimization algorithms. One technique to address this variability and study the uncertainty in the model is to look into the ensemble behavior of many training. For a targeted response, the ensemble average response can be calculated, and the standard deviation gives an estimate of confidence. In Figure 11, the ensemble average estimations corresponding to the two experimental cycles are shown. The Confidence Interval (CI) with 1 standard deviation is created based on the variability from the 50 training of ${\mathbb{F}}_{\mathrm{ML}}$ . The CI has a smaller range in the major cycle regions and a wider range in the minor cycle regions. This indicates that the model reproduces the major cycle responses used in the training with better confidence, and the minor cycle regions have higher uncertainty as they are predictions beyond the training region. The average response from the ensemble matches exactly with the upper minor cycle, as shown in Figure 11(a), and also captures a similar response in the case of the lower minor cycle, shown in Figure 11(b). Moreover, in comparison with the predictions in Figure 10, it can be concluded that the particular ${\mathbb{F}}_{\mathrm{ML}}$ training presented gave predictions close to the ensemble average response.

Figure 11.

Prediction of minor cycle responses in Ni_50.3Ti_29.7Hf₂₀ SMA at 395 MPa: (a) upper minor cycle and (b) lower minor cycle. The Confidence Interval (CI) showing the uncertainty due to the variability in the training of ${\mathbb{F}}_{\mathrm{ML}}$ . An ensemble containing 50 training of the ${\mathbb{F}}_{\mathrm{ML}}$ model is used. The CI marked 1 standard deviation from the mean. With ensemble training of the ML model, uncertainty in the predictions are captured.

5.4. Simulating complex thermal paths

A specific training of ${\mathbb{F}}_{\mathrm{ML}}$ is employed to simulate the SMA responses under complex cooling-heating trajectories. The selection is made to ensure that the prediction closely approximates the ensemble average and is subsequently utilized for the following predictions. For the selected Ni_50.3Ti_29.7Hf₂₀ SMA, new major cycles, minor cycles and inner cycles responses are simulated. Figure 12 shows predictions of upper and lower minor cycles with varying temperature ranges at stress levels 250 and 300 MPa. The temperature paths simulated at these stress levels are shown at the top of each figure. First the major cycle response and then the lower minor cycles and the upper minor cycles are simulated. Different minor cycles are simulated reducing the temperature range by 2°C in the adjacent cycles. It is seen that in all the cases, the minor cycle responses are confined within the major cycle because of the physics based constraints from Section 4.2. In the absence of these constraints, the minor cycle predictions will cross through the major cycle response as seen in Section 5.1. Also, these predictions of minor cycles have similar nature to the experimental minor cycle responses.

Figure 12.

Additional actuation responses in Ni_50.3Ti_29.7Hf₂₀ SMA calculated using the ${\mathbb{F}}_{\mathrm{ML}}$ model for stress levels: (a) 250 MPa and (b) 300 MPa. Major cycles and minor cycles produced during the respective thermal history are shown. Presented ML model can predict upper and lower minor cycle responses in a wide range.

Inner partial cyclic responses with partial heating and partial cooling are simulated for 300 MPa. The cycles are repeated seven times to simulate the evolution in multiple cycles. Cycles with initiation during cooling (Figure 13) and initiation during heating (Figure 14) in the major cycle are shown. The evolution of the inner cycles at three different sizes are simulated varying the size of temperature range. The inner cycles evolve significantly at the initial repetitions, and the difference gets reduced over succeeding cycles and reaches a steady cyclic response.

Figure 13.

Validation for inner minor cycles with initiation during cooling in Ni_50.3Ti_29.7Hf₂₀ SMA at 300 MPa. The corresponding temperature history is shown on top of each response. Thermal hysteresis between (a) 135°C–170°C, (b) 145°C–170°C, and (c) 155°C–170°C are studied. Presented ML model can predict complex thermal responses involving lower inner minor cycling.

Figure 14.

Validation for inner minor cycles with initiation during heating in Ni_50.3Ti_29.7Hf₂₀ SMA at 300 MPa. Corresponding temperature history is shown on top of each response. Thermal hysteresis between (a) 170°C–140°C, (b) 165°C–135°C, and(c) 160°C–140°C are studied. Presented ML model can predict complex thermal responses involving upper inner minor cycling.

The inner cycles show different evolution depending on whether its initiation is during cooling (Figure 13) or during heating (Figure 14). In Figure 13, where the initiation is during cooling, the strain value shifts higher in the successive cycles. This upward shift is consistent with experimental observations on evolution of inner cycle initiated during cooling reported in Amengual et al. (1996). In Figure 14, where the initiation is during heating, the strain shifts to lower value in the successive cycles. In the current model, the corresponding evolution of microstructure and transformation surface during partial phase transformation are not modeled and therefore cannot reason well this change of directions. When compared with the experiments of minor cycles (Section 3), where only either partial heating or partial cooling is considered, the evolution is only seen to increase with successive cycles. Therefore, the higher magnitude and different direction of shift in these predictions could be an artifact in the ${\mathbb{F}}_{\mathrm{ML}}$ model, and may differ from reality. In order to improve the model to capture this phenomenon accurately, extra constraint parameters will be needed that describe the microstructure and transformation evolution during the phase transformation.

The current analysis showed that the ${\mathbb{F}}_{\mathrm{ML}}$ model can predict the three types of minor cycles (upper, lower, inner) with realistic behavior. Although the training involved only major cycle responses, the model successfully predicted complex minor cycle responses outside of its training domain. The incorporation of physical constraints in the modeling framework is essential to ensure a stable and convergent response. When constraints are imposed, the response remains confined within the major cycle and closely follows experimental major and minor cycles. The quality of these complex minor cycle predictions validate the effectiveness of the constraints approach for extending the model predictions. As the presented ${\mathbb{F}}_{\mathrm{ML}}$ is not aware of the micromechanics, the model can be improved incorporating the evolution of microstructure with additional physics based constraints. Nevertheless, at high strain levels beyond the training stresses, the model may exhibit divergence due to extrapolation. Additionally, the current framework does not account for transformation-induced plasticity (TRIP), preventing the evolution of the strain derivative with the number of cycles. Ongoing work aims to incorporate TRIP into the model to enhance its predictive capability of cyclic evolution.

6. Summary

In this work, a data-based and physics-constrained neural network framework is proposed to predict the experimentally observed nonlinear actuation responses of Shape Memory Alloys (SMAs). The framework takes four state variables as input to describe the rate of thermal actuation, which are current stress, current strain, current temperature and thermal loading direction. Further, we proposed physics-based constraints in the ML framework to capture complex minor cycle responses in the SMA actuation resulting from partial phase transformation. With the introduction of constraints, the framework captured the upper and lower minor cycle responses close to the experiments, and their effectiveness is discussed.

Actuation responses in a high-temperature NiTiHf SMA is considered for modeling and testing predictions. The neural network model is targeted to fit the nonlinear variation of strain rate in the SMA response during thermally induced transformation. The strain rate at the reversal points of the minor loop are specified as potential physical constraints. These constraints captured the activation of thermoelasticity while changing the loading direction away from the transformation surfaces. The constrained model is then trained using the data of major actuation cycles at different stress level. The following conclusions are made based on the model’s capabilities.

• As expected, the ${\mathbb{F}}_{\mathrm{ML}}$ model without constraints can accurately predict the major cycle responses. However, the responses of minor cycles with this model deviated from reality.

• The model with physical constraints gets capabilities to predict beyond the training responses. The resulted model can predict accurately the major and minor cycles even though the training uses only the major cycle experimental data.

• An uncertainty quantification on the predictions is performed using ensemble training of the model. The average response of the ensemble matched the experiments, and the confidence interval for the predictions are estimated from the ensemble variations.

• The resulting model is providing an efficient computational platform to simulate the phase transformation behaviors under the complex thermal loading paths.

Two perspectives for the extension of the presented framework are: (i) accounting phase transformation behaviors under coupled thermomechanical loading and (ii) accounting the effects of plastic deformations such as Transformation Induced Plasticity (TRIP) for cyclic loading. For the first perspective on coupled thermomechancial loading, the current approach must be modified accounting the stress driven phase transformation with additional physical constraints of thermoelasticity at the initiation of partial transformation during the mechanical loading. For the second perspective, additional internal state variables and physical constraints accounting for the microstructural changes during phase transformation will be required.

In the realm of physics-based modeling, identifying the material parameters through calibration with experimental data is crucial, whereas in the ML approaches, the governing relationships are learned directly from the data. However, an inherent drawback of the ML approaches is their inability to predict beyond the training data. This can be resolved by introducing physics-based constraints which can improve the predictive capability in the ML approaches.