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Experimental investigation and modeling of the loading rate and temperature dependent behavior of a high performance SMA

Emre Acar1,3, Osman E. Ozbulut2 and Haluk E. Karaca1

1Department of Mechanical Engineering, University of Kentucky, Lexington KY 40506 USA

2Department of Civil and Environmental Engineering, University of Virginia, Charlottesville,

VA 22901 USA 3Department of Aircraft Engineering, Erciyes University, Melikgazi, Kayseri, 38039 Turkey

E-mail: ozbulut@virginia.edu

Abstract

This study explores the superelastic behavior of a recently developed Ni45.3Ti29.7Hf20Pd5 alloy

that has favorable mechanical properties (high strength and hysteresis) over the well-known

shape memory alloys. The effects of aging on shape memory properties of Ni45.3Ti29.7Hf20Pd5

polycrystalline alloys are revealed first. Next, the dependence of the superelastic response of an

aged Ni45.3Ti29.7Hf20Pd5 alloys on the strain amplitude, loading rate and test temperature are

examined via uniaxial compression tests. Then, the superelastic response of a solutionized

sample is compared with that of the aged sample. Finally, a soft computing approach that

employs neural networks and fuzzy logic is used to model the highly nonlinear behavior of

Ni45.3Ti29.7Hf20Pd5 alloys by considering the loading rate and temperature effects. The tests

results show that solutionized sample has wider stress hysteresis, larger energy dissipation and

the equivalent viscous damping than the aged sample. It is found that loading rate does not

significantly influence the superelastic behavior of NiTiHfPd. In addition, an increase in

temperature shifts the hysteresis loops upward but result in no considerable change in damping

characteristics.

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Keywords: Shape memory alloys, Superelasticity, High strength, Fuzzy model, loading rate,

Temperature

1. Introduction

Smart materials can change their properties in response to externally applied stimuli such as applied stress, magnetic or electric field, or change in temperature [1]. These unusual materials include, but are not limited to, shape memory alloys (SMAs), piezoelectric materials, magnetostrictive materials, magnetorheological and electrorheological materials [2]. SMAs are metallic materials that can remember their original shapes. Two unique properties of SMAs are shape memory effect and superelasticity [1]. These distinct features are due to reversible thermo-elastic martensitic transformations [3]. These transformations are induced by a change in temperature in case of shape memory effect whereas triggered by a change in stress in superelasticity.

Superelasticity can be defined as the ability to recover nonlinear strains reaching up to a certain limit depending on the material type. When a superelastic material is strained at a temperature above its austenite finish (Af) temperature, it starts transforming to martensite phase (stress induced martensite) at a critical stress level. Austenite fully transforms to martensite as the transformation continues if sufficient external stress is applied. Upon unloading, the stress induced martensite transforms back to austenite and the material returns to its original shape.

Shape memory alloys are promising candidates for reversible solid-state actuators due to their ability to recover large shape changes against high stresses resulting in high work-output values. SMAs can be exploited in aerospace and civil engineering industries for energy dissipation purposes such as for dampening the acoustic energy and vibration from aircraft engines [4] and for absorbing seismic energy in civil structures [5-7]. Energy dissipation stems from phase transformations (either austenite to martensite or martensite to austenite) and energy is dissipated by release of heat energy. In addition to the abovementioned industries, the other potential and current application areas for such alloys include electronic devices, medical tools and home appliances [8]. Besides their superior mechanical properties, SMAs are lightweight, frictionless, quiet, environment-friendly (no hydraulic liquids), and require little maintenance [9-11] compared to other systems. Thus, SMAs have a great potential to be used in many engineering applications. However, the stability of SMAs is crucial since they are usually operated under cycling forces or temperatures in real applications [12]. Thus, it is important to investigate the influence of loading frequency and temperature on the superelastic behavior of SMAs.

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A large number of alloys have been investigated for shape memory behavior. Compared to Fe-based and Cu-based SMAs, NiTi alloys have received the most interest because of their excellent physical and mechanical properties (e.g. good fatigue life, oxidation and corrosion resistance, high ductility and strain) [1, 13]. However, NiTi alloys have some drawbacks such as low transformation temperatures (TTs) (< 100oC) and low strength [1, 14]. The use of special pre-operation processing such as thermo-mechanical treatments and precipitation strengthening can enhance the material properties [13]. In addition to these two pre-operation methods, alloying could also be used to improve the shape memory and mechanical properties (e.g. TTs, strain and strength) [13, 15, 16]. The authors have actively used this approach to investigate the effects of Pd addition on the mechanical and shape memory properties of NiTiHf-based materials. Previously, polycrystalline [17-19] and single crystal [20-23] NiTiHfPd shape memory alloys were systematically studied. Ni45.3Ti29.7Hf20Pd5 polycrystalline alloys show transformation strain up to about 4% in constant stress thermal cycling experiments at 1000 MPa [17]. An aged [111]-oriented Ni45.3Ti29.7Hf20Pd5 single crystals showed about 2% strain at constant stress levels as high as 1500 MPa [22] while a solutionized [111]-oriented single crystals showed superelastic behavior under extremely high stress levels (2.5 GPa) with negligible plastic deformation resulting in a very wide mechanical hysteresis (>1200 MPa) [23] and consequently huge damping capacity (44 J.cm-3). The orientation dependence of the shape memory properties in Ni45.3Ti29.7Hf20Pd5 alloys were also recently revealed [20].

This paper presents a systematical investigation on the effects of strain amplitude, loading frequency, and temperature on the compressive superelastic behavior of a polycrystalline Ni45.3Ti29.7Hf20Pd5 shape memory alloy. Besides, the effects of aging on the phase transformation characteristics and the superelastic response are explored. The compressive tests are conducted at various strain amplitudes at temperatures from 23 to 90 and loading frequencies from 0.05 Hz to 2 Hz. The dependence of mechanical properties on strain amplitude, loading rate, and temperature is discussed. In addition to the experimental investigations, a soft-computing approach is used to model the rate- and temperature- dependent superelastic behavior of NiTiHfPd alloys.

2. Materials and experimental procedure

Fabrication of the Ni45.3Ti29.7Hf20Pd5 (at. %) shape memory alloy into a 25 mm in diameter by about 100 mm long ingot was accomplished by induction melting. The ingot was homogenized at 1050 for 72 hours followed by extrusion at 900°C with an area reduction ratio of about 7:1. Polycrystalline samples were electro discharge machined (EDM) from the extruded rod into 4mm×4mm×8mm rectangular prism shaped compression samples. Extensive uniaxial

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compression tests were conducted on a sample heat treated at 400 C° for an hour to investigate the dependence of superelastic response on strain amplitude, loading rate, and temperature. This sample is named as “aged NiTiHfPd” in the remainder of manuscript. Mechanical tests of the alloys were performed using an MTS Landmark servo hydraulic test frame with a capacity of 100 kN in compression. The samples were trained to have stable behavior by conducting 10 superelastic cycles with the frequency of 0.05Hz at 23 C° . Loading and unloading were performed separately with the frequencies of 0.05 Hz, 0.1 Hz, 0.5 Hz, 1 Hz, 1.5 Hz and 2 Hz at temperatures of 23 , 50 , 70 and 90 by using the same sample. The measured strain amplitudes were 1.8%, 2.7%, and 3.6% in the tests. The loading frequencies and strain amplitudes considered here corresponds to strain rates ranging from 0.0018 s-1 to 0.144 s-1.

An MTS high temperature extensometer with gage length of 12 mm was used to measure strains. The ceramic legs of the extensometer were attached to the compression grip faces. The samples were heated by mica band heaters embedded in the compression grips. The temperatures of the specimens were measured using K-type thermocouples attached directly to the samples while two additional thermocouples were attached to the top and bottom grips to control the thermal gradients during mechanical testing.

Another sample was solutionized at 1050°C for 4 h (in a quartz tube under vacuum to avoid oxidation) using a Lindberg/Blue M BF5114841 furnace followed by fast quenching in water to explore the effect of aging on superelastic response. This sample is named as “solutionized NiTiHfPd”. The superelastic tests were conducted at room temperature under two different loading frequencies (0.05 Hz and 2 Hz) and various strain amplitudes.

In addition to the mechanical tests, phase transformation temperatures were determined on NiTiHfPd samples aged at temperatures ranging from 400°C to 900°C for an hour by using a Perkin-Elmer Pyris 1 Differential Scanning Calorimeter (DSC) with a heating-cooling rate of 10

/min.

3. Experimental results

3.1. Effects of aging on the phase transformation temperatures

Figure 1 shows the DSC responses of the NiTiHfPd specimens aged for 1 hour at selected temperatures from 400 to 900 . To compare the TTs and determine the effect of aging temperature, the result of the as-extruded (without aging treatments) is added to the plot. In each thermal cycle, the peaks at higher temperatures indicate the endothermic transformation (martensite to austenite (reverse) transformation) while the peaks at lower temperatures show the exothermic reaction in the austenite to martensite (forward) transformation. Some of the thermal

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cycles are conducted several times to observe the effects of cycling on the TTs change (stability) of NiTiHfPd alloys.

Figure 1. DSC responses for the NiTiHfPd alloy after 1 hour as a function of aging temperature. Figure 1b shows the change in A peak, which is the middle point of Austenite start (As) and

Austenite finish (Af) temperatures, and M peak, which is the middle point of Martensite start (Ms) and Martensite finish (Mf) temperatures. The maximum and the minimum A peak is 97 and 0

, respectively while the maximum and the minimum M peak is 47 and -67 , respectively. The peak temperatures in Figure 1b are calculated from the second DSC curves at each aging

temperature in Figure 1a. An initial drop observed in the peak temperatures (Figure 1b) can possibly be linked to formation of very small size and inter particle distance precipitates in the microstructure of NiTiHfPd alloy. Since the interparticle distance is short, martensite needs extra energy (further undercooling in the course of forward transformation) to nucleate and, consequently the TTs are decreased [24]. After the initial decrease, the TTs increase as the aging temperature increases up to 700 due to the change in the composition of the matrix owing to the formation of precipitates. It is known that as the Ni-rich precipitates increase in a matrix, the TTs increase in NiTi-based alloys [25, 26]. The second decrease after aging at 700 can be attributed to a reduction in precipitate volume fraction with the aging temperature.

In addition to the TTs, thermal hysteresis behavior is also shown in Figure 1b, which is defined as the difference of M peak and A peak. The temperature hysteresis is observed to increase with increasing aging temperature up to 450 followed by a decrease for the aging temperatures reaching up to 700 . A second increase is then observed for the aging

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temperatures of 800 and 900 . The minimum temperature hysteresis is 49 for the aging temperature of 700 while the maximum hysteresis is 68 in the specimen aged at 450 .

3.2. Effect of loading rate on the superelastic response

SMAs will be subjected to high loading rates if they are used in dynamic applications. In this subsection, the effects of loading rate on the material behavior are examined. Figure 2 illustrates the compression stress-strain curves of aged NiTiHfPd under the selected loading frequencies at 3.6% strain amplitude and at selected temperatures. The corresponding loading rates are ranging from 0.0036 s-1 to 0.144 s-1. It can be seen that an increase in loading rate does not affect the forward transformation path whereas it somewhat increases the reverse transformation stresses. At higher loading rates, the latent heat released during loading did not have enough time to transfer out completely. This leads to a higher temperature in the specimen at the start of unloading, and, in return, increases the reverse transformation stresses. Therefore, the unloading path moves upward and the hysteresis loop narrows as the frequency is increased. The same effect of the loading rate on the material response is observed at all test temperatures. To enable the evaluation of the test results in a quantitative way, the energy dissipated per cycle (ED), the equivalent viscous damping (ζeq ), and the secant stiffness (Ks) are calculated. The equivalent

viscous damping and secant stiffness are defined as:

ζeq =ΔW4πW

(1)

Ks =Fmax − Fmindmax − dmin

(2)

where ΔW is the energy dissipated per cycle (hysteresis area), W is the maximum strain energy calculated for the same cycle, Fmax and Fmin are the maximum and minimum forces attained for the maximum and minimum cyclic displacements dmax and dmin. Figure 3 shows the variation of ED, ζeq , and Ks as a function of loading frequency at different temperatures. An increasing

loading rate decreases the dissipated energy by 29% to 33% for the selected temperatures. A similar decrease is observed for equivalent viscous damping. However, the secant stiffness appears not to be affected by loading rate. There is an increase of only 2% in the secant stiffness when the loading frequency increases from 0.05 Hz to 2 Hz.

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(a) (b)

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Figure 2. Experimental stress-strain curves of the aged NiTiHfPd at various loading frequencies and at (a) 23 , (b) 50 , (c) 70 and (d) 90 .

3.3. Effects of temperature on the superelastic response

The stress-stra change in the in curves of the aged NiTiHfPd at 3.6% strain and at loading frequencies of 0.05 Hz and 2 Hz for various temperatures are shown in Figure 4. Similar to other types of SMAs, an increase in temperature shifts both the forward and reverse transformation stresses to higher stress values. The influence of the temperature on superelastic response at slow and high loading rates is observed to be comparable.

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Figure 3. Dissipated energy, equivalent viscous damping, and secant stiffness of the aged NiTiHfPd as a function of loading frequency at selected temperatures.

The variations of the dissipated energy, equivalent viscous damping, and secant stiffness as a

function of temperature are given in Figure 5. For both ED and ζeq , the observed general trend is

a reduction in the range of 10% with increasing temperature. Note that the reduction in the ED and ζeq with the increasing loading rate is in the range of 30% as discussed above. The

increasing temperature moves both forward and reverse transformation stresses upward, whereas an increase in loading rate only shifts the reverse transformation stresses upward while the forward transformation stresses remain almost unchanged. That is the main reason for larger decrease in the ED and ζeqwith increasing loading rates. However, the effect of temperature on

the secant stiffness is slightly more pronounced as compared to the influence of loading rate. As the test temperature increases, the secant stiffness increases about 5% at different loading rates.

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Figure 4. Experimental stress-strain curves of the aged NiTiHfPd at various temperatures at (a) 0.05 Hz and (b) 2 Hz

Figure 5. Dissipated energy, equivalent viscous damping, and secant stiffness as a function of temperature at 0.01 Hz and 2 Hz.

Figure 6 shows the critical stress for austenite to martensitic transformation values of the aged NiTiHfPd at 0.05 and 2 Hz loading frequencies as a function of test temperature. The critical stress values were measured by the commonly used intersection of the lines method. As expected, the critical stress increases as the test temperature increases, which follows the Clausius-Clapeyron (C-C) relationship [27];

ΔσΔT

= −ΔHToεtr

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where ∆σ, ∆T and ∆H are the differences in crtical stresses, test temperatures and transformation

enthalpies, respectively, To is the equilibrium temperature and εtr is the transformation strain.

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Figure 6. Critical stresses for austenite to martensitic transformation of the aged NiTiHfPd polycrystalline alloys tested at 0.05 and 2 Hz loading frequencies.

It can be also seen that critical stress increases with the increasing loading frequency, which is also observed for NiTi alloys previously [28]. In particular, the critical stresses are 435 and 490 MPa for the loading frequencies of 0.05 Hz and 2 Hz, respectively, at 23 . As the tests temperature increases to 90 , the critical stresses increase to 645 and 695 MPa for 0.05 Hz and

2 Hz, respectively. The C-C slopes for the aged NiTiHfPd are 3.1 and 3 for the 0.05 Hz and 2 Hz, respectively. Note that the C-C slopes for Ni50Ti50 polycrystalline alloys

were reported to be 5 in tension [29] and 12 in compression [30] at quasi-static loading rates. That indicates stress-strain behavior of NiTiHfPd alloys is less temperature dependent as compared to that of NiTi.

3.4. Effects of strain amplitude on the superelastic response

Figure 7 shows the stress-strain curves of the aged NiTiHfPd as a function of strain amplitudes at 0.05 Hz and 2 Hz. It can be seen that initial slope of stress-strain diagram during loading slightly decreases after about 1% strain as the forward phase transformation starts; however, the material does not exhibit significant softening beyond this point, which is typically observed in NiTi SMAs. As can be seen from Figure 8, the equivalent viscous damping is about 2% and almost constant for various strain amplitudes at 0.05 Hz loading frequency. For higher loading frequency of 2 Hz, the equivalent viscous damping attains its lowest value at 1.8% strain and then is about 1.5% for higher strain amplitudes.

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Figure 7. Experimental stress-strain curves of the aged NiTiHfPd at various strain amplitudes at (a) 0.05 Hz and (b) 2 Hz

Figure 8. Dissipated energy, equivalent viscous damping, and secant stiffness of the aged NiTiHfPd as a function of strain amplitude at 0.01 Hz and 2 Hz.

3.5. Effects of aging on the superelastic response

In order to investigate the effects of aging on the superelastic response of the NiTiHfPd SMAs, the stress-strain curves of the solutionized NiTiHfPd are shown in Figure 9 for selected strain amplitudes at 0.05 Hz and 2 Hz. The limits of the axes are kept the same with those of Figure 7, which shows the response of aged NiTiHfPd SMA, to allow easier comparison. Furthermore, Figure 10 provides the variation of the dissipated energy, equivalent viscous damping, and secant stiffness with the strain amplitude for the solutionized NiTiHfPd tested at 0.05 Hz and 2 Hz. From Figures 7 and 9, it can be seen that the hysteresis loops for the solutionized material are wider compared to the aged specimen. They also exhibit a flatter stress-plateau during both forward and reverse phase transformations, which can be attributed to the

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effects of precipitates on martensite morphology and transformation characteristics [17-19]. For slow loading frequency of 0.05 Hz, both the energy dissipation and equivalent viscous damping significantly increase at large strain amplitudes. In particular, the energy dissipation and the

equivalent viscous damping are 14.1 J cm-3 and 4.7% for the solutionized NiTiHfPd that is tested

up to 3.4% strain. The same parameters are 7.8 J cm-3 and 2.0%, respectively, for the aged NiTiHfPd that is tested up to 3.6% strain.

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Figure 9. Experimental stress-strain curves of the solutionized NiTiHfPd at various strain amplitudes at (a) 0.05 Hz and (b) 2 Hz

Figure 10. Dissipated energy, equivalent viscous damping, and secant stiffness of the solutionized as a function of strain amplitude at 0.01 Hz and 2 Hz.

For a higher loading frequency of 2 Hz, the energy dissipation is 5.4 J cm-3 for the aged

NiTiHfPd whereas it is 11.9 J cm-3 for the solutionized NiTiHfPd. The corresponding equivalent viscous damping ratios for both materials are 1.4% and 3.3%, respectively. Compared to the

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aged NiTiHfPd, there is an important increase in the energy dissipation and in the equivalent viscous damping ratio for the solutionized NiTiHfPd. It can be also seen that the secant stiffness decrease rapidly with the increasing strain amplitude when the material is solutionized, which indicates the softening of the upper phase transformation plateau. From Figures 7 and 8, it can be seen that the phase transformation plateau is stepper and the secant stiffness does not considerably vary with strain amplitude for the aged NiTiHfPd.

4. Modeling behavior of NiTiHfPd SMAs

A simple and reliable model that describes the complex nonlinear behavior of the NiTiHfPd SMAs is needed to explore their potential applications. Several models were developed to predict the behavior of SMAs by using either microscopic [31, 32] or macroscopic [33-35] approaches. Macroscopic models employ phenomology to capture the SMA response at the macroscopic level. Some of these models rely heavily on the thermodynamic principles, while others are developed by setting material constants of a model to match experimental data. It was also shown that neural networks and fuzzy systems can be used as an alternative modeling approach at macroscopic level rather than the use of complex differential equations to describe the behavior of the SMAs [36]. Due to its simplicity and relative accuracy, a neuro-fuzzy modeling approach is used in this study to capture response of NiTiHfPd SMAs considering temperature and rate effects.

An adaptive neuro fuzzy inference system (ANFIS) is an artificial intelligence system that integrates fuzzy logic and neural network principles [37]. Fuzzy logic can encode expert knowledge of its application domain and uses a reasoning procedure to map a set of inputs to outputs without employing any quantitative analyses. Neural networks enable tuning of fuzzy parameters from input output data pairs and can significantly reduce the model development time while improving the performance. ANFIS uses learning ability of neural networks to develop a fuzzy inference system (FIS) whose parameters (membership functions and fuzzy rules) cannot be predetermined by user’s knowledge. More information on the ANFIS architecture and its learning algorithm can be found in [38]. Figure 11 shows the flowchart of building a system model using ANFIS. Each of these steps is described for developing a model of NiTiHfPd SMAs in the following sections.

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Figure 11. Flowchart of ANFIS modeling

4.1. Generating Initial FIS

To model the mechanical response of NiTiHfPd using ANFIS, a FIS needs to be created first. This initial FIS has random parameters and no knowledge about the behavior of NiTiHfPd SMAs. Here, a Sugeno-type FIS, which employs weighted average to compute the crisp output, is developed. A typical rule in a Sugeno fuzzy model has the form:

Rule i: IF X1 is λ1 and X2 is λ2 …and Xn is λn

THEN Y = µ0 + µ1 X1 + µ2 X2 + … + µn Xn

where X1, X2, …, Xn are input variables; Y is the output variable; λ1, λ2, …, and λn are the fuzzy sets defined over the domains of the respective inputs; and µ0, µ1, …, µn are the constant coefficients that characterize the linear relationship of the ith rule in the rule set (i = 1, 2, … , r). The strain, strain rate and temperature are selected as input variables for the FIS. The stress on the material is used as the only output variable.

Three different partitioning methods are used to create a Sugeno-type FIS as initial fuzzy system. Those methods are: (i) grid partition (corresponding fuzzy inference system named as

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FIS1), (ii) subtractive clustering (FIS2), and (ii) fuzzy c-means clustering (FIS3). In grid partition, the data space is divided into rectangular subspaces called “grids” based on the number of membership functions and their types. When a grid partition on the data is used, the number and type of input and output membership functions are explicitly specified. After several trials, three Gaussian membership functions are assigned to strain and two Gaussian membership functions are chosen for both strain rate and temperature for the FIS1. The Gaussian membership function is defined as:

λ(xi ) = exp−(x − c)2

2σ 2

"

#$

%

&' (4)

where c and σ are the parameters that control the shape of the membership function, and X is the input parameter. For the chosen number of membership functions, a total of 12 if-then fuzzy rules are employed to map input characteristics to a single-valued output. The number of membership functions assigned to the single output stress is the same as the number of rules. The output membership function type is selected to be linear.

Subtractive clustering [39] and fuzzy c-means clustering [40] are the most commonly used fuzzy clustering techniques to recognize and classify the similar patterns from a large data set into several groups. Fuzzy c-means clustering needs a priori knowledge of the number of clusters and initial guess positions for each cluster center. Subtractive clustering employs a search technique to determine the number and initial location of the cluster centers. In subtractive clustering, each data point is considered as a potential cluster center. Then, based on the density of surrounding data points, the likelihood of each selected point to define the cluster center is determined. Here, FIS2 and FIS3 are created using subtractive clustering and fuzzy c-means clustering on the data, respectively. For both fuzzy inference systems, the input and output membership function types are selected to be Gaussian and linear, respectively. When data clustering is used, membership functions and rules are automatically generated from the data. Subtractive clustering and fuzzy c-means clustering resulted in a total of 9 membership functions for each input variables in FIS2 and FIS3, respectively. Data clustering allows the development of a much simpler neuro-fuzzy model by extracting a set of rules that model the data behavior even when a relatively large number of membership functions is used to fuzzify input variables. For example, FIS2 and FIS3 employ only 9 fuzzy rules after the data is clustered. However, using 9 membership functions for each input variable with grid partition would produce up to 729 fuzzy rules.

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4.2. Data Selection and Training of Initial FIS

ANFIS modifies the membership functions and rules of the generic FIS during a procedure called training in order to predict correct output for given inputs. ANFIS uses a hybrid optimization method, a combination of least squares and back propagation gradient descent method, to learn information about the data set and build a model that approximates the real data. During ANFIS simulation, training data set is needed to tune parameters of initial FIS and validation data set is used to validate the final FIS. In addition to training and validation data sets, another set of data, called checking data, are employed to avoid over fitting of model during training. Data obtained from the above-described experimental tests on the aged NiTiHfPd specimen is concatenated to form training, checking and validation data sets for ANFIS simulations. Upon concatenation of these data sets, every third data is used for training, and the rest similarly is saved as checking and validation data sets. As the tests are conducted using a linear ramp loading and unloading, the strain rate data is calculated from the loading frequency and strain amplitude for each test. Each of training, checking and validation data sets is composed of 17,806 data points. A total of 200 training epochs is used to adjust parameters of the initial FIS.

To assess the performance of the models developed using different input space partitioning methods, the root mean squared error (RMSE) and the Pearson correlation coefficient (R) is calculated for each fuzzy inference system as follows:

RMSE =ti − yi( )

2

i=1

N

∑N

(5)

R2 =1−ti − yi( )

2

i=1

N

∑

ti − tave( )2

i=1

N

∑ (6)

Figure 12 illustrates the variation of RMSE as the number epochs are increased for fuzzy models generated using different partitioning techniques. In addition, Table 1 provides the RMSE and R values for the developed final fuzzy inference systems. For all fuzzy models, the correlation between experimental and predicted stresses is found to be high. However, the fuzzy inference system developed using fuzzy c-means clustering has the lowest training error. Therefore, it has been selected for modeling the response of NiTiHfPd SMAs.

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Figure 12. Training error versus epochs curves for different fuzzy systems

Table 1. Comparison of RMSE and R values for different fuzzy systems

RMSE (MPa) R FIS 1 – Grid Partition 33.01 0.9957 FIS 2 – Subtractive Clustering 36.52 0.9948 FIS 3 – Fuzzy C-Means Clustering 28.90 0.9967

The plots in Figure 13 show the inputs of the developed FIS together with the experimental

and predicted stress. Note that two subplots provide the zoom of the selected regions on the output stress data to better evaluate performance of the fuzzy model. It can be seen that ANFIS prediction of stress closely follows experimental training data.

0 50 100 150 20020

25

30

35

40

45

50

Epoch

RM

SE (M

Pa)

FIS1 − Grid PartitionFIS2 − Subtractive ClusteringFIS3 − Fuzzy C−Means Clustering

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Figure 13. Experimental input data and, measured and predicted stress

4.3. Validating Optimized FIS

Model validation is the process of presenting a data set that is not used during training of the developed FIS to determine the accuracy of the trained FIS. Figure 14 illustrates several hysteresis loops of NiTiHfPd SMAs for both experimental tests and fuzzy model at different temperatures, loading rates and strain amplitudes. In particular, experimental test results obtained for (i) a loading frequency of 1.5 Hz at 23ºC and 3.6% maximum strain, (ii) a loading frequency of 0.5 Hz at 50ºC and 1.8% maximum strain, (iii) a loading frequency of 2 Hz at 70ºC and 3.6% maximum strain, and (iv) a loading frequency of 1 Hz at 90ºC and 2.7% maximum strain are compared with the results of fuzzy model. It is clear that neuro-fuzzy model of NiTiHfPd successfully reproduces the experimental hysteresis loops at each condition. The developed model offers a viable approach to evaluation of NiTiHfPd SMAs for control or other applications.

3000 6000 9000 12000 15000 178000

1

2

3

Data Point

Stra

in(%

)Inputs of Fuzzy Model

3000 6000 9000 12000 15000 17800−5

0

5

Data Point

Stra

in ra

te (%

/s)

0 3000 6000 9000 12000 15000 1780020406080

Data Point

Tem

pera

ture

(ºC

)

0 3000 6000 9000 12000 15000 178000

500

1000

1500Stress Predicted by Fuzzy Model versus Experimental Data

Data Point

Stre

ss (M

Pa)

3000 3500 4000 4500 5000 5500 60000

500

1000

1500

Data Point

Stre

ss (M

Pa)

3800 4000 4200 4400 46000

500

1000

1500

Data Point

Stre

ss (M

Pa)

Training DataANFIS Prediction

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Figure 14. Stress-strain curves at various conditions for experimental results and ANFIS prediction.

5. Conclusions

This paper investigates the superelastic behavior of Ni45.3Ti29.7Hf20Pd5 as functions of strain amplitudes, loading frequencies and temperatures. Sytematical uniaxial compression tests are conducted at selected frequencies and temperatures. The energy dissipated per cycle, the equivalent viscous damping and the secant stiffness are calculated to quantitatively compare the test results. It is shown that increasing the loading rate or temperature decreases the equivalent viscous damping while slightly increases the secant stiffness. The effects of loading rate are more pronounced on the equivalent viscous damping as compared to the temperature variation. On the other hand, the secant stiffness increases more with the increasing temperature rather than the loading rate.

In addition, the effects of aging on the phase transformation temperatures and superelastic behavior of NiTiHfPd are explored. It is shown that aging the material for an hour at a desired temperature can easily alter the phase transformation temperatures. The M peak can be adjusted to be between -67C to 47 after suitable aging. It is also shown that the hysteresis loops of NiTiHfPd SMAs become wider when the material is solutionized. The solutionized NiTiHfPd also shows a flatter stress-plateau during superelasticity. That indicates the higher energy dissipation capacity and lower secant stiffness of the solutionized material.

Finally, a neuro-fuzzy modeling technique, which combines the knowledge illustrations of fuzzy logic systems and learning capabilities of neural networks, is employed to create a model

0 1 2 3 40

500

1000

1500

23°C − 1.5 Hz

Strain (%)

Com

pres

sive

Stre

ss (M

Pa)

0 1 2 3 40

500

1000

1500

50°C − 0.5 Hz

Strain (%)

Com

pres

sive

Stre

ss (M

Pa)

0 1 2 3 40

500

1000

1500

70°C − 2 Hz

Strain (%)

Com

pres

sive

Stre

ss (M

Pa)

0 1 2 3 40

500

1000

1500

Strain (%)C

ompr

essi

ve S

tress

(MPa

)

90°C − 1 Hz

ExperimentalFuzzy Model

20

to simulate the response of NiTiHfPd SMAs. Three different fuzzy inference systems are developed, each one of them based on different data partitioning method: (i) grid partition, (ii) subtractive clustering, and (iii) fuzzy c-means clustering. The results indicate that the developed fuzzy model is more effective when the fuzzy c-means clustering is applied to the experimental data prior to training. It is shown that the fuzzy model can successfully predict hysteresis loops of superelastic NiTiHfPd SMAs for given loading frequency and temperature. The developed model can be easily implemented into simulations to investigate the potential of NiTiHfPd SMAs for envisioned applications.

Acknowledgements

This work was supported by the NASA EPSCOR program under Grant NNX11AQ31A.

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