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A one-dimensional strain-rate-dependent constitutive model for superelastic shape memory

alloys

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2007 Smart Mater. Struct. 16 191

(http://iopscience.iop.org/0964-1726/16/1/023)

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INSTITUTE OF PHYSICS PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 16 (2007) 191–197 doi:10.1088/0964-1726/16/1/023

A one-dimensional strain-rate-dependentconstitutive model for superelastic shapememory alloysWenjie Ren1,2, Hongnan Li1 and Gangbing Song3

1 State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology,Dalian 116024, People’s Republic of China2 School of Civil Engineering, Hebei University of Technology, Tianjin 300132,People’s Republic of China3 Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4006,USA

Received 17 July 2006, in final form 13 October 2006Published 22 January 2007Online at stacks.iop.org/SMS/16/191

AbstractRecently, there is increasing interest in using superelastic shape memoryalloys (SMAs) in civil, mechanical and aerospace engineering, attributed totheir large recoverable strain range (up to 6–8%), high damping capacity, andexcellent fatigue property. In this research, an improved Graesser’s model isproposed to model the strain-rate-dependent hysteretic behavior ofsuperelastic SMA wires. Cyclic loading tests of superelastic SMA wires arefirst performed to determine their hysteresis properties. The effects of thestrain amplitude and the loading rate on the mechanical properties are studiedand formulated using the least-square method. Based on Graesser’s model,an improved model is developed. The improved model divides the full loopinto three parts: the loading branch, the unloading branch before thecompletion of the reverse transformation and the elastic unloading branchafter the completion of reverse transformation, where each part adopts itsrespective parameters. Numerical simulations are conducted using both theoriginal and the improved Graesser’s models. Comparisons indicate that theimproved Graesser’s model accurately reflects the hysteresis characteristicsand provides a better prediction of the SMAs’ actual hysteresis behavior thanthe original Graesser’s model at varying levels of strain and loading rate.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Shape memory alloys (SMAs) are unique alloys that havethe ability to undergo large deformations and return to theirundeformed shapes by heating (the shape memory effect) orthrough removal of the stress (the superelastic effect). Over thepast decades, advances in different engineering branches havecreated more and more opportunities for applications of SMAs,which makes it necessary to have a better understanding of thespecial properties of SMAs.

Modeling of the mechanical behavior of SMAs has beenan active area of research. It is difficult to establish aconstitutive model which is appropriate for the design of SMAdevices due to the complexity of the material behavior. Some

approaches have been proposed. Tanaka [1, 2] developed a setof explicit equations in terms of stress, strain, temperature anda set of exponential equations for evolution of kinematics phasefraction. Liang and Rogers [3] extended Tanaka’s model byusing a cosine function to describe the transformation kineticsinstead of the exponential function. Brinson [4] improvedthe model of Liang and Rogers so that it could represent theSMAs’ behavior over the full range of temperature. It hasbeen shown that the Tanaka-type models are more versatile andsimpler to use, but they do not take into account the effectof loading rate. Sun and Rajapakse [5] modified Brinson’smodel by assuming that the slopes of the forward and reversephase transformation lines in the phase transformation diagramwere strain-rate-dependent, however, the simulation results

0964-1726/07/010191+07$30.00 © 2007 IOP Publishing Ltd Printed in the UK 191

W Ren et al

Figure 1. Experimental setup for mechanical testing of SMA wire.

disagreed with the test results [6]. Other models, such as Boydand Lagoudas’s thermodynamics-based model [7], Nae et al’sgrain-based microscopic model [8], and Schmidt’s plasticitybased model with a focus on strain-rate dependence [9], areinconvenient for practical use due to the complex expressions.

Graesser [10] modified Ozdemir’s one-dimensional hys-teretic model to include the macroscopic characteristics ofSMAs. Graesser’s model not only has the advantages of arelatively simple formulation with parameters that can be eas-ily acquired, but also can approximately express the effect ofthe loading rate on hysteresis without considering the com-plex mechanism of phase transformation. However, Graesser’smodel uses identical parameters during both the loading andthe unloading branches, which results in an obvious differencebetween the prediction and the experimental results. It is nec-essary to improve Graesser’s model by using different parame-ters during the different loading branches.

Recently, there is increasing interest in using superelasticSMAs in civil, mechanical and aerospace engineering,attributed to their large recoverable strain range (up to 6–8%), high damping capacity, excellent fatigue property andstrain hardening. In this paper, one-dimensional cyclicloading tests of Nitinol superelastic shape memory alloy wiresare first performed at varying strain levels and strain rates.Then, the results are analyzed to evaluate the effects of thestrain level and the loading rate on the hysteresis behavior.Next, Graesser’s model is improved by using different modelparameters during the different loading branches. Finally,numerical simulations are conducted to demonstrate theaccuracy of the improved Graesser’s model.

2. Experimental setup of superelastic SMA wires

2.1. Test specimens and equipment

The test specimens are 0.8 mm diameter Ni–Ti SMAwires. A batch of wires are supplied by ShenzhenSuper-Line Technology Co., Ltd. The austenite finishtemperature (Af) is 0 ◦C, so the material showed superelaticityat room temperature. The tests are performed usingan electromechanical universal testing machine. The

σms

σasσaf

εaf

Figure 2. Schematic sketch of the superelastic hysteresis behavior ofSMAs.

experimental setup is shown in figure 1. Two speciallydesigned grips are used to hold the SMA wire specimen toprevent both slippage and stress concentration. SMA wiresamples, 500 mm in full length, are used for testing, witha 100 mm test length between the machine’s grips. Theaxial force is measured by a load cell located in the middlecrosshead, which can stand a 5.0 kN ultimate load. Theaxial displacement is monitored through the displacement ofthe middle crosshead. The elongation is measured by theextensometer with an initial gauge length of 25 mm. Themean strain, ε, and the stress, σ , applied on the specimen arecalculated from the measured elongation and axial force. Thedata are acquired automatically by a computer with 30 sampleseach second. The room temperature is 29 ◦C.

2.2. Test procedure

Prior to testing, each specimen has been cycled 30 times toreach a steady-state condition. Then a series of tests areconducted as follows:

(1) Cyclic loading tests at different strain levels, rangingfrom 1% to 6% by an increment of 0.5%, under a fixeddisplacement rate of 3 mm min−1.

(2) Cyclic loading tests at various displacement rates of 3, 15,35, 60 and 80 mm min−1, under a fixed strain amplitudeof 5.5%.

These experiments are all performed using displacementcontrol under an isothermal condition. The wires are pre-stressed between 20 and 30 MPa before testing to ensure a firmgrip of the wire by the extensometer.

In order to acquire reliable hysteresis behavior of SMA,three specimens are applied in each testing.

3. Experimental results and analysis

3.1. Definition of mechanical parameters

A typical superelastic SMA stress–strain curve is shown infigure 2. Before the stress reaches the forward transformationstress (σms), the SMA behaves elastically with a Young’smodulus (Da). Subsequently, the stress-induced martensitetransformation takes place and results in a large deformationwith little increase of stress. The inelastic modulus (Dy)

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A one-dimensional strain-rate-dependent constitutive model for superelastic shape memory alloys

Figure 3. Experimental hysteresis loops of SMA wires at variousstrain amplitudes.

is used to characterize this segment of the loading process.Upon release of the stress, the material unloads elastically withan elastic modulus (D). When the stress reaches a reversetransformation stress (σas), martensite begins to transform toaustenite with an inelastic modulus that has almost the samevalue as Dy . Once this transformation is completed, i.e. thestress reaches the austenite finish stress (σaf), there is a finalelastic unloading in its austenite phase. The area enclosed bythe loop represents the energy dissipated through the loading–unloading cycle.

3.2. Effects of the strain amplitude on the superelasticbehavior

The results of test series (1) are shown in figure 3. Thespindle shaped hysteresis loops are observed without residualdeformation, which shows that SMA is a good candidatefor a passive damper. Increase in the strain amplitude ledto an increase in the area of the hysteresis loop, and thusalso an increase in the energy dissipation or the dampingcapacity. The forward transformation stress (σms) almostremains unchanged with an increasing strain level, along withthe slope of the elastic loading branches (Da) and the inelasticmodulus (Dy). The experiments indicate that the elasticmodulus during unloading has a strong dependence on thestrain amplitude. With the strain level increasing, the slopeof the elastic unloading curve decreases, and Da is generallygreater than D, which may be the result of a larger martensitefraction at a higher level of strain.

Studies of the dependence of elastic modulus on strainlevel during unloading are not reported in the literature. Onlyan assumption for the unloading modulus function of SMAs,as suggested by Liang [11], is

D(ξ ) = Da + ξ(Dm − Da) (1)

where ξ is the martensite fraction of the material, which can becalculated according to the temperature and stress. However,Brinson [4] pointed out that equation (1) lacks adequateexperimental background and is only qualitatively correct forpartial hysteresis loops and thus should be expanded to accountfor return point memory for more accuracy. Therefore, toachieve a better accuracy, this paper tries to apply the least-square method to model the relationship between the slope ofthe elastic unloading curve and the maximum strain level based

Figure 4. Variation of the elastic modulus of SMA during theunloading at various strain amplitudes.

Figure 5. Experimental hysteresis loops of SMA wires at variousloading rates.

on experimental data, instead of the complex mechanism ofphase transformations. The fitted curve is shown as a solid linein figure 4, and the formulation is as follows:

D = 18 324 + 32 657 × exp(−ε/0.028) (2)

where D is the elastic modulus of SMA during unloading andε represents the one-dimensional strain level.

A correlation coefficient is found to be 0.990, which isadequate to demonstrate an observed trend for the unloadingmodulus versus the strain amplitude.

3.3. Effects of the loading rate on the superelastic behavior

The results of test series (2) are shown in figure 5. Sincethe testing machine is designed only for almost static loadingconditions, the data can only be obtained up to a maximumdisplacement rate of 80 mm min−1. Figure 5 shows that theforward transformation stress (σms) is almost insensitive to theloading rate. However, the reverse transformation stress (σas)increases with the increasing loading rate. At a lower loadingrate, the dissipated energy may initially increase, varying from5.7591 MJ m−3/cycle at 3 mm min−1 to 6.2088 MJ m−3/cycleat 15 mm min−1, but as the loading rate increases above15 mm min−1, the dissipated energy becomes obviously less,

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Figure 6. Variation of the inelastic modulus of SMA duringtransformation at various loading rates.

5.0457 MJ m−3/cycle at 80 mm min−1, which is about 88% ofthe energy dissipation at the low loading rate. The increasein the stress level may be the result of an increase in thetemperature in the specimen during a cyclic loading at a higherrate [12]. The self-heating of the specimen during a cyclecauses a decrease in the dissipated energy. Furthermore,figure 5 clearly indicates that, with an increasing loadingrate, the inelastic modulus (Dy) increases and the elasticunloading’s slope (D) decreases, which are the most distinctcharacteristics for rate-dependent responses of SMA wires.

In references [12–14], the rate-dependent characteristicsof SMA wires are studied in a large number of experiments,only emphasizing the experimental phenomena. In this paper,the inelastic modulus is taken as a measure of the ratedependence of the hysteresis behavior and its dependenceon loading rate is formulated. Figure 6 shows a solid linerepresenting a least-squares linear fit of the experimental data,which has the following expression:

Dy = 3673 − 3406 × exp(−v/12.46) (3)

where v is the displacement loading rate.A correlation coefficient is found to be 0.994, which

indicates the expression can well represent the relationshipbetween the inelastic modulus and the loading rate.

4. The improved Graesser’s model

4.1. An introduction to Graesser’s model

Based on Ozdemir’s model, Graesser proposed a one-dimensional hysteretic model that produced the generalmacroscopic stress–strain characteristics of SMAs. Theequations are given as

σ̇ = E ×[ε̇ − |ε̇|

(σ − β

Y

)n](4)

β = E × α × {εin + fT |ε|c erf (aε) [u (−εε̇)]

}(5)

where σ is the one-dimensional stress, ε is the one-dimensionalstrain, E is the elastic modulus, Y is the yield stress, β isthe one-dimensional backstress, n is a constant controlling

Figure 7. Comparison of SMA’s stress–strain curve betweenGraesser’s model and the experiment [10].

the sharpness of transition from the elastic state to the phasetransformation, σ̇ or ε̇ denotes the ordinary time derivative ofthe stress or strain, and α is a constant controlling the slope ofthe σ–ε curve, given by

α = Dy

(E − Dy)(6)

εin is the inelastic strain, given by

εin = ε − σ

E(7)

u( ) is the unit step function, defined as

u(x) ={

+1 x � 0

0 x < 0(8)

erf( ) is the error function, defined by

erf(x) = 2√π

∫ x

0e−t2

dt (9)

fT , a, and c are material constants. Other unnoted parametershave already been defined.

In fact, Graesser’s model only adds the term fT |ε|cerf(aε)

to Ozdemir’s model during the unloading process. During theunloading process, the unit step function activates the addedterm, and the error function causes the backstress to decreaseas zero strain is approached. By a proper choice of fT , a, andc, the inelastic strain can be fully recovered upon completionof the unloading to σ = 0, thus producing the superelastic typeof response [10].

Graesser’s model has a relatively simple expression withthe parameters that can be easily acquired and is easy toimplement; however, its modeling accuracy is not perfect.Figure 7 [10] shows a comparison of the model predictionand the experimental data, in which the dashed line is thepredicted result, and the solid line is the experimental result.It is clearly seen that the shape of the hysteresis loops basedon the model and the experiment do not match very well.The main reasons may be that the SMA material’s modulusdecreases with an increasing strain level. However, Graesser’smodel ignores the effect of loading path on the elastic modulus

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A one-dimensional strain-rate-dependent constitutive model for superelastic shape memory alloys

and uses the identical parameters during a cycle, which causesthe hysteresis loop predicted by the model to be fatter thanthe experimental loop. In addition, there exists an inherentdifference in the tensile and compressive material behavior ofSMA; however, Graesser’s model can only model a symmetricbehavior, which leads to the obvious discrepancy in thecompressive hysteresis shape between the experiment and themodel. In our experiments, only SMA wire’s tensile behavioris taken into account, so the model is improved consideringthe effects of loading path on the model parameters, but nottouching upon the symmetry problem. What’s more, it shouldbe noted that in the paper by Graesser [10], the model isbased on and compared with specimens that are not previouslystabilized, whereas the improved model is established on thebasis of the stabilized mechanical properties of SMAs.

4.2. The improved Graesser’s model

In order to more accurately simulate the hysteretic behavior ofsuperelastic SMAs, a full cycle is divided into three parts: theloading branch, the unloading branch before the completion ofthe reverse transformation, and the elastic unloading branchafter the completion of the reverse transformation, whichrespectively correspond to the OAB phase, BCE phase, andEO phase in figure 2. Each segment is determined according tothe values of SMA’s strain and its first derivative in a loadingcycle, and each part adopts its respective parameters. Since theprediction in the loading branch of Graesser’s model is nearlycoincident with the experimental result, the same characteristicparameters as Graesser’s model are applied in the OAB phase.Trial-and-error shows that, except for E , Y , n and fT , the BCEphase has identical parameters with Graesser’s model, and theparameters in Graesser’s model are similarly suitable for theEO phase. According to equations (4) and (5), the equationsare now rewritten as follows:

(1) If εε̇ > 0 (OAB phase):

σ̇ = Da ×[ε̇ − |ε̇|

(σ − β

σms

)n](10)

β = Da × α × εin. (11)

(2) If εε̇ < 0 and ε > εaf (BCE phase):

σ̇ = D ×[ε̇ − |ε̇|

(σ − β

Y

)n′](12)

β = D × α × {εin + fT ′ |ε|c erf (aε) [u (−εε̇)]

}(13)

where εaf is the strain when the reverse transformation iscompleted, and the yield stress Y is evaluated as

Y = σms × D

Da. (14)

(3) If εε̇ < 0 and ε < εaf (EO phase):

σ̇ = Da ×[ε̇ − |ε̇|

(σ − β

σms

)n](15)

β = Da × α × {εin + fT |ε|c erf (aε) [u (−εε̇)]

}. (16)

Other unnoted parameters have already been defined.

Figure 8. Comparison of SMA’s stress–strain curve betweencalculations and experiment.

The MATLAB program is compiled to calculate theimproved model.

It should be noted that the constants Da , D, σms, and α

are defined to match the elastic loading modulus, the elasticunloading modulus, the forward transformation stress, and theinelastic modulus index, respectively. The remaining constantsare chosen to fit the model response to the experimentalhysteresis loop of SMA wires. The improved Graesser’s modelaccounts for the effects of the strain level and the loading rateon the hysteresis behavior of SMA wires. As previously noted,the unloading modulus and the inelastic modulus are regardedas measures of the strain dependence and rate dependence ofthe tensile behavior of SMA wires, respectively. Therefore, ata fixed loading rate, the modulus during an unloading processis calculated according to equation (2), and at a fixed strainlevel, the inelastic modulus during a transformation is chosenon the basis of equation (3). For the sake of convenience forpractical use, in modeling the hysteresis curve of SMA at eachloading rate, we take the mean value of the unloading moduliunder diverse strain levels as the constant D.

5. Numerical simulations

In this section, Graesser’s model and its improved version areutilized to simulate the hysteretic responses of the SMA wiresfor comparison purposes. Comparisons of predictions usingthese two models and the experimental results are conductedfor situations with different peak strains and loading rates. Theexperimental studies used for the comparisons are from thesame tests used to develop equations (2) and (3).

5.1. Comparisons of SMA’s stress–strain curves: modelpredictions versus experiments

Figure 8 shows the comparison of the model predictionsand the experimental responses of the SMA wire at 4.5%strain with a loading rate of 3 mm min−1. The characteristicparameters used in the models are: σms = 477.07 MPa,Da = 49 840 MPa, D = 24 107 MPa, α = 0.0205, a = 158,c = 0.001, n = 5, n′ = 3, fT = 0.8, and fT ′ = 1.18.

Figure 8 reveals that the hysteresis loops based on theimproved Graesser’s model and experimental data do match

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Table 1. Comparisons of the model predictions and the experiments for varying peak strains.

Peak From the Predicted by Predicted by the Difference between Difference betweenstrain experiments Graesser’s model improved model Graesser’s model and the improved model(%) (MJ m−3/cycle) (MJ m−3/cycle) (MJ m−3/cycle) the experiments (%) and the experiments (%)

3.5 2.9025 3.1950 2.9015 10.08 −0.0354.5 4.3633 4.5262 4.2829 3.73 −1.845.5 5.7591 5.8568 5.7166 1.7 −0.74

Table 2. Comparisons of the model predictions and the experiments for varying loading rates.

Loading From the Predicted by Predicted by the Difference between Difference betweenrate experiments Graesser’s model improved model Graesser’s model and the improved model(mm min−1) (MJ m−3/cycle) (MJ m−3/cycle) (MJ m−3/cycle) the experiments (%) and the experiments (%)

3 5.7591 5.8568 5.7166 1.7 −0.7415 6.2088 7.0937 6.4266 14.3 3.5135 5.4235 6.1991 5.6210 14.3 3.6460 5.1541 5.9118 5.3618 14.7 4.03

with close accuracy. Compared with Graesser’s model, theimproved model can accurately reflect the characteristicsof the mechanical behavior of the SMA, for example, theforward transformation stress, the reverse transformationstress, the elastic loading’s modulus, the inelastic modulus,and the unloading modulus. The improved model showsmore desirable predictions coincident with the experimentalresponse during loading and unloading.

To make an additional quantitative observation regardingthe shape of the analytical and the experimental hysteresisloops, we compare the area enclosed within each loopbetween the predictions and the experimental response.The area enclosed by the experimental hysteresis loop is4.3633 MJ m−3/cycle. The energy dissipation per cyclepredicted by Graesser’s model is 4.5262 MJ m−3/cycle, beingmore than the experimental result by 3.73%. The energydissipation is predicted by the improved Graesser’s model upto 4.2829 MJ m−3/cycle, which is only 1.84% less than theexperimental value. Therefore, the results indicate that theimproved Graesser’s model gives a better prediction of theactual energy absorbed.

5.2. Comparisons of the model predictions and theexperimental data for varying peak strains

Let us next proceed to compare the model predictions withthe experimental data at various strain amplitudes of 3.5%,4.5% and 5.5%. Here, the applied loading rate in all tests isset at 3 mm min−1. The models utilize the parameters usedin section 5.1. Energy absorbed per cycle is chosen as aquantitative index for a comparison. The results are listed intable 1.

Table 1 shows that the absorbed energy per cycle predictedby the improved model is within 2% of that obtained in theactual experiments. However, the same index predicted byGraesser’s model is more than the experimental results byapproximately 10.08%. We can clearly see that compared withGraesser’s model, the improved Graesser’s model has a betterprediction of the actual energy dissipation in the full cycle.

5.3. Comparisons of the model predictions and theexperimental data for varying loading rates

Table 2 compares several separate cases at different levelsof loading rate, however, with a constant peak strain of5.5%. Each case uses its own model parameters. Notethat the differences in the enclosed areas of experimentaland theoretical hysteresis loops increase with the increasingloading rate. The energy dissipation per cycle predictedby the improved model is within approximately 4% of theexperimental results: however, the energy absorbed percycle is predicted by Graesser’s model to be more than theexperimental values by approximately 14.7% at 60 mm min−1.Table 2 shows again that the improved Graesser’s modelprovides better modeling accuracy.

6. Concluding remarks

This study involves the modeling of the hysteretic behavior ofsuperelastic SMA wires using the improved Graesser’ model.The influences of the strain amplitude and the loading rate onthe mechanical properties are tested and studied. In particular,the elastic modulus during the unloading process and theinelastic modulus during the transformation take account oftheir strain dependence and rate dependence. Graesser’s modelhas a relative simple formulation, whose parameters can beeasily acquired; however, it ignores the effects of the loadingpath on the model parameters, which increases its modelingerror. In this paper, based on Graesser’ model, an improvedversion is proposed and numerical simulations are conductedto validate the proposed model. The results from the study aresummarized as follow:

(1) As the strain amplitude increases, an increase in energydissipation is observed. The forward transformation stressalmost remains unchanged, along with the elastic modulusduring a loading process and the inelastic modulus duringa transformation. However, the modulus in an unloadingprocess shows an obvious decrease with an increasingstrain level.

(2) As the loading rate increases, the forward transformationstress remains relatively unchanged; however, the reverse

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A one-dimensional strain-rate-dependent constitutive model for superelastic shape memory alloys

transformation commences at a higher stress value.At a low loading rate, the dissipated energy initiallyincreases. However, as the loading rate increases above15 mm min−1, the energy dissipation decreases. The mostdistinct property for the rate-dependent response of SMAwire is the increasing inelastic modulus with an increasingloading rate.

(3) By using the least-square method, two relationships arederived. One expresses the relationship between the slopeof the unloading branch and the strain level, and thecorrelation coefficient is found to be 0.990. The otherrelationship is between the inelastic slope and the loadingrate with a 0.994 correlation coefficient.

(4) Based on Graesser’s model, an improved version isdeveloped. For the modeling, a full cycle is divided intothree parts: the loading branch, the unloading branchbefore the completion of the reverse transformation andthe elastic unloading branch after the completion of thereverse transformation. Each part uses its own parameters,which overcomes the deficiency of the original Graesser’smodel. A MATLAB program is compiled to simulate theimproved model.

(5) Comparisons of model predictions and experiments for thevarying levels of strain and loading rate are performed.The simulation results show that the improved Graesser’smodel provides a better prediction of a superelastic SMA’shysteretic behavior than the original Graesser’s modeldoes.

At last, it should be noted that the improved modelwith a given set of constant shape-defining parameter valueswill produce only one specific hysteretic loop, and theseparameters show apparent differences under different loadingrates. Combined with equations (2) and (3), the improvedmodel can reflect SMA’s hysteresis at different strain levelsunder a fixed loading rate, and both strain rate and strain leveleffects cannot be accounted for at the same time. Futureresearch will be conducted to advance the model consideringthe effects of strain rate and strain level at the same time, andsimulations of the partial loading cycles or internal hysteresisloops of the superelastic SMA will also be conducted.

Acknowledgments

This work has been supported by the Outstanding YouthScience Foundation of NSFC through Grant No. 50025823

and the Overseas Youth Scholar Research Foundation ofNSFC through Grant No. 50328807. These grants are greatlyappreciated.

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