# Use the Force: Review of High-Rate Actuation of Shape Memory Alloys

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermo-Mechanical Design Guidelines

#### 2.1. Force and Work Outputs

#### 2.2. Input Electric Energy

#### 2.3. High-Rate SMA Actuators

#### 2.4. Actuation Travel per Length of the Wire

#### 2.5. Summary

- (1)
- High-rate SMA actuators can have a response time (the time it takes to reach the maximal stress value) as short as few tens of microseconds.
- (2)
- The overall actuation duration is determined by $1/\omega $, where $\omega $ is the appropriate mechanical natural frequency.
- (3)
- To obtain high work per volume and high energy efficiency, high levels of stress in the SMA wire are required. Thus, in accordance with Equation (2), during the phase transformation the temperature of the SMA wire should be much higher than the stress-free transition temperatures (i.e., much higher than ${A}_{f}$). (Effects of fatigue and durability of the wire under high levels of stress and temperature are discussed in Section 6.) The desired levels of stress and temperature in the SMA wire are determined by a compromise between energy efficiency and actuator durability.
- (4)
- The required input energy per volume of the SMA wire, ${U}_{in}$, is determined by the desired temperature, via Equation (3).
- (5)
- After choosing a desired stress level, the required force, F, determines the cross section area A.
- (6)
- (7)
- After determining ${U}_{in},\phantom{\rule{0.166667em}{0ex}}A$ and ${L}_{0}$, the required input electric energy can be calculated by ${E}_{in}={U}_{in}A{L}_{0}$.

## 3. Selecting and Designing the Source of the Electric Pulse

## 4. Typical Experimental Results

#### 4.1. Materials and Methods

^{®}NiTi wires (${A}_{f}$ ≈ 80–90 ${}^{\circ}$C) with various diameters and lengths. This material was subjected to a thermo-mechanical preconditioning treatment by the supplier and has a significant two-way shape memory effect. At room temperature, this material is completely at the martensite phase (the martensite finish temperature is approximately 40 ${}^{\circ}$C [43]), and the reverse martensitic transformation does not involve the formation of the intermediate R-phase [43]. The grain size of Flexinol 90 ${}^{\circ}$C

^{®}is about 100 nm [44].

^{®}. The crimps were then covered by a two-part heavy-duty epoxy resin and glued into small ABS plastic cups that were then fitted into two intermediate alumina connectors which allowed for both thermal and electrical isolation of the wire from its mechanical frame and a comfortable gripping of the wire edges (cf. Ref. [33] for further details). In-fact, at extremely large stresses (>1.6 GPa), failure typically occurred in the crimp itself rather than the NiTi wire.

#### 4.2. Typical Results

## 5. Kinetics and Thermodynamics of the Phase Transformation

## 6. Phenomenological Approximations of the Clamp-Free Actuator Performance

#### 6.1. Energy Efficiency

#### 6.2. Duration of Actuation

#### 6.3. Repeatability and Fatigue

## 7. Detailed Design Tool: Full Dynamic Simulations of Clamp-Free Actuators

#### 7.1. Model Formulation

#### 7.1.1. Kinetics of the Transformation

#### 7.1.2. Force Equilibrium and Kinematics

#### 7.2. Actuator Response after the Phase Transformation Has Ended

## 8. Simulation Results

#### 8.1. Comparison with a Typical Experiment

#### 8.2. Effect of Characteristic Pulse Length

## 9. Discussion

- ${\tau}_{e}<\tau <\frac{\pi}{2{\omega}_{eff}}$; $\phantom{\rule{2.em}{0ex}}{t}_{peak\phantom{\rule{0.166667em}{0ex}}stress}\sim \tau $; $\phantom{\rule{2.em}{0ex}}{t}_{actuation}\sim \frac{\pi}{2{\omega}_{eff}}$;
- $\tau <{\tau}_{e}<\frac{\pi}{2{\omega}_{eff}}$; $\phantom{\rule{2.em}{0ex}}{t}_{peak\phantom{\rule{0.166667em}{0ex}}stress}\sim {\tau}_{e}$; $\phantom{\rule{2.em}{0ex}}{t}_{actuation}\sim \frac{\pi}{2{\omega}_{eff}}$;
- $\tau <\frac{\pi}{2{\omega}_{eff}}<{\tau}_{e}$; $\phantom{\rule{2.em}{0ex}}{t}_{peak\phantom{\rule{0.166667em}{0ex}}stress}\sim {\tau}_{e}$; $\phantom{\rule{2.em}{0ex}}{t}_{actuation}\sim {\tau}_{e}$.

## 10. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SMA | Shape Memory Alloys |

ODE | Ordinary Differential Equation |

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**Figure 1.**Schematics of a conceptual tablecloth-trick latch mechanism, based on fast SMA actuation, designed to reduce the required work output. The friction force at the contact between the deployed structure (large mass), M, and the latch (small mass), m, is greatly reduced by disengaging the masses as result of rapid actuation, compared with slow release [24].

**Figure 2.**Schematics of an SMA actuator in a spring-mass configuration. (

**a**) The SMA wire is connected to a mass and spring at its free-end. In ①, the wire is completely in its detwinned martensitic phase with a strain of ${\epsilon}_{0}$ relative to its original length in the austenite phase. In ②, the wire is completely in its austenite phase with an elastic strain of ${\epsilon}_{el}^{A}$. (

**b**) Diagram of austenite-martensite transformations in the stress-strain space. The upper flag diagram (thin solid lines) is the superelastic transformation path at temperatures above ${A}_{f}$. The lower curve (bold solid line) is the detwinning path at temperatures below ${M}_{f}$, Points ① and ② correspond with the diagrams in (

**a**). The dashed lines denoted by ${\sigma}_{QS}\left(\epsilon \right)$ and ${\sigma}_{Dyn}\left(\epsilon \right)$ are the shape-memory transformation paths under quasi-steady and rapid heating, respectively.

**Figure 3.**Schematics of the mechanical set-ups of the two experimental configurations: (

**a**) clamp-free; (

**b**) clamp-clamp.

**Figure 4.**Typical results of experiments in the clamp-free configuration of NiTi wires. The wire diameter was 0.38 mm and its length 93 mm, its free end was attached to a mass large enough to keep it aligned prior to actuation. (

**a**) Heating pulse of electric power transferred to the wire. (A zoomed-in view of the shape of the electric pulse is presented in the top panel of Figure 6.) (

**b**) Displacement measurements taken at the bottom end of the mass. (

**c**) Stress evolution in the wire. Points ①–④ signify the dead time of the wire, peak stress, onset of stress decrease in the wire, and the end of the transformation, respectively. The dashed line represents the equilibrium stress averaged over the first 1000 μs. (

**d**) Stress-strain curve plotted based on the data from figures (

**b**,

**c**). Points ③ and ④ are the same as in (

**c**). The linear dotted line has a slope equal to the elastic modulus of austenite, ${E}_{A}$, signifying the elastic relaxation of the austenitic wire at the end of the transformation.

**Figure 5.**Typical macroscopic stress results from experiments in the clamp-clamp configuration. The onset of the electric pulse was chosen as $t=0$. The curves show the measured tensile stress, in units of MPa, at each end of the SMA wire over a 4 ms time-period. The inset shows a zoom-in into the initial 100 μs.

**Figure 6.**Typical results of experiments in the clamp-clamp configuration. (

**Top**) Average wire temperature evolution in time, based on Ref. [34]. The dotted line shows the ∼3 μs pulse of electric power transferred to the wire, in normalized units, the onset of the electric pulse was chosen as $t=0$. (

**Bottom**) The blue squares are the integrated XRD intensity ${I}^{A}\left(t\right)$ of the austenite 110${}^{A}$ peak, normalized with respect to its mean final value (i.e., mean of plateau after 20 μs), corresponding to an estimated austenite volume fraction of 0.261 [46]. The solid curve is the mean measured tensile stress from all experiments shown in units of MPa, marked on the right vertical axis. The horizontal dashed line represents the equilibrium stress value averaged over the interval [1.4, 5] ms after the pulse (See Figure 5). The vertical dashed lines denote the different stages of the transformation. Reproduced from Dana et al. (2021) [34] with permission from Elsevier.

**Figure 7.**Effect of temperature on typical experiments in the clamp-clamp configuration. The wire temperatures immediately after the pulse are in the range $T\in [100,120,142,168,195,209]$. (

**a**) Zoom-in on the beginning of stress versus time curves measured by one of the force sensors, using the same wire and same initial stress, under different input energies. (

**b**) Normalized stress profiles (the left hand side of Equation (17)) for the curves presented in (

**a**). (

**c**) The stress versus time curves in (

**a**) presented over a larger period in the millisecond scale, showing the change in equilibrium stress.

**Figure 8.**Equilibrium stress, ${\sigma}_{eq}$ (left axis), as a function of temperature for experiments performed on a wire with a diameter of 0.2 mm and a length of 48 mm (the same as shown in Figure 7). In all tests, the initial stress was set to ${\sigma}_{0}=200$ MPa. The corresponding austenite volume fraction, ${x}_{eq}$ (right axis), for each point, was calculated using Equation (15).

**Figure 9.**Transition temperature as a function of the austenite volume fraction, measured at equilibrium under a constant stress of 20 MPa. The curved lines represent a fit to Equation (20a,b) with ${T}_{1}=24{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${T}_{2}=72{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${T}_{3}=71{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${T}_{4}=88{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${\lambda}_{1}=0.073$ and ${\lambda}_{2}=0.03$.

**Figure 10.**Measurements of the equilibrium stress, ${\sigma}_{eq}$, under different applied initial stress values, ${\sigma}_{0}$. The dashed lines represent calculations of ${\sigma}_{eq}$ as a function of ${\sigma}_{0}$ for constant T values, as indicated near each line.

**Figure 11.**Energy efficiency of a clamp-free actuator. Mechanical work output, ${U}_{out}$, presented as a function of the electrical input energy, ${U}_{in}$, based on Equation (3). The dashed lines represent different energy efficiencies based on the ratio between the specific output and input energies.

**Figure 12.**Approximated linear model for the stress-strain profile of the clamp-free experiment presented in Figure 4.

**Figure 14.**(

**a**) Evolution of the stress profile over 20 consecutive rapid heating actuations. (

**b**) Strain recovery under slow-rate heating in a thermal bath as a function of the temperature following the corresponding actuation cycle in (

**a**). The dotted line represents a virgin wire that was never actuated prior to the heating test.

**Figure 15.**Results of a simulation of our typical experiment presented in Figure 4. (

**a**,

**b**) The time evolution of the energy input into the wire normalized by its maximum value; ${U}_{in}\left(t\right)/{U}_{0}$, is presented in the blue dashed curve, and the evolution of the volume fraction of austenite, $x\left(t\right)$, in the solid red curve. (

**c**,

**d**) The temperature evolution of the wire in time, $T\left(t\right)$, is presented in the red solid curve, and the evolution of the volume-fraction-dependant transition temperature, $\tilde{T}\left(x\right)$, in the blue dashed curve. (

**a**,

**c**) A zoom-in into the initial 100 μs after the onset of the heating pulse.

**Figure 16.**Comparison of experimental results with our simulation. (

**a**) Time evolution of the measured and simulated displacement. The horizontal thin dashed line represents the maximal displacement of the wire due to the original strain of the detwinned martensite phase. (

**b**) Stress evolution in time. The experimental solid curve oscillates around the simulated curve. (

**c**) Stress-strain curves of both the experimental measurements and simulated results. The linear dotted line signifies the elastic relaxation of the austenitic wire with slope ${E}_{A}$, corresponding to its elastic modulus.

**Figure 17.**Simulated results of NiTi wire actuation at different values of characteristic heating time, ${\tau}_{e}\in [{10}^{-6},{10}^{-4},{10}^{-3}]$ s. Calculations considered a wire with diameter $d=0.2$ mm and length ${L}_{0}=50$ mm subjected to an initial strain of ${\epsilon}_{0}=0.03$, an attached mass of $0.1$ kg, and a discharged energy density ${U}_{0}=0.9$ J/mm${}^{3}$. (

**a**,

**b**) Stress evolution in time, $\sigma \left(t\right)$. (

**c**,

**d**) Normalized discharged energy onto the wire, ${U}_{in}\left(t\right)/{U}_{0}$. (

**e**,

**f**) Austenite volume fraction, $x\left(t\right)$. (

**g**,

**h**) Wire temperature evolution in time, $T\left(t\right)$. (

**a**,

**c**,

**e**,

**g**) A zoom-in into the initial 200 μs after the onset of the heating pulse. The heating time, ${\tau}_{e}$, increases in the direction of the annotated arrows.

**Figure 18.**Actuation performance of a NiTi wire at different values of ${\tau}_{e}$. The simulated configuration corresponds to that presented in Figure 17. (

**a**) Stress evolution as a function of strain. (

**b**) Energy output of the actuation cycle for different values of ${\tau}_{e}$. The output energy was normalized relative to the energy fraction invested in the phase transformation, ${U}_{in}(x=1)$. A corresponding actuation voltage for each value of ${\tau}_{e}$ was calculated based on Equations (9) and (10). (

**c**) Actuation duration, ${t}_{actuation}$, and response time, ${t}_{peak\phantom{\rule{0.166667em}{0ex}}stress}$, for different values of ${\tau}_{e}$.

**Figure 19.**Contour plots of key measures for actuator performance as a function of the heating time, ${\tau}_{e}$, and the mechanical response time, $\pi /2{\omega}_{eff}$. (

**a**) Total actuation time, ${t}_{actuation}$, in ms. (

**b**) Energy output (percent), calculated relative to the invested energy for heating up to $x=1$. The dashed lines represent boundaries of the different regimes denoted by I, II, and III. In (

**a**), lower values (brighter colors) are preferred, while, in (

**b**), larger values (darker colors) are preferred.

**Figure 20.**Comparison of previous experimental studies based on the different actuation regimes (see Figure 19).

**Table 1.**Comparison of a selection of previous studies of the thermo-mechanical response of SMA wires under electric pulse heating with respect to heat pulse duration, power source and boundary conditions.

Reference | Actuator Scale | Mechanical Boundary Conditions | Characteristic Times | Source of Electric Pulse | Voltage Range | |||
---|---|---|---|---|---|---|---|---|

Heat Pulse Duration (${\mathit{\tau}}_{\mathit{e}}$) | Phase Transformation Kinetics ($\mathit{\tau}$) | Mechanical Response Time ${}^{1}$ $\left(\frac{\mathit{\pi}}{2{\mathit{\omega}}_{\mathit{eff}}}\right)$ | Actuation Time ${}^{2}$ (${\mathit{t}}_{\mathit{actuation}}$) | |||||

Volkov et al. (2018) [35] | mm | Free. No loads. | ∼1 μs | ∼1 μs | Irrelevant ${}^{3}$ | Irrelevant | Discharged capacitor | ∼1 kV |

Vollach et al. (2017) [38,39] | mm | Clamped at both ends | ∼1 μs | ∼28 μs | Irrelevant | Irrelevant | Discharged capacitor | 2–4 kV |

Dana et al. (2021) [34] | mm | Clamped at both ends | ∼1 μs | ∼30 μs | Irrelevant | Irrelevant | Discharged capacitor | 2–4 kV |

Vollach et al. (2016) [33] | mm | Hanged mass | ∼1 μs | ∼28 μs | 1–5 ms | 1–5 ms | Discharged capacitor | 2–4 kV |

Malka and Shilo (2017) [24] | mm | Hanged mass and friction force | 80–300 μs | Dominated by heating rate | ∼200 μs | ∼500 μs | Discharged capacitor | 200–330 V |

Motzki et al. (2018) [36] | mm | Movable mass and spring | 1.5–40 ms | Dominated by heating rate | ∼20 ms | 20–50 ms | DC power supply | 24–125 V |

Barnes et al. (2006) [19] | mm | Hanged mass and friction force | 3–5 ms | Dominated by heating rate | ∼0.36 ms | 3–5 ms | Discharged capacitor | 90 V |

Otten et al. (2013) [9] | mm | Movable mass and spring and friction force | 6 ms | Dominated by heating rate | ∼2 ms | 6 ms | Discharged capacitor | 50 V |

Redmond et al. (2010) [21] | mm | Movable mass and spring and friction force | 20–50 ms | Dominated by heating rate | ∼6 ms | 20–50 ms | DC power supply | 40 V |

Knick et al. (2019) [30] | m | Movable mass and spring | 0.3 ms | Dominated by heating rate | ∼25 μs | ∼0.33 ms | DC power supply | 0.5 V |

**Table 2.**Collection of the main variables, equations and parameters used in the formulation of the model.

Variable | Description | Variable | Description |
---|---|---|---|

$\sigma \left(t\right)$ | Stress [Pa] | $T\left(t\right)$ | Temperature [K] |

$x\left(t\right)$ | Volume fraction [-] | t | Time [s] |

$u\left(t\right)$ | Displacement [m] | ||

(#) | Equation | (#) | Equation |

(3) | ${U}_{in}\left(t\right)=\rho {C}_{p}[T\left(t\right)-{T}_{R}]+Hx\left(t\right)+Q\left(t\right)$ | (18) | $\tau ={\left(\mu E{\epsilon}_{0}^{2}\right)}^{-1}$ |

(4) | $F=\sigma A={F}_{0}+ku+m\ddot{u}$ | (19) | ${\sigma}_{eq}=\tilde{\sigma}+\frac{H}{{\epsilon}_{0}}ln\left(\frac{T}{\tilde{T}\left(x\right)}\right)$ |

(9) | ${E}_{in}=\frac{1}{2}C{V}_{0}^{2}$ | (20a) | $\tilde{T}\left(x\right)={T}_{2}-\left({T}_{2}-{T}_{1}\right)exp\left(-\frac{x}{{\lambda}_{1}}\right),\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{2.em}{0ex}}x\left(t\right)\u2a7d0.5$ |

(10) | ${\tau}_{e}=\frac{1}{2}{R}_{tot}C$ | (20b) | $\tilde{T}\left(x\right)={T}_{4}-\left({T}_{4}-{T}_{3}\right)exp\left(-\frac{{\lambda}_{2}}{1-x}\right),\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{2.em}{0ex}}x\left(t\right)>0.5$ |

(13) | $\dot{x}\left(t\right)=\mu \left[{\sigma}_{eq}\left(T\left(t\right)\right)-\sigma \left(t\right)\right]{\epsilon}_{0}$ | ||

Parameter | Description | Parameter | Description |

H | Latent heat, 156,560,850 [J/m${}^{3}$]. | $\tilde{\sigma}$ | Reference stress, 20 [MPa]. |

${C}_{p}$ | Heat capacity, 837 [J/kg K]. | $\tilde{T}\left(x\right)$ | Reference transformation temperature, [K]. |

$\rho $ | Density, 6450 [kg/m${}^{3}$]. | m | Actuated mass, [kg]. |

${T}_{R}$ | Room temperature, 298 [K]. | g | Gravitational acceleration, 9.81 [m/s${}^{2}$]. |

${\tau}_{e}$ | Pulse rise time, [s]. | ${\omega}_{0}$ | Natural mechanical frequency of wire, [s${}^{-1}$]. |

$\tau $ | Characteristic time of the phase transformation, 28 [μs]. | ${\omega}_{s}$ | Natural mechanical frequency of spring, [s${}^{-1}$]. |

${t}_{1}$ | Minimal time to complete transformation, [s]. | ${\omega}_{eff}$ | Effective natural mechanical frequency, [s${}^{-1}$]. |

E | Young’s modulus, 60 [GPa]. | k | Spring stiffness, [Pa m]. |

${\epsilon}_{0}$ | Transformation (detwinning) strain, [-]. | ${F}_{ext}$ | External force, [N]. |

${L}_{0}$ | Wire length (austenite), [m]. | ${V}_{0}$ | Applied voltage, [V]. |

A | Wire cross-section, [m${}^{2}$]. | ${U}_{0}$ | Energy density, [J/mm${}^{3}$]. |

${R}_{tot}$ | Total resistance of the system, [$\mathsf{\Omega}$]. | ${E}_{in}$ | Input electrical energy, [J]. |

C | Discharged capacitance, [F]. | $\mu $ | Mobility coefficient, [m s/kg]. |

${T}_{1}$ | 297 [K], see Section 5. | ${T}_{4}$ | 361 [K], see Section 5. |

${T}_{2}$ | 345 [K], see Section 5. | ${\lambda}_{1}$ | 0.073 [-], see Section 5. |

${T}_{3}$ | 344 [K], see Section 5. | ${\lambda}_{2}$ | 0.03 [-], see Section 5. |

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**MDPI and ACS Style**

Dana, A.; Vollach, S.; Shilo, D. Use the Force: Review of High-Rate Actuation of Shape Memory Alloys. *Actuators* **2021**, *10*, 140.
https://doi.org/10.3390/act10070140

**AMA Style**

Dana A, Vollach S, Shilo D. Use the Force: Review of High-Rate Actuation of Shape Memory Alloys. *Actuators*. 2021; 10(7):140.
https://doi.org/10.3390/act10070140

**Chicago/Turabian Style**

Dana, Asaf, Shahaf Vollach, and Doron Shilo. 2021. "Use the Force: Review of High-Rate Actuation of Shape Memory Alloys" *Actuators* 10, no. 7: 140.
https://doi.org/10.3390/act10070140