# Theoretical and Experimental Study of a Thermo-Mechanical Model of a Shape Memory Alloy Actuator Considering Minor Hystereses

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## Abstract

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## 1. Introduction

## 2. Design Concept of the SMA Actuator

_{n}. Initially, the free ends of the SMA wire and the spring are connected at point A. At room temperature, the SMA wire is in martensite state and has maximum deformation x

_{m}= CA while the spring is at its minimum deformation state AE = x

_{n}− x

_{m}. This position is the equilibrium state of the actuator at room temperature T

_{∞}, which is assumed to be not higher than the final martensitic temperature M

_{f}.

_{m}to x

_{0}. The hardening forces the SMA wire to recover its original shape, which is its original undeformed length. The coordinate x

_{0}is the equilibrium position of the wire in the austenitic state corresponding to the maximally stretched spring. Once this coordinate is reached, the rise in wire temperature does not change the displacement of point A because the final austenitic temperature is reached. The whole volume of the wire is transformed into austenite, and after reaching this temperature, there are no structural-phase changes. After the voltage is switched off, the wire gradually lowers its temperature, turns into martensite and returns to the maximum stretched position corresponding to the point x

_{m}.

## 3. Development of One-Dimensional Dynamic Model of the Actuator

#### 3.1. Mathematical Modelling of the SMA Actuator Dynamics

_{0}= BC (see Figure 1), and its longitudinal deformation is x. Then, the length s and the strain ε of the wire are

_{s}(x,T) is the force in the SMA wire, ρ

_{s}is the density of the SMA wire, V

_{s}(x) is the volume of SMA wire, c

_{p}is the heat capacity of the wire, A

_{s}(x) is the surface area of the wire, h

_{c}is the convective heat transfer coefficient, ${T}_{\infty}$ is the room temperature; R(x,T) is the resistance of SMA wire and u is the voltage applied to the ends of the wire. The temperature dependence of h

_{c}is neglected.

_{c}is the electrical resistivity of the SMA wire and A

_{w}is its cross-sectional area.

_{0}is the initial diameter of the SMA wire and μ is the Poisson ratio. Considering the above relations, the system (3) is rewritten in the form

_{R}(T) is the temperature-dependent stiffness of the SMA wire presented by the following equation

_{m}, E

_{T}and E

_{d}are Young’s modulus of fully twined, partly twined, and detwinned martensite, respectively; ${\epsilon}_{m}^{y}$ is the yield strain of the twined martensite; ${\epsilon}_{m}^{d}$ is the minimum strain of detwinned martensite; E

_{a}is Young’s modulus of austenite.

_{s}is about 8%, the manufacturer recommends no more than 2.5% for extending the wire lifecycle [27]:

_{1}≈ 1.0165, k

_{2}≈ 1.042 and k

_{2}/k

_{1}≈ 1.025. These results show that the errors due to simplification of the equation are 1.065% for the time constant and 2.5% for the limit temperature, which values are entirely acceptable.

_{s}V

_{0}c

_{p}, one obtains:

^{E}and ${T}^{E}$ of the system (27) are found accepting $\ddot{x}=0$, $\dot{x}=0$ and $\dot{T}=0$:

#### 3.2. Mathematical Modelling of the Minor and Sub Minor Hystereses

_{m}is the volume of the martensite in the total volume vs. of the SMA wire. According to Madill’s model [41]:

_{0}are the current and tensile stress at $\theta =0$, ${k}_{m}^{C}$, ${k}_{m}^{H}$ referred to as temperature constants of cooling and heating, respectively, and M

_{s}, M

_{f}, A

_{s}, A

_{f}are the start martensite, final martensite, start austenite and final austenite temperatures correspondingly. The functions ${R}_{ma}^{C}\left(t\right)$, ${R}_{mb}^{C}\left(t\right)$, ${R}_{ma}^{H}\left(t\right)$, and remain constant if the wire is just being heated or just being cooled, and changes only if the sign of the fluctuation of the temperature changes.

_{s}, M

_{f}] or [A

_{f}, A

_{s}], the functions depend on the values of the martensite fraction ${R}_{m}^{f}$ and temperature T

_{f}at which the sign of the temperature fluctuation changes. If the sign of the temperature fluctuation changes from positive to negative, i.e., from heating to cooling, then

_{s}, A

_{s}]. In this case, the hysteresis does not exist, but a jump discontinuity in the point T

_{f}of change of the fluctuation sign of the relative martensite fraction is avoided. In Figure 3, a case is illustrated where a discontinuity is possible. If M

_{s}< T

_{f}< A

_{s}then, after moving from points A to B during heating, a cooling appears at point T

_{f}. If (14) is applied, the cooling process starts from point B, and the jump BC appears, followed by a smooth cooling along the curve DEA.

_{m}can be considered if cooling from a temperature T > A

_{f}is present and for ${T}_{f}\in \left[{M}_{s},{A}_{s}\right]$ a heating appears.

_{0}is when the SMA changes from austenite to martensite, i.e., the cooling process is present, and the SMA is in austenite phase.

_{i}(i = 1, 2, …, 14). A numerical algorithm for determining the fluctuation points is developed, and the results are depicted in Figure 5b. According to the algorithm, the sign of the temperature derivative as a function of time is calculated by the function $sd{T}_{i}=sign\dot{T}$ and its values are ±1. An additional function $sdd{T}_{i}=sd{T}_{i+1}-sd{T}_{i}$ that successively finds the differences of two adjacent signums of the temperature derivatives indicates that a fluctuation occurs if it obtains a value of ±2. Depending on where the point of the temperature fluctuation is, the appropriate equation for calculating the relative martensitic fraction is applied. The two graphs in Figure 6 show both cases for possible calculations of martensite fraction. If the model (36) is used (Figure 6a) it can be distinguished 10 cases of jump points. The relative martensite fraction is a smooth function only for the points T

_{12}÷ T

_{14}which lies outside the temperature interval [M

_{f},A

_{f}]. In Figure 6b, the points obtained using the improved model are shown. As one can see, the degenerated hysteresis loop appears for points T

_{0}÷ T

_{8}. Points T

_{9}and T

_{10}describe a sub minor hysteresis loop, and a minor loop starts from point T

_{11}.

## 4. Numerical Study of the System Behaviour Using Pulse Width Modulation Control

_{0}is the amplitude of the rectangular pulse, f is the frequency of the waveform, n = trunc(t × f) is the consecutive number of the period, $0<z<1$ is the duty cycle of the waveform. The function trunc(k) returns the greatest integer less than or equal to k.

_{p}= 900 J/(kg·°C), h

_{c}= 18 W/(m

^{2}·°C). The dashed line (pos.1) depicts the temperature time evolution without using PWM. As can be seen, the temperature reaches its limit value (19). In the same graph, the temperature when PWM is applied is shown (pos.2). The parts of the curve in which the wire is heated are shown in red, and the parts in which the cooling is performed are shown in blue. The limit temperature (19) cannot be reached when PWM is used due to cooling sections depending on duty cycle value. Maximum wire temperature and temperature increasing rates are controlled by changing the duty cycle value, which can improve the control of the actuator, especially in cases where a smoother operation is required. The limited cooling rate is a major factor that limits the actuator’s performance [44].

## 5. Experimental Studies and Validation of the Model

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the SMA actuator: 1—SMA wire, 2—bias spring, 3—point mass, 4—force-deformation diagram of the spring, 5—force—deformation diagram of the wire in the austenitic state, 6—force—deformation diagram of the wire in the martensitic state, F

_{sp}—spring force, F

_{d}—damping force, F

_{s}—SMA wire force.

**Figure 2.**Diagram of relative martensitic fraction R

_{m}vs. temperature

**T**: (

**a**) Major hysteresis if the sign of the temperature fluctuation changes when M

_{f}> T > A

_{f}; (

**b**) Minor hystereses if the change of the sign of the temperature fluctuation is between M

_{f}and M

_{s}or A

_{s}and A

_{f}.

**Figure 3.**A jump discontinuity in the martensite fraction when the change of the sign of the temperature fluctuation is in the interval [M

_{s}, A

_{s}].

**Figure 5.**Graph of the test function: (

**a**) graph of the temperature; (

**b**) temperature indicators sdT and sddT.

**Figure 6.**Calculated relative martensitic fraction R

_{m}: (

**a**) The case without considering the minor and sub minor loops; (

**b**) The case with minor and sub minor loops.

**Figure 7.**Time evolution of the SMA wire temperature: (

**a**) at f = 0.05 Hz, z = 0.5; (

**b**) at f = 0.1 Hz, z = 0.5.

**Figure 9.**Parametric study of the temperature T (

**a**), displacement x (

**b**) and SMA force F (

**c**) for change of duty cycle from 0.1 to 0.9 and f = 0.1 Hz.

**Figure 10.**Parametric study of the temperature T (

**a**), displacement x (

**b**) and SMA force F (

**c**) for change of duty cycle from 0.1 to 0.7 and f = 1 Hz.

**Figure 11.**Parametric study of the temperature T (

**a**), displacement x (

**b**) and SMA force F (

**c**) for change of duty cycle from 0.1 to 0.7 and f = 10 Hz.

**Figure 12.**Parametric study of the temperature T (

**a**), displacement x (

**b**) and SMA force F (

**c**) for different frequencies and z = 0.5.

**Figure 13.**Comparison of experimental and numerical results: (

**a**) time evolution of the displacement x and temperature T of the wire; (

**b**) temperature hysteresis.

Parameter | Notation | Value | Unit |
---|---|---|---|

Diameter of SMA wire | d | 0.00035 | m |

Initial length of SMA wire | s_{0} | 0.160 | m |

Voltage | u | 5 | V |

Resistance | R | 52 | Ω |

Density of SMA | ${\rho}_{s}$ | 6450 | kg/m^{3} |

Specific heat | ${c}_{p}$ | 200 | J/(kg·°C) |

Convection heat transfer coefficient | ${h}_{c}$ | 70 | W/(m^{2}·°C) |

Room temperature | 26 | °C | |

Martensite Young’s Module | ${E}_{m}$ | 21.7 × 10^{9} | Pa |

Young’s modulus of NiTi at partly twinned martensite | ${E}_{T}$ | 0.56 × 10^{9} | Pa |

Young’s modulus of NiTi at detwinned martensite | ${E}_{d}$ | 11.1 × 10^{9} | Pa |

Austenite Young’s modulus | ${E}_{a}$ | 55.5 × 10^{9} | Pa |

Yield strain of twined martensite | ${\epsilon}_{m}^{y}$ | 0.0024 | - |

Minimum strain of detwinned martensite | ${\epsilon}_{m}^{d}$ | 0.044 | - |

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**MDPI and ACS Style**

Mitrev, R.; Todorov, T.; Fursov, A.; Ganev, B.
Theoretical and Experimental Study of a Thermo-Mechanical Model of a Shape Memory Alloy Actuator Considering Minor Hystereses. *Crystals* **2021**, *11*, 1120.
https://doi.org/10.3390/cryst11091120

**AMA Style**

Mitrev R, Todorov T, Fursov A, Ganev B.
Theoretical and Experimental Study of a Thermo-Mechanical Model of a Shape Memory Alloy Actuator Considering Minor Hystereses. *Crystals*. 2021; 11(9):1120.
https://doi.org/10.3390/cryst11091120

**Chicago/Turabian Style**

Mitrev, Rosen, Todor Todorov, Andrei Fursov, and Borislav Ganev.
2021. "Theoretical and Experimental Study of a Thermo-Mechanical Model of a Shape Memory Alloy Actuator Considering Minor Hystereses" *Crystals* 11, no. 9: 1120.
https://doi.org/10.3390/cryst11091120