A study on the contraction and cooling times of actively cooled shape memory alloy wires

Abstract

Shape memory alloys (SMAs) are thermally activated and show a strong thermomechanical coupling (i.e. the relationship between strain, stress and temperature). SMA wires are commonly used to develop SMA-based actuators. The most important characteristics of SMA wires is their capability to exert linear forces with strokes up to 8% of their original length. This make SMAs an attractive smart material for actuation purposes. However, the main weakness of SMAs is their limitation on actuation bandwidth. This limitation comes from the rate at which the SMA wires are able to heat up and cool down, releasing heat energy to the surroundings. This paper focuses on the effects that different heating and cooling rates have on the SMA wire’s working rate that can be attained. An SMA wire has been experimentally tested under different conditions of heating, cooling and applied load to find the influence of these parameters on the contraction and cooling times of an SMA wire and, therefore, on its attainable frequency. In addition, a model for SMAs has been implemented in a finite element analysis software and the experiments have also been simulated, which has been used to corroborate the conclusions drawn from the experiments.

Keywords

SMA wires active convective cooling working rate SMA based actuator

Introduction

Shape memory alloys (SMAs) are one of the so-called smart materials that have attracted the attention of scientists and engineers in the last decades due to their unique features. These metals can recover from stress-induced strains of up to 8% (Callister and Rethwisch, 2010; Pons, 2005) upon heating due to transformations between two solid phases: austenite, which is the preferred phase at high temperatures and low stresses; and martensite, which is the preferred phase at low temperatures and high stresses. Both phases can coexist when the phase transformation occurs. In this case, each phase fraction is determined by the state of temperature, stress and previous phase composition of the SMA. Two resultant effects of such transformations are observable on the macroscopic scale. Shape memory effect, which is used for actuation, is observed upon strain-cycling the SMA below the austenite temperature T_A . Superelasticity is observed upon strain-cycling it above T_A . Thus, SMAs have the ability to recover their memorized shape upon heating after a stress-induced deformation (see Figure 1).

Owing to their interesting features, novel devices are integrating SMA in their designs. Based on SMAs, there is a wide variety of industrial (Wu and Schetky, 2000), automotive (Stoeckel, 1990) and robotic (Cho and Asada, 2005) applications. Given its bio-compatibility, SMAs are also used in medical applications (Petrini and Migliavacca, 2011). SMAs have some features that make them particularly interesting for novel aerospace applications (Hartl and Lagoudas, 2007). In addition to other smart materials, SMAs have been used in research to develop adaptive wings (Sofla et al., 2010), which are able to adapt their shape to different flight conditions by means of sensors, controllers and actuators. They are used to increase the lift-to-drag ratio, which contributes to a reduction of fuel consumption (Monner et al., 2000).

For actuation purposes, the most important characteristics of SMA wires are that they exert linear forces with strokes up to 8% and possess a high force density (Pons, 2005). Their main weakness is their low working rate. This low working rate makes SMA’s working frequency be very low, normally lower than 1 Hz, when it is activated cyclically. This relatively low frequency is determined by the duration of the transformation from the austenite to the martensite phase. This in turn depends on the heat exchange rate with the surrounding materials, normally air. Whereas SMAs can be quickly Joule heated, the cooling rate by convection is much lower, thus limiting its working frequency. These features make SMA an attractive smart material to develop smart actuators for low-frequency actuation purposes in which space, weight, stroke and force are critical design parameters, like in aerodynamic load control devices for aerospace purposes.

The wires studied here are part of the SMA-based actuator presented in Lara-Quintanilla et al. (2013). This actuator is intended to work as an aerodynamic load control actuator on aircraft wings. It consists of a flat plastic surface with SMA wires that are embedded off the neutral axis. The contractions of the SMA wires makes the flat surface perform out-of-plane deformations. This actuator is intended to be placed on the trailing edge, thus influencing the lift on a span-wise section of the wing while keeping attached flow. The SMA wires are eccentrically embedded along channels that go through the beam-like surface. The proof-of-concept (design, manufacturing process and control strategy of the actuator) can be found in Lara-Quintanilla et al. (2013). Lara-Quintanilla et al. observed a satisfactory performance of the actuator upon tracking different types of signals. The actuator was able to track sinusoidal input signals of amplitude up to 12 mm peak-to-peak at frequencies of 0.6 Hz with a relatively small error (less than 10%) between the set point and the actual deflection of the tip of the actuator. This article shows the importance of using an active cooling system to cool the SMA wires in order to increase their bandwidth and controllability. At this point, a deeper study into the thermomechanical performance of the SMA wires becomes necessary in order to improve the design and the capabilities of this actuator in terms of maximum attainable working frequency.

The cooling aspects of the SMA wires are of great importance in the design of SMA-based actuators. An active cooling system allows to design more accurate position control for SMA-based actuators (Reynaerts and Brussel, 1998) as well as to increase its working frequency (Howe et al., 1995; Lara-Quintanilla et al., 2013; Tadesse et al., 2010). SMAs are also known for their self-sensing capabilities by monitoring their electrical resistance during actuation. Lewis et al. (2013) found a strong relationship between the power supplied to an SMA wire under ambient temperature zero airspeed and the excess of power that must be supplied to the wire in order to maintain a similar resistance and strain when the wire is exposed to different values of constant airspeed, thus using the SMA wire as an airspeed sensor.

In this paper, we focus on the effects that the heating and the cooling rates have on an SMA wire’s working rate. The results presented in this article are based on experiments and model simulations. These results show the capabilities of a single SMA wire in terms of attainable working rate, which is determined by the thermal limitations of an SMA wire that is Joule heated and convectively cooled by a controlled airflow along and around it.

The rest of the paper is organized as follows. The basics of the model, as well as its implementation in COMSOL, are explained in the section “Theory”. In the first place, the “Experimental details” section presents the validation of the model and, subsequently, the description of the experimental setup and the followed methods to determine the convective heat transfer coefficient, contraction and cooling times and the SMA wire’s attainable frequency. The following sections are the “Results and discussion” and, finally, the “Conclusions”.

Theory

SMAs show strong thermomechanical coupling (i.e. the relationship between strain, stress and temperature). Moreover, their behavior is highly nonlinear and hysteretic, what makes their modeling complex. Many thermomechanical constitutive models have been suggested for the prediction of the SMAs’ behavior. A summary of existing one- and three-dimensional SMA models is given in Lagoudas (2008), which highlights the main features of each model.

The present study is based on the SMA model developed by Achenbach (1989) and Seelecke and Müller (2004). This model has shown a high accuracy and computer efficiency (Heintze, 2004). In addition, the parameters needed to initialize this model are directly related to experimental data (Crews et al., 2012). This model has been used to model different kinds of SMA-based actuators; for instance, Crews and Buckner (2012) used the model together with a genetic algorithm to optimize the design of a SMA-actuated robotic catheter, and Yang and Seelecke (2009) also based the analysis of an SMA-based bone–joint system on this model.

In this article, we focus on the one-dimensional version of the Achenbach–Müller–Seelecke SMA model to study the limitations of the SMA wire in terms of attainable frequency and stroke, limitations that strongly depend on the thermal performance of the SMA wires, especially on their cooling rate.

The model

The model uses phase transition rates to evaluate the time evolution of the different phase fractions (x_A , x_M+ , x_M ₋, with 0 ≤ x ≤ 1), where x_A represents the austenite fraction and x_M ₊ and x_M ₋ the martensite fraction induced by tensile and compressive stresses, respectively. Subsequently, the phase fractions and the temperature are used to calculate the current stress–strain state of the SMA. Since the phase fractions quantify the presence of each phase in the SMA, the austenite fraction x_A is calculated as

x_{A} = 1 - x_{M +} - x_{M -}

(1)

It is assumed that the stress and the temperature, and therefore also the phase fractions, are uniform along the wire. The time evolution of the phase fraction is determined by

\begin{matrix} {\overset{\cdot}{x}}_{M +} = x_{A} p_{A \to M +} - x_{M +} p_{M + \to A} \\ {\overset{\cdot}{x}}_{M -} = x_{A} p_{A \to M_{-}} - x_{M -} p_{M_{-} \to A} \end{matrix}

(2)

where p_i _→j denote the transition rates and represent the likelihood of a lattice particle to surpass the energy barriers that separate the wells in the Gibbs energy landscape (i, j can be two contiguous phases in the Gibbs landscape x_M ₋, x_A or x_M+ ), shown in Figure 2. A detailed description of these phase transition probabilities can be found in Massad and Smith (2005). We use the method proposed by Crews et al. (2012), where the transition rates are assumed to depend on the barriers in the Gibbs energy landscape (see Figure 2).

The transition rates are governed by

p_{i \to j} = \frac{1}{τ} e^{- \frac{V_{LE} Δ G_{i \to j}}{k_{B} T}}

(3)

where V_LE is the layer volume, τ the relaxation time and k_B the Boltzmann constant. The term ΔG_i _→j refers to the barriers that a particle needs to overcome to change its phase. This term, represented in Figure 2, is calculated as the difference between consecutive local minima and maxima. The Gibbs energy determines the equilibrium phase composition of the SMA:

G (T, ε, σ) = ψ (T, ε) - ε σ

(4)

The Gibbs landscape is a continuous and differentiable piecewise function represented by five quadratic polynomials. The Helmholtz energy function ψ(T, ε) in (4) is

\begin{matrix} ψ (ε, T) = \\ {\begin{matrix} \frac{1}{2} E_{M} {(ε + ε_{T})}^{2} & ε < - ε_{M} (T) \\ A (T) ε^{2} + B (T) ε + C (T) & - ε_{M} (T) \leq ε < - ε_{A} (T) \\ \frac{1}{2} E_{A} ε^{2} + Δ β (T) & | ε | \leq ε_{A} (T) \\ A (T) ε^{2} - B (T) ε + C (T) & ε_{A} (T) \leq ε < ε_{M} (T) \\ \frac{1}{2} E_{M} {(ε - ε_{T})}^{2} & ε > ε_{M} (T) \end{matrix} \end{matrix}

(5)

where E_A and E_M are the elastic modulus of the austenite and martensite phase respectively, and ε_T is the maximum recoverable strain. These polynomials represent the three phases and the two barriers between them, in which the unknown coefficients can be calculated assuring C ¹ continuity. The inflection points ε_A and ε_M are calculated as

\begin{matrix} ε_{A} (T) = \frac{σ_{A} (T)}{E_{A}} \\ ε_{M} (T) = \frac{σ_{M} (T)}{E_{M}} + ε_{T} \end{matrix}

(6)

where σ_A and σ_M are the stresses at which the transition from A to M+ and from M+ to A occurs respectively. These stresses can be calculated as

\begin{matrix} σ_{A} (T) = σ_{L} + Δ σ_{T} (T - T_{L}) \\ σ_{M} (T) = σ_{A} (T) - σ_{R} \end{matrix}

(7)

where σ_R is the height of the stress hysteresis loop, Δσ_T is the temperature dependence of the stress hysteresis loop, and σ_L is the value that σ_A takes at the low-temperature (T_L ) isothermal test. The parameter T_L is explained in detail in the section “Mechanical parameters”. The parameters commented here are represented in Figure 3 for a better understanding.

Since martensite and austenite phases have different elastic modulus, the strain of an element depends on its equilibrium phase composition and its respective strains:

\begin{matrix} ε (t) & = x_{A} (t) ε_{A} + x_{M +} (t) ε_{M +} + x_{M -} (t) ε_{M -} \\ = x_{A} (t) \frac{σ}{E_{A}} + x_{M +} (t) (\frac{σ}{E_{M}} + ε_{T}) + x_{M -} (t) (\frac{σ}{E_{M}} - ε_{T}) \end{matrix}

(8)

Solving (8) for σ yields the relationship between the stress and the strain, which is nonlinear as a consequence of the presence of the two crystalline phases:

σ (t) = \frac{ε (t) - ε_{T} (x_{M +} (t) - x_{M -} (t))}{\frac{1 - x_{M +} (t) - x_{M -} (t)}{E_{A}} + \frac{x_{M -} (t) + x_{M +} (t)}{E_{M}}}

(9)

With regard to the heat, there are several processes and assumptions that need to be taken into account to calculate the time evolution of the temperature. First, it is assumed that all of the heat losses are due to convective heat transfer, what was also experimentally verified. Second, the temperature is raised by means of Joule heating. Third, the latent heats associated to the phase transformations also have an influence on the temperature. Fourth, given the small diameter of the SMA wire (0.4 mm), its temperature is assumed to be uniform through its diameter. This last assumption is supported by the fact that the thermal time constant of the used SMA wire (τ = ρv_SMAc_p /hA_SMA ) is much lower than the resulting contraction and elongation times (given in the section “Results and discussion”). Consequently, the time evolution of the temperature is determined by

\begin{matrix} ρ_{SMA} c_{p} \overset{\cdot}{T} (t) & = - h {\frac{A_{SMA}}{v_{SMA}} |}_{cil} (T - T_{\infty}) + j (t) \\ - (h_{M +} - h_{A}) {\overset{\cdot}{x}}_{M +} (t) - (h_{M -} - h_{A}) {\overset{\cdot}{x}}_{M -} (t) \end{matrix}

(10)

where T is the current temperature of the SMA, ρ_SMA is the SMA’s density and c_p its heat capacity; h is the convective heat transfer coefficient between the SMA wire and the surrounding air, which is at room temperature T_∞ ; ${\frac{A_{SMA}}{v_{SMA}} |}_{cil}$ is the surface-to-volume ratio; j(t) is the Joule-heating power per volume unit provided by the power supply; finally, the terms h_M ₊ − h_A and h_M ₋ − h_A are the latent heats related to the phase transformations. A further explanation of the latent heats is given in Massad and Smith (2005) and Li (2006).

Finally, the stress along the wire is considered to be uniform because the wire diameter is also uniform. The load is applied at one end of the wire while the other end is gripped. Therefore, Equations (2) and (10) and the assumption of uniform stress along the wire yield the system of equations that govern the thermomechanical behavior of the SMA wire:

{\begin{matrix} \frac{\partial σ (x)}{\partial x} = 0 \\ {\overset{\cdot}{x}}_{M +} = x_{A} p_{A \to M_{+}} - x_{M +} p_{M_{+} \to A} \\ {\overset{\cdot}{x}}_{M -} = x_{A} p_{A \to M_{-}} - x_{M -} p_{M_{-} \to A} \\ ρ_{SMA} c_{p} \overset{\cdot}{T} = - h {\frac{A_{SMA}}{v_{SMA}} |}_{cil} (T - T_{\infty}) + j \\ - (h_{M +} - h_{A}) {\overset{\cdot}{x}}_{M +} - (h_{M -} - h_{A}) {\overset{\cdot}{x}}_{M -} \end{matrix}

(11)

COMSOL implementation

The model described in the previous section was implemented in COMSOL multiphysics^® (COMSOL Inc., Burlington, MA, USA), a finite element analysis software. The system of ordinary differential equations (11) was implemented as partial differential equations on its general form

e_{a} \frac{\partial^{2} u}{\partial t^{2}} + d_{a} \frac{\partial u}{\partial t} + \nabla Γ = f

(12)

where, according to (11), the dependent variables are

u = [\begin{matrix} u_{x} \\ x_{M +} \\ x_{M -} \\ T \end{matrix}]

the mass matrix is

e_{a} = 0

the damping matrix is

d_{a} = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & h_{M +} - h_{A} & h_{M -} - h_{A} & ρ_{SMA} c_{p} \end{matrix}]

the flux vector is

Γ = [\begin{matrix} σ \\ 0 \\ 0 \\ 0 \end{matrix}]

and the source term is

f = [\begin{matrix} 0 \\ x_{A} (t) p_{A \to M_{+}} (T, ε, σ) - x_{M +} (t) p_{M_{+} \to A} (T, ε, σ) \\ x_{A} (t) p_{A \to M_{-}} (T, ε, σ) - x_{M -} (t) p_{M_{-} \to A} (T, ε, σ) \\ - h {\frac{A_{SMA}}{v_{SMA}} |}_{cil} (T - T_{\infty}) + j (t) \end{matrix}]

In addition, proper boundary conditions must be defined to these equations. One end of the SMA is fixed and the other one is free, which is specified in COMSOL by Dirichlet conditions. On the fixed end, displacement is constrained. The boundary conditions on the free end differ according to the type of simulation. In isothermal simulations, which are strain controlled, the Dirichlet boundary condition is defined by the wire’s length difference caused by the applied strain (ΔL = L ₀ ε(t)). In isobaric simulations, the boundary condition is defined by a flux term that represents the constant and uniform stress that the weight exerts on the wire. Considering that phase fractions and temperature are assumed to be uniform along the wire and that they only depend on time, boundary conditions are not necessary for them. Finally, appropriate initial conditions must also be introduced for the initial strain, temperature and phase fractions in order to obtain an accurate behavior of the model.

Experimental details

This section is divided into four subsections. The first subsection describes the setup that was used to determine the only thermal parameter that needed to be found for the model, the convective heat transfer coefficient. The second subsection shows the method used to find the mechanical parameters for the model. Afterwards, the validation of the model is presented. The last subsection explains the method used to determine the contraction and cooling times of the SMA wire.

Convective heat transfer coefficient

The work presented in this article is based on the SMA-based actuator developed by Lara-Quintanilla et al. (2013). One important feature of that actuator is the use of a controlled airflow along the SMA wire that cools it down rapidly, thus achieving higher working frequencies (also called working rates throughout this document). Since the wire is cooled by an airflow, the convective heat transfer coefficient between the SMA wire and the airflow needs to be found. Experimental and analytical data showed that conduction was negligible compared with convection. Therefore, only convective heat losses have been considered here.

In a convection process, the heat loss by convection $\overset{\cdot}{q}$ is defined by

\overset{\cdot}{q} = - hA (T_{SMA} - T_{\infty})

(13)

where h is the convective heat transfer coefficient that is to be found, A is the lateral surface area of the wire and T_SMA is the temperature of the wire. The airflow speed is high enough to assume that the temperature of the airflow T_∞ remains at room temperature when passing along the wire. Therefore, it is also assumed that the temperature is uniform throughout the wire. However, the wire’s temperature changes rapidly over time as a consequence of the heat loss. The T_SMA evolution is defined by

\frac{{dT}_{SMA}}{dt} = \frac{\overset{\cdot}{q}}{v_{SMA} ρ_{SMA} c_{p}}

(14)

where v_SMA is the volume of the wire. Equation (14) can be solved for T_SMA , yielding the following expression

T_{SMA} (t) = T_{\infty} + (T_{0} - T_{\infty}) e^{\frac{- {hA}_{SMA}}{v_{SMA} ρ_{SMA} C_{p}} t}

(15)

where T ₀ is the temperature at t = 0. The temperature decay described by Equation (15) can be experimentally obtained by measuring the temperature decay upon cooling a wire that was initially at high temperature T ₀. Therefore, the data from a thermographic camera can be fitted to an exponential equation of the form

y = y_{0} + {ae}^{bt}

(16)

All of the parameters in (15) are known except for h, which is deducted from the coefficient b in (16).

In order to carry out this measurement, the setup shown in Figure 4 was used. Similar thermal conditions to those during the real performance of the SMA-based actuator wanted to be recreated. For that reason, a similar beam made out of the same material was manufactured. The only difference between the beam used here and the one used for the proof-of-concept in Lara-Quintanilla et al. (2013) was the shape of the channels. In this case, the section of the channels was rectangular instead of circular, and two wires were placed in each channel. In order to maintain the same channel section per wire, the cross-section of the new rectangular channels was the double that the section of the previous version The width of the channels was 10 mm and the height 0.5 mm and, therefore, the cross-section of the channels was 5 mm². A looped SMA wire was placed throughout one of the channels of the flexible beam in order to have two SMA wires inside the channel. The two ends of the looped wire were attached above and out of the beam and a weight was hung from the loop at the bottom of the wire, as indicated schematically in Figure 4. The airflow inlet was at the top of the beam and the outlet at the bottom. The electrical connections were made at the two ends of the wire, as close to the beam as possible to ensure that the heated portion of wire was the same that the portion of wire to be cooled afterwards. A FLIR SC7000 thermographic camera was used to measure the temperature of the wire and a linear variable differential transformer (LVDT) was used to measure the vertical displacement of the hanging weight. The LVDT was not used in this experiment but it was used in the contraction and cooling times experiment. The setup is sketched in Figure 4.

Ideally, the temperature of the wire should be measured inside the actuator. Since it was not feasible in this setup, the thermographic camera was set on the upper part of SMA out of the bottom of the beam, as shown in Figures 4 and 5. The temperature decay measured by the thermographic camera on this point of the SMA wire was used to calculate the convective heat transfer.

This experiment consisted of two steps. First, the wire was heated by means of an electric current of 1.5 A supplied by a SM 120-50 DC Power Supply from Delta Elektronika BV. The current was kept constant until the wire reached a constant temperature of roughly 393 K, which guaranteed that the SMA had reached complete transformation into austenite. Subsequently, the current was cut off at the same time that the valve was opened. The electronically controlled valve used here was a 2/2-way proportional valve Type 2826 from Bürket Fluid Control Systems. A timing diagram of this procedure is shown in Figure 6. The setup was controlled and the data was acquired by a computer running LabVIEW^®(National Instruments Corporation, Austin, TX, USA).

The convective heat transfer coefficient was calculated for 11 values of airflow. The aperture of the valve was set at voltage values ranging from 0 to 5 V in steps of 0.5 V. The values of airflow corresponding to the different values of aperture of the valve are shown in Table 1 and plotted in Figure 7. The fitting values for Equation (16), shown in Table 1, exhibited an excellent agreement with lab data (R ² = 0.99 for all the fits). The heat transfer coefficient was deducted from b in (16) by matching it with the exponential term in Equation (15).

Table 1.

Exponential curve fit to the measured data.

Valve [V]	Airflow [l/min]	y ₀	a	b	R ²	h [W/(m² K)]
0	0.0	45.22	66.08	−0.36	0.99	116
0.5	1.4	48.13	61.18	−0.40	0.99	128
1	5.9	45.43	65.39	−0.90	0.99	290
1.5	9.1	44.17	70.13	−0.95	0.99	308
2	11.4	45.50	70.33	−1.95	0.99	629
2.5	14.3	45.81	65.56	−2.32	0.99	748
3	16.3	48.64	62.31	−2.43	0.99	783
3.5	18.6	46.89	67.60	−2.64	0.99	851
4	20.8	47.06	62.45	−2.87	0.99	926
4.5	22.6	47.46	66.79	−3.05	0.99	985
5	24.1	47.46	56.38	−2.95	0.99	952

The time evolution of the temperature was monitored on three different points, shown in Figure 5. The maximum temperature along line 1 was used to calculate h. Line 2 was set to compare visually the temperature measured on line 1 with the temperature on the other part of the wire to ensure reliability of the measurements. In addition, the temperature of the beam, measured by line 3, was used between experiments to guarantee that the beam was at room temperature at the beginning of each experiment. The relationship between the heat transfer coefficient, the airflow and the valve aperture are shown in Figure 7. The airflow increases almost linearly with the aperture of the valve (embedded graph in Figure 7), what corresponds with the expected behavior of a proportional valve. Similarly, the relationship between the airflow and the heat transfer coefficient is increasing and fairly linear, except for the values of valve aperture 1.5 and 5 V for which the values of h are slightly below the trend line. The importance and efficacy of the active cooling is clearly seen in Figure 7. The heat transfer coefficient is 8.5-times higher at the maximum tested airflow 985 W/(m² K) than at natural convection 116 W/(m² K).

Mechanical parameters

There are several material properties and behavioral parameters that must be identified in order to initialize the model. The method to obtain the necessary parameters used here was proposed by Crews et al. (2012). An important aspect of this method is the ease with which the required parameters can be determined directly from experimental data. It consists of running two isothermal experiments at two different temperatures T_L and T_H , both above the austenite temperature T_A , in accordance with T_A < T_L < T_H . The wire used here is a 0.4 mm diameter SmartFlex wire from SAES Getters (SAES Smart Materials, New Hartford, NY, USA). The phase diagram for this wire was obtained experimentally and it is shown in Figure 8.

The isothermal experiments were run using a 10 kN MTS testing machine in a strain-controlled configuration together with a Thermotron FR-1-CH-LN2 environmental chamber. The SMA wire was strained at two different temperatures, T_L = 368 K and T_H = 383 K, both temperatures sufficiently above T_A (363 K according to manufacturer data, 345 K according to the experimental results shown in Figure 8). A maximum strain of 7% was applied to assure that the alloy reached the full stress-induced martensite phase, as indicated by the vertical arrows in Figure 8. The clamps that grip the wire were also placed inside the environmental chamber, thus assuring that they did not act as heat sinks for the wire and to avoid thermal inhomogeneities along the wire. The consequences of thermal inhomogeneities along an SMA wire were studied by Furst et al. (2012). This allows to make the assumption mentioned beforehand that the temperature and the phase fractions are uniform along the wire. The characterization of the SMA wire yielded the model parameters listed in Table 2. Finally, the density ρ_SMA = 6450 kg/m³ and the heat capacity c_p =500 J/kg K were both provided by the manufacturer.

Table 2.

Model parameters for a SmartFlex wire.

Parameter	Value
E_A	45.3 GPa
E_M	41.5 GPa
T_L	378 K
σ_L	413 MPa
Δσ_T	5.16 MPa/K
σ_R	276 MPa
ε_T	0.0513

Model validation

Once all of the needed parameters were gathered, several simulations were run and compared with experimental data in order to validate the model. With that purpose, data obtained from isothermal experiments and the corresponding COMSOL simulations have been put together in Figure 9, which collects the validation graphs. Figure 9 shows the results of the experiments run under strain control with a maximum strain of 7% on a wire of 196 mm length. As it can be seen in Figure 9, the simulations in COMSOL show good agreement with experimental data.

Some inaccuracies can be detected regarding the curvature of the graphs on the plateaus, where phase transformations happen. This is reasonable because the model implemented here is a single crystal model whereas the used SMA wire is polycrystalline. Strain–stress validation for a polycrystalline version of this model can be found in Crews and Buckner (2012). Despite those inaccuracies, the model captures the most important features of the SMA wire’s behavior for actuation purposes. The maximum stress is well predicted in most of the cases.

Some permanent plastic deformation was induced to the wire during the test shown in Figure 3. As can be seen, the wire does not return to the origin after relaxation. In order to stabilize the wire, it was initially cycled thermally for 6000 times at a constant stress of 234 MPa (3 kg) using an isobaric configuration (see Figure 4). However, in the isothermal test shown in Figure 3 the wire was stressed up to 750 MPa, which caused the irrecoverable plastic transformation and might contribute to yield a slightly high value of maximum recoverable strain (ε_T ).

The stress hysteresis loop’s width is also well captured by the model. A small difference between model and experiments can be detected on the tests at 368 and 373 K. This is due to a small inaccuracy in the temperature controller of the environmental chamber, which was unable to keep a constant temperature throughout the experiment, causing slight temperature variations. Although those variations were small, they were sufficient to slightly decrease the stress level at which the reverse transformation happened.

Contraction and cooling times

This experiment was performed in order to determine the elongation and contraction rates of the SMA wire under different loads. The contraction time t_h of an SMA wire is defined as the time that it takes for the wire to recover from the elongated shape (stress-induced martensite) to the memorized shape (austenite). In the same way, the cooling time t_c is the time that it takes for the wire to elongate from the memorized shape to the elongated shape. This can be considered a working cycle of an SMA wire and, therefore, the working frequency f_w of an SMA wire can be defined as the inverse of the sum of the contraction and the cooling times:

f_{w} = \frac{1}{t_{h} + t_{c}}

(17)

In order to measure the contraction and cooling times of the SMA wire, the procedure shown in Figure 6 was applied to the isobaric setup, schematically represented in Figure 4. To simplify the procedure, the tests were split into two stages to measure the cooling and contraction times. The experiment to measure the cooling times was similar to the experiment used to calculate the heat transfer coefficient. A constant current of 1.5 A was passed through the SMA wire to reach a constant temperature around 393 K, thus assuring full transformation to austenite. Subsequently, the current was cut off and the valve opened. The same 11 values of airflow shown on Table 1 were tested, and the temperature and strain decays data were recorded. The experiments to measure the contraction times followed a similar procedure. First, the valve was set at 5 V to pass a high airflow through the wire and guarantee that the initial temperature of the wire and the beam were room temperature before heating. Subsequently, the valve was closed and the power supply turned on, applying a constant electric power to the wire. Eight values of power were applied to the wire ranging from 5 to 40 W is steps of 5 W. Power control was used instead of voltage or current control because the resistance of the SMA wire changes during the SMA’s phase transformations (Ma et al., 2004). Therefore, if a constant voltage or current were applied to the wire during a phase transformation, the power supplied to the system would not be constant over time. For this reason, power control is the easiest way of ensuring a constant supply of energy to an SMA wire over time.

The temperature signal was measured on a single point of the wire (see Figure 5), which did not ensure that the whole wire was exactly at the same temperature. The strain, however, exhibits the overall effect of the local temperatures throughout the wire. Furthermore, strain is normally the controlled variable during actuation and it does not change significantly at temperatures above T_A or below T_M , as can be seen in Figure 11. For these reasons, the cooling and contraction times have been calculated in the present study as the time that it takes for the wire to elongate and contract respectively between the 10% to the 90% of its full strain range (see Figure 10). In a similar manner, the isobaric test was simulated with the model to predict the working rate of the SMA wire. The tests described in this section were repeated for three different hanging weights of 1, 2 and 3 kg.

Results and discussion

This section shows and discusses the resulting contraction and cooling times, which are shown in Figure 12. There is an overall disagreement between the simulated and experimental results. This disagreement can be explained by comparing one simulation and one experimental isobaric test. As shown on the left-hand side of the Figure 11, the transformations temperatures—and therefore the temperature’s hysteresis loop—do not coincide. Here T_A is about 343 K for the simulation as well as for the experiment. However, there is a difference of about 15 K between T_M from the simulations and from the experiment. Most importantly, on the right-hand side of Figure 11 it can be seen that the transformations occur more rapidly in the simulation that in the experiment. This difference is more noticeable in the transformation from austenite to martensite. This is the reason why the cooling and contraction times gathered from the simulations are considerably shorter than those from the experiments. The overall disagreement in general and, in particular, the disagreement between the maximum strain observed on both cases comes from the use of a single-crystal model, whereas the SMA wire is polycrystalline. This means that the single-crystal model considers that the wire consists of a single grain that is oriented on the direction in which the elongation and contraction happens. However, the tested SMA wire has a polycrystalline structure and, therefore, grains that are not perfectly oriented. Furthermore, the SMA wire used to deduce the model parameters was polycrystalline. The overall SMA strain is the result of the addition of the local strains of each grains. For this reason, the full strain range is noticeably higher in the case of the single-crystal model. Moreover, the wire used in the experiment was previously cycled thermally in order to limit the effects of the functional fatigue (Scire-Mammano and Dragoni, 2014). We have experimentally observed that the recoverable strain of a cycled SmartFlex wire decreases to a 77% after 6000 cycles in comparison with the original recoverable strain. The recoverable strain does not decrease noticeably after 6000 cycles and therefore the wire can be considered to be stabilized in terms of functional fatigue.

Although the simulations cannot be trusted quantitatively, the qualitative outcomes are relevant for the study of the effect of heating and cooling rates in the contraction and cooling times of SMA wires. Despite this mismatch between the simulations and the experiments, some conclusions can be drawn from the results shown in Figure 12.

In both cases, the cooling times decrease as the applied airflow is increased. The cooling times are roughly five times shorter for large airflows than for natural convection or small airflows. It can also be observed that from a certain value of airflow onwards, increasing the airflow does not have a significant effect on the cooling time. The maximum effective airflow for the actuator and channel geometry discussed here is 16.3 l/min according to the experimental data and 14.3 l/min according to the simulations. The corresponding experimental cooling time is 7 s. This limitation on the maximum attainable cooling time is determined by the time that it takes for the alloy’s structure to transform from austenite to martensite. Furthermore, the influence of the applied load is also observable. The heavier the weight is, the shorter the cooling time is. This comes from the fact that the weight applies a tensile load on the wire. The temperature of transition to martensite increases when the stress on the wire increases (see Figure 8) and, therefore, a lower heat loss is required to achieve martensitic transformation.

The obtained contraction times draw similar conclusions. A quantitative mismatch exists between model and experimental data. However, they are qualitatively comparable. In both cases, the contraction time decreases as the supplied power increases, as expected. Similarly to the cooling times, there is a value of electrical power above which its increase does not have a noticeable effect on the resulting contraction time. According to the results, this value is 20 W on the experiments and about 10 W on the model. The influence of the applied load is also shown. Unlike the cooling case, the contraction times are shorter for lighter hanging weight. This happens because the austenite transition temperature also increases upon increasing the stress on the wire (see Figure 8) and more heat is needed to achieve transformation to austenite. For this reason, lighter loads lead on to shorter contraction times.

Finally, the working rates (actuation frequency of the SMA wire) resulting from linking together the different contraction and cooling times as described in Equation (17) are shown in Figure 13 for experimental results and in Figure 14 for simulations. These figures show the attainable working rates at different applied values of power and airflow. The maximum working rates of the experimental setup are 0.12 Hz for a load of 1 kg, 0.14 Hz for 2 kg and 0.15 Hz for 3 kg. The results observed here show that the limiting factor for low working rates (below 0.06 Hz) is the electrical power while the airflow has barely an influence on the resulting working rate. However, when higher working rates want to be attained, the airflow around the wires becomes the limiting factor. The results from the COMSOL simulations draw the same conclusions, as was previously discussed, although their results cannot be trusted quantitatively. The isolines have a horizontal trend on the bottom of the contour graphs, what indicates that the working rate increases as the power increases. However, after 15 W, the vertical isolines prevail over the horizontal isolines, which indicates that increasing the airflow is the only way of increasing the working rate above 0.05 Hz. The applied stress does not affect the time that it takes for the wire to heat up or cool down. However, it has a noticeable impact on the contraction and elongation times as shown in Figure 12. Figures 13 and 14 show that the overall effect of increasing the stress on the SMA wire results in higher attainable working rates.

The results shown in Figure 13 should be quantitatively taken into account for any SMA application in which the working rate plays a crucial role.

Conclusions

SMA wires have unique features that make them very interesting for actuation purposes. However, their bandwidth is very low. This article has focused on using an active cooling system in order to increase the working rate of an SMA-based actuator. An SMA wire has been Joule heated and convectively cooled by means of a controlled airflow around the wire. Different heating and cooling rates have been applied to the wire and the times that it takes for it to contract and elongate have been measured. In addition, a finite element model for SMAs has been implemented in COMSOL. The simulations capture the SMA behavior, but were unable to yield reliable quantitative results about the contraction and the cooling times. However, the results are qualitatively useful. There are two reasons for this disagreement between simulation and experimental results. First, the model simulates a single crystal SMA wire whereas the SMA used to gather the parameters was polycrystalline as well as the wire used in the experiments. Second, the parameters for the model come from experimental measurements and are calculated by linear fitting what intrinsically brings inaccuracies along with it.

The efficacy of using a controlled airflow to cool the SMA wire down has been proven. This airflow has shown to increase the convective heat transfer coefficient up to eight-fold compared with natural convection.

Afterwards, several values of power and airflow have been applied to heat and cool the SMA wire at different rates. The time that it takes for the wire to contract and elongate has been measured. Both experimental and simulated results drew similar conclusions. The increment of the heating and the cooling rates results in a reduction of the contraction and cooling times. There are values of power and airflow from which their increments do not results in a significant reduction of the contraction and cooling times. In addition, the experiment has been repeated for different hanging weights of 1, 2 and 3 kg, which changes the transformation temperatures. On the one hand, increasing the weight results in decreasing the cooling time since the weight cooperates with the elongation of the wire. On the other hand, increasing the weight increases the contraction time because it counters the contraction of the wire. It must be remarked that all of the data presented in this work were gathered from simulations and experiments of a 0.4 mm diameter SMA wire. Given the convective heat transfer nature of the cooling process, a global decrease of the cooling times can be expected for smaller wire’s diameters for any given hanging weight.

Finally, the contraction and cooling times have been put together in order to find the resulting working rate of the SMA wire when it is heated and cooled at different rates. It has been found that the heating is the limiting factor for the lowest working rates. However, when higher working rates want to be reached, the airflow becomes the limiting factor and, therefore, increasing it turns into a higher working rate. The overall effect of increasing the stress on the SMA wire results in higher attainable working rates.

This paper presents a procedure to find the limitations of SMA wires in terms of actuation working rate. Joule heating and convective cooling have been used here but this procedure can be extended to any other heating and cooling methods. Different SMA devices require different working rates, and the use of this procedure in situ can help to find the optimum contraction and cooling times for the required actuation rate, what results in a reduction of the overall energy consumed by the SMA based actuator.

Footnotes

Acknowledgements

The authors would like to thank David Donado-Cortés for his careful experimental work. This work has been performed within the framework of the European Community funded project Clean Sky Joint Technology Initiative (JTI): “Smart Fixed Wing Aircraft - Integrated Technology Demonstrator” (SFWA-ITD).

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research has been supported by CSJU-GAM-SFWA-2008-001, within the European framework of JTI Clean Sky.

References

Achenbach

(1989) A model for an alloy with shape memory. International Journal of Plasticity 5(4): 371–395.

Callister

Rethwisch

(2010) Materials Science and Engineering: An Introduction, Eigth Edition Binder Ready Version. New York: John Wiley & Sons.

Cho

Asada

(2005) Multi-axis sma actuator array for driving anthropomorphic robot hand. In: Proceedings of the 2005 IEEE international conference on robotics and automation (ICRA), pp. 1356–1361.

Crews

Buckner

(2012) Design optimization of a shape memory alloy-actuated robotic catheter. Journal of Intelligent Material Systems and Structures 23(5): 545–562.

Crews

Smith

KP (2012) Data-driven techniques to estimate parameters in the homogenized energy model for shape memory alloys. Journal of Intelligent Material Systems and Structures 23(17): 1897–1920

Furst

Crews

Seelecke

(2012) Numerical and experimental analysis of inhomogeneities in SMA wires induced by thermal boundary conditions. Continuum Mechanics and Thermodynamics 24(4–6): 485–504.

Hartl

Lagoudas

(2007) Aerospace applications of shape memory alloys. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 221(4): 535–552.

Heintze

(2004) A Computationally Efficient Free Energy Model for Shape Memory Alloys - Experiments and Theory. PhD Thesis, North Carolina State University, Raleigh.

Howe

Kontarinis

Peine

(1995) Shape memory alloy actuator controller design for tactile displays. In: Proceedings of the 34th IEEE conference on decision and control, vol. 4. IEEE, pp. 3540–3544.

10.

Hulskamp

(2011) The Smart Rotor Concept on Wind Turbines. PhD Thesis, Delft University of Technology.

11.

Lagoudas

(2008) Shape Memory Alloys: Modeling and Engineering Applications. New York: Springer.

12.

Lara-Quintanilla

Hulskamp

Bersee

(2013) A high-rate shape memory alloy actuator for aerodynamic load control on wind turbines. Journal of Intelligent Material Systems and Structures 24(15): 1834–1845.

13.

Lewis

York

Seelecke

(2013) Experimental characterization of self-sensing SMA actuators under controlled convective cooling. Smart Materials and Structures 22(9): 094012.

14.

(2006) Modeling and Finite Element Analysis of Smart Materials. PhD Thesis, North Carolina State University.

15.

Song

Lee

(2004) Position control of shape memory alloy actuators with internal electrical resistance feedback using neural networks. Smart Materials and Structures 13(4): 777.

16.

Massad

Smith

(2005) A homogenized free energy model for hysteresis in thin-film shape memory alloys. Thin Solid Films 489(1–2): 266–290.

17.

Monner

Breitbach

TB (2000) Design aspects of the adaptive wing - the elastic trailing edge and the local spoiler bump. Aeronautical Journal 104(1032): 89–95.

18.

Petrini

Migliavacca

(2011) Biomedical applications of shape memory alloys. Journal of Metallurgy 2011: 501483. DOI: 10.1155/2011/501483.

19.

Pons

(2005) Emerging Actuator Technologies: A Micromechatronic Approach. New York: John Wiley & Sons.

20.

Reynaerts

Brussel

(1998) Design aspects of shape memory actuators. Mechatronics 8(6): 635–656.

21.

Scire-Mammano

Dragoni

(2014) Functional fatigue of Ni–Ti shape memory wires under various loading conditions. International Journal of Fatigue 69: 71–83.

22.

Seelecke

Müller

(2004) Shape memory alloy actuators in smart structures: Modeling and simulation. Applied Mechanics Reviews 57(1–6): 23–46.

23.

Sofla

Meguid

KT (2010) Shape morphing of aircraft wing: Status and challenges. Materials and Design 31(3): 1284–1292.

24.

Stoeckel

(1990) Shape memory actuators for automotive applications. Materials and Design 11(6): 302–307.

25.

Tadesse

Thayer

Priya

(2010) Tailoring the response time of shape memory alloy wires through active cooling and pre-stress. Journal of Intelligent Material Systems and Structures 21(1): 19–40.

26.

Schetky

(2000) Industrial applications for shape memory alloys. In: Proceedings of the international conference on shape memory and superelastic technologies, pp. 171–182.

27.

Yang

Seelecke

(2009) Fe analysis of SMA-based bio-inspired bone–joint system. Smart Materials and Structures 18(10): 104020.