This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 129.81.226.78

This content was downloaded on 06/09/2014 at 11:33

Please note that terms and conditions apply.

Thermo-mechanical characterization of shape memory alloy torque tube actuators

View the table of contents for this issue, or go to the journal homepage for more

2000 Smart Mater. Struct. 9 665

(http://iopscience.iop.org/0964-1726/9/5/311)

Home Search Collections Journals About Contact us My IOPscience

Smart Mater. Struct. 9 (2000) 665–672. Printed in the UK PII: S0964-1726(00)15580-6

Thermo-mechanical characterizationof shape memory alloy torque tubeactuators

Andrew C Keefe and Gregory P Carman

Department of Mechanical and Aerospace Engineering, University of California,Los Angeles, 320 Westwood Plaza, Los Angeles, CA 90095, USA

Received 6 September 1999, in final form 24 May 2000

Abstract. A parametric study was performed on NiTiCu shape memory alloy torque tubes.Four samples with varying wall thickness values were evaluated for their thermo-mechanicalresponse. The torque against the angular displacement was measured for each sample whenpurely martensitic and purely austenitic. The recovery torque was measured as a function ofthe pretwist applied during loading in the martensite phase. The trends observed werecompared to three models with relatively good agreement. The recovery torque for biaxialtension/compression–torsion tests were measured with values comparable to pure torsionalloadings. During the biaxial loading, a time-dependent phenomenon was observed. Toevaluate the time dependence, standard creep tests were performed to elucidate the influenceof the load on the temporal response of the material.

(Some figures in this article are in colour only in the electronic version; see www.iop.org)

1. Introduction

Shape memory alloy (SMA) torque tubes have been proposedfor use as actuators in a number of engineering applications.SMAs are solid state, durable, and largely unaffected by theenvironment. They can be used where electrical motors,hydraulic, or compressed air systems are either too bulkyor too complex. SMAs also have the largest energy densityof any active material. However, the thermo-mechanicalbehavior of SMA tubes is not well understood because themajority of research has focused on wires, springs, or plates.In this research we provide an initial attempt at understandingthe response of SMA torque tubes.

Applications for SMA torque rods were first proposedby Schetky [1]. Davidson et al [2] investigated solidSMA rods, for their blocking torque and cyclic behavior.They suggested that the central section of the SMA rodremains elastic and acts as an internal bias spring, therebycreating a two-way effect. Jardine et al [3] proposedSMA tubes for wing twist applications in military aircraft.The authors found that tubes were superior to solid rodsbecause of the lower weight, higher output torque, andsuperior frequency response. More recently, attention hasturned to the behavior of superelastic SMA tubes undercombined loading. Lim and McDowell [4] performedcombined axial–torsional experiments on superelastic NiTitubes, and found rate-dependent behavior (i.e. creep) thatthey attributed to the heat generation produced during stress-induced phase changes. Sittner et al [5] carried outcombined tension–compression experiments on superelasticCuAlZnMn SMA polycrystalline tubes. The authors found

the material response to be asymmetric and the stress-inducedmartensite (SIM) was sensitive to the applied stress state.While all of these papers provided useful data, they didnot look at the influence of the geometry on the torsionalresponse.

A number of analytical models for SMA materials havebeen proposed. Constitutive models routinely cited in theliterature include Tanaka [6], Tanaka and Nagaki [7], Tanakaand Iwasaki [8], Liang and Rogers [9] and Bo et al [10]. Thedevelopment of these constitutive relations was guided by testdata for SMA plates and wires. A torsional model for solidSMA rods was proposed by Shishkin [11]. While Shishkinprovided an analytical framework he did not corroborate theanalysis with any experimental data.

This paper describes results obtained from a parametricstudy on NiTiCu SMA torque tubes with different wallthicknesses [12]. The results include the torsion response ofthe tubes with respect to varying wall thickness, comparisonswith three different simple analytical models, biaxial loading,and a preliminary study of the creep phenomena [13].

2. Experimental test set-up

The load frame used is an MTS 858 tension–torsionsystem. This system has two control modes that canapply either torsional/rotation or axial tension/displacementloading simultaneously. Unless otherwise specified, the loadframe is operated in angle control and axial load control whenloading. This control mode provides a precise manner toproduce a defined twist angle in the sample without arbitrarilyintroducing an axial load (i.e. pure torsion). The use of the

0964-1726/00/050665+08$30.00 © 2000 IOP Publishing Ltd 665

A C Keefe and G P Carman

-7500

-5000

-2500

0

2500

5000

10 30 50 70 90 110

Temperature [deg C]

DSC

[uW

]

Heating

Cooling

AsAf

MsMf

Figure 1. DSC results for a representative sample obtained from atube.

load control to reach a defined elongation or twist angle isproblematic due to the nonlinear response of the material.Furthermore, when twisting a sample beyond the linearregime, the tube will naturally tend to elongate (Poyntingeffect). When unloading, the load frame is operated inload control. This permits the load to be removed from thesample without additional deformation. Custom hydraulicgrips (pressure 1.5 ksi) hold the sample in the load frame.The temperature is measured by a thermocouple adhered tothe sample near the grip region, unless otherwise specified.The temperature at the grip region of the sample is a minimumbecause of the heat conduction to the large hydraulic grips. Tobe certain that the entire sample has changed to the austenitephase, the minimum temperature (i.e. at the grips) must beabove the austenite finish temperature. On the other hand, thelargest temperature occurs in the middle of the tube section,farthest from the heat-sink grips. The thermocouple isconnected to a Partlow PID process controller. Heating tape iswrapped tightly around the sample and connected to the PIDcontroller solid state relay (SSR) driver. The MTS softwarecontrols the PID temperature set point, which controls thecurrent to the heating tape such that temperature can be heldto within ±2 ◦C at the thermocouple. This tolerance is validonly at the thermocouple and cannot be assumed to be validthroughout the specimen. Since temperature varies in thesample, we measured the torsional response at temperatureseither below Mf (martensite finish temperature) or above Af

(austenite finish temperature).After each heating cycle (either a recovery torque test

or an austenite test), the sample is quenched in cold water,then annealed at 200 ◦C for a 10 min hold time. Thesample dimensions and properties are listed in table 1. Thesample numbers are ordered by increasing thickness, suchthat sample one is the thinnest and sample four the thickest.The parameters given for each sample are the outer and innerradii (Ro and Ri), wall thickness (T ); the gauge length (L),which is an inch shorter than the sample length; and theexperimentally determined shear moduli in the martensiteand austenite phases (GM and GA). The shear moduli weremeasured from the unloading portion of the torque againstangle curves.

The approximate transformation temperatures for these

Load

Unload

Heat

Angular Deformation

Tor

que

RecoveryTorque

Pretwist

Set-point

Figure 2. Schematic diagram of the mechanical testing.

samples are listed below, as obtained by the differentialscanning calorimetry (DSC) experiment shown in figure 1:

• Ms = 62 ◦C• Mf = 49 ◦C• As = 65 ◦C• Af = 74 ◦C.

3. Test procedures

Each sample is initially cold-worked in torsion to eliminatedata drift. The cold-working process involves twisting thesample through small angles (i.e. 1% shear strain), unloadingthe sample, holding the torsional rotation fixed, and thenheating to above Af . This is repeated a minimum of fivetimes. This process is then repeated at higher strain levels(i.e. up to the maximum value sample will be subjectedto), until the stress–strain behavior and recovery torque isrepeatable and reproducible.

There are four different tests used in this investigation(martensite, austenite, biaxial, and creep). The first set oftests is performed on martensitic samples (see figure 2). The(room-temperature) martensitic sample is twisted in anglecontrol to a set point at a strain rate (at the outer radius) of2.37 × 10−4 s−1. During loading, the axial load is controlledto be zero, but the displacement is non-zero. This axialdisplacement seen during a pure-shear loading may be dueto several phenomena, such as the loading history or the pathdependence of the material (Sittner et al [5] attributed thisto material anisotropy due to cyclic loading). However, webelieve that it is due to the ‘Poynting effect’ produced by thelarge shear deformations introduced into the material. Onceat the defined set point, the torque is ramped to zero underload control (see figure 2); when unloading, the materialbehavior is fairly linear and the slope is used to define theshear modulus reported in this paper. When unloaded, somedeformation, or ‘pretwist’, remains in the material. Thesample is held torsionally fixed and the axial load is set tozero while the temperature of the tube is ramped to 200 ◦C at arate of 20 ◦C min−1. As the temperature increases, the torquetube exerts a recovery torque. While all the torsional data arerecorded, only the maximum recovery torque is reported inthis paper.

The second set of tests is designed to measure theresponse of the torque tube in the austenite phase. The PID

666

Characterization of shape memory alloy torque tube actuators

Table 1. Sample dimensions and properties.

Ro Ri T L gage G martensite G austeniteSample (in) (in) (in) (in) (psi) (psi)

One 0.153 0.125 0.028 2.7 1.28 × 10+06 2.21 × 10+06

Two 0.205 0.127 0.078 2.5 1.43 × 10+06 2.52 × 10+06

Three 0.225 0.127 0.099 2.3 1.43 × 10+06 2.52 × 10+06

Four 0.250 0.135 0.116 2.7 1.72 × 10+06 2.83 × 10+06

0

200

400

600

800

1000

0 1 2 3 4 5

Twist [deg/in]

Axi

al F

orce

[lbf

]

0

20

40

60

80

100

120

Torq

ue [i

n-lb

f]

ε =3%ε =1%

AB

C

DTorsion

Tension

Figure 3. Illustration of the combinedtension/compression–torsion tests.

controller initially ramps the unloaded tubes to 200 ◦C at20 ◦C min−1. Once the tube is at 200 ◦C for 10 min, a torsionalramp loading is applied at a strain rate of 2.37×10−4 s−1. Thetube is loaded and unloaded in this state while the axial forceremains zero (i.e. load control). The unloading portion of thecurve is used to measure the shear modulus in the austenitephase. During loading and unloading, the thermocouplereadings indicate that the temperature does not vary morethan ±2 ◦C from the 200 ◦C set point.

The third set of tests examined the material behaviorunder combined tension–torsion (biaxial) loading. Loadswere introduced to the martensitic specimens in four steps.The first step applies an axial (tensile) strain of 1 or 3% axialtension to the specimen (strain rate of 1.9×10−4 s−1). Duringthis axial loading the torque is set to zero. Once the desiredaxial strain is reached, the second step twists the specimento a specified angle (strain rate of 2.37 × 10−4 s−1). Thissecond step is performed under angle control while the axialdisplacement is held fixed. Once the rotation is complete,the third step ramps the torque to zero at 20 in lb f s−1 whilethe axial displacement is held fixed. The final step ramps theaxial force to zero at 50 lb f s−1 while the torsional load isheld at zero.

An example biaxial loading sequence is provided infigure 3 as a function of the twist angle. The four set ofcurves in the figure represent two complete loading sequencefor either a 1% or 3% axial strain. Each loading sequence isdivided into a tension and a torsional plot, thus four curvesare presented. The tension curves correspond to the verticalaxis on the left while the torsional curves correspond to theaxis on the right. As an example, look at the curve with atensile strain of ε = 1%. During step A the axial load isapplied while the torsional load is held fixed at zero. Themechanical force ramps up to 800 lb, but there is also a

small twist imparted to the sample due to previous cold-working. In step B, the torsional load is applied and thesample rotates and simultaneously the axial load begins todecay to approximately 500 lb. In step C, the torsional loadis removed and the axial load remains fairly constant. In thefinal step, D, the axial load is removed and the load ramps tozero with a plastic twist of approximately 2◦. Similarly, thetwo torsional curves in the figure could be described. Notethat the two torsional curves (i.e. different axial strains) arealmost identical. For each level of axial strain, the torsionalload was applied several times (see the martensite tests) withdifferent angular deformations. The reader should be awarethat during all of the torsional loadings (step B in figure 3),the axial load reduced substantially due to apparent creepeffects. Following this deformation, the recovery torque wasmeasured by heating the sample (see the first test) and holdingthe angular displacement fixed.

The fourth set of tests investigated the room-temperaturecreep behavior found during the biaxial tests. These testswere performed in axial tension. The thermocouple wasadhered to the tube surface in the middle of the gagesection, where the temperature change should be a maximum.The grips at the specimen ends provide a thermal sinkand if a temperature increase is incurred due to a phasetransformation one would expect it to be largest at the centerof the tube, furthest removed from the grips. Two tests wereconducted. In the first test, the sample was pulled to 3.5%axial strain (strain rate of 1.9 × 10−4 s−1) at which timethe axial displacement was held fixed. The axial stress wasobserved to decrease as a function of time. When the changesin the axial force were indiscernible, the test was stopped. Inthe second experiment, the sample was loaded to an axialstress of 6.9 ksi (load rate of 100 lb f min−1), at which pointthe stress was held constant. The sample was observed toelongate to over 5% strain before the test was aborted.

4. Analytical model

Three models were compared to the experimental test data.The first is an approximation based on extrapolating the testdata using a polar moment of inertia. The second is based ona simple nonlinear constitutive relation proposed by Shishkin[11]. The third is an extension of the second, which assumes anonlinear displacement distribution through the thickness ofthe tube. These models are best classified as curve-fit modelsand are intentionally simple in comparison to other morecomplicated constitutive relations models. The advantageof the simple models are that practicing engineers can easilyuse them.

Linear extrapolation is a common method used to predicttrends in many material properties and behaviors. For the

667

A C Keefe and G P Carman

torsion of cylinders, the constant of proportionality is thepolar moment of inertia J , where J = (r4

o − r4i )π/2. From

linear elasticity, the shear stress in a cylinder is given by

τ = Gγ = G( rϕ

l

)= Grθ (1)

where G is the shear modulus, γ is the shear strain, ϕ is thetwist angle of the end section, r is the radius, l is the specimenlength, and θ is the relative twist angle defined as (ϕ/l). Theexternal torque in the sample can be related to the shear stressby

T =∫

τrdA = GJθ. (2)

Nonlinearities associated with the SMA stress–strainbehavior cannot be represented by equation (1) and (2)directly. If we assume that the intrinsic nonlinearity of theSMA can be represented by G = f (θ), equation (2) can berewritten as

T = Jθf (θ). (3)

By using experimental data obtained from one tube todefine f (θ) for either austenite or martensite, the responseof the other tubes can be extrapolated with J . Anotherapproach used to model the torsional response of SMA tubeswas published by Shishkin [11]. He assumed a nonlinearconstitutive relation of the following form:

τ = Bshγn = Bsh(rθ)n (4)

where Bsh and n are constants to be determined fromexperiments. Shishkin derived the governing equations fora solid rod, while it is derived here for a hollow tube.Substituting equation (4) into equation (2) we arrive at

T =∫ 2π

0

∫ ro

ri

rτdA = Bshθn

∫ ro

ri

rn+1dA

= 2Bshθnπ

∫ ro

ri

rn+2dr = 2(rn+3o − rn+3

i

)π

n + 3Bshθ

n. (5)

The constants (Bsh and n) are temperature-dependentquantities that have different values in the austenite andmartensite phases. Consequently, equation (5) is actuallya function of temperature or, similarly, of the martensitevolume fraction.

Shishkin’s model assumes that the rotation is a linearfunction through the thickness. An extension of his modelcan be obtained by proposing a nonlinear displacementdistribution through the tube wall thickness. The assumedshear strain distribution takes the form

γ = Arxθ (6)

where A and x are constants to be determined. Demandingthat the rotation at the exterior of the tube (r0θ) must equalthe applied rotation we have the following constraint:

A = r1−xo . (7)

Using equations (6) and (7) in equation (4), we write the shearstress as

τ = Bγ n = B(Arxθ)n (8)

and the external recovery torque T is

T = 2πBθnr(1−x)no

3 + nx

(r3+nxo − r3+nx

i

). (9)

0

50

100

150

200

250

300

350

0 2 4 6 8Angular Deformation [deg/in]

Tor

que

[in-

lbf]

I

II

III

IV

Figure 4. Experimental data from the martensite tests.

0

100

200

300

400

500

0 1 2 3

Angular Deformation [deg/in]

Tor

que

[in-

lbf]

I

II

III

IV

Figure 5. The austenite load against the angular deformation.

5. Results

The initial tests were conducted on the four samples at roomtemperature. Figure 4 shows the differences in the torqueagainst the angular deformation curves for all four samples.Obviously, the thicker tubes require much more torque thanthe thinner tubes for the same amount of angular deformation.Notice that sample one needs less than 25 in lb f torque todeform 5◦ in−1 while sample four requires 325 in lb f (apolar moment of inertia, J , ratio of 12:1). The increase inthe torque values is just slightly higher than the ratio of thepolar moment of inertia values. The loading portion of thecurve is nonlinear whereas the unloading portion is linear.This trend is consistent for all four tubes. Each sample wastested a number of times (a minimum of three) with generallyrepeatable results. That is there is little deviation whensuperposing load–deformation curves for different amountsof twist

The load–deformation curves for these samples werealso produced in a hot state, where the material is fullyaustenitic. Figure 5 is a torque against angular deformationplot for the four samples, when completely austenitic. Again,the first observation is the nonlinear relationship betweenthe sample geometry, load, and angular deformation. Itis possible that the nonlinearity in the torque against the

668

Characterization of shape memory alloy torque tube actuators

0

100

200

300

400

500

600

700

800

0 2 4 6 8 10

Angular Deformation [deg/in]

Tor

que

[in-

lbf]

I

II

III

IV

Figure 6. Experimental data on the torque against the angle fortubes in the austenite phase.

angular deformation is caused by a phase transformation inthe outer radii, where the stress is highest; however, there is noconclusive evidence to support that stress-induced martensiteforms at this region. Notice the torsional difference betweenthe first and fourth samples. For 2◦ in−1 sample one requires45 in lb f while sample four requires 450 in lb f. Using thepolar moment of inertia and extrapolating the data for sampleone would lead to the conclusion that 545 in lb f would beproduced for tube four, which predicts a value larger thanmeasured.

The test results from the blocking torque against thepretwist experiments are presented in figure 6. Ideally,the recovery torque should be identical to the austeniticload–deformation curve in figure 5. Samples one to threebehaved as expected; however, the fourth sample consistentlygave results visibly lower than expected. The first threesamples have a smooth curve trend, with a consistentlydecreasing slope, whereas the fourth sample has a jaggedtrend. We emphasize that this was a repeatable anomaly. Webelieve that sample four may be large enough that detwinningpreferentially occurs in the grip region rather than throughoutthe specimen. Note that the jagged portion of the curve is atthe approximate location that significant detwinning beginsfor this specific tube. Once passed this initial detwinningregion, the recovery torque curve appears to become betterbehaved.

The experimental results for the martensite tests (×)were compared to the polar moment of inertia model (◦)in figure 7. A linear extrapolation was performed relative tomeasurements from sample one, which is exact by definition.The predictions for the martensite have mixed results. Thedata from sample three agrees well for small deformations,but deviates steadily (over predicts) as the deformationincreases. For sample two, the model over predicts theresponse of the tube. For sample four, the model underpredicts the response of the tube by up to 30% for somepoints. The discrepancies maybe due to the fact that themartensite curve is better represented by a elastic–perfectlyplastic model.

The experimental results (×) for the recovery torqueagainst the extrapolation (◦) of the polar moment of inertiaare presented in figure 8. Clearly, sample three shows the

0 0.05 0.1 0.15 0.2 0.250

10 0

20 0

30 0

40 0

TheoryExperiment IV

III

II

I

Angular Deformation [rad/in]

Tor

que

[in-

lbf]

Figure 7. Analytical against the experimental data using linearextrapolation (J -ratio) for the martensite phase.

Figure 8. Analytical against the experimental curves for therecovery torque using a linear extrapolation (J -ratio).

best agreement with the experimental data as was the casefor the martensite results. Here the range of agreement issubstantially broader than obtained for the martensite results.Samples two and four again show the greatest deviation fromthe measured values, from 15–40%. However, this time bothsamples over predict the experimental data. The results forsample four deviate greatly at all points and become worseas the experimental curve begins to knee. Because the thick-walled sample follows a different trend than sample one, thelinear extrapolation error is exacerbated.

The other two analytical models were also used to predictthe behavior of the tubes in the martensite phase (see figure 9).Different combinations of the constants were tried until abest fit was found for a broad range of test data. The valuesfor the constants in Shishkins model (◦) are B = 40 000,and n = 0.39, while for the nonlinear displacement model(– – –) are B = 60 000 ksi, x = 0.15, and n = 0.53.The nonlinear displacement shear strain model does give asomewhat better approximation than the linear strain model;both in magnitude and shape, the nonlinear displacement

669

A C Keefe and G P Carman

Figure 9. Comparison of the Shiskin and nonlinear displacementmodels for the experimental data for the martensite phase.

Figure 10. Comparison of the Shishkin and nonlineardisplacement models for the experimental recovery torque.

approximation is closer to the experimental data points.Notwithstanding, all models overestimate the load levels ofsample two and underestimate the load levels for samplefour. None is exact for more than two samples, althoughthe nonlinear model is marginally the best.

When the constants are changed to fit the recovery torqueagainst deformation data (figure 10), we observe that the cor-relation is much better than in the martensitic phase. Thevalues for Shishkin’s model are B = 200 000 and n = 0.26(◦) while the values for the nonlinear displacement modelare B = 230 000, x = 0.15, and n = 0.61 (– – –). In allcases both models closely track the experimental data. Thedifferences between the two models are nearly indistinguish-able. In general, it appears as though both of these modelsprovide accurate predictions of the recovery torque.

In figure 11 the recovery torques for combined tension–torsional loads are presented. Superimposed at the bottom

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10

Angular Deformation [rad/in]

Tor

que

[in-

lbf]

1% compression3% tension

pure torsion

Figure 11. Recovery torque data for biaxial loading.

of this figure are the martensite torsional loading curves. Asone can see, the martensite loading curves are identical andindependent of the axial load. There were three differentaxial loads investigated; that is, 1 and 3% tensile strainsand 1% compression strain. The tests results in figure 11are for sample three. For both the 1 and 3% axial strainand the 1% axial compression, the recovery torque measuredwas marginally higher than that measured in simple torsion.The tests conducted at 1% compression and 3% axial tensionshowed nearly equal amounts of torque increase above thepure torsion tests. While a marginal increase was observedin specimen three, an increase in recovery torque was notobserved in the three remaining samples. In light of theseresults, the increase in the recovery torques is attributed to thetraining of the sample and not to a combined tension torsionalloading. That is, sample number three had the largest numberof cycles performed on it. After repeated load cycles,the martensite begins forming in a preferred direction (intorsion). The axial load could possibly reorient the variantscausing the torsional loading path to be different. Duringthese biaxial experiments, the temperature was measuredat the center of the gage section in order to observe anyheating from the deformation. It was reported by Lim andMcDowell [4] that during the loading of a superelastic SMAthe heat generated during the loading (i.e. up to 30 ◦C) cangreatly affect the stress–strain behavior. On the contrary, thetemperature measured in our experiments remained fairlyconstant during all mechanical testing (varied only a fewdegrees Celsius).

The axial unloading phenomenon observed during thecombined loading experiments led to an investigation of theroom-temperature creep. Please note that the term ‘creep’ isused somewhat loosely, in that the phenomenon responsiblefor the very ‘creep-like’ behavior is the detwinning of themartensite variants under a large constant stress, as opposedto typical creep in metals at high temperature. As mentionedin the previous tests, a thermocouple attached to the specimenindicated that the temperature did not change during thesetests. In figure 12, a 3.5% axial strain is applied to the thirdtube, at which time the displacement is held fixed. The axialstress decreases by approximately 1500 psi (1.5 ksi or 162 lb ffor this sample). This decrease is comparable to the decreaseobserved during the combined axial–torsional loading curves

670

Characterization of shape memory alloy torque tube actuators

0

1

2

3

4

5

6

7

8

0 1 2 3 4

% Axial Strain

Axi

al S

tres

s (k

si)

2

3

4

5

0 100 200 300Time (sec)

% A

xial

Str

ain

0

2

4

6

8

0 2 4 6

% Axial Strain

Axi

al S

tres

s (k

si)

0

1

2

3

4

5

6

0 200 400 600

Time (sec)

% A

xial

Str

ain

Loading

Constant Stress6.9 ksi

Steady statecreep

Constant Strain

Constant Stress

a b

cd

Figure 12. Results of the room-temperature creep experiments for sample three; (a) decrease in the axial stress when the axial strain is heldfixed, (b) increasing axial strain caused by holding the axial stress constant (6.9 ksi), (c) the axial strain against time for the entire creepexperiment, including both loading and unloading and (d) the axial strain against time during constant axial stress, showing the steady-statecreep.

shown in figure 3. We note that the rate of unloading atconstant displacement is non-uniform, but decays with time.

Typically, creep experiments are performed at constantstress, rather than under constant strain. A constant stresscreep test was also performed on sample three. The specimenwas loaded to 7000 psi (i.e. 7 ksi) in force control at 20 lb f s−1,which is the same stress level reached in the constant straintest. At 7 ksi, the stress was held constant and the samplebegan to steadily elongate. The test was stopped at 5.5%strain, see figure 12(b). The entire strain–time curve is plottedin figure 12(c). At time zero, the sample is stress free andcompletely martensitic. From 0 to 300 s, the strain increasesduring the ramp-loading phase. After 300 s, the stress isheld constant and the material continues to deform due todetwinning [13]. Figure 12(d) shows only the portion ofthe experiment where the stress is constant. The time scalehas been shifted such that at time zero, the stress becamea constant value. The slope is clearly decreasing until anearly linear or ‘steady-state creep’ begins. Had the testbeen allowed to continue, this steady-state creep would likelyproceed until the martensite was completely detwinned.

6. Conclusions

The material behavior is nonlinear in both the martensiticand austenitic phases when loading, but is fairly linear inunloading. The test results indicate that wall thickness playsa prominent role in the results. The linear extrapolationusing the J -ratio method has been shown to be fairlyinaccurate for estimating the material behavior in themartensite phase and only partially accurate for the austenitephase. Shishkin’s model reproduces the austenitic behaviorwell but only marginally for martensite. The nonlineardisplacement model provides marginally better results than

Shishkin’s model. Based on its simplicity it appears as thoughShishkin’s model is the most useful. Because the martensitebehavior of the SMA is elastic–perfectly plastic, these simpleexponential curve fitting models may not accurately representthe physical phenomena, especially as the wall thicknessincreases and the stress gradient becomes significant throughthe thickness.

The results also suggest that biaxial loading does notappreciably influence the recovery torque. Tests wereconducted in both tension–torsion and compression–torsion.The differences in the responses were attributed to thedetwinning paths rather than the external load profile. Weprovide the caution that this data was for a very specificloading scenario and these results may change for otherloading scenarios.

We found that time-dependent detwinning plays aprominent role. The time-dependent detwinning orcreep behavior appears to continue until the martensiteis completely detwinned under sufficiently high levels ofconstant stress. The lack of a significant temperature changeduring these experiments indicates that the behavior is dueto a mechanical phenomena rather than a thermal one. Theresults of these experiments should serve as an indicationof the behavior one can expect from SMA torque tubes ofvarying thickness and under different loading parameters.The models presented, although not exact, do serve as a toolto estimate the mechanical behavior of SMA torque tubes asa function of their geometry and temperature.

Acknowledgments

The authors of the paper gratefully acknowledge thepartial support from Army Research Office (contractnumber DAAH04-95-1-0095) contract monitor John Prater,

671

A C Keefe and G P Carman

Air Force Office of Scientific Research grant/contractnumber F49620-98-1-0058 managed by Brian Sanders, andDARPA/Northrop-Grumman Smart Wing Project.

References

[1] Schetky L McD 1991 Shape memory alloy applications inspace Mater. Design 12 29–32

[2] Davidson F M, Liang C and Lobitz D 1997 Investigation oftorsional shape memory alloy actuators Proc. SPIE 2717672–82

[3] Jardine A P et al 1997 Shape memory TiNi actuators for twistcontrol of smart wing designs Proc. SPIE 2717 160–5

[4] Lim T J and McDowell D L 1995 Path dependence of shapememory alloys during cyclic loading J. Intell. Mater. Syst.Struct. 6 817–30

[5] Sittner P, Hara Y and Tokuda M 1995 Experimental study onthe thermoelastic martensitic transformation in shapememory alloy polycrystal induced by combined externalforces Metall. Mater. Trans A 26 2923–35

[6] Tanaka K 1986 A thermomechanical sketch of shapememory effect: one dimensional tensile behavior Res.Mechan. 18 251–63

[7] Tanaka K and Nagaki S 1982 A thermomechanicaldescription of materials with internal variables in theprocess of phase transitions Ingenieur-Archiv 51 287–99

[8] Tanaka K and Iwasaki R 1985 A phenomenological theory oftransformation superelasticity Eng. Fracture Mechan. 21709–20

[9] Liang C and Rogers C A 1990 One-dimensionalthermomechanical constitutive relations for shapememory materials J. Intell. Mater. Syst. Struct. 1 207–34

[10] Bo Z, Lagoudas D C and Miller D 1999 Materialcharacterization of SMA actuators under nonproportionalthermomechanical loading J. Eng. Mater. Technol. 12175–85

[11] Shishkin S V 1994 On theoretical interrelations betweenthermomechanical diagrams in tension, compression, andtorsion for alloys possessing the shape memory effectIndustrial Laboratory 60

[12] Keefe A C 1994 Thermo-mechanical characterization ofshape memory alloy torque tube actuators Masters ThesisUniversity of California at Los Angeles

[13] Keefe A C and Carman G P 1999 Thermo-mechanicalcharacterization of shape memory alloy torque tubes SPIE6th Int. Symp. on Smart Materials and Structures (March1–6, 1999) 3324 58–67

672