Modeling and Analysis of Resistance-Sensing Characteristics for Two-Way Shape Memory Alloy-Based Deep-Sea Actuators

Abstract

Deep-sea actuators based on shape memory alloys (SMAs) are an emerging frontier field of multidisciplinary crossover, and the resistive sensing characteristics are the basis for the drive control of SMA deep-sea actuators. The resistance and resistivity of SMAs are complex and highly dependent on temperature and stress, and there is no complete description of SMAs for extreme environments of high pressure, low temperature, and high salinity in the deep sea. In this study, the logistic function is introduced to improve the kinetic equation of phase transition, and the macromechanical model, the law of resistance, and the resistivity mixing rule are integrated to model and analyze the resistive self-awareness characteristics of two-way shape memory alloy deep-sea actuators. The complex coupling relationships among resistance, strain, stress, resistivity, and temperature under constant load conditions are investigated, and the validity of the resistance-sensing model is verified by the water bath cycling test. The results show that the predicted values of the model agree well with the measured values. The self-perceived relationship between the resistance and deformation of the two-way shape memory alloy can be effectively expressed, which provides theoretical model support for the design of memory alloy deep sea actuators and sensorless drive control.

1. Introduction

The deep sea is the area of the ocean with a depth of more than 500 m. As the largest unknown region on the earth, it contains a variety of natural resources needed for the future development of human society, and it is the core area of the “smart ocean” and “transparent ocean” [1]. In recent years, with the deep integration of emerging foreword technologies in the marine field, the limitations of traditional deep-sea actuator systems, such as large volume, high cost, low reliability, and poor biocompatibility, have been infinitely amplified. The new concept of actuators based on smart materials has the advantages of miniaturization, intelligentization, and low energy consumption and has gradually become one of the frontiers of cross-application of marine technology and multi-disciplines in the new period [2].

Shape memory alloys (SMAs) are new functional materials with similar biological characteristics, which are closest to the practical application of marine engineering at present and have excellent thermomechanical properties [2,3]. Among them, nickel–titanium (Ni–Ti) SMAs are widely used in engineering practice due to their excellent shape memory effect (SME) and superelasticity [4,5]. At present, the application of SMAs in deep-sea engineering is less. Naganuma et al. [6] developed an automatic submarine hydrothermal sampler with an SMA spring as a high-temperature sensor and actuating element. Angilella et al. [7] developed a buoyancy-heat engine with SMA wire as the driving element, which improved the endurance and voyage of an unmanned underwater vehicle (UUV) and filled an important technical gap in this field. SMAs can generate a two-way SME through thermomechanical cycle training, and they can realize spontaneous and reversible bidirectional deformation with temperature rise and fall without additional auxiliary devices. As an actuator driving element, two-way SMA has the advantages of a light weight, simple structure and a high power-to-weight ratio. Resistance, displacement, and force can all be used as feedback signals for the drive control of SMAs, but displacement and force signals require additional external sensors, which are not conducive to the realization of full dynamic control of driving elements and miniaturized engineering applications. Therefore, resistance is usually used as a feedback signal. The resistance-sensing characteristics of SMAs are sensitive to temperature and stress. On the basis of its own resistance changes, it can sense the load, position, deformation, ambient temperature, and crystal structure of the driving element, while its complex nonlinear characteristics increase the difficulty of analysis and modeling, hindering the engineering applications of SMAs in deep-sea actuators. Currently, there are fewer studies on the application of the resistance-aware properties of two-way SMAs in the ocean, and most resistance models ignore the complex dependence of temperature and stress, which do not meet the application scenarios of deep-sea actuators [8,9,10,11]. Additionally, the relationship between resistance and resistivity with temperature is not uniform, and there are relatively few modeling studies for engineering applications. Therefore, it is necessary to investigate the resistive sensing characteristics of two-way SMAs to provide theoretical support for the design of deep-sea actuators and sensorless drive control.

Aiming at the extreme application scenarios in the deep sea, the logistic function was introduced to improve the traditional cosine phase-transition dynamic equation, and a simple and effective engineering application model of two-way SMA resistance-sensing was constructed. The complex relationship between resistance, resistivity, deformation, and temperature under constant load is revealed. The relevant characteristic parameters of the resistance model were identified through the water bath temperature cycle experiment under constant load, and the experimental data were used to verify the validity of the model.

2. Experimental Setup

The SMA wire is a two-way Nitinol with diameters of 0.3 mm (Ni 50.8 at%, Ti 49.2 at%; Beijing Jiyi Technology Co., Ltd., Beijing, China). The water bath temperature cycle test system was built with a constant temperature digital water bath box (Shanghai Lichen-BX Instrument Technology Co., Ltd., Shanghai, China), an 8½-digit digital multimeter (Keysight Technologies, Santa Rosa, CA, USA), and a constant load test platform of SMA wires to study the resistance-sensing characteristics of the two-way SMA wire under constant load thermal cycling. This is shown in Figure 1 and Figure 2.

The two ends of the SMA wire are connected to Teflon wires; one side is connected to the mounting weight by the inelastic nylon wire supported by the pulley, and the other side is fixed on the insulating seat of the device platform, which is insulated from the metal base platform as a whole. The initial length of the SMA wire is 330 mm. The SMA wire was stabilized at −4 °C for 2 h before the test so that the sample was in a completely stable martensitic state. During the experiment, a 50 g weight was mounted on one end of the SMA wire. The shrinkage of the SMA wire is driven by heating through a water bath, and the maximum heating temperature is 99 °C. The heating rate was 5 °C/min, and each instantaneous temperature was maintained for 2 min. After heating, the SMA wire was recovered by adding ice cubes to a water bath for cooling, and the lowest temperature was 12 °C. The resistance change of the SMA wire during the temperature cycle was measured with a high-speed digital multimeter at 60 readings per second and transferred to the computer for storage. The effective length of the wire is read through the scale of the device platform.

3. Resistance-Sensing Model of SMA

The resistance of the SMA wire is directly related to the resistivity and the change in length of the material. Resistivity is strongly dependent on the volume fraction of martensite, which is a result of the joint action of temperature and stress. Therefore, the resistance-sensing characteristic of an SMA is a function of temperature, stress, and strain.

3.1. Macromechanical Characterization of SMAs in the Deep-Sea Environment

SMA wires are often used as the driving elements of deep-sea microactuators, and their special nonlinear characteristics of thermodynamic coupling can be described by a macroscopic phenomenological constitutive relation. In the special environment of the deep sea, the deformation of isotropic SMA wires with small diameters and compact structures under the action of omnidirectional hydrostatic pressure can be ignored, and the multidimensional load state can be simplified to one dimension [12,13]. According to the different dynamic functions of phase transformation, the widely used SMA macromechanical models mainly include cosine function and exponential function [14,15].

In response to the drawback that the Tanaka and Liang-Rogers model cannot describe the martensite selective orientation, the Brinson model extends the phase transformation causative factor into two parts: temperature and stress. It is assumed that the transformation modulus and elastic modulus of SMAs have a linear relationship with the volume fraction of the martensite. These can be expressed as follows:

$$\xi ={\xi }_T+{\xi }_S$$

(1)

$$D(\xi )=\xi D_M+(1-\xi )D_A$$

(2)

$$\mathrm{\Omega }(\xi )=-{\epsilon }_{L}D(\xi )$$

(3)

where $\xi$ is the volume fraction of martensite and ${\xi }_T$ and ${\xi }_S$ are the volume fraction of martensite induced by temperature and stress, respectively; $D(\xi )$ is the elastic modulus of SMAs, and $D_M$ and $D_A$ are the elastic moduli of the complete martensite and austenite phases, respectively; $\mathsf{\Omega }$ is the phase transformation tensor and ${\epsilon }_L$ is the maximum recoverable residual strain of the SMA.

Combining Equations (1)–(3), the constitutive relation of SMA described by the Brinson model can be expressed as follows:

$$\sigma -{\sigma }_0=D(\xi )\epsilon -D({\xi }_0){\epsilon }_0+\mathsf{\Omega }(\xi ){\xi }_S-\mathsf{\Omega }({\xi }_0){\xi }_{S0}+\mathsf{\Theta }(T-T_0)$$

(4)

where $\sigma$ is the stress, $\epsilon$ is the strain, T is the temperature, $\mathsf{\Theta }$ is the thermoelastic tensor of SMAs, ${\xi }_{S0}$ is the initial value of the detwinned martensite volume fraction, and the subscript “0” represents the initial value of the corresponding variables.

The phase transition temperature of SMA is directly related to the magnitude of the external stress, but there is no significant correlation with the direction of the stress [16,17]. Figure 3 shows the rate of change of stress with temperature as follows:

$${\displaystyle \frac{d\sigma }{d{T}_{\sigma }}}=C_i$$

(5)

Integrating Equation (5) results in the following:

$$T_{\sigma }={\displaystyle \frac{\sigma }{C_{\mathrm{i}}}}+T_0$$

(6)

where $\sigma$ is the applied external stress, $T_{\sigma }$ is the transformation temperature under stress, T₀ is the transformation temperature when the stress is zero and $C_i$ is the rate of change of the stress of the martensite transformation (i = M) or austenite transformation (i = A).

According to previous research [17], the variation trend of martensite volume fraction is a typical S-shaped nonlinear growth curve. By replacing the traditional kinetic equation of phase transformation with an improved logistic function, it can completely describe the slow change process at the beginning and end stages of phase transformation, which is more consistent with the experimental results [18]. Then, the kinetic equation of the phase transition of an SMA can be expressed as follows:

$$\xi ={\displaystyle \frac{1}{1+e^{k_i(T-T_i- \frac{\sigma }{C_i})}}}$$

(7)

Martensitic transformation:

$$\left\{\begin{array}{c}{T}_M={\displaystyle \frac{M_s+M_f}{2}}\begin{array}{ccc}& & \stackrel{•}{T}<0\end{array}\\ k_M={\displaystyle \frac{\tau }{M_s-M_f}}\begin{array}{ccc}& & \stackrel{•}{T}<0\end{array}\end{array}\right.$$

(8)

Reverse martensite transformation:

$$\left\{\begin{array}{c}{T}_A={\displaystyle \frac{A_s+A_f}{2}}\begin{array}{ccc}& & \stackrel{\cdot }{\stackrel{•}{T}>0}\end{array}\\ k_A={\displaystyle \frac{\tau }{A_f-A_s}}\begin{array}{ccc}& & \stackrel{•}{T}>0\end{array}\end{array}\right.$$

(9)

where T_i is the characteristic temperature of the martensite (i = M) or austenite (i = A) phases under a state of zero stress, and k_i is the transformation constant of the SMA; $\tau$ is the material constant, which is related to the composition ratio of Nitinol and can be obtained by the DSC experiments.

The internal crystal dislocation structure introduced during the cyclic training of the SMAs will generate a directional residual stress field, which will drive the two-way SME. Martensite in the two-way SMA exists in the detwinned form, and the internal stress generated by training can automatically transform the cooled twinned martensite into the detwinned form [19]. The specific 2D crystal phase structure and stress–strain–temperature curves are shown in Figure 4.

During thermal cycling under constant load, the external stress is constant ($\mathsf{\Delta }\sigma =\sigma -{\sigma }_0=0$). The initial condition of the low-temperature detwinned martensite crystal phase can be expressed as follows:

$$\left\{\begin{array}{c}\xi ={\xi }_s\\ {\xi }_0={\xi }_{s0}=1\\ {\epsilon }_0={\epsilon }_L\\ D({\xi }_0)=D_M\\ \mathsf{\Omega }({\xi }_0)=-{\epsilon }_L{D}_M\end{array}\right.$$

(10)

The mechanical properties of SMA can be expressed as follows:

$$\epsilon ={\epsilon }_L\xi -{\displaystyle \frac{\mathsf{\Theta }(T-T_0)}{D(\xi )}}$$

(11)

Given that the Young’s modulus of SMAs is much larger than the product of the thermoelastic tensor and temperature difference, the effect of thermoelasticity on SMA strain can be neglected [20,21]. Therefore, the complex nonlinear relationship between the SMA strain and martensite volume fraction can be simplified to a linear relationship:

$$\epsilon \approx {\epsilon }_L\xi$$

(12)

3.2. Temperature and Stress Dependence of Resistivity

SMAs have different resistivities in different crystal phase states. The changes in the driving process are complex, and there is a strong direct or indirect dependence on temperature and stress [22]. The influence relationship is shown in Figure 5.

With the change in temperature, the resistivity response of the SMA wires has three different regions. Among them, the resistivity of SMAs has a linear relationship with temperature in the non-transformation region (pure martensite and pure austenite), while it shows a nonlinear relationship in the mixed transformation region [23]. In the application of deep-sea actuators, the SMA wire is in a low-stress state, and the effect of stress on the resistivity of the SMA wire is much smaller than that of temperature, so it can usually be ignored in practical applications [22]. In the water bath experiment, the average tensile stress on the cross-section of the SMA wire with diameters of 0.3 mm is only 7.1 MPa (g = 10 N/kg) when a 50 g weight is mounted. This is also a small stress state, so only the effect of temperature on the conductivity is considered. In the non-phase transformation region, the temperature dependence of the resistivity of each phase can be uniformly expressed as follows:

$${\rho }_i={\rho }_{0i}+{\lambda }_i(T-T_{0i})+{\delta }_i\sigma ={\rho }_{0i}+{\lambda }_i(T-T_{0i})$$

(13)

where ${\rho }_i$ is the resistivity of martensite phase (i = M) or austenite phase (i = A), T is the temperature, ${\rho }_{0i}$ is the resistivity of each phase measured at temperature T_0i, ${\lambda }_i$ is the temperature coefficient of resistivity of each phase, and ${\delta }_i$ is the resistivity stress coefficient of each phase.

Based on the crystalline phase composition of the SMA material, the total resistivity can be expressed by the linear mixing rule as [17] follows:

$$\rho ={\rho }_M{\xi }_M+{\rho }_A(1-{\xi }_M)$$

(14)

3.3. Resistance-Sensing Characteristics of SMAs

As a temperature-controlled smart material, the SMA mainly includes two crystal phase structures of austenite and martensite (ignoring the intermediate R phase). With the change in the environmental temperature, there is martensitic transformation during cooling and reverse martensitic transformation during heating. Each crystal phase has a different resistance, and the specific structure is shown in Figure 6.

By analogy to the sensing principle of ordinary metal resistance wire, the resistance of SMA wire is a function of cross-sectional area, resistivity, and length. The resistance of each micro-length unit on the SMA wire can be expressed as follows:

$$d{R}_{M/A}={\displaystyle \frac{{\rho }_{M/A}}{S{\xi }_{M/A}}}dl$$

(15)

As shown in Figure 6, the sublayers of different crystal phases of SMAs can be considered to be formed in parallel [24,25], and the resistance of the SMA wire can be expressed as follows:

$$\begin{array}{l}R={{\displaystyle {\int }_0^L(d{R}_A^{-1}+d{R}_M^{-1})}}^{-1}={{\displaystyle {\int }_0^L(\frac{{\xi }_{A}S}{{\rho }_A}+\frac{{\xi }_{M}S}{{\rho }_M})}}^{-1}dl={({\displaystyle \frac{{\xi }_{A}S}{{\rho }_A}}+{\displaystyle \frac{{\xi }_{M}S}{{\rho }_M}})}^{-1}L\\ \end{array}$$

(16)

$$\left\{\begin{array}{c}{R}_{M/A}={\displaystyle \frac{{\rho }_{M/A}L}{S}}\\ {\xi }_M+{\xi }_A=1\end{array}\right.$$

(17)

where $R_{M/A}$ is the resistance of pure martensite or austenite phases, respectively; ${\rho }_{M/A}$ is the resistivity of pure martensite or austenite phases, respectively; ${\xi }_{M/A}$ is the volume fraction of martensite or austenite phases, respectively; L is the length of the SMA wires and S is the cross-sectional area of the SMA wire.

Combining Equations (16) and (17), the resistance (R) of the SMA wire can be expressed as follows:

$$R^{-1}={\xi }_M{R}_M^{-1}+{\xi }_A{R}_A^{-1}={\xi }_M{R}_M^{-1}+(1-{\xi }_M)R_A^{-1}$$

(18)

Combining Equations (12) and (18), the resistance (R) of the SMA wire can be expressed as follows:

$$R={\displaystyle \frac{R_M{R}_A}{{\xi }_M(R_A-R_M)+R_M}}={\displaystyle \frac{R_M{R}_A}{\frac{\epsilon }{{\epsilon }_L}(R_A-R_M)+R_M}}$$

(19)

According to the resistance characteristics of resistivity to stress and temperature, the resistance of SMA wires can be expressed as a function of temperature and stress. In practical engineering applications, the effect of small stress on the resistance of SMA wires is usually ignored because it is much smaller than the effect of temperature, so the resistance of each crystal phase can be expressed as follows:

$$R_i=R_{0i}+{\alpha }_i(T-T_{0i})+{\beta }_i\sigma =R_{0i}+{\alpha }_i(T-T_{0i})$$

(20)

where $R_i$ is the resistance of martensite phase (i = M) or austenite phase (i = A), T is the temperature, $R_{0i}$ is the resistance of each phase measured at the temperature T_0i, and T_0M and T_0A are the minimum and maximum temperatures of the experiment, respectively; ${\alpha }_i$ is the temperature coefficient of resistance and ${\beta }_i$ is the resistance stress coefficient.

4. Results

The relationship between the resistance, strain, and temperature of SMA wires under constant load was simulated and predicted by the resistance-sensing model, and the validity of the model was verified by the experimental data of the water bath.

4.1. Numerical Simulation and Analysis of SMA Resistance-Sensing Model

Temperature cycling under constant load was a typical application scenario for deep-sea actuators. SMAs, as a driving element, usually work under load conditions, so it was more in line with the actual situation to study the thermodynamic properties of SMA under load. The resistance-sensing characteristics of SMAs under constant load were numerically simulated and predicted using MATLAB R2015b, and the material parameters involved are shown in

Table 1. Performance parameters of SMAs.

Phase Transformation Temperature	Resistance Parameters	Young’s Modulus	Phase Transformation Parameters	Maximum Residual Strain
Mf = 34.5 °C	${\alpha }_M$ = 0.007 Ω/°C	DM = 67,000 MPa	CM = 8 MPa/°C	${\epsilon }_L$ = 0.04
Ms = 45 °C	${\alpha }_A$ = 0.003 Ω/°C	DA = 26,300 MPa	CA = 13.8 MPa/°C
As = 73 °C	R0M = 3.84 Ω	Θ = 0.55 MPa/°C	$\tau$ = 7
Af = 82 °C	R0A = 3.47 Ω
	T0M = 12 °C
	T0A = 99 °C

. The phase transformation temperature parameters were obtained by DSC (TA Instruments, Newcastle, DE, USA), the mechanical properties parameters refer to the experimental data in [26,27], and the resistance parameters were obtained by water bath experiments.

During the thermal cycles with a constant load, the initial conditions of this experiment were T₀ = 12 °C, ${\sigma }_0$ = 7.1 MPa, $\xi ={\xi }_S$, ${\xi }_0={\xi }_{S0}=1$, and ${\epsilon }_0={\epsilon }_L$. Figure 7 is the numerical simulation relationship between martensite volume fraction and temperature, which shows that SMA had obvious nonlinear and phase transformation hysteresis characteristics. During heating, when the temperature was lower than As, the martensite volume fraction remained constant ($\xi =1$), and no phase transformation occurred. When the temperature was between A_s and A_f, the volume fraction of martensite decreased suddenly, and the interior of the SMA was a mixed state of martensite and austenite. When the temperature was higher than A_f, the phase transformation ended ($\xi =0$), and the martensite was completely transformed into austenite. During cooling, the martensite volume fraction remained constant in the non-transformed regions with temperatures above M_s (pure austenite state) and below M_f (pure martensite state), while a sudden increase occurred in other regions.

The steady-state thermomechanical response of an SMA can be numerically simulated by this resistance-sensing model. Figure 8a,b shows that, in the temperature region of the pure martensite and austenite states, the resistance of the SMA wire exhibits a linear relationship with the temperature change, and the resistance-change rate of the martensite phase was significantly greater than that of the austenite mutually. In the temperature region of the mixed phase state, the resistance changed with temperature in a nonlinear relationship, and the resistance–temperature curves during heating and cooling did not completely coincide. There was an obvious hysteresis response phenomenon. Resistivity is closely related to temperature and phase transformation, and the change trend is synchronized with resistance. Figure 8c,d shows that the deformation and resistance of the SMA wire were not strictly linear, and the non-transformation region exhibits obvious nonlinear characteristics with increasing deformation amount. Although the amount of deformation in the high-temperature and low-temperature regions was basically constant, the resistance value was still changing. This trend indicates that the resistance change in the phase transformation region was dominated by the deformation effects, while the resistance change in the non-transformation region was dominated by the temperature effects.

4.2. Thermomechanical Response in the Water Bath Environment

The measured relationship between the resistance, strain, and temperature of the SMA wire in the water bath environment under constant load is shown in Figure 9. Figure 9a,b shows that the two-way SMA wire resistance–temperature curves do not overlap during temperature cycling. There was an obvious hysteresis effect, and the resistance changed by about 10%. During the reverse martensitic transformation process, the two-way SMA resistance first increased linearly with temperature and reached a maximum value at A_s (75 °C), then the resistance changed abruptly and decreased rapidly until it reached a minimum value at A_f (83 °C), and then expanded linearly and slowly. During the martensitic transformation, the two-way SMA resistance first decreased linearly with temperature, then increased rapidly and suddenly, and finally decreased linearly and slowly. The temperature reached a minimum and a maximum at M_s (46 °C) and M_f (35.5 °C), respectively. The resistivity also had a comprehensive characterization of linearity and nonlinearity in the temperature cycle process, and its variation characteristics were basically synchronized with the resistance. Approximate linear relationships were represented in the non-transformation region, while nonlinear relationships were presented in other regions. Figure 9c,d shows that the abrupt point of deformation of the two-way SMA wire with the temperature cycle is basically consistent with the characteristic points of the four phase transition temperatures. There is an obvious strain difference in the process of temperature rise and fall cycling. The maximum residual strain generated by the training gradually decreased with increasing temperature and then gradually recovered with decreasing temperature. The deformation and resistance of the SMA wire were not strictly linear. In the non-transformation region, the strain was roughly constant, and the resistance changed linearly with temperature. In the phase transformation region, the resistance changed approximately linearly with the deformation.

5. Discussions

In this study, a resistance self-sensing model of an SMA deep-sea actuator is constructed for extreme application environments in the deep sea. At present, there are fewer marine application studies on the resistance-sensing characteristics of two-way SMAs, and there is a lack of simple and effective engineering application models. Most of the resistance models ignore the complex dependence of temperature and stress, which is not suitable for the application scenarios of deep-sea actuators. The improved SMA resistance-sensing model based on logistic function can effectively describe the resistance-sensing characteristics of two-way SMA wires, especially in the beginning and end regions of phase transition, which has better superiority and adaptability. The effectiveness of the SMA resistance-sensing model was verified by the water bath cycle experiment, and the comparison results are shown in Figure 10. The results show that the simulation prediction results of the steady-state thermomechanical response of the SMA were in good agreement with the experimental data, and there was only a small deviation in the end temperature region of the martensitic transformation. The model can basically effectively reflect the actual resistance-sensing characteristics of the SMA wire, which indirectly shows that the resistance can be used as a feedback signal for the precise control of deep-sea actuators.

The specific error between the resistance-sensing model and the experimental data was quantified by introducing a root mean square error function (RMSE). The equation was as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{{\displaystyle \sum _{i=1}^n(y_e-y_m)}}^2}$$

(21)

where n is the number of samples of the experimental data, y_e is the experimental data, and y_m is the model output.

The closer the RMSE is to 0, the closer the simulated value is to the experimental value, and the better the fit of the curve. The resistance-sensing characteristic calculated by (21) shows that the predicted value and the simulated value agree well, as shown in Figure 11.

The resistance change of the SMA in the non-phase change region is mainly dominated by temperature, and the resistance characteristics of the SMA are consistent with ordinary metals. The resistance change of the SMA in the phase-change region is dominated by deformation, and the resistance characteristics of the SMA are completely opposite to ordinary metals. The main reason is that the change in the internal crystal structure of the SMA leads to the shrinkage deformation of the material, and the influence of deformation on SMA resistance is far greater than that of temperature, which is consistent with the experimental and numerical fitting results. On the other hand, the resistance of the SMA does not drop sharply, but it is a slow process. This phenomenon can be well described by the improved model of logistic function, and it is a good supplement to the existing model.

6. Conclusions

A two-way Ni–Ti SMA is an ideal application material for the two-way reciprocating drive of deep-sea actuators. The self-sensing coupling relationship between the resistance and deformation of the two-way Ni–Ti SMA was revealed by constructing a resistance-sensing model, which fills the gap in the engineering application model of deep-sea actuators and effectively promotes the application of SMAs in the marine field.

(1) The change in the internal crystal structure of SMAs has a prominent effect on their thermomechanical response. In the non-transformation region, the two-way SMA wire resistance and resistivity are dominated by temperature effects and exhibit an approximately linear relationship. In the phase transformation region, the deformation effect dominates. It exhibits nonlinear characteristics and has obvious hysteresis.

(2) The deformation and resistance of the SMA wire are not strictly linear and show obvious nonlinear characteristics in the non-transformation region.

(3) The SMA resistance-sensing model improved by the logistic function can effectively describe the resistance-sensing characteristics of the two-way Ni–Ti SMA wire and the predicted value agrees well with the experimental data. This will provide important theoretical support for the structural design and sensorless drive control of SMA deep-sea actuators.