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

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## Abstract

**:**

## 1. Introduction

## 2. Variable Speed Phase Transformation Constitutive Model

_{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.

#### 2.1. Phase Transformation Equation

_{A}and C

_{M}are coefficients related to the phase transformation critical stress and the phase transformation temperature of SMAs.

#### 2.2. Constitutive Equation

_{0}= 0, σ

_{0}= 0, ξ

_{TM}

_{0}=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 Table 1 and Table 2, which are identified by experiment.

## 3. Resistance Characteristic Model

_{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

_{0}is affected by the volume fraction of twinned martensite, detwinned martensite and austenite, ρ

_{0}can be described as

_{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 Table 1 and Table 2.

## 4. Parameter Identification

_{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.

## 5. Numerical Simulation

_{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

_{DM}, E(ξ

_{A}, ξ

_{DM}), and Ψ are obtained from Section 2. The relative change of resistance ΔR/R

_{r}can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ΔR/R

_{r}and ε are plotted in Figure 5.

_{0}= α

_{A}T

_{0}− α

_{M}T

_{r}, σ

_{0}= 0, ρ

_{0}= ρ

_{A}. Assuming that the temperature is constant during the experiment, the relative change in resistance in this case is

_{A}, ξ

_{DM}, E(ξ

_{A}, ξ

_{DM}), α(ξ

_{A}), and Ψ are obtained from Section 2. The relative change in resistance ΔR/R

_{r}can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ΔR/R

_{r}and ε are presented in Figure 6.

_{0}= 117 MPa, T

_{0}=25 °C (T

_{0}< M

_{f}

_{0}), ξ

_{A}

_{0}= 0. Combining constitutive Equation (7) with phase transformation Equation (5), ε

_{0}is

_{A}, ξ

_{DM}, E(ξ

_{A}, ξ

_{DM}), E(ξ

_{A}

_{0}, ξ

_{DM}

_{0}), α(ξ

_{A}), and Ψ are obtained from Section 2. The relative change in resistance ΔR/R

_{r}can be simulated by MATLAB. Combined with the simulation results of strain ε, the relations between ΔR/R

_{r}and ε are depicted in Figure 7.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Symbol | Meaning | Symbol | Meaning |
---|---|---|---|

T | Temperature | k | Crystal variable speed coefficient |

M_{s} | Martensite start temperature | K | Crystal variable speed function |

M_{f} | Martensite finish temperature | A | Austenite |

A_{s} | Austenite start temperature | DM | Detwinned martensite |

A_{f} | Austenite finish temperature | TM | Twinned martensite |

M_{p} | Martensite peak temperature | Ψ | Transformation modulus |

A_{p} | Austenite peak temperature | E_{A} | Elastic modulus of austenite |

σ | Stress | E_{TM} | Elastic modulus of TM |

σ_{s} | Starting stress | E_{DM} | Elastic modulus of DM |

σ_{f} | Finishing stress | ξ_{A} | Volume fraction of austenite |

σ_{p} | Peaking stress | ξ_{DM} | Volume fraction of DM |

ε | Strain | ε_{L} | Maximum residual strain |

α_{M} | Thermal expansion coefficient of martensite | α_{A} | Thermal expansion coefficient of austenite |

ν | Poisson’s ratio | a | Temperature coefficient of R |

S | Cross-sectional area. | l | Length |

ρ | Resistivity | ρ_{TM} | Resistivity of TM |

Ρ_{DM} | Resistivity DM | ρ_{A} | Resistivity DM |

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M_{f} (°C) | M_{s} (°C) | A_{s} (°C) | A_{f} (°C) | σ_{s} (MPa) | σ_{f} (MPa) |

35.8 | 62.8 | 47.6 | 81.9 | 26.5 | 326.8 |

E_{A} (GPa) | E_{TM} (GPa) | E_{DM} (GPa) | C_{M} (MPa/°C) | C_{A} (MPa/°C) | ε_{L} |

54.64 | 17.96 | 31.52 | 10.4 | 8.0 | 0.0645 |

α_{M} (°C^{−1}) | α_{A} (°C^{−1}) | l_{0} (mm) | k_{1} | k_{2} | k_{3} |

2.73 × 10^{−7} | 9.16 × 10^{−7} | 100 | 1.15 | 1.28 | 7.88 |

ν | a (°C^{−1}) | ρ_{TM} (Ω∙m) | ρ_{DM} (Ω∙m) | ρ_{A} (Ω∙m) |
---|---|---|---|---|

0.3 | 8.75 × 10^{−4} | 0.87 × 10^{−6} | 0.82 × 10^{−6} | 0.72 × 10^{−6} |

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**MDPI and ACS Style**

Lu, Y.; Zhang, R.; Xu, Y.; Wang, L.; Yue, H.
Resistance Characteristics of SMA Actuator Based on the Variable Speed Phase Transformation Constitutive Model. *Materials* **2020**, *13*, 1479.
https://doi.org/10.3390/ma13061479

**AMA Style**

Lu Y, Zhang R, Xu Y, Wang L, Yue H.
Resistance Characteristics of SMA Actuator Based on the Variable Speed Phase Transformation Constitutive Model. *Materials*. 2020; 13(6):1479.
https://doi.org/10.3390/ma13061479

**Chicago/Turabian Style**

Lu, Yifan, Rongru Zhang, Ye Xu, Lei Wang, and Honghao Yue.
2020. "Resistance Characteristics of SMA Actuator Based on the Variable Speed Phase Transformation Constitutive Model" *Materials* 13, no. 6: 1479.
https://doi.org/10.3390/ma13061479