Study and Modeling Behavior of Shape Memory Alloy Wire

Transcription

1 Study and Modeling Behavior of Shape Memory Alloy Wire Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar, Pune Institute of Computer Technology Pune Institute of Computer Technology ABSTRACT From early 20s till today, advancement in the technology is highly speeded up. Every innovation talks about the incorporation of intelligence and smartness into the system. This smartness is added by various materials categorized as smart materials. One of the materials from the same category is Shape Memory Alloy (SMA). Proper expected functioning of the material and system, it is necessary to know the behavior of the material. Hence the material is studied and later mathematically modeled to track the response of material for an input. The type of excitation and excitation duration plays a very crucial role, Hence a material is excited with various signals such as DC, triangular and trapezoidal wave for which a respective temperature profile is obtained. Change in temperature from 50 to C leads to phase transformation. The temperature dependent transformation is mathematically modeled using different mathematical functions and Liang and Rogers, Duhem models are compared with each other. This paper finds that transformation temperature by Liang and Rogers agrees well with datasheet provided by manufacturer while there is ± 5 0 C changes in Duhem Model. Hence, the behavior of the Nickel-Titanium (NiTi) shape memory alloy wire with respect to temperature and its phase transformation is mathematically modeled in MATLAB. The results of this model can be used to decide the type of excitation, to obtain a desired response with phase transformation temperatures according to the application. Keywords Shape Memory Alloy (SMA); Nickel-Titanium (NiTi); phase transformation; thermal behavior INTRODUCTION Smart materials are designed materials that have one or more properties that can be significantly changed in a controlled fashion by external stimuli, such as stress, temperature, electric or magnetic fields. There are various materials under this category. Few of them are piezoelectric, thermoelectric, photo-chromic, thermo chromic, etc. Amongst these materials, there is one more material called as Shape Memory Alloy (SMA). SMAs justify their presence in the list of smart material because of their two phenomenal properties of Shape Memory Effect (SME) and super elasticity. SMAs are drawn in various forms such as wire, sheets, cylinders, hollow rods, etc. They have applications in various fields such as arch wires in orthodontics, stents to open the collapsed arteries and the most popular as actuators for robots in medical as well as non medical applications [1-5]. These smart alloys possess temperature dependent structural changes. When material is at low temperature (Martensite phase), it can be easily deformed and it holds the deformation at low temperature. Once the material temperature is increased i.e. high temperature (Austenite phase/ Parent phase), it regains its original (memorized) shape, irrespective of stress applied to the material. This property of material is called as SME. Again, when the material is cooled down by natural convection like free air or 774 Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar,

2 forced convection like fan, it can be deformed and when again it is heated to a specific temperature, it regains its original shape. This cycle can be performed for number of times without losing the material property. This is called as super elasticity. In martensite phase, there are two crystalline structure: twinned martensite i.e. when material is just cooled down and other is detwinned martensite, where a load is applied to twinned martensite leading to plastic deformation in the material. When the material in detwinned martensite is heated it acquires third crystalline structure, called as austenite structure in austenite phase. Austenite phase is also called as parent phase as it consists of memory of the material. This type of SMA is called as one way shape memory alloy[6]. Other type of SMA is two way shape memory alloy. As in one way SMA, the material exibits SME only upon heating, similarly in two way SMA, the material exibits SME but, during both heating and cooling. In this paper, we consider only one way SMA. SELECTION OF SHAPE MEMORY ALLOY MATERIAL Today in market, there are various alloys available commercially in various forms with different combinations of metals, at different prices. For the study, the wire form of the material is more suitable as it is easy to cut, excite and to measure the deformation in the material. To obtain the stress-strain profile of material, the material has to undergo stress conditions. As wire form is chosen, the applied stress will be of tensile form. Hence to bear with the applied tensile stress, material with high tensile strength and maximum recoverable strain needs to be chosen. Amongst all, the availability of the material in specific form and quantity is important. All these conditions are met by NiTi alloy. Hence in this paper, for study and modeling behavior we consider NiTi SMA wire from Dynalloy [7] with specifications as mentioned in table (1). Table 1: Specifications of selected material. Parameters Symbol Values(Units) Alloy material NiTi Nickel-Titanium Chemical composition (by atomic wt.) - Ni 50% Ti 50% (approx.) Form - Wire Density ρ 6450 (kg/m 3 ) Specific heat cp 873 (J/kg - 0 C) Diameter d 0.5 (mm) Length l 500 (mm) Convection heat transfer coefficient h 110 (W/m 2 -K) Resistance R 4.3(Ω/m) Stroke for dead weight bias S 4(%) Current for contraction in 1 sec Ic 4(A) MATHEMATICAL MODELING BEHAVIOR OF NITI SMA WIRE Temperature dependent behavior of the NiTi SMA wire can be achieved by modeling its two physical phenomenons that is possessed by the material. According to [7], material possesses four physical phenomena: Heat transfer based on joule heating effect; Phase 775 Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar,

3 transformation; Strain-Martensite volume fraction relation; Electrical resistance. Amongst them the temperature dependent phenomena are heat transfer which is due to the resistive heating of the wire, commonly called as joule heating effect and the other one is temperature dependent diffusion-less structural phase transformations in the material. In this section, each mentioned physical phenomenon is modeled independently. Following is the system model as shown in figure (1). Vi Ii Heat Transfer (Temperature profile) T Phase transformation Hysteresis M s M f A s A f Fig 1: System model to characterize temperature dependent phenomena 1. HEAT TRANSFER BY JOULE HEATING EFFECT In this subsection, we describe thermal behavior of the NiTi wire with its mathematical model. When excitation is applied to the wire, wire gets heated up due to the joule heating effect. This causes rise in overall temperature of the wire. The temperature profile of the wire can be plotted using the following mathematical model as given in [8]. The thermal model is given by first order equation (1).. = h( ) (1) where ρ is density of wire, c is specific heat coefficient, d is diameter of wire, l is length of wire, v is input voltage, i as input current, h is convection heat transfer coefficient and Ta is ambient temperature. For temperature T, the solution to equation (1) obtained is as below. = (1 ) + (2) Where, = is a constant. 2. PHASE TRANSFORMATION HYSTERESIS Shape memory alloy wire undergoes structural transformations under certain conditions like stress and temperature. These alloys have two phases called as martensite and austenite. The former one is a low temperature phase defined by two temperatures, martensite start (Ms) and martensite finish (M f). The later one is a high temperature phase defined by two temperature austenite start and finish as As and Af respectively. With change in temperature it undergoes structural phase transformations. The phase transformation from Af As Ms Mf (cooling) or from Mf Ms As Af (heating) is characterized by a variable that defines the amount of specific phase transformation present in the material with respect to temperature. That variable is called as martensite volume fraction and represented as Rm. The values that Rm can take are between 0 Rm 1. Rm is 1 implies that the material is in its fully martensite phase while if it takes value as 0, then material is in its austenite phase. Similar to martensite volume fraction Rm, there is one more variable used in the literature known as 776 Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar,

4 austenite volume fraction represented as Ra. The relation between Rm and Ra is given by equation (3).At any given temperature, both the phases do existsimultaneously in the material. Rm+Ra=1 (3) Such hysteresis is been mathematically modeled by different researchers using different mathematical functions, aiming at the exact behavior modeling of the material. Amongst them, 2.1 Duhem Model This model uses Gaussian probability distribution function to model the hysteresis curve[8]. Model consists of two variables, input variable is x(t) and output variable is y(t). Now the assumptions are as follows, If x(t) increases, y(t) increases along one path If x(t) decreases, y(t) decreases along another path The slopes of these paths are represented by Gaussian PDF, with slopes as g+ for increasing curve and g- for decreasing curve. Mathematically, these slopes are given as in following equation (4). g / (x) = exp μ / σ (4) / π σ / where μ is mean and σ 2 is variance. Integrating the Gaussian probability density function gives the major hysteresis loop as given in equation (6). Here + sign indicates increasing curve and sign indicates decreasing curve. y / = f / (x) = g / (x)dx (5) / ( ) = Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar, / / Using the above equations (4) and ( 6), the major hysteresis is plotted in Matlab software platform. The parameters used in simulations are listed in table (2). 2.2 Modified Liang and Rogers model This model uses cosine function proposed by Linag and Rogers model the hysteresis curve [9]. In this the martensite and Austenite volume fraction is defined for cooling and heating cycle independently. Following are the notations. Rmh = martensite volume fraction during heating process Rmc = martensite volume fraction during cooling process Rah= austenite volume fraction during heating process Rac = austenite volume fraction during cooling process All of these parameters are related to each other by following equations (7) and (8). Rac = 1-Rmc (7) Rah = 1-Rac (8) The classical Liang and Rogers model is simplified by eliminating the term as the pretension τ is very small, causing negligible effect in the experiment [9]. Hence, the new modified Liang and Rogers model categorizing the austenite volume fraction for heating and cooling process is as given in equation (9) and (10). (6) /

5 1. Heating process 1 T <Af Rah= As T Af (9) 0 T >As 2. Cooling process 1 T <Ms Rac= Mf T Ms (10) 0 T >Mf Based on equation ( 9) and ( 10), with appropriate parametric values as listed in table ( 3), the model is simulated in MATLAB. SIMULATIONS AND RESULTS 1. HEAT TRANSFER BY JOULE HEATING EFFECT The heat transfer in the material is due to the resistive heating of the material to the applied current. This physical phenomenon is described by equation (1). Further this equation is solved as in equation (2) to obtain the temperature profile for applied current. The same is simulated in MATLAB Simulink R2013a. The system model is as given in figure (2). Fig. 2: Simulated Heat transfer model The parameters required for simulation are as listed in table (1) and (2). Firstly, the model is validated to the datasheet of selected material i.e. datasheet of Dynalloy Inc. for Ni-Ti SMA wire [7]. This is done by applying 4A current for 1sec to the model as shown in figure (3) the datasheet says that for applied 4A current for 1sec, the wire contracts and attains its original length. As in figure (4), the maximum attained temperature in 1sec is C, which is nearer to Af of wire. The wire recovers its original length at its austenite finish temperature, implying for applied 4A current for 1sec, the wire has contracted to most of it to its original length, thus validating the model. 778 Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar,

6 Fig.3: Input Current signal Fig.4: Temperature profile for 4A current for 1sec Now, as model is validated, various shapes of signals are applied to the model. Details of various applied current signalsfor heat transfer model are as given in table 2. Table 2: Various inputs for heat transfer model Parameters Symbol Values Ambient temperature Ta 26 0 C Input Voltage V 5 V Input Current Signal I DC Triangular Trapezoidal Peak amplitude Ip 1A 1A 1A Observation duration Tob 100 sec 100 sec 100 sec Results for the input parameters for heat transfer model are as given in figure (5a), (5b) and (5c). The rise time also known as heating time is necessary to predict the frequency that can be applied to the material. Heating time can be reduced by increasing the applied current and voltage [12]. (a) (b) (C) Fig.5: Temperature profile for (a) DC 1A input current,(b) triangular 1A input current (c) trapezoidal 1A input current From the above results, few observations are listed as below. 1. Temperature profile obtained follows the shape of input signal. 779 Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar,

7 2. Irrespective of the shape of input voltage, for constant current amplitude of 1A and duration of 100sec, the maximum attainable temperature is C. 3. Rise time for DC input signal is 38sec. While rise time for triangular and trapezoidal input signal is signal dependent. In figure (5b), it is 50 sec and 40 sec in figure (5c). 4. Rise time cannot be improved for DC input signal, whereas for triangular and trapezoidal input signal, rise time can be improved by increasing the slope of rate of increase in applied current amplitude. 2. PHASE TRANSFORMATION HYSTERESIS Duhem model given by equation (6), Modified Liang and Rogers s model by equations (9) and (10),are simulated in MATLAB. Simulation parameters for different phase transformation hysteresis models are as given in table (3). Table 3: Simulation parameters for phase transformation hysteresis model Model Parameters Value μ C Duhem model μ C σ C σ C Ms 54 0 C Modified Liang and Rogers model Mf 70 0 C As 90 0 C C Results for the input parameters for Duhem, Modified Liang and Rogers is discussed in this section as below. For Duhem model, the model is validated with the parameters of table (1) in [14]. The selected material is modeled and hence the respective parameters are listed in table (3). The equation (4) is simulated and the figure (6) shows the two Gaussian curves for increasing and decreasing functions. When these curves are integrated, the obtained martensite phase transformation is as shown in figure (7). Af Fig.6 Gaussian PDF as slope function Fig.7 Martensite phase transformation hysteresis based on Duhem model 780 Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar,

8 For modified Liang and Rogers model, the obtained austenite and martensite phase transformation hysteresis by simulating equation ( 9) and ( 10) are shown in figure ( 8) and ( 9) respectively. Obtained all the transformation temperatures are exact without and positive or negative shift. Fig.8 Austenite phase transformation hysteresis based on Liang and Rogers model Fig.9 Martensite phase transformation hysteresis based on Liang and Rogers model Table 4: Comparison of temperature for different phase transformation hysteresis models Parameters Modified Liang and Rogers Duhem model model Ms 50 0 C 54 0 C Phase Transformation Mf 75 0 C 70 0 C Temperatures As 85 0 C 90 0 C Af C C Shift in temperatures Ms, As -5 0 C 0 0 C Mf,Af +5 0 C 0 0 C From above results and observations, hysteresis obtained from Duhem shows ±5% temperature shift than stated values in manufacturer datasheets, while that of modified Liang and Rogers are almost exact with no shift in temperature. CONCLUSION Shape memory alloys have been in limelight since its invention in 1954, due to its unique property of shape memory effect. This led to the temperature dependent behavior of the material. Hence in this paper, the temperature dependent behavior of the NiTi shape memory alloy of diameter 0.5mm is aimed and presented. The temperature dependent physical phenomenon i.e. heat transfer and phase transformation hysteresis are mathematically modeled and simulated in MATLAB. Hysteresis is modeled by two different models: Duhem, modified Liang and Rogers,. Amongst them, the heat transfer and Duhem model is validated with the literature. In heat transfer model, the model is excited with three different current signals: DC, triangular and trapezoidal. Obtained temperature profile followed the shape of input signal. Rise time for each signal was 38 sec, 50 sec and 40 sec respectively. Rise time can be reduced by further increasing 781 Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar,

9 the slope of the signal. Further, all the two hysteresis models are compared with each other. For Duhem model, there is a temperature shift of ± 5 0 C, whereas for modified Liang and Rogers s model shows no shift of temperatures. REFERENCES [1] D. W. Raboud, M. G. Faulkner and A. W. Lipsett. Superelastic response of NiTi shape memory alloy wires for orthodontic applications. Smart materials and structures, vol.9, no.5, pp. 684, 2000 Oct;9(5):684. [2] Y. Haga, M. Mizushima, T. Matsunaga andm. Esashi. Medical and welfare applications of shape memory alloy microcoil actuators. Smart materials and structures, vol. 14, no. 5, pp. S226, Aug 24;14(5):S266. [3] J. Van Humbeeck. Non-medical applications of shape memory alloys. Materials Science and Engineering: A Dec 15;273: [4] Y. Furuya and H. Shimada. Shape memory actuators for robotic applications. Materials & Design Feb 1;12(1):21-8. [5] M. Ho andj. P. Desai. Characterization of SMA actuator for applications in robotic neurosurgery. InEngineering in Medicine and Biology Society, EMBC Annual International Conference of the IEEE 2009 Sep 3 (pp ). IEEE. [6] E. Patoor, D. C. Lagoudas, P. B. Entchev, L. C. Brinson andx. Gao. Shape memory alloys, Part I: General properties and modeling of single crystals. Mechanics of materials Jun 30;38(5): [7] Dynalloy Inc., Flexinol Wire specifications < [accessed ]. [8] S. M. Dutta and F. H. Ghorbel. Differential hysteresis modeling of a shape memory alloy wire actuator. IEEE/ASME Transactions on Mechatronics Apr;10(2): [9] Li, Junfeng andh. Harada. Modeling of an SMA actuator based on the Liang and Rogers model. International Journal of Applied Electromagnetics and Mechanics, vol.43, no. 4, pp , [10] D. Madill and D. Wang, Modeling and L2-Stability of a Shape Memory Alloy Position Control System, IEEE Transactions on Control Systems Technology, vol. 6, no. 4, pp , [11] R. Romano, E. A.Tannuri. Modeling, control and experimental validation of a novel actuator based on shape memory alloys. Mechatronics, vol. 19, no. 7, pp , [12] P. L.Potapovand E. P. Da Silva. "Time response of shape memory alloy actuators." Journal of intelligent material systems and structures, vol. 11, no. 2, pp , Sweta A. Javalekar, Prof. Vaibhav B. Vaijapurkar,