High-speed response of SMA actuators

Transcription

1 International Journal of Applied Electromagnetics and Mechanics 12 (2000) IOS Press High-speed response of SMA actuators Jinhao Qiu a, Junji Tani a, Daisuke Osanai a, Yuta Urushiyama b and David Lewinnek c a Institute of Fluid Science, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai , Japan Tel.: ; Fax: ; qiu@ifs.tohoku.ac.jp b Honda R&D Co., Ltd., Tochigi R&D Center, Haga-gun, Tochigi , Japan c Platform Products, I/CD, Teradyne, Inc., 179 Lincoln St., MS-L21, Boston, MA 02111, USA Received 25 August 2000 Accepted 19 December 2000 Abstract. The main problem with Shape Memory Alloy (SMA) actuators is their slow response speed. Current SMA actuators work in tens of seconds though they can generate large force and displacement outputs. The objective of the research in this paper is to improve the response speed of SMA actuators to 10 milliseconds or less. SMA actuators are driven by thermal energy and their slow response is attributed to the inefficient heat transfer. To improve the response, direct Ohmic heating was used in the experiments. The SMA actuators used in this study are Ti-Ni-Cu columns of φ5 50 mm with residual strain produced upon unloading after 5.3% axial compression. The transient displacement response of the SMA actuators without axial constraint and the transient force response of the actuators under axial constraint were measured with direct heating. The shortest response time achieved for unconstrained actuators is 4.6 ms and that for constrained actuators is 6.5 ms. The average value of maximum recovery force is 10.8 kn, which corresponds to 550 MPa of recovery stress. It was also found that constrained actuators need about 50% more energy for heating than unconstrained actuators. 1. Introduction Shape recovery of a deformed Shape Memory Alloy (SMA) is induced due to the Austenitic phase transformation when it is heated above the reverse phase transformation finish temperature (A f )of the material. Since the recoverable strain of SMAs can be more than 10% and the shape recovery process can cause larger force generation if strain is constrained, SMAs have attracted wide attention for their application as actuator materials in smart structural systems [1 3]. Although SMA actuators have high energy density and larger displacement stroke, they have the disadvantage of slow response. This disadvantage has limited the application of SMA as actuators for dynamical systems where fast response is necessary. For example traditional SMA actuators can still not be used to control vibration over 1 Hz though many studies have performed to raise their response [4,5]. Since SMA actuators are driven by thermal energy, their response speed is mainly dependent upon the heating process. For cyclic actuation, the cooling process is also very important [6]. In this study, we only consider actuators for applications such as the control of dynamic buckling of automobile frame during crash accident, in which actuators only need to be heated once so that response speed depends only on heating time. There are several ways to heat SMA actuators, such as conductive heat transfer [7], radiative heat transfer through microwaves or infrared light [8], inductive heating with oscillating magnetic fields [9] and direct Ohmic heating [10]. The first two methods involve the heat conduction, which depends essentially on the temperature gradient in the actuators, and are therefore inefficient. In the case of sufficient power rate of conductive heat transfer, the temperature gradient will /00/$ IOS Press. All rights reserved

2 88 J. Qiu et al. / High-speed response of SMA actuators Table 1 Properties of Ti-40.8wt%Ni-9.9wt%Cu Density (g/cm 3 ) 6.5 Specific heat (J/(g deg)) 0.49 Heat of transformation (J/g) 25 Electrical resistance (µω cm) 100 A s ( C) 51.9 A f ( C) 64.9 M s ( C) 44.6 M f ( C) 30.0 be extremely large and the temperature on the surface of the actuator would exceed the melting point of the material. The third method uses an oscillating magnetic field to induce eddy current which in turn produce heat due to the resistance of the materials. Since magnetic field can penetrate a metal, the actuator can be uniformly heated. The generated heat by eddy current varies with the square of flux density and frequency of magnetic field. For paramagnetic materials, it takes an extraordinary strong magnetic field to generate adequate magnetic field. Increasing the frequency of magnetic field increases the efficiency of heat generation, but it also makes the magnetic field more difficult to penetrate into the depths of a metal. Another major problem with eddy current heating is the necessity of a large electromagnet and the related equipment for the generation of strong magnetic field. In this study, high-speed response of SMA actuators was achieved by direct electric heating. The influence of various parameters on the response time and generated force was investigated. The objective value of response time was 10 ms in this study. 2. Circuit design and experimental setup The experiments involved heating SMA by discharging larger capacitors across the columns. Phase transformation, which starts at temperature As and finishes at A f, causes shape recovery in the actuators. The shape recovery is accompanied by large displacement output, or force output if displacement is constrained Actuator material and size In this study, Ti-40.8wt%Ni-9.9wt%Cu columns of φ5 50 mm were used as test pieces of actuators. The material properties and the phase transformation temperatures of the material are shown in Table 1. The physical and geometrical parameters of the actuators are shown in Table 2. The test pieces were compressed by 5.3% (about 2.65 mm) in the axial direction and then unloaded to obtain approximately 2 mm residual deformation. The energy applied to the actuator for finishing the phase transformation can be divided into two parts, one for raising temperature of the actuator to A f and the other for phase transformation, that is, it can be expressed in the following form: E t = E h + E p = m (λ T + µ), (1) where E h is the energy for raising temperature, E p the energy for phase transformation, m is the mass of the actuator, µ is the energy needed for phase transformation of unit mass, λ is the specific heat, T is the temperature increment.

3 J. Qiu et al. / High-speed response of SMA actuators 89 Table 2 Parameters of actuators Original length of the actuators (mm) 50 Original diameter of the actuators (mm) 5 Residual strain (%) 4 Mass of the actuators (g) 6.38 Resistance of the actuators (mω) 2.55 Room temperature T 0 ( C) 22.9 Temperature increment T ( C) 42 Energy for raising temperature (J) 134 Energy for phase transformation (J) 128 Total required energy (J) 262 Fig. 1. Circuit for heating actuators. The room temperature T r was supposed to be 22.9 C, and therefore the temperature increment from T r to A f is 42 C. The energies E h, E p and E t can then obtained from Eq. (1) to be 134 J, 128 J and 262 J, respectively, that is, totally 262 J of energy is required to drive the actuator. The parameters of the actuator, temperature conditions and required energies are also summarized in Table Response time and circuit design In the direct Ohmic heating, the power rate can be expressed in the following form: P a = R a I 2 = V 2 a /R a, (2) where R a is the resistance of the actuator, I is the current and V a is the voltage on the actuator. A response time of 10 ms requires that 262 J of energy be supplied within 10 ms, that is, the average power rate be larger than 26.2 kw (P a > 26, 200 W). On the other hand, the resistance of the actuator R a is about 2.55 mω. This means that the current must be larger than 3204 A. Since this strong current has to be sustained for only 10 ms, it is more practical to use capacitors than the traditional power supply. The electric circuit shown in Fig. 1 is used to heat the actuator. The total resistance of the circuit R includes 3 parts that are respectively from the actuator, the capacitors and the connecting cables and joints, and can be expressed in the following form: R = R a + R c + R w. (3) The total resistance of capacitors R c is determined by the number of capacitors and their connection (parallel or serial), and the total resistance of cables and joints R w depends on the configuration of the

4 90 J. Qiu et al. / High-speed response of SMA actuators Model Type Capacitance Maximum voltage Parasitic resistance Height Diameter Weight Table 3 Specifications of capacitors LNRIC105MSM Aluminum electrolytic capacitor 1 F 20 V 2 mω 220 mm 90 mm 1700 g circuit. As to be discussed in the next section, the resistance of the switch and all joints are negligible. Copper cables of 00 gage (266 µω per meter, consisting of 19 strands of 2 mm wire) were used in the experiment. The total length of the cables is about 2 m, and therefore resistance R w is about 0.53 mω. The following relationship holds between the total power of capacitors P and the power consumed on the actuator P a : P a = R a R P. (4) The power P a is determined by the required response time of the actuator. In order to reduce the energy consumed by the capacitors and connecting cables, and increase the efficiency of heating, R c and R w must be as small as possible. In this study, capacitors made by Nichicon Corporation were used. The specifications of the capacitors are shown in Table 3. The parasitic internal resistance of a capacitor is2mω. A single capacitor can store 200 J of energy, which is not enough to heat an actuator to over the A f temperature. When 2 capacitors are connected in parallel, the stored energy is doubled to 400 J and the total resistance of the capacitors is reduced to 1 mω. But since a part of the stored energy will be consumed by the connecting cables and the capacitors themselves, the energy to be supplied to the actuator is only 250 J, which is still smaller than the required energy 262 J. Hence, at least three capacitors are necessary. In this study, multiple capacitors are always connected in parallel. Parallel connection of capacitors decreases the total resistance of the circuit and increase the efficiency of heating. It also decreases the response time due to smaller total resistance of the circuit. When capacitors are discharged across the actuator, the electric current, which decays exponentially, can be expressed in the following form: I = V 0e t RC R, (5) where t is time, V 0 is the initially charged voltage of the capacitors, C is the total capacitance of the used capacitors. When R, C, V are given, the time needed for supplying the required energy E t can be obtained through the integration of Eq. (5): τ = 1 ( 2 RC log 1 2RE t R a CV0 2 ). (6) On the other hand the response time is defined as the time for complete phase transformation, which is the same as the time for the actuator to reach the maximum recovery displacement. If the speed of phase transformation is infinite, that is, phase transformation is finished as soon as the required energy is applied, then time τ defined in Eq. (6) equals the response time. Since the speed of phase transformation is unknown, Eq. (6) was used in the calculation of theoretical response time in this study.

5 J. Qiu et al. / High-speed response of SMA actuators 91 Fig. 2. Theoretical relationships between voltage and response time. There are three parameters V 0, C and R in Eq. (6) that influence the response time. However, C and R are both dependent upon the number of capacitors. When n capacitors are connected in parallel, the total capacitance becomes n times, but the total resistance of the capacitors becomes 1/n of a single capacitor. Hence, if the total capacitance C is given, then resistance R c and therefore R are consequently determined. Figure 2 shows how the capacitance and voltage influence the response time when E t = 262 J. There is a lower bound (denoted by V l ) of charging voltage for each capacitance, at which the theoretical response time trend to infinity. For example V l for the capacitance of 3Fisabout 16 V. This lower bound corresponds to the condition, at which the stored energy equals the total required heating energy (equal to 262 J at this study). When the charging voltage is much higher than V l (for example at 20 V), the response time is not very sensitive to either the voltage or the capacitance. But when the charging voltage is only slightly larger than V l, the error in charging voltage will significantly influence the response time Experimental setup and procedure The experimental setup shown in Fig. 3 was used in the measurement of the displacement response. The pre-compressed test piece was integrated to a closed circuit, which includes a number of capacitors connected in parallel, a switch, connecting copper cables, and two connectors used to connect the actuator to the cables. In order to raise the response speed of the actuator and the efficiency of heating, the resistance of the cables, connectors and switch should be as small as possible. The switch should have a closing time of less than 1 ms and be able to handle several thousand Amperes of current with a negligible voltage drop. In this study, a switch was designed and made, as shown in Fig. 3. The switch consists of two blocks of copper, which contacts on a surface area of 15 cm 2. The two blocks are held together by springs with a maximum force of 300 kn. In the open position, the two copper blocks were held apart by a wire. To close the switch, the wire was cut. The ramp-up in current when closing the switch was less than 1 ms. Since the joints between the cables and blocks of the switch and the joints between the cables and the blocks of connectors were soldered, their resistances are negligible. The resistance of the cables has been discussed in the preceding section. The actuator was connected to the copper blocks by clamps with an

6 92 J. Qiu et al. / High-speed response of SMA actuators Fig. 3. Experimental setup for displacement measurement. Fig. 4. Experimental setup for force measurement. overlap of 5 mm at either end. The resistance is also negligible, but the clamp imposes mechanical and thermal constraints on the actuator and reduces the recoverable deformation as to be discussed later. A laser sensor was used to measure the transient response of the displacement. The output voltage of capacitors and the displacement signal from the laser sensor were recorded with a FFT analyzer. The experimental setup for force measurement is shown in Fig. 4. The same setup as that for displacement measurement was used with an additional constraint of displacement and a load cell (LM- 2TS, manufactured by Kyowa Electronic Instruments Co., Ltd.) for force measurement. The load cell can measure up to kn with errors less than 2% of the rated output. Its natural frequency is 50 khz. Before test, the SMA columns were compressed in the axial direction by about 5.5% and then unloaded. A die with a hole of 5.2 mm in diameter and 40 mm high was used to prevent plastic buckling during

7 J. Qiu et al. / High-speed response of SMA actuators 93 (a) (b) Fig. 5. Time history of displacement and voltage. (a) Displacement (b) Voltage. compression. A residual deformation of about 2 mm (or 4%) remains in the columns after unloading. In the experiment, a predetermined number of capacitors were connected in parallel to constitute the desired capacitance. They were charged to the desired voltage and then the switch was turned on. The actuators were quickly heated by the discharging current and the transient response of the displacement or force was measured. Measurements were performed with different combinations of the capacitances and charging voltages to investigate their influence on the maximum displacement/force output and the response time. 3. Displacement response of SMA actuators 3.1. Displacement and needed energy The displacement response of the SMA actuator was measured under different combination of capacitances and charging voltages and some of the results are shown in Fig. 5(a). The higher the charging

8 94 J. Qiu et al. / High-speed response of SMA actuators Fig. 6. Displacement versus energy relationship. Fig. 7. Illustrative relationship between displacement/force and energy. voltage and the larger the capacitance is, the faster the response of the actuators is. The time history of output voltage of the capacitors is shown Fig. 5(b). The output voltage is the sum of the voltage on the cables and that on the actuator. If it is assumed that the resistance of the actuator does not vary with temperature, then the energy consumed on the actuator at any time can easily be obtained through the integration of the output voltage. After the energy is obtained, the displacement versus energy relationship can be plotted as shown in Fig. 6. Obviously, the displacement versus energy curve can be divided into three regions, which are labeled with A, B and C in the illustrative curve in Fig. 7. Region A is for heating actuator before phase transformation and the extension of the actuator due to thermal expansion is negligible. Phase transformation is induced in region B and large displacement occurs. Further supply of energy in region C causes the temperature rise of actuator to continue, but the displacement is also negligible. By drawing tangent lines

9 J. Qiu et al. / High-speed response of SMA actuators 95 Table 4 Energies, response times and recovery displacements No. Condition Total stored Theoretical Experimental Phase trans. Phase trans. Theoretical 95% disp. Recovery Energy energy consumed energy consumed start energy finish energy response response disp. (J) on actuator (J) on actuator (J) (J) (J) time (ms) time (ms) (mm) 1 3 F, 15 V * 2 3 F, 17.5 V F, 20 V F, 15 V F, 20 V DG of region B and GC of region C, two intersecting points D and G can be obtained. The energy E s at point D is the energy needed for heating the actuator before phase transformation and is called phase transformation start energy. The extension of length L f at point G is defined as the maximum recovery displacement. The corresponding energy E f at this point is the required energy for heating actuator and is called phase transformation finish energy. Since the thermal expansion in region C is very small, the maximum output displacement L u is almost the same as the maximum recovery displacement L f. When the supplied energy is smaller than the required energy the curve will end in region B. For these case the maximum recovery displacement is defined as the final displacement. The maximum recovery displacements L f, the phase transformation start energy E s, the phase transformation finish energy E f and the experimental energy consumed on the actuator E u (corresponding to point H in Fig. 7) were estimated for different conditions and summarized Table 4. The phase transformation finish energy is not available for the case of 3 F and 15 V since the phase transformation is not finished due to too small supplied energy. The theoretical values of total energy stored in the capacitors and the energy consumed on the actuator for different conditions are also listed in the second and third columns of Table 4. From Table 4, it can be seen that the phase transformation start energy E s varies in a wide range from 118 J to 189 J. The variation can be attributed to several factors. In all of them, the time needed for phase transformation and the higher potential needed for driving faster phase transformation are the two most important ones since the starting energy E s increases with heating rate. The average value of E s is 156 J. The phase transformation finish energy E f varies in a relatively narrow range of about 10% and the average value is 277 J, which is a few percent larger than the theoretical value of 262 J. It can also be seen that the final displacement is significantly smaller than residual deformation, that is, a part of the residual strain was not recovered. It can be attributed to the constraint at the two ends of the actuator by the connectors used to clamp the actuator to the cables. The connectors impose mechanical constraint to preventing the actuator from shape recovery and also thermal constraint to prevent it from being heated. The average of maximum recovery displacements for the cases with complete phase transformation (the value labeled with * in Table 4 is not included) is about 1.73 mm, about 87% of the residual deformation Response time In order to quantitatively evaluate the response speed of the actuators, the experimental response time should be estimated from the response curve for different heating conditions. As discussed in Section 2.2, the response time is the time needed for the actuator to extend to the maximum recovery displacement, at which the shape recovery is complete. The maximum recovery displacement corresponds to the displacement L f at point G in Fig. 7. In order to reduce the influence of saturation near L f, the

10 96 J. Qiu et al. / High-speed response of SMA actuators experimental response time is defined as the time when the displacement reaches 95% of L f. The estimated experimental response times for different conditions are also listed in Table 4, together with the theoretical values of response time calculated from Eq. (6). In the case of 3 F and 15 V, the theoretical response time is not available because the energy supplied to the actuator is too small and phase transformation will never be complete. The experimental response for the same condition is finite since it has been defined as the time for the actuator to reach 1.14 mm, 95% of the maximum recovery displacement 1.2 mm. The experimental response times are about 10 20% larger than the theoretical ones except for the case of 4 F and 15 V. The difference between theoretical and experimental results can be attributed to the difference between the theoretical and experimental values of required energy (difference between E f and E t ). Despite of the errors, Eq. (6) still gives a good estimation of the response time of SMA actuators and can be used in the future design of circuits for heating SMA actuators. It can also be seen that the shortest response time is about 4.61 ms, which is less than one half of the objective response time of 10 ms. 4. Recovery force of SMA actuators 4.1. Recovery force and needed energy When SMA materials are used as actuators, the output force is very important. In the study, the transient response of generated force was measured when the displacement of actuators was completely constrained. Measurement was performed for different combination of capacitances and charging voltages and the results are shown in Fig. 8. The small peaks at the beginning of heating in some of the curves are due to the interference of the electromagnetic field due to the sudden rising of electric current in the circuit with the signal of load cell. The actual force should be zero at the beginning of heating. Since the problem was solved in the later measurement, there is no peak in most of the curves. From Fig. 8, it can be seen that the response speed and the maximum output force varies significantly with the capacitance and charging voltage. It seems that the larger the energy stored in the capacitors is, the faster the response and the larger the recovery force is. In order to clarify their relationship, the recovery force is plotted versus energy as shown in Fig. 9. The energy consumed in the actuator is obtained by integrating the time history of output voltage with respect to time on the assumption that the resistance of actuators is constant. In the same way as used in Section 3.1, the force versus energy relationship can also be divided into three regions (Fig. 7), and the phase transformation start energy E s, phase transformation finish energy E f, total consumed energy on the actuator E u, maximum recovery force F f, and maximum output force F u can be defined. These values were estimated and summarized in Table 5. Significant increase in output force can be observed in region C. This increase can be attributed to the combinative influence of thermal expansion and the variation of Young s modulus with temperature. For the cases in which the phase transformation finish energy E f is labeled by, the input energy is too small so that the phase transformation is not finished. From Table 5, it can be seen that the phase transformation start energy E s varies in a wide range from 174 J to 254 J. The variation can be attributed to the same factors as those for the transformation start energy of unconstrained actuators. The phase transformation finish energy Ef varies in a relatively narrow range of about 10%. An important phenomenon is that more energy is needed for heating constrained actuators than for heating unconstrained ones. The average of phase transformation start energy E s for constrained actuators

11 J. Qiu et al. / High-speed response of SMA actuators 97 (a) (b) Fig. 8. Time history of recovery force under different conditions. is about 212 J, compared to 156 J for actuators without constraint, and the average of phase transformation finish energy E f for constrained actuators is 410 J, about 50% larger than 277 J for actuators without constraint. The increase of required energy can be attributed to the rising of phase transformation temperatures induced by the stress in the SMA columns due to the mechanical constraint [11]. Although the energies of case 2, 3, 4, 7 in Table 5 are enough for complete phase transformation of unconstrained actuators, they are smaller than the required energy for constrained actuators and smaller forces were generated than in the other cases due to incomplete phase transformation. For the cases with complete phase transformation (the cases labeled with * are not included), the average of recovery force is about 10.8 kn. When the recovery force in Table 5 is plotted against the experimental energy consumed on the actuators for different cases, Fig. 10 can be obtained. The figure confirms that at least 410 J of energy is required to drive constrained actuators. When the applied energy is smaller than this value, the output force decreases with the decreasing energy due to incomplete phase transformation. If the applied energy

12 98 J. Qiu et al. / High-speed response of SMA actuators (a) (b) Fig. 9. Recovery forces versus energy. exceeds this value, then the maximum recovery force is almost the same for different supplied energy Response time of recovery force In the similar way as used in Section 3.2, the response time of recovery force is defined as the time when the output force of the actuator reaches 95% of the maximum recovery force listed in Table 5. The response time for different conditions were estimated from the response curve of recovery force and also listed in Table 5. Obviously, the response time of recovery force is longer than that of displacement due to constraint. For example, case 6 in Table 5 and case 5 in Table 4 have the same capacitance and charging voltage, but the response time of the former is 7.23 ms, over 50% larger than 4.61 ms, the response time of the later. The increase is simply due to the increase of required energy for complete phase transformation due to constraint and then more time is needed for supplying the required energy. The shortest response time of recovery force is 6.5 ms, which is smaller than the objective value of 10 ms.

13 J. Qiu et al. / High-speed response of SMA actuators 99 Fig. 10. Maximum recovery forces versus energy. Table 5 Energies, response times and recovery forces No. Condition Total stored Theoretical Exp. energy Phase trans. Phase trans. 95% recovery Maximum Maximum energy energy consumed starting finishing force response recovery output (J) consumed on on actuator energy energy time force F r force E f actuator (J) (J) (J) (J) (ms) (kn) (kn) 1 3 F, 15 V * F, 17.5 V * F, 20 V * F, 15 V * F, 17.5 V F, 20 V F, 12 V * F, 17.5 V F, 20 V In the practical applications, a SMA actuator will subject to reactive force from the object, which it is used to control, and the displacement will be smaller than that of constrained actuators. Under such a condition, the required energy will be smaller than that for constrained actuators, but larger than that for unconstrained actuator. The response time and displacement will also lie between the two cases. Hence, the circuit used for heating and the response time achieved in this study are suitable for practical applications in the future. 5. Summary A circuit using capacitors with low parasitic resistance for heating high-speed SMA actuators was designed and the responses of recovery displacement and recovery force were measured for different combination of capacitance and charging voltage in the designed circuit. The following conclusion can be drawn from the results. 1. The response times of displacement of unconstrained actuators were estimated from the response curves and compared the theoretical value. Although the theoretical response times are 10 20%

14 100 J. Qiu et al. / High-speed response of SMA actuators smaller than the experimental ones, they are still good prediction of the response time in the design of circuit. The shortest achieved response time is 4.61 ms, less than one half of the objective value 10 ms. 2. About 50% more energy is needed for heating constrained actuators than for heating unconstrained actuators. The response time of recovery force is also longer than that of recovery displacement under the same heating condition. The shortest achieved response time for constrained actuators is 6.5 ms, smaller than the objective value of 10 ms. 3. A maximum recovery force of about 11.3 kn (or 576 MPa of recovery stress) was achieved for the SMA actuator used in the study. References [1] J. Tani, T. Takagi and J. Qiu, Intelligent Material Systems: Application of Functional Materials, Applied Mechanics Reviews, ASME 51 (1998), [2] Y. Furuya, A. Sasaki and M. Taya, Enhanced Mechanical Properties of TiNi Shape Memory Alloy Fiber/Al Matrix Composite, Materials Trans, JIM [3] C. Dickinson and J.T. Wen, Feedback Control Using Shape Memory Alloy Actuators, Journal of Intelligent Material Systems and Structures 9 (1998), [4] Y. Nakamura, S. Nakamura, L. Buchchaillot, M. Ataka and H. Fujita, Two Thin film Shape Memory Alloy Microactuators, Transaction of IEE of Japan 177 (1997), [5] Y. Luo, T. Takagi, S. Maruyama and Y. Kohama, Thermo-mechanical Response of a Novel Shape Memory Alloy Actuator with Peltier Elements, in: Non-Linear Electromagnetic Systems, P. Di Barba and A. Savini, eds, IOS Press, 2000, pp [6] A.R. Shahin, P.H. Meckl and J.D. Jones, Enhanced Cooling of Shape Memory Alloy Wires Using Semiconductor Heat Pump Modules, Journal of Intelligent Material Systems and Structures 5 (1994), [7] A. Bhattacharyya, D.C. Lagoudas, Y. Wang and V.K. Kinra, On the role of thermoelectric heat transfer in the design of SMA actuators: theoretical modeling and experiment, Smart Materials and Structures 4 (1995), [8] S.W. White and J.B. Berman, Thermomechanical Response of SMA Composite Beams with Embedded Nitinol Wires in an Epoxy Matrix, Journal of Intelligent Material Systems and Structures 9 (1998), [9] T. Takagi, J. Tani, S. Suzuki and M. Matsumoto, Electromagneto-thermo-structural Analysis of an ARSME Plate, Transactions of the Japan Society of Mechanical Engineering, Ser. C (in Japanese) 64 (1998), [10] S. Chonan, Z.W. Jiang, J. Tani, S. Orikasa, Y. Tanahashi, T. Takagi, M. Tanaka and J. Tanikawa, Development of an Artificial Urethral Valve Using SMA Actuators, Smart Materials and Structures 6 (1997), [11] K. Tanaka, H. Tobushi and S. Miyazaki, Mechanical Properties of SMA, (in Japanese), Yokendo Ltd., Tokyo, 1993, pp