Resistance Characteristics of SMA Actuator Based on the Variable Speed Phase Transformation Constitutive Model

The shape memory alloy (SMA)-based actuators have been increasingly used in different domains, such as automotive, aerospace, robotic and biomedical applications, for their unique properties. However, the precision control of such SMA-based actuators is still a problem. Most traditional control methods use the force/displacement signals of the actuator as feedback signals, which may increase the volume and weight of the entire system due to the additional force/displacement sensors. The resistance of the SMA, as an inherent property of the actuator, is a dependent variable which varies in accordance with its macroscopic strain or stress. It can be obtained by the voltage and the current imposed on the SMA with no additional measuring devices. Therefore, using the resistance of the SMA as feedback in the closed-loop control is quite promising for lightweight SMA-driven systems. This paper investigates the resistance characteristics of the SMA actuator in its actuation process. Three factors, i.e., the resistivity, the length, and the cross-sectional area, which affect the change of resistance were analyzed. The mechanical and electrical parameters of SMA were obtained using experiments. Numerical simulations were performed by using the resistance characteristic model. The simulation results reveal the change rules of the resistance corresponding to the strain of SMA and demonstrate the possibility of using the resistance for feedback control of SMA.


Introduction
The shape memory alloy (SMA) is a kind of alloy which can remember its original shape. When deformed by an external force, it can return to the pre-deformed shape by heating. SMAs have shape memory effect and superelasticity, as well as good physical and chemical properties and biocompatibility [1]. As the actuators based on SMA have the advantages of simple structure, small size, light quality, and low cost, the SMA is increasingly used in different fields [2]. For example, advanced SMA-based devices have been designed to improve the aerodynamic performance of vehicles [3], to actuate light grippers and reusable non-explosive lock release mechanisms [4,5], and to generate constant force components [6], etc.
However, how to accurately and effectively control the output force and displacement of the SMA is challenging. The resistance characteristics of SMA is important for the closed-loop control of SMA actuators. When displacement or force is used as the feedback signal in the control of the SMA, it is necessary to add a corresponding displacement sensor or force sensor. However, in practical engineering applications, the addition of external sensors will increase the volume of the structure the resistivity, length and cross-sectional area of SMA on the relative change of resistance are fully considered. In order to simulate the relationship between resistance and strain, the characteristic parameters of the Ni-Ti SMA wire were identified using experiments, including the DSC experiment, the loading experiment and the thermal cycling experiment. The Ni and Ti contents of the SMA wire were 54.94 wt % and 45.06 wt %, respectively. Finally, the simulation results were analyzed and the possibility of using resistance for feedback control was proved.

Variable Speed Phase Transformation Constitutive Model
The phase composition of SMA changes with its stress and temperature. When the stress increases above the start critical stress (σ s ), twinned martensite is transformed into detwinned martensite, and when the stress reaches the finish critical stress (σ f ), the transformation is complete. However, when the stress decreases, the phase composition does not change. When the temperature increases to the austenite start temperature (A s ), martensite (including twinned and detwinned martensite) is transformed into austenite, and when the temperature reaches the austenite finish temperature (A f ), the transformation is complete. When the temperature decreases to the martensite start temperature (M s ), the austenite is transformed into twinned martensite, and when the temperature reaches the martensite finish temperature (M f ), the transformation is complete. In this study, a new variable speed phase transformation model is used to describe the phase composition change of SMA, including the phase transformation equation and the constitutive equation.

Phase Transformation Equation
In all the thermomechanical behaviors of SMA, the crystal phase transformation process of SMAs can be divided into three categories: martensite (M) to austenite (A), austenite (A) to martensite (M), and twinned martensite (TM) to detwinned martensite (DM). By referring to the derivation process of Liang and introducing the crystal variable speed function K, a new constitutive model was obtained in our previous research [23]. For a specific phase transformation process of the SMA, the crystal variable function K can be determined by the material parameters of SMA and a crystal variable speed coefficient k. The coefficient k is a constant determined by experiments, which reflects the rate distribution of the volume fraction of martensite/austenite with temperature or stress during the transformation process. Please note that all the variables and abbreviations mentioned in this article and their meanings are shown in Appendix A.
For the M → A transformation process, the volume fraction of austenite can be derived as where For the A → M transformation process, the volume fraction of austenite is derived as where where C A and C M are coefficients related to the phase transformation critical stress and the phase transformation temperature of SMAs. For the TM → DM transformation process, the volume fraction of detwinned martensite is derived as where

Constitutive Equation
The total strain of the SMA can be divided into three parts: the strain induced by elasticity, the strain induced by phase transformation and the strain induced by thermo [24]. According to the above strain classification, the constitutive equation can be derived as where The volume fraction of each crystal phase in the transformation process was derived in Section 2.1. A numerical simulation of the constitutive model was performed with the initial state setting as ε 0 = 0, σ 0 = 0, ξ TM0 =1, T 0 = 0 • C. The results are presented in Figure 1, which reveals the relationship between temperature, stress, and strain of the SMA. The parameters used in the simulation can be found in Tables 1 and 2, which are identified by experiment.

Constitutive Equation
The total strain of the SMA can be divided into three parts: the strain induced by elasticity, the strain induced by phase transformation and the strain induced by thermo [24]. According to the above strain classification, the constitutive equation can be derived as The volume fraction of each crystal phase in the transformation process was derived in Section 2.1. A numerical simulation of the constitutive model was performed with the initial state setting as ε0 = 0, σ0 = 0, ξTM0 =1, T0 = 0 °C. The results are presented in Figure 1, which reveals the relationship between temperature, stress, and strain of the SMA. The parameters used in the simulation can be found in Tables 1 and 2, which are identified by experiment.
With the above-mentioned constitutive model, the resistance characteristic model of SMA will be derived in the next section, which can predict the thermomechanical behaviors of SMA in accordance with its resistance.

Resistance Characteristic Model
In this part, a one-dimension SMA wire is considered. The resistance R of the SMA can be described as where ρ is the resistivity, l is the length, and S is the cross-sectional area. By taking the natural logarithm of Equation (11), and then applying the total differential operator, the resistance equation can be derived as dR Thus, the three factors affecting the change of resistance of SMA are separated and can be expressed separately. Among them, the resistivity varies with temperature, which can be described as where ρ 0 is the resistivity when temperature is 0 • C, and a is the temperature coefficient of resistance. Therefore, the resistivity equation of the decomposed form can be obtained as Since ρ 0 is affected by the volume fraction of twinned martensite, detwinned martensite and austenite, ρ 0 can be described as where ρ TM is the resistivity of twinned martensite, ρ DM is the resistivity of detwinned martensite, and ρ A is the resistivity of austenite. Since both ξ A and ξ DM are functions of temperature and stress, ρ 0 is the function of temperature T and stress σ, too. Combining with the phase transformation equation of the constitutive model from Equations (1)-(6), the relation between ρ 0 , T and σ can be obtained. A numerical simulation of the resistivity of SMA was carried out. The relationship between temperature, stress, and resistivity of the SMA is shown in Figure 2. The parameters used in the simulation are collected in Tables 1 and 2. Substituting Equations (14) into Equations (12), the resistance equation is derived as where r is the radius of the cross section, and ν is Poisson's ratio. Convert Equation (16) into incremental form, the resistance characteristic model can be finally derived as Materials 2020, 13, x FOR PEER REVIEW 6 of 12 Figure 2. Relationship between temperature, stress and resistivity of the SMA.
Substituting Equations (14) into Equations (12), the resistance equation is derived as where r is the radius of the cross section, and ν is Poisson's ratio. Convert Equation (16) into incremental form, the resistance characteristic model can be finally derived as

Parameter Identification
The parameters of the SMA wire are identified by experiments. The diameter of the SMA wire is 0.4 mm, with 54.94 wt % Ni and 45.06 wt % Ti. Firstly, the DSC experiment using a DSC synchronous thermal analyzer (LINSEIS STA PT1000) was performed, which gave the phase transformation temperatures, and the coefficients k1 and k2. Then, the loading experiments at 25 and 100 °C were performed with a tensile testing machine and a miniature high-low temperature chamber, as shown in Figure 3. The elastic moduli, critical stresses, maximum residual strain and the coefficient k3 were obtained. Finally, the thermal cycling experiment was conducted using a high-low temperature chamber (GDW/GDJS-100) and a laser displacement sensor (Keyence LK-G5000), as shown in Figure 4, based on which the coefficient CA as well as CM were obtained. The thermal and mechanical parameters of the SMA wire are collected in Table 1. The electrical parameters of the SMA were measured by an Agilent digital multimeter (344450A) and listed in Table 2.

Parameter Identification
The parameters of the SMA wire are identified by experiments. The diameter of the SMA wire is 0.4 mm, with 54.94 wt % Ni and 45.06 wt % Ti. Firstly, the DSC experiment using a DSC synchronous thermal analyzer (LINSEIS STA PT1000) was performed, which gave the phase transformation temperatures, and the coefficients k 1 and k 2 . Then, the loading experiments at 25 and 100 • C were performed with a tensile testing machine and a miniature high-low temperature chamber, as shown in Figure 3. The elastic moduli, critical stresses, maximum residual strain and the coefficient k 3 were obtained. Finally, the thermal cycling experiment was conducted using a high-low temperature chamber (GDW/GDJS-100) and a laser displacement sensor (Keyence LK-G5000), as shown in Figure 4, based on which the coefficient C A as well as C M were obtained. The thermal and mechanical parameters of the SMA wire are collected in Table 1. The electrical parameters of the SMA were measured by an Agilent digital multimeter (344450A) and listed in Table 2.

Numerical Simulation
Numerical simulations of three cases were carried out: the loading experiments at 25 °C (100% martensite in the SMA), the loading experiments at 100 °C (100% austenite in the SMA), and the thermal cycling experiment. The parameters used in the numerical simulation are shown in Table 1 and Table 2. The initial condition of loading at 25 °C is ε0 = 0, σ0 = 0, ρ0 = ρTM. Assuming that the temperature is constant during the experiment, the relative change in resistance in this case is

Numerical Simulation
Numerical simulations of three cases were carried out: the loading experiments at 25 • C (100% martensite in the SMA), the loading experiments at 100 • C (100% austenite in the SMA), and the thermal cycling experiment. The parameters used in the numerical simulation are shown in Tables 1 and 2. The initial condition of loading at 25 • C is ε 0 = 0, σ 0 = 0, ρ 0 = ρ TM . Assuming that the temperature is constant during the experiment, the relative change in resistance in this case is where ∆ρ 0 = ρ 0 − ρ TM . Combined with the variable speed phase transformation constitutive model, the three-segment function of the loading process is The function of the unloading process is where ξ DM , E(ξ A , ξ DM ), and Ψ are obtained from Section 2. The relative change of resistance ∆R/R r can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ∆R/R r and ε are plotted in Figure 5. The initial condition of loading at 100 Assuming that the temperature is constant during the experiment, the relative change in resistance in this case is where Combined with the variable speed phase transformation constitutive model, the three-segment function of the loading process is The function of the unloading process is where ξDM, E(ξA, ξDM), and Ψ are obtained from Section 2. The relative change of resistance ΔR/Rr can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ΔR/Rr and ε are plotted in Figure 5. The initial condition of loading at 100 °C is ε0 = αAT0−αMTr, σ0 = 0, ρ0 = ρA. Assuming that the temperature is constant during the experiment, the relative change in resistance in this case is Combined with the variable speed phase transformation constitutive model, the three-segment function of the loading process is The three-segment function of the unloading process is The three-segment function of the unloading process is where where ξ A , ξ DM , E(ξ A , ξ DM ), α(ξ A ), and Ψ are obtained from Section 2. The relative change in resistance ∆R/R r can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ∆R/R r and ε are presented in Figure 6.
Materials 2020, 13, x FOR PEER REVIEW 9 of 12 where 0 0 ( ) ( , ) where ξA, ξDM, E(ξA, ξDM), α(ξA), and Ψ are obtained from Section 2. The relative change in resistance ΔR/Rr can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ΔR/Rr and ε are presented in Figure 6. The initial condition of the thermal cycling experiment is σ0 = 117MPa, T0 =25 °C (T0< Mf 0), ξA0=0. Combining constitutive Equation (7) with phase transformation Equation (5), ε0 is The initial condition of the thermal cycling experiment is σ 0 = 117 MPa, T 0 =25 • C (T 0 < M f 0 ), ξ A0 = 0. Combining constitutive Equation (7) with phase transformation Equation (5), ε 0 is The stress changed little during the whole experiment, so it can be assumed that the stress was constant. The relative change of resistance is where Combined with the variable speed phase transformation constitutive model, the three-segment function of the heating process is The three-segment function of the cooling process is where where ξ A , ξ DM , E(ξ A , ξ DM ), E(ξ A0 , ξ DM0 ), α(ξ A ), and Ψ are obtained from Section 2. The relative change in resistance ∆R/R r can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ∆R/R r and ε are depicted in Figure 7.
where ξA, ξDM, E(ξA, ξDM), E(ξA0, ξDM0), α(ξA), and Ψ are obtained from Section 2. The relative change in resistance ΔR/Rr can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ΔR/Rr and ε are depicted in Figure 7. It can be observed from Figure 5 to Figure 7 that for different cases, the relative change in resistance of the SMA varies corresponding to the strain. In fact, for most applications of SMA-based actuators, we just need to pay close attention to a particular phase transition process of the SMA. Thus, it is possible to monitor the resistance instead of the strain of the SMA for precision control of It can be observed from Figure 5 to Figure 7 that for different cases, the relative change in resistance of the SMA varies corresponding to the strain. In fact, for most applications of SMA-based actuators, we just need to pay close attention to a particular phase transition process of the SMA. Thus, it is possible to monitor the resistance instead of the strain of the SMA for precision control of the SMA-based actuator. The proposed SMA resistance characteristic model can therefore be used to accurately calculate the output displacement or force of the actuator.

Conclusions
A new resistance characteristic model of SMA was proposed in this work based on the variable speed phase transformation constitutive model. The influences of different parameters, such as the resistivity, the length, the cross-sectional area of the SMA, and the temperature on the resistance, were fully considered. The coupling relationship of stress-strain-temperature-resistance of SMA was revealed. Compared with the existing models, the proposed model further considers the influence of the phase transformation rate on the change of SMA's resistivity, which can greatly enhance the accuracy of the model, especially for those with distinct asymmetry phase transformation process. Therefore, the proposed model can describe the one-to-one correspondence between resistance and strain of SMA in a particular phase transformation process more accurately and comprehensively; namely, the resistance can be used as an internal variable to reflect the strain of SMA. The model serves as an important theoretical basis for precision control of SMA-based actuator with resistance signal feedback, which will further promote the study on the control of SMA-based systems, and is conducive to the lightweight and miniaturization of SMA-based structures.

Conflicts of Interest:
The authors declare no conflict of interest.