A phenomenological constitutive model for one-dimensional shape memory alloys based on artificial neural network

Abstract

Shape Memory Alloys (SMA) have become a material with great application prospects because of their unique characteristics and superior properties. A phenomenological constitutive model for SMA is constructed in the current paper. The proposed constitutive model is based on the phenomena observed in the experiment, and Artificial Neural Networks (ANN) are used to simulate part of the characteristics of SMA. The parameter identification method is also proposed, where Back-Propagation (BP) algorithm and the nonlinear optimisation algorithm are used at the same time. The numerical experiment has been carried out, which can well capture the constitutive relationship curve obtained from uniaxial tension and compression experiments of SMA, thus the model can be verified. The model can also describe the phase transformation characteristics of SMA well.

Keywords

Shape memory alloy phenomenological constitutive model artificial neural network

1. Introduction

Shape Memory Alloys (SMA) have unique properties because of their unique ability to reversibly transform when subjected to appropriate thermal and/or mechanical loads (Lagoudas, 2008). Therefore it has considerable application prospects in many fields (Mohd Jani et al., 2014), such as oil and gas industry (Patil and Song, 2016), autonomous air vehicles (Pellone et al., 2017), wind energy (Das et al., 2019), surgical tools (Roshan et al., 2019), space exploration (Costanza et al., 2017) and so on.

For the application of SMA in various fields, a comprehensive and correct understanding of its characteristics is essential (Paiva and Savi, 2006), that is, an accurate constitutive model is required. The characteristics of SMA originate from the microscopic changes in the crystal lattice and is related to factors such as stress, temperature and loading history. The coupling of these factors brings many difficulties to accurately describe the deformation of SMA. Different constitutive models of SMA have been established. Falk (1983) established a single crystal constitutive model, based on Landau theory and considering the Helmholtz free energy function of SMA. Abeyaratne and Knowles (1993) proposed a one-dimensional mathematical constitutive relationship of SMA, based on a mechanical model of Erieksen stress-induced solid–solid phase transition. Boyd and Lagoudas (1996) proposed a constitutive model of mesomechanics, based on mesomechanics and thermodynamics.

Based on macro observations, many phenomenological constitutive models describing SMA have also been proposed. The macroscopic phenomenological constitutive model is based on experimental phenomena and introduces internal variables that characterize the degree of the phase transition to describe the macroscopic properties of materials. The form of this type of model is relatively simple, the parameter is easy to determine and the model obtain experimental verification, so it plays a huge role in engineering applications. Tanaka et al. (1986), Liang and Rogers (1990) and Brinson (1993) proposed the classic phenomenological constitutive model.

Tanaka et al., (1986) proposed a constitutive model, which assumes that the volume fraction of martensite $ξ$ only changes with temperature T in SMA under uniaxial strain $ε$ , and the stress $σ$ is a function of these two variables. The change of martensite volume fraction $ξ$ is used to describe the process of phase transformation, and an exponential function is chosen to describe the change of martensite volume fraction $ξ$ . The constitutive relationship of its differential form is as follows:

\overset{\cdot}{σ} = \frac{\partial σ}{\partial ε} \overset{\cdot}{ε} + \frac{\partial σ}{\partial T} \overset{\cdot}{T} + \frac{\partial σ}{\partial ξ} \overset{\cdot}{ξ} = E \overset{\cdot}{ε} + Θ \overset{\cdot}{T} + Ω \overset{\cdot}{ξ}

Integrate the above equation, one can obtain:

σ - σ_{0} = E (ε - ε_{0}) + Θ (T - T_{0}) + Ω (ξ - ξ_{0})

where E is the elastic modulus of SMA, which is a function of martensite volume fraction $ξ$ , and $Θ$ is the thermal elastic coefficient, $Ω$ is the phase transformation coefficient. There is a significant difference between the elastic modulus E of martensite and austenite. When the material is in the mixed state of the two phases, the overall elastic modulus of the material is related to the percentage of the two. Therefore, the elastic modulus of the material is generally expressed by the following formula:

E (ξ) = E_{A} + ξ (E_{M} - E_{A})

The phase transformation coefficient of the material:

Ω (ξ) = - ε_{L} E (ξ)

where $ε_{L}$ is the maximum residual strain.

In the Tanaka model, the martensite volume fraction $ξ$ is expressed in the form of an exponential function. In the process of SMA transformation from austenite to martensite, the martensite volume fraction is as follows:

ξ = 1 - e^{a_{M} (M_{S} - T) + b_{M} σ}

In the process of SMA transformation from martensite to austenite, the martensite volume fraction is as follows:

ξ = e^{a_{A} (A_{S} - T) + b_{A} σ}

where $a_{A} = \frac{\ln 0.01}{A_{S} - A_{F}}$ , $b_{A} = \frac{a_{A}}{C_{A}}$ , $a_{M} = \frac{\ln 0.01}{M_{S} - M_{F}}$ , $b_{M} = \frac{a_{M}}{C_{M}}$ .

In the Tanaka model, an exponential function is used to describe the change in the martensite volume fraction. Since the exponential function will produce meaningless points, Liang and Rogers (1990) used a trigonometric function to describe the change of martensite volume fraction. In the process of SMA transformation from austenite to martensite, the martensite volume fraction is as follows:

ξ = \frac{1 - ξ_{A}}{2} \cos [a_{M} (T - M_{F}) + b_{M} σ] + \frac{1 + ξ_{A}}{2}

In the process of SMA transformation from martensite to austenite, the martensite volume fraction is as follows:

ξ = \frac{ξ_{M}}{2} \cos [a_{A} (T - A_{S}) + b_{A} σ] + \frac{ξ_{M}}{2}

where $a_{A} = \frac{π}{A_{F} - A_{S}}$ , $a_{M} = \frac{π}{M_{S} - M_{F}}$ .

The Tanaka model and Liang–Rogers model both describe the transformation between austenite and martensite, but neither of them distinguishes whether martensite is temperature-induced or stress-induced. And stress-induced martensite is the basis for the shape memory effect of SMA. Brinson (1993) proposed a model that distinguishes stress-induced martensite from temperature-induced martensite, and divided the martensite volume fraction $ξ$ into stress-induced martensite $ξ_{S}$ and the temperature-induced martensite $ξ_{T}$ , which better describes the characteristics of the material. In Brinson model, the martensite volume fraction is expressed as:

ξ = ξ_{S} + ξ_{T}

The constitutive relationship of its differential form is as follows:

\overset{\cdot}{σ} = E \overset{\cdot}{ε} + Θ \overset{\cdot}{T} + Ω_{S} {\overset{\cdot}{ξ}}_{S} + Ω_{T} {\overset{\cdot}{ξ}}_{T}

Integrate the above equation, one can obtain:

\begin{matrix} σ - σ_{0} = E (ε - ε_{0}) + Θ (T - T_{0}) + Ω_{S} (ξ_{S} - ξ_{S 0}) \\ + Ω_{T} (ξ_{T} - ξ_{T 0}) \end{matrix}

Based on the classic Brinson model, the researchers also tried to more accurately describe the characteristics of SMA. In order to solve the inaccuracy of describing the compressive thermo-mechanical behaviour of SMA caused by tension–compression asymmetry, Yoo et al. (2015) derived new transformation kinetics which precisely predicts the compressive behaviour of SMA. Three new parameters are incorporated into the suggested transformation kinetics, and the suggested transformation kinetics can predict the thermo-mechanical behaviour of SMAs more precisely under compression. Kim et al. (2018) developed a numerical model of an SMA plate reflecting the tension–compression asymmetry of axial stress under pure bending. To achieve the goal, the Brinson model with modified martensite transformation kinetics was proposed with the Drucker-Prager yield criterion to realize the tension–compression symmetry of SMA. These are very useful attempts, but the model may be complicated, making calculations inconvenient.

Another commonly used approach to construct the constitutive model for SMA is to apply the time-dependent Landau theory. The essence is to characterize the martensite and austenite through the Landau free energy function while considering the dissipation term. This approach usually uses a symmetric free energy function, so the deformation of the material is the same in the process of tension and compression (only the signs are different). In (Wang and Melnik, 2012), a polynomial with only odd-order terms are used to express potential energy function. In fact, this mapping relationship can be well fitted by higher-order polynomials.

With the development of various neural networks, researchers began to apply this tool to the description of the constitutive relationship of SMA (Yu et al. 2017; Zhou et al., 2016). Li et al. (2006) developed a BP neural network model for prediction of the mechanical properties of porous NiTi SMA prepared by the thermal explosion. The predicted results agree with the actual data within reasonable experimental error. Mai et al. (2016) proposed a time-delayed dynamic neural network for modelling hysteresis of SMAs in online applications, by introducing a time delay between the input and output response. The proposed network was applied to an SMA wire actuator, and the experimental results demonstrate the effectiveness of the model. Different forms of neural networks have also been adopted. Peng et al. (2008) proposed an elasto-plastic constitutive model based on clustering radial basis function neural network. Gao et al. (2018) proposed a constitutive model of SMA wire based on a chaotic Gaussian bat optimized wavelet neural network.

What makes neural network methods so effective is that it is a ‘universal approximator’ (Hornik et al., 1989). In the mathematical theory of neural networks, the universal approximation theorem states that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact subsets of $R$ , under mild assumptions on the activation function. Neural networks work well in practice because they compactly express nice, smooth functions that fit well with the statistical properties of data we encounter in practice, and are also easy to learn using our optimisation algorithms (e.g. gradient descent). Similarly, the fact that deeper networks (with multiple hidden layers) can work better than a single-hidden-layer network is an empirical observation, despite the fact that their representational power is equal.

In the current paper, a phenomenological constitutive model for SMA is proposed, where Artificial Neural Networks (ANN) are used to simulate part of the characteristics of SMA. The constitutive model is proposed based on the constitutive relationship curve obtained from uniaxial tension and compression experiments of SMA, where viscous terms and elastic terms are introduced to describe the stress of SMA. The parameter identification method is also proposed in the current paper, where Back-Propagation (BP) algorithm is first used to obtain an initial solution, and then nonlinear optimisation algorithm is used to optimize the parameters. Finally, the model is verified through numerical experiments.

The paper is organized as follows. In section ‘Typical Experimental Curves’, the curves observed in uniaxial tension and compression tests are presented, and their characteristics are analyzed, which directly inspires the establishment of models. In section ‘Phenomenological Constitutive Model’, based on the experimental phenomenon, a phenomenological model is obtained, which is divided into two parts, viscous term and elastic term, and the elastic term is expressed by an artificial neural network. In section ‘Parameter Identification’, the parameter identification method of the proposed model is described, which is divided into two steps. The first step uses a rough method to estimate the parameters, whose result is used as the initial value for further optimisation. In section ‘Numerical Experiment’, the proposed model and method are used to carry out numerical experiments, including uniaxial tension and compression experiment, and a theoretical phase transformation experiment. The chart explaining the strategy and approach is shown in Figure 1.

2. Typical experimental curves

So far, a large number of experimental studies have been conducted on SMA under uniaxial condition. Miyazaki et al. (1981) found that the tension stress–strain curves are significantly different at different temperatures. Orgéas and Denis (1998) and Liu et al. (1998) found that when induced by stress, the mechanical properties of tension and compression are asymmetry. Leo et al. (1993) and Gadaj et al. (2002) found that the material temperature changes significantly under different loading rates.

The typical experimental constitutive relationship of SMA in uniaxial tension and compression (one-dimensional case) is first studied. Figure 2 shows the experimental stress versus strain curves for SMA in tension and compression, where the experimental data from (Thamburaja and Anand, 2001) is used here. The experiment used a TiNi material and was carried out at a temperature of 298 K, where an extensometer was used to obtain the macroscopic strain. The experiment was conducted at a low constant true strain rate of $5 \times 10^{- 4} s^{- 1}$ to ensure isothermal testing conditions.

From the experimental curves, we can find two characteristics of SMA. One of the characteristics is that the loading and unloading processes do not coincide, the stress during the loading process is greater, and the stress during the unloading process is smaller, that is, there is a hysteresis effect. Another characteristic is that the tension and compression processes are asymmetric, which is obvious when symmetrical the compression curve from the third quadrant to the first quadrant. These two characteristics are expected to be taken into account in the model we are going to propose.

3. Phenomenological constitutive model

For the non-coincidence characteristics of the loading and unloading processes, the derivative of strain with respect to time $\overset{\cdot}{ε}$ is introduced into the model. This part is called the viscous term of stress in the current paper. For the loading process, $\overset{\cdot}{ε}$ is positive, so the stress is increasing; for the unloading process, $\overset{\cdot}{ε}$ is negative, so the stress is decreasing. Except for the viscous term, the rest of the stress term is called elastic term in the current paper. So the constitutive model proposed in the current paper can be expressed as:

σ = μ \overset{\cdot}{ε} + f (ε, T)

(1)

where $μ \overset{\cdot}{ε}$ is the viscous term of stress, $μ$ is a constant; $f (ε, T)$ is the elastic term of stress, which is a function of strain $ε$ . The elastic term $f (ε, T)$ actually characterizes the elastic deformation of SMA when there is no phase transformation. If the asymmetry of tension and compression is not considered, the elastic term should be symmetric, but when we consider the asymmetry of tension and compression, it should be asymmetric. According to Wang and Melnik (2012), the physical meaning of the elastic term is the derivative of the potential energy function.

3.1. Viscous and elastic term

For a set of experimental stress–strain data, $\overset{\cdot}{ε}$ can be obtained according to experimental conditions, or by numerical differentiation:

{\overset{\cdot}{ε}}_{i} = {\begin{matrix} \frac{- 3 ε_{i} + 4 ε_{i + 1} - ε_{i + 2}}{2 Δ t} i = 0 \\ \frac{3 ε_{i} - 4 ε_{i - 1} + ε_{i - 2}}{2 Δ t} i = n \\ \frac{ε_{i + 1} - ε_{i - 1}}{2 Δ t} else \end{matrix}

(2)

where n is the number of data points obtained from the experiment. And the constant $μ$ can be obtained by linear regression:

μ = {({\overset{\cdot}{ε}}^{T} {\overset{\cdot}{ε}})}^{- 1} {\overset{\cdot}{ε}}^{T} {σ}

(3)

where ${\cdot} is$ the data column vector.

In other words, for a set of experimental stress–strain data, the derivative of the strain $\overset{\cdot}{ε}$ and the constant $μ$ can be obtained.

Now that the derivative of the strain $\overset{\cdot}{ε}$ and the constant $μ$ are obtained, the elastic term $f (ε, T)$ can be obtained through $f (ε, T) = σ - μ \overset{\cdot}{ε}$ . Perform the above operations on the data in Figure 2, and the dashed curve in Figure 3 can be obtained, that is, the curve of the elastic term of stress to the strain. It can be seen that when the viscous term $f (ε, T)$ is introduced, the hysteresis effect produced by the loading and unloading process is almost eliminated, that is, the loading and unloading curves almost coincide. And what we have to do next is to obtain a curve such as the solid line in Figure 3 to express the elastic term of stress, so that a phenomenological constitutive model of SMA can be established. It should be pointed out that the solid line (elastic term of stress) in Figure 3 is not symmetrical about the origin, which can explain the asymmetry of tension and compression.

3.2. Artificial neural network

For the elastic term $f (ε, T)$ , it can be expressed by polynomials, splines, etc., which is essentially a mapping relationship. As mentioned earlier, the neural network is considered to be a more general mapping relationship, rather than a specific mathematical expression. ANN provides a very complex nonlinear mapping relationship. Therefore, it can express not only simple polynomial and spline relationships, but also very complex relationships that cannot be expressed with clear mathematical formulas. In the current paper, ANN is used to express elastic term of stress, and its structure is shown in Figure 4, where strain $ε$ is the input, the elastic term $f (ε, T)$ is the output, there is only one hidden layer, and the number of hidden units is 3.

According to Figure 4, the mathematical expression of the ANN used can be written as:

\hat{f} (x) = v sigmoid (wx + b_{1}) + b_{2}

(4)

where $w$ and $v$ are weights in neural network, $b_{1}$ and $b_{2}$ are biases in neural network, $w$ is a $n \times 2$ matrix, $b_{1}$ is a $n \times 1$ matrix, $v$ is a $1 \times n$ matrix, $b_{2}$ is a $1 \times 1$ matrix (scalar), n is the number of hidden units, $n = 3$ , and $sigmoid (x) = \frac{1}{1 + e^{- x}}$ . The matrix definitions are also shown in Figure 5.

Another neural network that is very effective for nonlinear function approximation is the Radial Basis Function (RBF) neural network, which is a neural network that uses RBF as the activation function. The neurons in the hidden layer of the BP neural network use the inner product of the input of the network and the weight as the input of the activation function, each parameter has the same influence on the network, so the BP neural network is a global approximation to the nonlinear mapping. The neurons in the hidden layer of the RBF neural network use the distance between the network input and the centre of RBF as the input of the activation function, and the farther the network input is from the centre of RBF, the lower the activation degree of the neuron. RBF neural network therefore has the characteristic of ‘local mapping’, which means that the approximation of the function is based only on the data near the point. BP neural network approximates the function using all data, not just the data near the point. In addition, local mapping can simplify the amount of calculation, making RBF neural network training faster.

4. Parameter identification

4.1. A rough estimation method

As mentioned above, based on the rough estimate of $μ$ and $\overset{\cdot}{ε}$ , the elastic term $f (ε, T)$ can be obtained, which is the dashed line in Figure 3. Therefore, we can input $f (ε, T)$ as the true value into the ANN to train the ANN. Mature training algorithms, such as BP algorithms, can be used to obtain various parameters of ANN.

The core idea of the BP algorithm is to use gradient descent method to search for the best weight and bias term. Specifically, the loss function is used to move to the negative gradient direction every time until the loss function reaches the minimum value. The loss function is defined as follows:

E = \frac{1}{n} {(\hat{f} - f)}^{T} (\hat{f} - f)

(5)

The partial derivative of the loss function with respect to the weight and bias term of each layer, the gradient of the loss function, is calculated according to the chain rule.

\begin{array}{l} \frac{\partial E}{\partial w} = \frac{\partial h^{'}}{\partial w} \frac{\partial h}{\partial h^{'}} \frac{\partial \hat{f}}{\partial h} \frac{\partial E}{\partial \hat{f}} = ε h ⊙ (1 - h) v (\hat{f} - f) \\ \frac{\partial E}{\partial b_{1}} = \frac{\partial h^{'}}{\partial b_{1}} \frac{\partial h}{\partial h^{'}} \frac{\partial \hat{f}}{\partial h} \frac{\partial E}{\partial \hat{f}} = h ⊙ (1 - h) v (\hat{f} - f) \\ \frac{\partial E}{\partial v} = \frac{\partial \hat{f}}{\partial v} \frac{\partial E}{\partial \hat{f}} = h (\hat{f} - f) \\ \frac{\partial E}{\partial b_{2}} = \frac{\partial \hat{f}}{\partial b_{2}} \frac{\partial E}{\partial \hat{f}} = (\hat{f} - f) \end{array}

(6)

where $⊙ means$ that two matrices are multiplied element by element.

The gradient of the loss function is used to update the weight and bias term:

\begin{matrix} w^{(k)} = w^{(k - 1)} - η \frac{\partial E}{\partial w}, b_{1}^{(k)} = b_{1}^{(k - 1)} - η \frac{\partial E}{\partial b_{1}} \\ v^{(k)} = v^{(k - 1)} - η \frac{\partial E}{\partial v}, b_{2}^{(k)} = b_{2}^{(k - 1)} - η \frac{\partial E}{\partial b_{2}} \end{matrix}

(7)

where $η$ is the learning rate, k is the number of iterations. This iterative process will always be executed until the loss function obtains the minimum value or the set number of iterations is completed. In this way, the optimal parameters (weights and biases) in ANN are obtained.

The solid line in Figure 3 is the curve of the elastic term of stress to the strain obtained after training with BP ANN. It can be seen that the obtained curve can describe the elastic term well, so it can be used in the constitutive model of SMA.

But it should be pointed out that in the above method, $μ$ and $\overset{\cdot}{ε}$ are only rough estimates, so $f (ε, T)$ is also an estimate. The accuracy of the result obtained by this method is not high. So here a more accurate method is adopted.

4.2. A more precise method

The parameters of the proposed model become $μ$ , $w$ , $b_{1}$ , $v$ and $b_{2}$ . And for the ANN structure in the current paper, there are a total of 13 parameters. For a set of model parameters, given the stress, the strain $ε$ can be solved according to:

μ \overset{\cdot}{ε} + v sigmoid (wx + b_{1}) + b_{2} = σ

(8)

In other words, for a set of parameters and input stress $σ$ , output strain $ε$ can be obtained by solving ordinary differential equation. And the Runge–Kutta method is used in the current paper.

Therefore, the difference between the experimental strain and the simulated strain can be compared. For the current investigation, the least square estimation method is used, whose strategy is to make the simulated strain as close as possible to the experimental ones, in the sense of least square error. Using this method, the parameter identification problem can be transformed into a nonlinear optimisation problem as follows:

min_{μ, w, b_{1}, v, b_{2}} \frac{1}{n} \sum_{i = 1}^{n} {({\hat{ε}}_{i} - ε_{i})}^{2}

(9)

where n is the number of experimental samples, $ε_{i}$ is the experimental value of strain at the i-th sampling point, and ${\hat{ε}}_{i}$ the simulated one.

The optimisation problem discussed here is a conventional nonlinear optimisation problem, and nothing is special. The real difficulty for the parameter estimation is caused by the fact that, if the optimisation algorithm starts with an arbitrary initial value, numerical convergence cannot be guaranteed, due to the very strong nonlinearity in the governing equation. The value of $μ$ obtained through the rough estimation and the value of ANN parameters obtained through BP ANN is chosen as the initial value of optimisation.

For the solution of optimisation problems, one can use algorithms for solving nonlinear programming problems, such as the Nelder–Mead method, or heuristic algorithms, such as simulated annealing algorithm, genetic algorithm, tabu search algorithm, ant colony algorithm, etc. The Nelder–Mead method is used in the current paper, as it is one of the most commonly used and effective algorithms for solving unconstrained optimisation problems.

Flowchart of the proposed algorithm is shown in Figure 6, which explains in detail the calculation process of the algorithm.

It should be pointed out that through the above method, one does not need to train the ANN, but turn the parameters of ANN into variables to be optimized. This is because the stress is divided into viscous and elastic terms, although the elastic terms are expressed in ANN, there are also parameters to be optimized in the viscous terms.

5. Numerical experiment

5.1. Uniaxial tension and compression experiment

Using the model and method presented above, the parameters in the model can be obtained, and then the simulation stress versus strain relationship can be obtained. By plotting the experimental results and simulated results in the same figure, the simulated results are compared with the experimental results in Figure 7, where the experimental results are shown by the solid lines and the simulated results by dashed lines. It can be seen that the simulated results are in good agreement with the experimental results, indicating that the proposed model can accurately simulate the stress versus strain relationship. In addition, the non-coincidence characteristics of the loading and unloading processes and the asymmetry characteristics of tension and compression processes mentioned above have also been captured.

Figure 8 shows the mean squared error changes during optimisation (iteration) process. Since the parameters obtained through the rough estimation method is chosen as the initial value of the more precise method, Figure 8 actually shows both the error of the rough estimation method and the more precise method. The error of the rough estimation method is at the beginning of the iteration process, about 50; and the error of the more precise method is after the iteration process ends, about 0.05. It can be seen that the result obtained with the more precise method is better than the rough estimation method.

As mentioned earlier, if the relationship obtained by ANN is numerically integrated, the energy function with respect to strain obtained by ANN can be obtained, as shown in Figure 9. It can be seen that the potential energy function is not symmetric about the energy axis, which explains the asymmetry of tension and compression.

5.2. Phase transformation

Now that the relationship between the elastic term and energy function is known, the data of the existing model can be used to train ANN. Figure 10 shows the energy function with respect to strain and temperature in (Wang et al., 2009). This two-dimensional function can be used to train the ANN to get the model parameters, and for the viscous term $μ = 2 \times 10^{6} kg / (m^{2} \cdot s)$ , which is the same as the one in (Wang et al., 2009). At this time, due to the increase of data, more hidden units are used to improve the accuracy, $n = 10$ . The phase transformation induced by both mechanical and thermal loads are simulated. The single hysteresis loop associated with mechanically induced martensite re-orientations and the double hysteresis loops associated with the pseudo-elastic effects are presented.

The first numerical experiment is carried out to illustrate that, the sharp jumps of strain caused by austenite–martensite transformation, and thermal hysteresis loops associated with the phase transformation caused by cooling and heating processes, can be simulated by the proposed model, as shown in Figure 11. For the chosen SMA material, it is in austenite when the temperature is 300 K, while in martensite when 200 K. The chosen SMA has an initial temperature 300 K and initial strain zero, where there is only austenite. SMA is cooled down to 200 K, and austenite-to-martensite transformation will be induced. Due to the offset stress $σ_{v} = 1 \times 10^{7} Pa$ , $M +$ will become the favourite variant (transformation during tension), and there is a sharp jump of strain accompanying the transformation. Then SMA is heated up again to 300 K. The heating process will induce a martensite-to-austenite transformation, and another sharp jump of strain will occur when the transformation takes place. The two sharp jumps of the strain during the cooling and heating processes will not be coincident. In other words, the temperatures for transformation during cooling and heating processes are different, as shown by the solid line in Figure 11. The same numerical experiment is also carried out with the offset stress $σ_{v} = - 1 \times 10^{7} Pa$ , so $M -$ will become the favourite variant (transformation during compression), and the result is shown by the dashed line in Figure 11. It can be concluded that the thermal hysteresis loops associated with the phase transformation caused by cooling and heating processes are different for $M +$ and $M -$ , due to the energetically non-equivalence of the two martensite variants.

The second numerical experiment is carried out to illustrate that, the single hysteresis loop induced by martensite variant re-orientations under mechanical load and the double hysteresis loops associated with the pseudo-elastic effects, can be simulated by the proposed model, as shown in Figure 12. Both are driven by a sinusoidal mechanical load $σ_{v} = σ_{m} \sin 2 ω t$ , but with different temperature. The first chosen temperature is 205 K, where martensite is the only stable phase. So SMA transforms between two martensite variants, and there is only one hysteresis loop, as shown in Figure 12(a). The second chosen temperature is 255 K, such that austenite is globally stable and martensite is locally stable. Austenite is globally stable, so the strain will always be zero when there is no mechanical load; martensite is locally stable, so when the mechanical load is large enough, martensite will be formed due to the austenite-to-martensite transformation. SMA transforms between austenite and the two martensite variants, so there are two sub-hysteresis loops, as shown in Figure 12(b). Both the single hysteresis loop and the double hysteresis loops are not symmetrical about the origin, also due to the energetically non-equivalence of the two martensite variants. For the single hysteresis loop, the stress required for the phase transformation from $M -$ to $M +$ is greater than that from $M +$ to $M -$ . This is because the free energy of $M -$ is greater and therefore it is a more stable state. For the double hysteresis loops, the area enclosed by the hysteresis loop that transforms between $M -$ and A is larger than that of $M +$ . This is also because the free energy of $M -$ is greater, so it consumes more energy.

It should be pointed out that, Figures 11 and 12 simulate a single crystal lattice (depend on the parameters used), thus resulting in the discrepancy between the simulated abrupt behaviour and the experimentally observed smoother behaviour. When considering a single crystal lattice, the strain changes suddenly. However, if consider all the crystal lattices in the entire material, some crystal lattices are close to the force end, the stress is greater, and therefore the phase transformation occurs first; while other crystal lattices are far away from the force end, the stress is smaller and no phase transformation occurs. When considering all the crystal lattices, we can superimpose the strain–stress curve of each lattice to get a smoother curve, which is the curve obtained by the material in the experiment. Because of this, these simulation results are difficult to verify through experiments, we can only compare with the simulation results in the literature. The simulation results of the proposed model in the current paper are consistent with the simulation results in literature (Wang et al., 2009), which shows the reasonability of the results obtained.

6. Conclusions

In the current paper, a phenomenological constitutive model for SMA is proposed. The constitutive relationship curve obtained from uniaxial tension and compression experiments of SMA is first studied. The stress is divided into two parts, viscous terms and elastic terms, in the proposed constitutive model, and ANN is used to simulate elastic terms of SMA. The use of ANN makes parameter identification difficult, so the parameter identification method is also proposed in the current paper. BP algorithm is first used to obtain an initial solution, and then the nonlinear optimisation algorithm is used to optimize the parameters. The numerical experiment has been carried out, which can well capture the constitutive relationship curve obtained from uniaxial tension and compression experiments of SMA, thus the model can be verified. With the physical meaning of the model, the model can also describe the phase transformation characteristics of SMA well. The proposed model is formulated as an ordinary differential equation, which is very convenient for system dynamic analysis and implementing of modern control technology.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Fan Wang

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