INSTITUTE OF PHYSICS PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 14 (2005) 1297–1308 doi:10.1088/0964-1726/14/6/022

An enhanced SMA phenomenologicalmodel: I. The shortcomings of the existingmodelsMohammad H Elahinia1 and Mehdi Ahmadian2

1 Department of Mechanical Industrial and Manufacturing Engineering,The University of Toledo, Toledo, OH 43606, USA2 Center for Vehicle Systems and Safety, Department of Mechanical Engineering,Virginia Tech, MC-0238 Blacksburg, VA 24061, USA

E-mail: mohammad.elahinia@utoledo.edu and ahmadian@vt.edu

Received 26 October 2004, in final form 14 July 2005Published 17 October 2005Online at stacks.iop.org/SMS/14/1297

AbstractThis paper provides an enhanced phenomenological model for shapememory alloys (SMAs), to better model their behavior in cases where thetemperature and stress states change simultaneously. The phenomenologicalmodels for SMAs, consisting of a thermodynamics-based-constitutive and aphase transformation kinetics model, are the most widely used models forengineering applications. The existing phenomenological models areformulated to qualitatively predict the behavior of SMA systems for simpleloadings. In this study, we have shown that there are certain situations inwhich these models are either not correctly formulated, and therefore are notable, to predict the behavior of SMA wires or the formulation is notstraightforward for engineering applications. Such cases most often occurwhen the temperature and stress of the SMA wire change simultaneously,such as the case of rotary SMA actuators. To this end, a rotarySMA-actuated robotic arm is modeled using the existing constitutivemodels. The model is verified against the experimental results to documentthat the model is not able to predict the behavior of the SMA-actuatedmanipulator, under certain conditions.

1. Introduction

Shape memory alloys (SMAs) are a group of metal alloys thatexhibit the characteristics of either large recoverable strains orlarge force due to temperature and/or load changes. The uniquethermomechanical property of the SMAs is due to the phasetransformation from the austenite (parent) phase to martensite(product) phase and vice versa. These transformations takeplace because of changes in the temperature, stress, or acombination of both. In the stress-free state, an SMA materialat high temperature exists in the parent phase. The parentor austenite phase usually is a body centered cubic crystalstructure. When the temperature of the material decreasesthe phase transforms into martensite which is usually a facecentered cubic structure. In the stress-free state the martensitephase exists in multiple variants that are crystallographicallysimilar but are oriented in different habit planes [1].

The observable macroscopic mechanical behavior ofSMAs can be separated into two categories: the shape memoryeffect and pseudoelastic (superelastic) effect. In the shapememory effect, an SMA material exhibits a large residualstrain after the loading and unloading. This strain can befully recovered upon heating the material. In the pseudoelasticeffect, the SMA material achieves a very large stain uponloading that is fully recovered in a hysteresis loop uponunloading [1]. SMAs show different mechanical behaviorsat different temperatures. Some of the SMA stress–strainbehaviors are illustrated in figure 1.

Since shape memory material behavior depends onstress and temperature and is intimately connected with thecrystallographic phase of the material and the thermodynamicsunderlying the transformation process, the formulationof adequate macroscopic constitutive laws is necessarilycomplex [2]. There has been an extensive body of research

0964-1726/05/061297+12$30.00 © 2005 IOP Publishing Ltd Printed in the UK 1297

M H Elahinia and M Ahmadian

σ

εheat

(a)

σ

εheat

(b)

σ

εheat

(c)

σ

ε

(d)

Figure 1. Mechanical behavior of a shape memory alloy. The nonlinear portion of the plot is due to detwinning the martensite variants ortransforming the austenite to martensite. (a) T < As at the beginning material is fully martensitic; (b) Ms < T < As at the beginningmaterial is fully austenitic; (c) As < T < Af at the beginning material is fully austenitic; (d) T > Af at the beginning material is fullyaustenitic [1].

on modeling the phase transformation and thermomechanicalbehavior of SMAs that is mostly devoted to one-dimensionalSMA components. Brinson et al have divided these modelsinto four categories [3]:

(1) models based on thermodynamics and derived from a freeenergy formulation [4–9],

(2) mathematical models for the dynamics of phase boundarymotion [10–12],

(3) constitutive laws based on micromechanics concepts usingenergy dissipation guidelines [13, 14], and

(4) phenomenological laws based on uniaxial stress–strain–temperature data [15, 16, 1].

These constitutive models are each aimed at describinga different aspect of shape memory behavior and addressthe thermomechanical behavior of SMAs due to the phasetransformation process on different scales.

Engineering applications have emphasized phenomeno-logical models, which avoid difficult-to-measure parameterssuch as free energy and use only clearly defined engineeringmaterial constants. The focus of this part of the paper is onthe adequacy of the phenomenological models for describingthe behavior of the SMA elements under complex thermome-chanical loads. This paper presents the shortcomings of theexisting phenomenological models in predicting the behaviorof SMA-actuated systems under complex thermomechanicalloadings. The second part of the paper introduces an enhancedphenomenological model to address these shortcomings.

2. SMA phenomenological models

In 1986, Tanaka [17] presented a unified one-dimensionalmartensitic phase transformation model. His studywas restricted to the stress-induced martensite phasetransformation. The basic assumption was that thethermomechanical process of the material is fully describedby three state variables: strain, temperature, and martensitefraction. The martensite fraction (ξ ), as an internal variable,characterizes the extent of the martensite phase transformation.Choosing the Helmholtz free energy and using the Clausius–Duhem inequality, Tanaka developed the constitutive equationin the rate form, showing that the rate of stress is afunction of the strain, temperature, and martensite fractionrates. If the expression for free energy is known, the free

energy minimization may determine the equilibrium stateand therefore the relation of the martensite fraction withapplied stress and temperature. Instead, based on the studyof the transformation kinetics, Tanaka assumed an exponentialkinetics function.

The kinetics equation describes the phase transformationfraction as an exponential function of stress and temperature:

ξM→A = exp[Aa(T − As) + Baσ ]

ξA→M = 1 − exp[Am(T − Ms) + Bmσ ](1)

where Aa, Am, Ba, and Bm are material constants in termsof transition temperatures, As, Af , Ms, and Mf . Tanakaqualitatively showed that the pseudoelasticity, ferroelasticity(partial pseudoelasticity), and shape memory effect can bedescribed using a combination of the presented constitutiveand kinetics models. The thermomechanical loadings thatwere considered are either isothermal mechanical loading orchanging the temperature under stress-free conditions. Anadvantage of this model is that the parameters are simplydetermined by mechanical experiments.

Motivated by the need for a unified and theoreticalconstitutive model for SMA material, in 1990, Liang andRogers developed a new phenomenological model based onthe Tanaka model [18]. In their model Liang and Rogersadopted Tanaka’s constitutive equation. For the phase kinetics,however, they assumed a cosine relationship to describe themartensite fraction as a function of the stress and temperature.Additionally, they assumed that the material properties areconstant. The phase transformation kinetics equation for theheating is

ξ = ξM

2cos[aA(T − As) + bAσ ] +

ξM

2. (2)

The phase transformation equation for the cooling is

ξ = 1 − ξA

2cos[aM(T − Mf) + bMσ ] +

1 + ξA

2(3)

where aA = πAf −As

, aM = πMs−Mf

are two material constants,and Af , As, Mf , and Ms are austenite final, austenite start,martensite final and martensite start temperatures, respectively.The two other material constants are defined as bA = − aA

CA

and bM = − aMCM

. CA and CM indicate the influence of stresson these four transformation temperatures. Furthermore, ξM

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An enhanced SMA phenomenological model: I

and ξA are the martensite fraction reached before heating andcooling, respectively.

The argument for the cosine functions should be between0 and π , which means that the transformation takes place onlyif the temperature is in the transformation range:

As T Af or Mf T Ms. (4)

Equivalently, the stress range for the martensite to austenitephase transformation is defined as

CA(T − As) − π

|bA| σ CA(T − As) (5)

and the corresponding stress range for the austenite tomartensite transformation is defined as

CM(T − Mf) − π

|bM| σ CM(T − Mf ). (6)

In 1993, Brinson [1] developed a one-dimensional modelfor SMAs based on the previous works of Tanaka and Liang.The same constitutive equation that was initially introducedby Tanaka was adopted considering the same thermodynamicsprinciples. For the phase transformation kinetics, however,Brinson used Liang’s model rewritten as

ξ = ξ0

2cos

[aA

(T − As − σ

CA

)]+

ξ0

2. (7)

for CA(T − Af) < σ < CA(T − As), and

ξ = 1 − ξ0

2cos

[aM

(T − Mf − σ

CM

)]+

1 + ξ0

2(8)

for CM(T −Ms) < σ < CM(T −Mf). Although it is not statedby Brinson, one can assume that equation (7) is for heating(martensite to austenite transformation) and equation (8) isfor cooling (austenite to martensite transformation). CA andCM are defined as material properties defining the relationshipbetween temperature and the critical stresses that induce thetransformation.

The Brinson model does not have one of the shortcomingsof Liang model. The Liang model cannot describe the shapememory effect, which takes place because of detwinning of themartensite, at temperatures below Mf . Brinson separated themartensite fraction into two fractions as the fraction inducedby stress and the fraction induced by temperature:

ξ = ξs + ξT. (9)

The constitutive equation was also modified accordingly:

σ = Dε + T + ξs. (10)

The phase transformation equations are similar to the onespresented by Liang written in terms of critical stresses.Therefore, the Brinson model is capable of the showing shapememory effect at lower temperature which takes place as theresult of detwinning, not as the result of the austenite tomartensite phase transformation. Furthermore, in the Brinsonmodel the elastic modulus and the transformation tensor wereassumed to be functions of the martensite fraction.

Brinson and Bekker developed a set of phase kineticslaws for the austenite to detwinned martensite and martensite

to austenite phase transformations [19, 20]. These phasetransformation models predict the martensite fraction as afunction of the loading path Ł, initial state of the material ξ0,stress σ , and temperature T at time t .

ξ(t) = (T (t), σ (t); ξ0)(T,σ)∈Ł (11)

where

(T (t), σ (t); ξ0) =

FA(T , σ ; T0, σ0; ξ0),

if Ł ∈ [A] and Ł ↑↑ nA

FM(T , σ ; T0, σ0; ξ0),

if Ł ∈ [M] and Ł ↑↑ nM

FD = ξ0,

otherwise.

(12)

The functions FA, FM, and FD are branches of the phasekinetic law, n is the normal vector to the phase transformationstrip. According to equation (12), the martensite to austenitephase transformation takes place if the loading path Ł and thenormal vector to the austenite start temperature line have acomponent in the same direction (Ł ↑↑ nA).

3. SMA-actuated robotic arm

In order to further investigate the SMA phenomenologicalmodels, an SMA-actuated robotic arm is studied in thissection. Using the existing phenomenological models, thesystem is modeled, the derived model is simulated, and thesimulation results are compared with experimental data. Theshortcomings of the SMA models in predicting the completebehavior of the SMA-actuated arm are studied using thesimulation results of the system. Using a dead-weight systemthat is actuated by an SMA wire, the model shortcomingsare experimentally studied, in part II. An enhanced modelis developed to address the shortcomings of the existingSMA phenomenological models. Furthermore, based on theenhanced phenomenological model, a general procedure formodeling systems that are actuated by shape memory alloys ispresented.

The one-degree-of-freedom shape memory alloy (SMA)actuated arm is shown in figure 2. The shape memory effect(SME) is not reversible; the SMA wire must be deformed bya bias force in martensite to repeat the movement. There aretwo ways of providing the bias force and therefore two typesof SMA actuator. One actuator, called bias type, is composedof an SMA element and a bias spring. The other, calleddifferential type, is made of two SMA elements. In our designthe robotic arm is actuated with a bias-type SMA wire actuatorwhere the bias force (torque) is provided by the linear springand the weight of the moving arm. Therefore, the generatingtorque is the difference between the bias torque and SMA wiretorque.

When the SMA wire is heated beyond the activationtemperature it contracts due to the phase transformation frommartensite to austenite. The temperature is raised using theresistive electrical (Joule) heating. Upon cooling, the wire’stemperature drops, causing the austenite to martensite phasetransformation. As a result the SMA wire is elongated underthe effect of the bias torque. While elongating, the twinned

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M H Elahinia and M Ahmadian

Shape Memory Alloy Wire

Sha

pe M

emor

y A

lloy

Wire

Pulley

Pulley

Payload

PulleyMoving arm

Base

Bias sprin

g

+θ O

Figure 2. The one-degree-of-freedom arm, actuated by SMA NiTi wire, with a bias spring.

(This figure is in colour only in the electronic version)

martensite changes to the stress-preferred martensite. Theinitial length of the SMA wire, that is guided by the pulleys, ischosen in a way that upon full contraction it can rotate the armfrom its initial position at −45 to the top position at +90.

The model for the SMA-actuated arm consists of phasetransformation kinetics, heat transfer, SMA wire constitutive,and arm dynamic model blocks. The input is voltage (to theSMA wire applied through an amplifier) and the outputs arethe arm angular position, arm angular velocity, the SMA wire’stemperature, stress, and the martensite fraction.

4. Heat transfer model

The SMA wire heat transfer equation consists of electrical(Joule) heating and natural convection:

mcpdT

dt= I 2 R − hc Ac(T − T∞). (13)

The SMA wire that is used in the system is Ni–Ti alloy. Itsdiameter is 150 µm and the parameters in the heat transferequation are: m = (ρπ d2

4 ) is mass per unit length of wire,ρ is density of wire, d is diameter of wire, Ac = (πd) iscircumferential area of the unit length of wire, cp is specificheat, I is electrical current, R is resistance per unit length ofwire, T is temperature of wire, T∞ is the ambient temperature,and hc is the heat convection coefficient. Although here weassumed that the hc and R are both constant, a detailed analysison heat transfer as well as resistance analysis of the SMA wirecan be found in the appendix.

5. Wire constitutive model

The wire constitutive model shows the relationship betweenstress, strain and temperature. We used the phenomenologicalmodel presented by Tanaka [17] and later completed byLiang [18] and Brinson [1]. The basic equation is

σ = E ε + θTT + ξ (14)

where E is the Young’s modulus, θT is the thermal expansionfactor, = −EεL is the phase transformation contributionfactor, and εL is the maximum recoverable strain.

6. Phase transformation model

For simulating the constitutive equation, the value of themartensite fraction derivative needs to be known at each instantof time. Based on the hysteresis behavior of SMA wires, acosine phase kinetics model is developed by Liang [16]. Thephase kinetics model needs temperature and stress to calculatethe fraction. Due to the hysteresis behavior of SMA wires theequations for heating and cooling are different.

7. Reverse transformation

The reverse transformation equation describing the phasetransformation from martensite to austenite (heating) is

ξ = ξM

2cos[aA(T − As) + bAσ ] + 1 (15)

where ξ is martensite fraction, which has a value between1 (martensite phase) and 0 (austenite phase). ξM is theminimum martensite fraction the wire reached during thecooling. aA = π

Af −As(C−1) is a curve-fitting parameter, T

is the wire’s temperature, σ is the wire’s stress, As is theaustenite phase start temperature, Af is the austenite phasefinal temperature, and bA = −aA

CAand CA are curve-fitting

parameters.Therefore, the derivative equation for heating can be

written as

ξ = −ξM

2sin[aA(T − As) + bAσ ][aA T + bAσ ]. (16)

If A′s = (As + σ

CA) < T < (Af + σ

CA) = A′

f .

Otherwise ξ = 0. Here A′s and A′

f are the stress modifiedaustenite start and final temperatures, respectively.

8. Forward transformation

The forward transformation equation describing the phasetransformation from austenite to martensite (cooling) is

ξ = 1 − ξA

2cos[aM(T − Mf) + bMσ ] +

1 + ξA

2(17)

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An enhanced SMA phenomenological model: I

Table 1. Modeling parameters and their numerical value.

Parameter Description Unit Value

m SMA wire’s mass per unit length kg 1.14 × 10−4

Ac SMA wire’s circumferential area per unit length m2 4.712 × 10−4

Cp Specific heat of wire kcal kg−1 C−1 0.2R SMA wire’s resistance per unit length 45T∞ Ambient temperature C 20hc Heat convection coefficient J m−2 C−1 s

−1150

EA Austenite Young modulus GPa 75.0EM Martensite Young modulus GPa 28.0θT SMA wire’s thermal expansion factor MPa C−1 0.55 Phase transformation contribution factor GPa −1.12σ0 SMA wire’s initial stress MPa 75.0ε0 SMA wire’s initial strain 0.04T0 SMA wire’s initial temperature C 20ξ0 SMA wire’s initial martensite faction factor 1.0As Austenite start temperature C 68Af Austenite final temperature C 78Ms Martensite start temperature C 52Mf Martensite final temperature C 42CA Effect of stress on austenite temperatures MPa C−1 10.3CM Effect of stress on martensite temperatures MPa C−1 10.3l0 Initial length of SMA wire mm 900r Pulleys diameter mm 8.25mp Pay load mass g 57.19ma Moving link mass g 18.7k Bias spring stiffness N m−1 3.871

0 5 10 15 20 25 30 35 40Time (sec)

θ (d

egre

e)

Experiment (V = 7.2 v)Experiment (V = 7.0 v)Simulation (V = 7.2 v)Simulation (V = 7.0 v)

– 50

– 40

– 30

– 20

– 10

0

10

20

30

Figure 3. Comparison of theoretical model simulations andexperimental results.

where ξA here is the minimum martensite fraction obtainedduring heating, aM = π

Ms−Mfis a curve-fitting parameter, Ms is

the martensite phase start temperature, Mf is the martensitephase final temperature, and bM = −aM

CMis a curve-fitting

parameter.Therefore, the derivative equation for cooling can be

written as

ξ = 1 − ξA

2− sin[aM(T − Mf) + bMσ ][aM T + bMσ ]. (18)

If M ′f = (Mf + σ

CM) < T < (Ms + σ

CM) = M ′

s.

Otherwise ξ = 0. Here M ′s and M ′

f are the stress modifiedmartensite start and final temperatures, respectively.

9. Kinematic model

The kinematic model describes the relationship between strainε and angular displacement θ . Measuring positive angleclockwise, the equation is

ε = −2r θ

l0(19)

where r is the pulley’s radius and l0 is the wire’s initial length.

10. Dynamic model

The nonlinear dynamic model of the arm including spring andpayload effects is

Ie θ + τg + τs + cθ = τw (20)

where τw, τg, and τs are the resulting torques from the SMAwire, gravitational loads, and spring, respectively. Ie is theeffective mass moment of inertia of the arm, and the payload,and c is the torsional damping coefficient.

11. Model simulation and verification

The material properties of the shape memory alloy areprimarily taken from Liang [16] and Waram [21], which areshown in table 1. Experimental evaluations, by Elahiniaand Ashrafiuon [22], have demonstrated reasonable accuracyof the actuator model, as is shown in figure 3. Thefigure compares simulation and experimental results at 7.0and 7.2 V. Some of the differences between the simulationand experimental results are due to parameter uncertaintiesand model simplifications. Specifically, the modulus,

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M H Elahinia and M Ahmadian

-50 0 50 10050

100

150

200

250

300

350

θ (degree)

σ (M

Pa)

40 60 80 100 1200

50

100

150

200

250

300

350

T(oC)

σ (M

Pa)

Ms

Mf

As

Af

σ

Shape Memory Alloy Wire

Sha

pe M

emor

y A

lloy

Wire

Pulley

Pulley

Payload

PulleyMoving arm

Base

Bias sprin

g

Maximum Stress Position

Figure 4. When the arm moves beyond the maximum stress position the transformation temperatures decrease.

0 2 4 6 8 10Time (sec)

θ (d

egre

es)

0 2 4 6 8 1050

100

150

200

250

300

Time (sec)

σ (M

Pa)

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Time (sec)

ξ

0 2 4 6 8 1020

40

60

80

100

120

Time (sec)

T (

C)

T

Mf

Ms

As A

f

– 50

– 40

– 30

– 20

– 10

0

10

20

Figure 5. SMA model open-loop simulation for a 7.2 V input.

transformational tensor, and thermal coefficient were allassumed to be constant. The assumptions of a linear springforce and viscous friction contributed to the discrepancy of theresults.

The main qualitative aspects of the SMA arm model can bederived from figures 5 and 6. In figure 5, a 7.2 V input is appliedto the actuator and the temperature of the wire rises. The initialmartensite fraction is equal to 1 and no phase transformationtakes place as the temperature increases past the martensitefinal temperature, Mf . As the wire continues to heat, the

temperature exceeds the austenite start temperature, As, andthe martensite fraction decreases—resulting in contraction ofthe wire—at a rate defined by the derivative of equation (16).In steady state, the temperature falls somewhere betweenthe austenite start and final temperatures, hence only partialtransformation is achieved.

As is shown in figure 6, only a slightly larger voltage, 7.3 V,results in a full transformation to austenite. The increasedvoltage heats the wire enough to cause the arm to rotate throughthe angle of maximum stress. At this angle, shown in figure 4,

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An enhanced SMA phenomenological model: I

0 2 4 6 8 10– 50

0

50

100

0 2 4 6 8 1050

100

150

200

250

300

350

0 2 4 6 80

0.2

0.4

0.6

0.8

1

0 2 4 6 8 1020

40

60

80

100

120

Mf

Ms

T T

(C

)σ

(MP

a)

θ (d

egre

es)

ξ

Time (sec) Time (sec)

Time (sec)Time (sec)

As A

f

Figure 6. SMA model open-loop simulation for a 7.3 V input.

the stress of the SMA wire due to gravitational and springtorques is maximum. Upon passing this angle, the stress in thewire begins to decrease rapidly, which results in a drop in thetransformation temperatures. This drop allows the temperatureof the wire to pass through the austenite final temperature,causing the martensite fraction to fall to zero and the wire toachieve maximum strain.

12. SMA phenomenological models: where they fail

While the SMA phenomenological models give proper resultsfor most cases, there are certain complex loading cases inwhich the output of these models does not match experimentalresults. An example of this shortcoming is presented in thissection. The phase transformation kinetics by Liang [18]and Brinson [1] are adopted in this study. Since in therotary SMA actuator the wire is always under stress, bythe bias spring, these two phenomenological models predictsimilar behaviors. It is worth noting that the Brinson andBekker phase model covers the general loading conditionsand their corresponding phase transformations and thereforedoes differentiate between the cases shown in figure 11. Thisphase transformation model, however, is not easy to implementfor engineering applications since it requires finding andupdating several points along the loading path. An easy-to-usephase transformation kinetics formulation is therefore neededto use in conjunction with the phase transformation modelsfor numerical simulation of the SMA-actuated systems thatundergo complex thermomechanical loadings. To address thisneed a phase transformation model is developed as presentedin the next section.

Figure 7 shows an example of the shortcomings of thesemodels. In this simulation, a sliding mode controller was usedto stabilize the rotary SMA actuator (for details see [23]). As

shown in the figure, the controller was not able to regulate thearm at θd = 85. As figure 8 shows, the controller applied highenough voltage such that the temperature of the wire exceededthe austenite final temperature, in an attempt to minimize theposition error. Even though the temperature of the wire reachedbeyond the austenite final temperature, according to the model,the wire did not go through the full phase transformation, asshown in figure 9. Therefore the arm did not reach the desiredangular position. Such a behavior has never been observed inthe experiments with the SMA-actuated arm; in other words ithas never been observed that the voltage to the SMA wire wasincreased but the arm did not move accordingly unless the SMAwire was already in the fully austenitic phase. Furthermore,according to the aforementioned phenomenological models, ifthe temperature of the SMA wire exceeds the austenite finaltemperature the material reaches the fully austenitic phase(ξ = 0), which clearly has not happened in this example.

Figure 10 illustrates why in this and similar cases theexisting phenomenological models cannot predict the behaviorof SMA wire: in the simulation, the temperature of theSMA wire exceeded the austenite final temperature whilethe temperature itself was decreasing. While the SMA wirewas cooling down, the stress of the wire decreased, whichcaused the austenite final temperature to decrease. The ratesof temperature decrease for the wire and the austenite finaltemperature are not equal: the SMA wire’s temperature decaysslower than the austenite final temperature; this way the wire’stemperature—while decreasing—exceeded the austenite finaltemperature. According to the existing phenomenologicalmodels, no phase transformation takes place between theaustenite start and final temperatures unless the temperatureis increasing.

Similar cases exist in which the existing phenomenologi-cal models cannot predict the behavior of SMA wires. Three

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M H Elahinia and M Ahmadian

0 1 2 3 4 5 6Time(sec)

θ (d

eg)

θdθ

– 50

0

50

85

100

Figure 7. Phase transformation model problem: arm does not reachthe desired position.

0 1 2 3 4 5 6Time(sec)

T (

o C)

AfmTAsmMsmMfm

20

40

60

80

100

120

140

160

Figure 8. Phase transformation model problem: temperature of theSMA wire and transformation temperatures.

such cases are illustrated in figure 11. For all the cases shownin the figure, according to these models, the amount of phasetransformation that takes place is the same. In all three cases,the stress of the wire decreases by the same amount, whichconsequently causes the austenite start and final temperaturesto drop. This effect is shown in the figure as similar shiftsin the hysteresis loop in the (T –ξ ) plane. In figure 11(a) thewire cools down but with a slower rate, and therefore the tem-perature exceeds the austenite final temperature. The examplewith the SMA-actuated robotic arm, as illustrated in figure 10,in which the arm did not reach the desired position, falls inthis category. In figure 11(b) the wire cools down such thatthe temperature remains between the austenite start and final.In the third case, as shown in figure 11(c), the temperatureremains constant and still exceeds the austenite final tempera-ture. For all these three cases with similar stress changes butdramatically different temperature changes the existing phe-nomenological models predict the same martensite fraction atpoint 2. In other words, no further phase transformation takesplace beyond point 1.

0 1 2 3 4 5 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time(sec)

ξ

Figure 9. Phase transformation model problem: martensite fraction.

20 40 60 80 100 120 140 160T (oC)

σ (M

Pa)

Mf Ms

As

Af

50

100

150

200

250

300

350

400

Figure 10. Phase transformation model problem: stress versustemperature.

Figures 12–15 depict the same simulation results for asimulation with SMA-actuated robotic arm. In this case thearm was to track a changing step. Since the SMA wire didnot undergo any of the complex loading patterns, the existingphase transformation models calculated the martensite fractionproperly.

13. Summary

The existing SMA phenomenological models were investi-gated, using an SMA-actuated robotic arm. The model wasverified against experimental data. Using simulations, it wasshown that for cases that involve complex thermomechan-ical loadings—in terms of simultaneous changes in stressand temperature—the existing phenomenological models haveshortcomings. In a following paper, these shortcomings will befurther highlighted using the experimental data with a dead-weight SMA actuator. Finally, an enhanced phenomenolog-ical model will be presented. This mode is able to predict

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An enhanced SMA phenomenological model: I

AfAs

AfAs

T

T

AfAsT

(a)

(b)

(c)

ξ

Τ

M'f

M's Ms

Mf A's As

AfA'f

ξ

Τ

M'f

M's Ms

Mf A's As

AfA'f

ξ

Τ

M'f

M's Ms

Mf A's As

AfA'f

1

2

1

1

2

2

2

2

2

1

1

1

Τ

time

Τ

time

Τ

time

Figure 11. The shortcomings of the phase transformation model in predicting the martensite fraction for complex force and heat loading, (a)exceeding the austenite final temperature while cooling; (b) remaining in the phase transformation range while cooling; (c) exceedingaustenite final temperature with constant temperature.

0 1 2 3 4 5 6Time(sec)

θ (d

eg)

θdθ

– 60

– 40

– 20

0

20

40

60

80

Figure 12. Phase transformation model works properly: armfollows the desired position.

the behavior of SMA wires under complex thermomechanicalloadings.

Appendix

A.1. Heat transfer analysis

There are number of studies on the heat transfer of theSMA materials. A representative example is the work of

0 1 2 3 4 5 6Time(sec)

T (o C

)

AfmTAsmMsmMfm

20

30

40

50

60

70

80

90

100

110

120

Figure 13. Phase transformation model works properly:temperature of the SMA wire and transformation temperatures.

Bhattacharyya et al [24] in which they studied the possibility ofusing the thermoelectric effect in order to improve the coolingresponse of the SMA actuators. Bhattacharyya et al [25] alsoworked on the characterization of the convection heat transfercoefficient for SMA wires. They modeled the coefficient asa linear function of the current of the wire and performedexperiments with SMA and non-SMA wires to verify themodel. Potapov and da Silva [26] used Liang’s kinematicsmodel for SMA elements with constant stress and constant

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M H Elahinia and M Ahmadian

0 1 2 3 4 5 60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Time(sec)

ξ

Figure 14. Phase transformation model works properly: martensitefraction.

20 30 40 50 60 70 80 90 100 110 1200

50

100

150

200

250

300

350

400

T (oC)

σ (M

Pa)

Mf Ms

As Af

Figure 15. Phase transformation model works properly: stressversus temperature.

strain in order to find both heating and cooling response times.They assumed the convection heat transfer to be constantand showed that the time constant is in agreement with theexperiments.

Here we have used the principle of the conservation ofenergy to model the heat transfer behavior of the SMA wire.Unlike the works of Bhattacharyya et al [25] and Potapov andda Silva [26] we have used a convection heat transfer coefficientthat is based on the thin cylinder theory. As explained next,the convection coefficient is found to be a function of wiretemperature.

The heat transfer model of the wire, using the principle ofconservation of energy, is written as

mCpdT

dt= I 2 R(ξ ) − h(T )A(T − T∞) − mH ξ (21)

where Cp is the specific heat, I and R are electric current andresistance, respectively, h(T ) is the convection heat transfer,A is the circumference area of the wire, T and T∞ are the

temperature of the wire and ambient, respectively, and H isthe latent heat associated with the phase transformation [26].This equation presents the effect of the Joule heating,convection heat transfer, and latent heat on the internal energyof the wire.

A.2. Convection heat transfer

Natural heat convection takes place because of buoyancyforce and the difference between cold and warm air densities.Because of the temperature difference between the verticalsurfaces, in this case a cylindrical surface, and the ambienta boundary layer flow develops over the lateral surface. Whenthe boundary layer thickness is smaller than the cylinderdiameter, the cylinder can be modeled as a vertical wall,ignoring its curvature. In the case of a thin wire, however,the diameter is not necessarily larger than the boundary layerthickness and hence a different Nusselt number and therefore adifferent convection coefficient should be used. The criterionfor considering a cylinder as a vertical wall is

D

l> Ra

− 14

l (22)

where D and l are the cylinder’s diameter and length,respectively, and Ral is the Rayleigh number. It can be shownthat for the SMA wire this condition is not satisfied for therange of temperature of interest. On the other hand for verticalthin cylinders the Nusselt number is written as

Nul = 4

3

[7Ral Pr

5(20 + 21Pr)

] 14

+4(272 + 315Pr)l

35(64 + 63Pr)D(23)

where Pr is the Prandtl number, Nul = hl/k, and Ral =gβTl3/αν. In these definitions h is the length-averagedheat transfer coefficient, and T is the temperature differencebetween the surface of the ambient. Also β is the volumeexpansion coefficient that for the ideal gas can be shown tobe β = 1

T , where T is the absolute temperature. k, α andν are thermal conductivity, thermal diffusivity and kinematicviscosity of the air calculated at the film temperature Tf =T +T∞

2 .Therefore the convection coefficient is a function of both

the temperature of the wire and the ambient temperature. Weassume that the ambient temperature is constant while thewire’s temperature changes considerably. Table A.1 shows theproperty of the air over the actuation temperature range [27].The ambient temperature is assumed to be 23 C. Usingthese properties the convection coefficient can be calculated.Figure A.1 shows the calculated convection heat transfercoefficient which increases as a function of the temperatureof the wire. As shown in the figure, we have approximated itwith a linear function as h(T ) = 0.1558T + 89.33, where T isthe temperature of the wire in centigrade.

It is worth noting that the cooling rate is proportionalto the ratio of the surface area to the heat capacity. Thusthe ratio of the surface area to the volume of the wire is aindicator of the cooling rate. The generating force on the otherhand is proportional to the sectional area of the wire. Thesurface area/volume is in inverse proportion to the diameter.Therefore, if the diameter is small, the cooling rate is fast.However, the generating force is much smaller.

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An enhanced SMA phenomenological model: I

30 40 50 60 70 80 90 100 110 12092

94

96

98

100

102

104

106

108

T (oC)

h (W

/m2 o C

)

Figure A.1. The convection heat transfer coefficient as a function ofthe SMA wire temperature.

– 40 – 20 0 20 40 60 80– 0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Angular position (degrees)

App

lied

curr

ent (

Am

p)

Moving upMoving down

Figure A.2. The current passing through the SMA wire changes asa hysteretic function of the angular position of the actuator.

Table A.1. Air properties as a function of the SMA wiretemperature; the ambient temperature is assumed to beT∞ = 23 C).

Tw Tf k α ν

(C) (K) (W m−1 C−1) (m2 s−1) (m2 s−1) Pr

31 300 0.0261 2.21 × 10−5 1.57 × 10−5 0.71251 310 0.0268 2.35 × 10−5 1.67 × 10−5 0.71171 320 0.0275 2.49 × 10−5 1.77 × 10−5 0.71091 330 0.0283 2.64 × 10−5 1.86 × 10−5 0.708

111 340 0.0290 2.78 × 10−5 1.96 × 10−5 0.707

A.3. Using SMA for sensing

Although SMAs are mostly used for actuation they alsohave a good sensing capability. Several properties ofan SMA element change as it undergoes martensite phasetransformation. Among these properties is the resistivity ofSMAs that decreases as the temperature of the wire increasesand hence its phase transforms to austenite. Using the SMArotary actuator we conducted an experiment to evaluate thechange in the resistance of the SMA wire.

– 40 – 20 0 20 40 60 80Angular position (degrees)

App

lied

volta

ge (

V)

Moving upMoving down

– 2

0

2

4

6

8

10

12

14

16

Figure A.3. The voltage of the SMA wire changes as a hystereticfunction of the angular position of the actuator.

– 40 – 20 0 20 40 60 8045

50

55

60

65

70

75

80

85

90

Angular position (degrees)

Res

ista

nce

(Ω)

Moving upMoving downLinear approximation

Figure A.4. The resistance of the SMA wire changes almost as alinear function of the angular position of the actuator.

In the experiment the applied voltage to the wire increasedincrementally while recording the current and angular positionof the moving arm. Phaser transformation is hysteretic,therefore the voltage, current and angular position were alsorecorded while the arm moved down due to the voltage drop.Figures A.2 and A.3 show the hysteretic plot of the voltage andcurrent versus angular position of the actuator.

While the voltage and current both change with angularposition in a hysteretic fashion the resistance of the wirechanges almost linearly with the angle. This is shown infigure A.4 along with the linear approximation of resistanceas a function of the angular position. The resistance isapproximated as

R = −2.914θ + 50.94. (24)

This experiment is an indication of sensing capabilities ofthe SMA elements that can be further utilized.

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