a comparative study of ni ti and ni ti hf · commercially available ni 49.9ti 50.1 and a...

A Comparative Study of Ni49.9Ti50.1 and Ni50.3Ti29.7Hf20

Tube ActuatorsJ.S. Owusu-Danquah, A.F. Saleeb, B. Dhakal, and S.A. Padula II

(Submitted November 19, 2014; in revised form January 21, 2015; published online February 18, 2015)

A shape memory alloy (SMA) actuator typically has to operate for a large number of thermomechanicalcycles due to its application requirements. Therefore, it is necessary to understand the cyclic behavioralresponse of the SMA actuation material and the devices into which they are incorporated under extendedcycling conditions. The present work is focused on the nature of the cyclic, evolutionary behavior of twowidely used SMA actuator material systems: (1) a commercially available Ni49.9Ti50.1, and (2) a develop-mental high-temperature Ni50.3Ti29.7Hf20 alloy. Using a recently developed general SMA modeling frame-work that utilizes multiple inelastic mechanisms, differences and similarities between the two classes ofmaterials are studied, accounting for extended number of thermal cycles under a constant applied ten-sile/compressive force and under constant applied torque loading. From the detailed results of thesimulations, there were significant qualitative differences in the evolution of deformation responses for thetwo different materials. In particular, the Ni49.9Ti50.1 tube showed significant evolution of the deformationresponse, whereas the Ni50.3Ti29.7Hf20 tube stabilized quickly. Moreover, there were significant differencesin the tension-compression-shear asymmetry properties in the two materials. More specifically, the Ni50.3-Ti29.7Hf20 tube exhibited much higher asymmetry effects, especially at low stress levels, compared to theNi49.9Ti50.1. For both SMA tubes, the evolution of the deformation response under thermal cycling typicallyexhibited regions of initial transients, and subsequent evolution.

Keywords asymmetry in tension-compression-shear, Ni49.9Ti50.1,Ni50.3Ti29.7Hf20, thermomechanical cycles, tubeactuators

1. Introduction

In recent years, SMA-based, solid-state actuators haveattracted considerable attention (i.e., in comparison to otherconventional actuation counterparts) in many engineering fieldssuch as aerospace, automotive, and energy. This is mainlyattributed to their high energy density which offers moreefficient actuation, providing high strokes with desirablefeatures of reduced weight, size, and complexity (Ref 1-4).

The design of SMA actuators is typically based on the shapememoryeffect triggered through the phase transformationbetweena high-symmetry Austenite (A—parent) phase and low-symmetryMartensite (M—daughter) phase of the SMA material systems.These actuators operate such that under a constant applied systemofmechanical loads (axial force, bendingmoment, and/or torque),cooling of the SMA actuator below the martensite finish (Mf)temperature produces a stroke (i.e., change in deformation), whichis recovered upon heating above the austenite finish (Af)temperature. The mode by which the mechanical load(s) is/areapplied determines the type(s) of deformation (i.e., axial, rotation-al, or both) produced in the actuation system.

The magnitude of the actuation stroke produced upon the firstforward phase transformation by an SMA actuator is a funda-mental factor in assessing the capabilities of the SMA actuatorsystem. The NiTi-based SMAs, especially the commerciallyavailable Ni49.9Ti50.1 (at.%), have proved to be viable materialsfor the design of solid-state actuator systems. Their ability toproduce large recoverable work output at a lower cost is a majorreason for their popularity (Ref 5, 6). However, there are anumber of limitations associated with the use of these materials,such as their moderate transformation temperature range(�90 �C to +100 �C), and their higher degree of dimensionalinstability (i.e., the continual increase in the accumulated strainsover the extended number of thermal cycles). This will presentsignificant difficulties in the field of aeronautics where manyactuator systems operate at temperature range higher than100 �C, and where there are severe operational restrictions onthe actuator space. Many efforts have been made to overcomethese undesirable limitations in the Ni49.9Ti50.1, including thedevelopment of several alternative training procedures andaddition of other alloys. For instance, thermal cycling anddifferent aging treatments (Ref 7) have been proved to influencethe dimensional stability and transformation temperature of thismaterial system. The inclusion of additional elements such as Hf,Zr, Pd, Pt, and Au in the amounts of 8-20% (at.%) increases thetransformation temperature of the NiTi alloy by more than 70 �C(Ref 8, 9). The NiTiHf alloy, relative to the other NiTiX (X = Pd,Pt, Zr etc.) ternary alloys, is considered to be a more favorableactuation alloy than the other high-temperature shape memoryalloys (HTSMAs). Its benefits include the relatively lower cost,reasonable ductility, and unique ability to withstand higher levelsof stresses (Ref 10).

In addition to the SMA material properties, the geometricalconfiguration of the actuator essentially affects the output of theactuator system. In recent years, a wide variety of geometrical

J.S. Owusu-Danquah, A.F. Saleeb, and B. Dhakal, Department ofCivil Engineering, The University of Akron, 302 Buchtel Common,Akron, OH 44325-3905; and S.A. Padula II, N.A.S.A. GlennResearch Center, 21000 Brookpark Rd., Cleveland, OH 44135.Contact e-mail: [email protected].

JMEPEG (2015) 24:1726–1740 �ASM InternationalDOI: 10.1007/s11665-015-1425-1 1059-9495/$19.00

1726—Volume 24(4) April 2015 Journal of Materials Engineering and Performance

structures (wire, plate, helical spring, rods, tube, honeycombs,etc.) have been experimentally studied to improve the perfor-mance of SMA actuators (Ref 11). Also there are more recentapplications involving mini/micro SMA actuators utilizing theunusual ‘‘snake-like’’ wire geometries (Ref 12).

SMAactuators existing in the forms ofwires, beams, cables, andcoils havebeen foundveryuseful for specificapplications.Althoughthe simple wire geometry actuators are efficiently able to providelinear actuation, applications (especially in the field of aerospace)requiring complex, high actuation forces rely upon the tubular-configuration actuators to provide such high strokes. In particular,the tubular geometry is aerodynamically favorable for applicationsrequiring large recoverable torsional deformations (Ref 1). Theseactuators are designed such that highmagnitudes of actuation twistsor strokes are produced when the tube is thermally cycled betweentemperatures above the Austenite finish and temperatures below theMartensite finish phases of the SMA material.

Among these, the tubular solid-state actuators havebeen shownto be well suited for the design of propellers (Ref 13) reconfig-urable rotor blade, and airplane wing morphing systems (Ref 14).In such tubular actuation systems, the thickness and length of theactuator tube determines themagnitude of force or torque requiredto produce a desirable amount of stroke in the SMA actuator. Inparticular, a thicker tubewould require a larger amount of torque orforce to generate the same amount of angular or axial deformationproduced in a relatively thinner tube (Ref 15).

Over the past decade, significant progress has been madetoward the development of micromechanical (Ref 16-18) andphenomenological (Ref 19-21) material models, to expound onthe underlying mechanisms responsible for the unique featuresthat are exploited in the SMA material�s application (Ref 22).The systematic investigations carried out using a suitable SMAmodel can provide a simplified and a guided means of studyingthe global thermomechanical responses of different SMAmaterials and structures without resorting to an extensive setof empirical experiments.

In our present study, a thin, tubular, solid-state actuator wasanalyzed using two different SMA material systems (i.e., acommercially available Ni49.9Ti50.1 and a high-temperature Ni50.3-Ti49.7Hf20), to determine their performance under the iso-force andiso-torque loading conditions.Ageneral 3DSMAmodel developedby Saleeb et al. (Ref 23), which has been utilized to calibrate theNi49.9Ti50.1 material system and has been implemented in the largescale simulation of a Ni49.9Ti50.1 helical spring actuator (Ref 24) isusedhere for the followingpurposes. First, themodeling frameworkwas re-parameterized to account for the behavior observed in theHTSMA Ni50.3Ti49.7Hf20 material. Subsequently, the calibratedSMA model for these two different actuator materials is imple-mented for the simulation of theNi49.9Ti50.1 and theNi50.3Ti49.7Hf20SMA tube actuators. Detailed qualitative and quantitative compar-isons (i.e., the similarities and differences) in the cyclic deformationresponse of these two different tube actuators, under differentmechanical loading conditions (axial tensile/compressive, torque),constitute the major effort in the present study.

2. Summary of the 3D multi-mechanism SMAmodel

The SMA model framework, developed by Saleeb et al. (Ref23), is a general constitutive model whose formulation is based

on the balance of the energy dissipated and the energy storedduring the thermomechanical deformation process of the SMAmaterial system. The model is designed to comprehensivelycapture the unique responses observed in ordinary SMAs aswell as the high-temperature SMAs. In the formulation, thetotal strain tensor eij (and its rate, _e) is compactly given aseij ¼ eeij þ eIij where eeij, i.e., the reversible or elastic componentimplicitly accounts for all possible rate dependencies ofindividual austenite and martensite phases and eIij, the irre-versible or inelastic component, accounts for all transforma-tion-induced strains. Due to the insignificant magnitude of thethermal strain in comparison to the transformation-inducedstrain, the thermal strain contribution to the total strain isneglected in the formulation. The corresponding stress tensor,rij, is also decomposed into effective stress, ðrij � aijÞ, and aninternal state tensorial variable aij ¼

PN

b�1aðbÞij , where N indicates

the number of inelastic mechanisms (denoted by superscript‘‘b’’), with the stress-like, and its conjugate strain-like, internalvariables defined as aij

(b) and cij(b), respectively. More specifical-

ly, for the present applications of the model, six inelasticmechanisms were utilized (i.e., b = 1-6) for the two SMAmaterials (Ni49.9Ti50.1 and Ni50.3Ti49.7Hf20). In particular, themechanisms, b = 1-3 are devoted to regulate the energy stored,and those for b = 4-6 correspondingly regulate the energydissipated during the material�s transformation process.

These mechanisms in the model (to different degrees) affectthe non-linear response of the material; as such their magni-tudes are selected based on the transformation characteristics ofa particular SMA material system. For instance, for thesuperelastic stress-versus-strain curve shown in Fig. 1, themechanism b = 1 primarily dictates the rapid development ofthe transformation strains (i.e., during the loading) toward thecritical state denoted as ‘‘a’’ in Fig. 1, which is the onset of thelimited stress transformation surface. The gradual hardening inthe transformation regime (i.e., from ‘‘a’’ to ‘‘b’’ in Fig. 1) isalso controlled by the mechanism, b = 2. These two mechan-isms, i.e., b = 1, and 2 also together determine the pattern andmagnitude of strains developed at latter part of the unloadingstage (i.e., ‘‘e’’-‘‘f’’ in Fig. 1) as well as the amount of residualstrains attained at the end of the unloading branch (i.e., stage‘‘f’’ in Fig. 1). Moreover, the storage mechanism, b = 3, alsoprovides an increasing hardening function that determines thelimiting internal forces regulating the completion of all phasetransformations; i.e., the rehardening regions ��b-c-d’’ in Fig. 1.The energy dissipation mechanisms b = 4, 5, and 6 mainly

Fig. 1 A schematic for a typical pseudoelastic stress vs. strainresponse of an SMA material

Journal of Materials Engineering and Performance Volume 24(4) April 2015—1727

control the height of the hysteresis loop and they play a veryimportant role in also regulating the evolutionary character ofthe SMA; i.e., changes in the details of such hysteresis loopswith repeated thermomechanical cycles.

The two fundamental energy potentials used in the modelformulation are the Gibb�s complementary function, U, anddissipation function, X. These thermodynamic/energy functions(stored energy and dissipation) are formulated in terms of thestrain-and-stress contributions from each of the six inelasticmechanisms. These tensorial variables in the model, includingthe stress tensor, as well as the inelastic strain- and internalstress-state variables are the key quantities responsible forenergy storage-dissipation partitioning that are needed tocapture such important SMA characteristics, such as supere-lasticity, pseudoplasticity, shape memory effect, and the cyclicevolutionary response under thermomechanical loads.

The resulting sets of mathematical equations in the modelformulation, confined to our present study are briefly stated inTables 1 (basic equations), and 2 (transformations and harden-ing functions). Note that all tensorial quantities in the model areformulated in the rotated configuration in order to accommo-date more complicated boundary-value problems, which con-sider the effect of large deformations (Ref 25). For the purposeof model calibration in section 3, as well as its application inthe numerical simulations of section 4, the present SMA modelwas implemented in ABAQUS (Ref 26) via the user materialsubroutine (UMAT) option.

3. Parameterization of the SMA model

In order to parameterize the current SMA model for the twoselected materials, i.e., the Ni49.9Ti50.1 and the Ni50.3Ti49.7Hf20alloys, a total of 25 material constants needed to be determined.These included

(a) 2 ‘‘empirical’’ (handbook value quantities; elastic modu-lus (E) and Poisson�s ratio (t) (see Table 3)).

(b) 5 material parameters to define the rate equation govern-ing the transformation (inelastic) strain. Among these,two parameters, i.e., exponent �n� and modulus �l,� ac-count for rate-dependency of the material. The nu-merical values were selected to be the same for the two

different SMA materials, and reflect their well-knownweak rate-dependency, i.e., relatively large �n� and small�l�, (see Table 3). The remaining 3 parameters com-prised two material constants (i.e., �c� and �d�), dedicatedto represent the degree of ATC, and the threshold �j�,that defines the critical ‘‘onset’’ stresses for A fi de-twinned M (at T>Af), and twinned M fi detwinnedM variants (at T<Mf).

(c) 18 non-linear hardening parameters distributed amongthe 6 inelastic mechanisms (3 for each mechanism), i.e.,threshold �j(b)�, exponent �b(b)� and modulus �H(b)� foreach of the b = 1-6.

In the specific parameterization of the model for each of thetwo materials studied here, the 13 material parameters (�d�, �H(b)�and �b(b)� for b = 1 to 6) were taken to be fixed as given in Table 3.However, the set of 8 remaining material constants (� j�, �c�, and�j(b)�for b = 1-6) is taken to be functionally dependent, to accountfor the possible temperature and/or stress-state dependencies ofthe thermomechanical SMA responses such as strain evolutionduring thermal cycling, ATC, transformation temperature-shifts,and any other unique behaviors observed experimentally for aparticular SMA material. In particular, in such cases, thetemperature and stress-state dependency will be assumed in amultiplicatively decoupled form written as follows:j ¼ jðrÞðTÞ � gðreÞ and jðbÞ ¼ jðrÞðbÞðTÞ � gðbÞðreÞ, where � jðrÞ�and � jðrÞðbÞ� are temperature dependent reference thresholdfunctions; � g�or � gðbÞ� is a non-dimensional factor dependenton the intensity of stress; � T � is the temperature, and � re� is theeffective (also known as von Misses/multiaxial stress intensity),given by re ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ðrijMijklrklÞ=2

p.

3.1 Ni49.9Ti50.1 SMA Material

Data collected from isobaric experiments, conducted intension on Ni49.9Ti50.1 specimens at different bias-stress levels(i.e. from 0 to 300 MPa), for 100 thermal cycles between alower cycle temperature (LCT) = 30 �C and an upper cycletemperature (UCT) = 165 �C, were used for the model pa-rameterization (Ref 27-29).The specimens were 10-mm di-ameter rods. These circular rods have varying lengths in thehot-rolled/hot-drawn and hot-straightened condition (Ref 27).From the differential scanning calorimetry (DSC), theNi49.9Ti50.1 material had an austenite start temperature,

Table 1 Summary of basic equations of the multi-mechanism SMA material model

Equation Set 1: Decomposition of stress and strain

eij ¼ eeij þ eIij; aij ¼PN

b¼1a bð Þij

Equation Set 2: Specific functional forms for stored energy and dissipation potentials

UR rij

� �¼ 1

2 rijE�1ijklrkl; UIR rij; a

bð Þij

� �¼ rijeIij þ

PN

b¼1�H bð Þ;

X rij � aij

� �;a bð Þ

ij

� �¼R

j2Fn

2l dF;

Equation Set 3: Evolutionary laws

_eij � _eIij ¼ ddt

@UR@rij

� �¼ E�1ijkl _rkl; _eIij ¼ @X

@rij;

_a bð Þkl ¼ @2UIR

@a bð Þij @a

bð Þkl

� ��1_c bð Þij ; _c bð Þ

ij ¼ � @X@a bð Þ

ij

;

Where j, l, and n are material constants, F is the transformation function, and �H bð Þ are hardening functions (see Table 2). Eijkl is the isotropicfourth-order tensor of elastic moduli (Young�s Modulus, E and Poisson�s ratio m)


As = 95± 5 �C, and austenite finish temperature, Af = 115 �Cunder stress-free conditions (Ref 30).

A typical tensile, isobaric experimental test result, showing thestrain-versus-temperature response of the Ni49.9Ti50.1 SMAmaterial at 150 MPa bias-stress, is shown in Fig. 2(a). Incalibrating the model for the Ni49.9Ti50.1 material, the intrinsicATC parameters (i.e., �c�, and �d�) were deactivated since onlytensile results (but not their compressive counterparts) fromisobaric tests were available at the time of calibration. However,under effects of large deformation, there will be differences in themagnitudes of displacement measured under same intensity ofcompressive and tensile loading (see Fig. 4f, in Ref 28). The keyfeatures observed in the experimental test response that served as a

benchmark for comparing the calibratedmodel resultswere (a) theinitial transient behavior occurring in the cooling branch of thefirst thermal cycle, and (b) the strain evolution in the Ni49.9Ti50.1material with increasing number of thermal cycles. The reader isreferred to our previous papers (Ref 28, 29) for more elaborateinformation on the procedures involved in the parameterization ofthe Ni49.9Ti50.1 SMA material, and extensive comparisonsbetween the model responses and the experimental measure-ments. For our purpose here, the model predicted strain versustemperature responses of the Ni49.9Ti50.1 material at 150 MPabias-stress level is shown in Fig. 2 in comparison to theexperimental results. The values of the material parameters usedin this model simulation are summarized in Tables 3, 4, and 5.

Table 2 Summary of transformation and hardening functions used in the multi-mechanism SMA material model

Equation Set 4: Transformation and hardening functions

F rij � aij

� �¼ 1

j2

1

2q2rij � aij

� �Mijkl rkl � aklð Þ

� �

;

�H bð Þ ¼j2

bð ÞR

1�h g bð Þð Þ dG

bð Þ; for b ¼ 1; 2; 3;

j2bð ÞR

1h G bð Þð Þ dG

bð Þ; for b � 4;

8<

:

�h g bð Þ� �

¼

q bð Þj bð ÞH bð Þffiffiffiffiffiffig bð Þp� � b bð Þ�1ð Þ

j bð ÞþH bð Þffiffiffiffiffiffig bð Þp� �b bð Þ

; for b ¼ 1; 2;

q bð ÞH bð Þ 1þffiffiffiffiffiffig bð Þp

j bð Þ=H bð Þ

b bð Þ" #

; for b ¼ 3;

8>>>>><

>>>>>:

h G bð Þ� �

¼ H bð Þ 1�ffiffiffiffiffiffiffiffiffiG bð Þp

q bð Þ

!b bð Þ

h Lð Þ

2

4

3

5; for ; b � 4;

whereG bð Þ a bð Þ

ij

� �¼ 1

2j2bð Þ

a bð Þij Mijkla

bð Þkl

� �; g bð Þ c bð Þ

ij

� �¼ c bð Þ

ij c bð Þij ; q ¼ 1þc

ffiffidp

1þcffiffiffiffiffiffiffiffidþk3p ; qðbÞ ¼ 1;

k3 ¼ cos 3h ; where h is Lode�s angle calculated from the invariants of the effective stress ( rij � aij) (Ref 33)

h Lð Þ = the Heaviside function with argument being the loading index L ¼ a bð Þij Cij ; where Cij ¼ @F

�@ rij � aij

� �

Mijkl ¼ 12 dikdjl þ dildjk� �

� 13 dijdkl; With dij = Kronecker delta

HðbÞ, bðbÞ, and jðbÞ = Material parameters for the individual hardening mechanism

c, d; = Material parameters for tension/compression asymmetry (Ref 23)

Table 3 Set of fixed material parameters used for simulated test cases

ParametersUnits

Material SystemNi49.9Ti50.1 Ni50.3Ti29.7Hf20 (EXT124)

‘‘Deflated’’ Elastic stiffnessmodulus, E

MPa 60000 75000

Poisson�s ratio, m … 0.3n … 5l MPa s 1.00E + 05Number of inelasticmechanisms, b

… 6

H(b), b(b) b = 1 MPa, Non-dimensional 4009 103, 1 4009 103, 1b = 2 3009 103, 1 3009 103, 1b = 3 200, 10 500, 1b = 4 419103, 10 409 103, 1b = 5 29 103, 1 309 103, 1b = 6 600, 2.5 1800, 5


3.2 Ni50.3Ti29.7Hf20 SMA Material

The experimental results from a series of isobaric tensionand compression tests carried out on 5 mm diameter, 17.8 mm

gage length-Ni50.3Ti29.7Hf20 dog-bone tensile specimens,(machined from extrusion-124) were used for the Ni50.3Ti29.7Hf20 SMA material model calibration. DSC results for

Fig. 2 Strain vs. temperature response of Ni49.9Ti50.1 for (a) Experimental results and (b) SMA model prediction at a constant bias stress of150 MPa for 100 thermal cycles between LCT = 30 �C and UCT = 165 �C (Ref 19)

Table 4 Temperature-dependency of material parameters j and j(b), for b = 1, 2, …, 6

Material system Temperature (�C)

Material Parameters (MPa)

j j(b), b = 1, 2 j(b), b = 3 j(b), b = 4 j(b), b = 5 j(b), b = 6

Ni49.9Ti50.1 T1 = 20 2.00 9.200 1.00E + 21 90.00 10.88 52.000T2 = 65 (50 �C for b = 4) 20.00 0.200 0.001 21.00T3 = 115 (120 �C for b = 4) 20.00 62.200 0.001 21.00T4 = 200 20.00 53.700 400.000 21.00

Ni50.3Ti29.7Hf20 T1 = 20 40.00 0.200 16.000 256.010 20.00 1.00E + 21T2 = 120 40.00 0.200 0.010T3 = 150 40.00 0.200 0.010T4 = 200 40.00 320.200 64.010T5 = 300 40.00 320.200 64.010

(For each parameters, values are interpolated linearly between the values given in the table at characteristic temperatures T1, T2, T3, and T4)

Table 5 Stress-dependency of the non-dimensional factors g and g(b).Values are linearly interpolated between shownstress levels

Material system Stress levels (MPa)

Scale factors

g g(b), b = 1, 2 g(b), b = 3 g(b), b = 4 g(b), b = 5 g(b), b = 6

Ni49.9Ti50.1 10 0.070 0.125 1.000 1.000 0.9000 0.10050 0.150 0.500 1.2056 0.304100 1.000 1.000 1.0000 1.000150 1.300 1.500 1.6667 1.500200 2.350 1.950 1.4000 2.500300 4.000 2.700 6.6667 3.000375 10.000 3.450 22.8333 5.872

Ni50.3Ti29.7Hf20 100 0.900 0.600 5.000 1.000 0.100 1.000200 1.679 1.300 3.000 1.000300 2.949 2.000 1.000 1.000400 5.129 2.250 1.000 1.000500 6.994 2.500 1.000 1.000


the Ni50.3Ti29.7Hf20 SMA material showed approximate trans-formation temperatures of Mf = 129 �C, Ms = 136 �C,As = 156 �C and Af = 165 �C, under stress-free conditions(Ref 8). As no extended thermal-cycling data were reported forthis extrusion-124, an assumption was made regarding theamount of cyclic strain evolution that would be allowed in thecalibrated model based upon the limited test data of anotherextrusion (extrusion-146) of the same target composition,which showed a rather small amount of accumulated open-loopstrains during 100 thermal cycles (Ref 31).

The criteria for comparison between the Ni50.3Ti29.7Hf20model calibrated results and the experimental response underuniaxial tension and compression tests were (a) the significantdifferences in the magnitudes of actuation strains at all thedifferent bias-stress levels, (b) the marked shifts in transfor-mation temperature regions with the increase of the stress levelsboth in tension as well as in compression, and (c) the strongasymmetry (ATC) in the material under tension-versus-com-pression load-biased stresses (Ref 8). Moreover, with regard tothe evolutionary response with thermal cycling, the modelparameterization was forced to exhibit a limited amount ofstrain evolution during 100 thermal cycles under each of theindividual bias stresses (tension and compression), in confor-mity with the discussion alluded to at the beginning of this sub-section.

To account for the significant ATC effects observedexperimentally in the Ni50.3Ti29.7Hf20 material, especially atlower stress levels (see Fig. 3), the intrinsic ATC parameters,�c�, and �d�, in the material model had to be activated. In

particular, a constant value of �d� = 1.05 and stress-dependentvalues of �c� was used (see Table 6). Furthermore, theexperimentally observed temperature-shifts were accountedfor in the model through the temperature-shift factors (Tshift)stated in Table 6; i.e., this simply amounts to the replacement ofthe functional temperature-dependency of jðTÞ and jðbÞðTÞ byjðT � TshiftÞ and jðbÞðT � TshiftÞ, respectively. The final valuesof the material parameters in this case are stated in Tables 3, 4,5, and 6. A comparison of the experimental and model results,showing the variation of strain with temperature at varyingstress levels from 100 to 500 MPa (for tension) and �100 to�500 MPa (for compression) at increments of 100 MPa isshown in Fig. 3.

4. Finite Element Simulation of the Ni49.9Ti50.1 andNi50.3Ti29.7Hf20 SMA Tube Actuators

4.1 Geometry, Boundary Conditions, and Mesh Convergence

A thin-walled tube cylinder (i.e., t/D = 0.0472) ofexternal diameter, D = 5.08 mm, internal diameter,d = 4.60 mm, length, L = 50 mm and thickness, t =0.24 mm with the associated boundary conditions as shownin Fig. 4, was used for the present study. During theanalysis, one end of the tube was fixed against translationaland rotational degrees of freedom and the other end wasmade free to deform.

Fig. 3 (a) Experimental results and (b) SMA model predictions at various stress levels in tension and compression for Ni50.3Ti29.7Hf20. Theresults taken from the 2nd thermal cycle at each stress level indicate significant stress-dependency and ATC

Table 6 Stress-dependency of distortion constant, �c� and temperature-shift factor, �Tshift�

Material system Stress levels (MPa)Distortion constants Temperature-shift (in �C)c Tshift

Ni50.3Ti29.7Hf20 100 0.560 �20.000200 0.480 �10.000300 0.450 0.000400 0.200 10.000500 0.200 20.000


The generalized 4-node, bilinear, axisymmetric stress ele-ment with twists (CGAX4 in ABAQUS) which provides radialtranslation, R, axial translation, Z, and twist angle, U degrees offreedom was selected for mesh generation and large scalesimulation of the tube actuator. Using the CGAX4 fullintegration elements allowed the tube to be modeled andanalyzed at a more reduced complexity and computationalexpense. The nodal degrees of freedom associated with theCGAX4 element enabled any of the mechanical load controls;i.e., tensile, compressive forces and more especially the torque,to be directly applied at the free-end nodes. A mesh convergencestudy was conducted to determine the suitable mesh size neededfor accurate simulation of the tube. Based on the convergence ofthe magnitude of angle of twist with increasing number ofelements, the axisymmetric section was discretized by 19 50elements; with 1 element in the thickness (due to the thinness ofthe wall of the tube), and 50 elements along the length of the tube(to enable the handling of the anticipated large spatial twistangles in the simulation of the tube actuators).

In each analysis case, the same tube geometry with thefixed-free boundary condition was used. Note however, thatthere are differences in the magnitude of applied mechanicaland thermal load corresponding to the selected Ni49.9Ti50.1 orNi50.3Ti29.7Hf20 tube actuators. Both SMA tube actuators werestudied under the iso-force and iso-torque conditions.

4.2 Loading Controls

The geometrical configuration of the thin tube gives moreaccessible options for it to be mechanically loaded in order toexperience the individual or combined stress states of tension,compression, shear, or internal pressure. By their very nature(i.e., of small thicknesses), stress distribution in thin tubes aremore uniform than in thick tubes. The type of load applied tothe tube dictates the final state of stress developed in the tube.In our present study, we restrict ourselves to the individualstress states of (i) shear (by applying torque), (ii) tension (byapplying tensile axial force), and (iii) compression (by applying

compressive axial force) in the thin SMA tube (see Fig. 5a).Loading case (i) involving the torque is referred to here as iso-torque loading condition, while cases (ii) and (iii) involvingtension and compression respectively, are for brevity termedtogether as the iso-force loading condition.

An important remark is made here to acknowledge thatalthough the present paper is focused on the individual stressstates of tension, compression, and shear, our future work willinvestigate the thermomechanical behavior of this same SMAtube actuator under more general conditions of combined stateof stresses (i.e., proportional and non-proportional loads oftorsion combined with tension or compression).

4.2.1 Thermomechanical Loading Procedure. A torqueor axial force was applied at the free end of the tube to initiate atorsional or an axial deformation at the initial lower cycletemperature (LCT) of 30 �C. For the two SMA tube actuators, adifferent magnitude of effective stress was targeted at the end ofthe load up during the mechanical phase of the loadingprocedure. This was made mainly to achieve a similar state ofdeformation of the two different SMA tubes in the twistactuation mode at the end of the first thermal cycle, where theSMA materials are known to exhibit the significant transientactuation strokes (starting the load up at the LCT and endingthe first cooling also at the LCT). This in turn will enable thefocus of the present study to be placed on the comparison of theevolutionary aspects (i.e., occurring between the 2nd and 50ththermal cycles) affecting the dimensional stability of the twodifferent SMA tubes. The specific selected values of the biastorques and/or axial forces (tension/compression) in the presentstudy were obtained from a prior study (not reported here)involving the following steps. First, choosing as a reference thevalue of 73� twist rotation (reached at the end of the firstthermal cycle) for the two SMA tubes, two different magni-tudes torques required to achieve this rotation were determinedfor the two SMA tubes. The resulting levels of the effectivestresses were 150 and 400 MPa, in the Ni49.9Ti50.1 tube andNi50.3Ti29.7Hf20 tube actuators, respectively. Typical distribu-

Fig. 4 Geometric details and load conditions for the SMA tube being utilized in the study


Fig. 5 Details of the ‘‘isobaric’’ loading programs considered in the study highlighting (a) a description of the loading paths taken for the dif-ferent tests (i.e., tension, compression and/or shear), (b) the nature of the isothermal loading phase, and (c) aspects of the subsequent thermal-cycling phase

Fig. 6 Distribution of the effective and shear stresses in (a) the Ni49.9Ti50.1, and (b) the Ni50.3Ti29.7Hf20 SMA tube at the end of the isothermal,torsional loading phase. Note the small nonuniformity/inhomogeneity in the distribution of the stress showing small stress gradient over the thinwall of the tube (here, the axisymmetric tube model is swept about 360� to show the full view of tube)


tion of the effective as well as the shear stress components,showing the very small stress gradient over the thin wall ofeach SMA tube is depicted in Fig. 6(a) and (b).

Next, when the axial actuation mode was subsequentlyconsidered, similar effective stress values were targeted at the

end of the load up at LCT under either tensile or compressiveaxial forces. Finally, keeping the applied torque/force constantafter reaching the targeted stress, the tubes were thermallycycled between the LCT and UCT range that was appropriatefor each of the individual SMA material systems. The heating

Table 7 Applied mechanical loads and the corresponding displacements at the ends of load up (i.e., do, Uo), first heatingbranch (i.e., dA, UA) and martensite displacement (dM) or twist (UM) occurring at the end of cooling branch of 1st and50th thermal cycle

Material system Load case Applied load

Response at end of Response at martensite

Load up 1st heating 1st cycle 50th cycle

Ni49.9Ti50.1 (LCT = 30 �C, UCT = 165 �C) Tension 547.6 N dA = 0.229 dA = 0.166 dM = 2.81 dM = 5.13Compression �547.6 N dA = �0.220 dA = �0.170 dM = �2.26 dM = �4.01Torque 700.0 N mm UA = 7.39 UA = 5.14 UM = 73.02 UM = 175.1

Ni50.3Ti29.7Hf20 (LCT = 30 �C, UCT = 300 �C) Tension 1430.0 N dA = 0.454 dA = 0.308 dM = 1.94 dM = 1.99Compression �1430.0 N dA = �0.409 dA = �0.30 dM = �1.45 dM = �1.50Torque 2030.0 N mm UA = 13.06 UA = 11.14 UM = 73.36 UM = 76.02

Table 8 The accumulated open-loop displacement (dM) or twist (UM) between selected thermal cycles

Ni49.9Ti50.1 Ni50.3Ti29.7Hf20

Iso-force (dM)Iso-torque (UM)

Iso-force (dM)Iso-torque (UM)

Cycles Tension Compression Torque Tension Compression Torque2nd-25th 55.64 52.25 90.34 2.46 2.37 7.1425th-50th 17.05 16.11 20.07 0 0.97 0.84Evolutionary behavior/dimensional stability Highly evolving More dimensionally stable

Fig. 7 Evolutionary response under tensile, iso-force for: (a, b) the Ni49.9Ti50.1 and (c, d) the Ni50.3Ti29.7Hf20 tube actuators, indicating the axialdisplacement vs. time and the axial displacement vs. temperature variations over 50 thermal cycles


and cooling rates of 0.25 and 0.125 �C/s, respectively, wereutilized. The magnitudes of applied forces or torques as well asthe corresponding LCT and UCT used for the different loadcases with respect to (w.r.t) the two SMA tube actuators arestated in Tables 7 and 8. Typical mechanical and thermal loadcontrol histories used for the simulation are also shown inFig. 5(b), and (c), respectively.

Under the iso-torque loading condition, torques of700 N-mm (at a rate of 0.875 N-mm/s) and 2030 N-mm (at arate of 2.534 N mm/s) were applied at the free end of theNi49.9Ti50.1 and Ni50.3Ti29.7Hf20 tube actuators, respectively.Similarly, for the case of iso-force loading conditions, tensile orcompressive forces of ±547.6 N (at a rate of ±0.684 N/s) and±1430 N (at a rate of ±1.788 N/s) were applied to theNi49.9Ti50.1 and Ni50.3Ti29.7Hf20 tube actuators, respectively.

4.3 Thermomechanical Response of the SMA Tube Actuator

The detailed thermomechanical response of the Ni49.9Ti50.1and Ni50.3Ti29.7Hf20 SMA tube actuators, i.e., the state ofdeformation at the end of load up, and the similarities anddifferences in their response patterns during thermal cycling(cycles 1-50) under the iso-force and iso-torque conditions arediscussed in this section. For clarity and elaborate discussion ofthe results, the responses under the iso-force loading conditionsare described first in section 4.3.1 followed by that under theiso-torque condition in section 4.3.2. To facilitate the subse-quent discussion, we denoted the axial displacements at LCT(martensite) and UCT (austenite) as � dM �and � dA�, respectively,and the corresponding angles of twist at the LCT and UCT as �UM �and � UA�. The axial and twist actuation strokes, whichdetermine the actuation capability of the SMA tube under theselected loading mode, were represented as � dACT � and � UACT �,respectively. Mathematically, dACT was simply defined as thedifference in the displacements measured at the end of heatingand cooling for each thermal cycle, i.e., d Nð Þ

ACT ¼ dðNÞM � dðNÞA ,where N is the thermal cycle number. Similarly, the UACT wasalso designated as U Nð Þ

ACT ¼ UðNÞM � UðNÞA .4.3.1 Response Under Iso-force Condition. A compar-

ison of the variation of the axial displacement with time andtemperature for the Ni49.9Ti50.1 and Ni50.3Ti29.7Hf20 tubeactuators, under the tensile and compressive forces is shownin Fig. 7 and 9, respectively. The two tube actuators under theiso-force (pure tension or compression) loading conditions

produced different magnitudes of axial displacements at the endof the cooling branch of the 1st thermal cycle. Although highermagnitudes of axial forces were applied to the Ni50.3Ti29.7Hf20tube actuator, the amount of axial displacement, and hence themagnitude of actuation, produced by the Ni50.3Ti29.7Hf20 tubeactuator was less than that observed in the Ni49.9Ti50.1 tubeactuator. In particular, the 1430 KN tensile force applied to theNi50.3Ti29.7Hf20 tube generated a dM of 1.94 mm (3.8% w.r.ttube length) at the end of the 1st thermal cycle. In contrast, alesser magnitude of 547.6 N tensile force applied to theNi49.9Ti50.1 tube produced a dM , of 2.81 mm (5.6% w.r.t tubelength) after the 1st thermal cycle cooling. These values (i.e.,the percentages) of displacements were inherently correspon-dent to the strains measured in the simple, homogeneous,uniaxial, tensile, isobaric experiment, and model test responsesat stress levels of 150 MPa in Ni49.9Ti50.1 and 400 MPa inNi50.3Ti29.7Hf20.

Considering the transient behavior of the two materialsduring the 1st thermal cycle, both exhibited qualitativelysimilar patterns, i.e., very small changes in displacementsduring the first heating branch, followed by a significantincrease in displacement during the subsequent cooling branch.To facilitate the quantitative comparisons, the measureddisplacements at the ends of the isothermal load-up stage (atroom temperature for both tubes), first thermal heating, andsubsequent cooling are reported in Table 7. In reference to theloading case involving the tubes under tension (see Fig. 7c andd), it is observed that the displacements of 0.229 mm in theNi49.9Ti50.1 tube and 0.454 mm in the Ni50.3Ti29.7Hf20 tubedeveloped at the end of the load-up stage reduced to 0.166 and0.308 mm, respectively, after the first heating branch. Thesemagnitudes of displacements markedly increased after thecompletion of the first cooling branch; i.e., to values of 2.81and 1.94 mm in the Ni49.9Ti50.1 and Ni50.3Ti29.7Hf20 tubeactuators, respectively. This signifies quantitatively distincttransient responses in the two actuator tubes.

Although these materials share qualitatively similar behav-ior in their initial (transient) response, there is a strikingdisparity in the displacement response occurring between the2nd and 50th thermal cycles in the Ni49.9Ti50.1 tube comparedto its Ni50.3Ti29.7Hf20 counterpart. Particularly, under thecompression stress state, the Ni49.9Ti50.1 tube actuator attainedan initial axial displacement of �2.26 mm after the 1st thermalcycle cooling; yet the measured displacement at the end of the

Fig. 8 Evolutionary response under compressive, iso-force for: (a, b) the Ni49.9Ti50.1 and (c, d) the Ni50.3Ti29.7Hf20 tube actuators, indicatingthe axial displacement vs. time and the axial displacement vs. temperature variations over 50 thermal cycles


50th cycle cooling was �4.00 mm, reflecting about 77% (i.e.,dð50ÞM �dð1ÞM

dð1ÞM

� 100) increase in axial displacement. This observed

difference in the magnitudes of displacements occurringbetween the end of the 2nd and 50th thermal cycles is also ameasure of the integrated open-loop displacement occurringbetween successive thermal cycles. Here, the open-loopdisplacement is defined as the difference in the axial displace-ment at the end of cooling between two successive thermal

cycles i.e., d Nð ÞOL ¼ dðNÞM � dðN�1ÞM for Nth cycle (with N = 2-50).

This clearly demonstrates the highly evolutionary character ofthe Ni49.9Ti50.1 with thermal cycling.

On the other hand, the Ni50.3Ti29.7Hf20 tube actuator undersimilar loading condition (i.e., compression) showed a ratherlimited amount of axial displacement increase of approximately

3.27% between the 1st and 50th thermal cycle cooling, which isremarkably less than that occurring in the Ni49.9Ti50.1 tubeactuator. The open-loop displacements between cycles keptreducing for higher number of thermal cycles in both theNi49.9Ti50.1 and the Ni50.3Ti29.7Hf20 tube actuators. The record-ed magnitudes of the accumulated open-loop displacementbetween 2nd and 25th thermal cycles and between 25th and50th thermal cycles are presented in Table 8.

Moreover, although the same intensity of force was appliedto each SMA tube under the two iso-force conditions, smallerdisplacements were observed for the SMA tube actuators undercompression compared to those under tension (refer to Fig. 7and 8). This was noticeable in the Ni49.9Ti50.1 tube actuator aswell as Ni50.3Ti29.7Hf20 tube actuator, hence indicating thedegree of ATC present in both actuation systems. In particular,

Fig. 9 Summary plots showing the cyclic evolution of axial displacement (in mm) at martensite (dM), austenite (dA) and the correspondingactuation stroke dACT = dM� dA for: (a) the Ni49.9Ti50.1, and (b) the Ni50.3Ti29.7Hf20 tube actuator under tensile, iso-force thermal cycling

Fig. 10 Summary plots showing the cyclic evolution of axial displacement (in mm) at martensite (dM), austenite (dA) and the correspondingactuation stroke dACT = dM� dA for: (a) the Ni49.9Ti50.1, and (b) the Ni50.3Ti29.7Hf20 tube actuator under compressive, iso-force thermal cycling


ratios of 1.24 for the Ni49.9Ti50.1 actuator and 1.34 for theNi50.3Ti29.7Hf20 actuator were produced (when comparingtension displacements to compression displacements) at theend of the 1st thermal cycle cooling. This signifies higher ATCeffect in the Ni50.3Ti29.7Hf20 tube actuator than its Ni49.9Ti50.1counterpart. It is important to recall that the ATC occurring inthe Ni49.9Ti50.1 tube actuator is as result of large deformationeffects only (alluded to the above statement in section 3.1).However, the ATC observed in the Ni50.3Ti29.7Hf20 tubeactuator comprised both the effects of large deformation andthe intrinsic material ATC effect which was accounted forexplicitly in the activated �c� and �d� model parameters for theNi50.3Ti29.7Hf20 material case. Also, the degree of the ATC inthe Ni50.3Ti29.7Hf20 tube actuator at the higher stress of

400 MPa is relatively lesser compared to its counterpart atthe lower stress level of 100 MPa (contrast case in Fig. 7c, dfor tension to their counterparts Fig. 8c, d for compression).This is evident in the experimental response (see Fig. 3a) usedfor the model calibration; the ATC was lesser for higher levelsof stresses. Further elaborations on the ATC aspects of theNi50.3Ti29.7Hf20 tube are given in section 4.3.3.

The cyclic variation of dM, dA, and dACT, as shown in Fig. 9and 10 shows that a stable axial displacement actuation i.e.,dACT was achieved by the Ni50.3Ti29.7Hf20 and the Ni49.9Ti50.1tube actuators, after the 10th thermal cycle. Nevertheless, thehigh evolution nature of the Ni49.9Ti50.1 material is reflective inthe slopes of the curves displaying the variation of dM and dAwith number of cycles. It is clearly seen that while the dM and

Fig. 11 Evolutionary response under iso-torque for: (a, b) the Ni49.9Ti50.1 and (c, d) the Ni50.3Ti29.7Hf20 tube actuators, indicating the angle oftwist vs. time and angle of twist vs. temperature variations over 50 thermal cycles

Fig. 12 Summary plots showing the cyclic evolution of angle of twist (in degrees) at martensite (UM),austenite (UA) and the correspondingactuation stroke UACT = UM�UA for: (a) the Ni49.9Ti50.1, and (b) the Ni50.3Ti29.7Hf20 tube actuator under iso-torque thermal cycling


dA produced by the Ni50.3Ti29.7Hf20 tube actuator remainedalmost constant after 10th thermal cycle, signifying dimen-sional stability in the Ni50.3Ti29.7Hf20 material, the dM and dAgenerated by its Ni49.9Ti50.1 counterpart would require extended(i.e., more than the 50) number of thermal cycles to observemeaningful stability or saturation.

4.3.2 Response Under Iso-torque Condition. Similar tothe iso-force loading cases in section 4.3.1, the plot of angle oftwist-versus-temperature shows an initial transient response afterthe 1st cooling under torque, followed by cyclic evolution duringthe subsequent thermal cycles. Although the two SMA tubes hadinitiated almost the same initial angle, Uð1ÞM , of 73�, the finalangles of twist at martensite,Uð50ÞM , attained at the end of the 50ththermal cycle were 175.10� (i.e., 139% increase) and 76.02� (i.e.,4.13% increase) in the Ni49.9Ti50.1 and Ni50.3Ti29.7Hf20 tube

actuators, respectively. Typical plots of the angle of twist-versus-time and angle of twist-versus-temperature for the Ni49.9Ti50.1and Ni50.3Ti29.7Hf20 tube actuators are shown in Fig. 11(a) and(b), respectively. The values of integrated open-loop angle oftwists, U Nð Þ

OL ¼ UðNÞM � UðN�1ÞM , for the Ni49.9Ti50.1 and Ni50.3-Ti29.7Hf20 SMA tube actuators between the 1st and 25th cycles,and between the 25th and 50th cycles, are depicted in Table 8.

Cyclic behavior of the angle of twist at LCT, UM , and UCT,UA, and the corresponding actuation angle of twist, UACT forthe Ni49.9Ti50.1 and Ni50.3Ti29.7Hf20 SMA tube actuators areshown in Fig. 12(a), and (b), respectively. From the plots, it isnoticed that although the angles of twist at LCT, UM , and UCT,UA, kept increasing with thermal cycles in the Ni49.9Ti50.1 tube,the amount of angular actuation, UACT , remained stable afterthe 10th thermal cycle (Fig. 12a).

Fig. 13 Plots of the secondary axial displacement response vs. the associated primary tangential displacement for: (a) the Ni49.9Ti50.1 and (b)the Ni50.3Ti29.7Hf20 tube actuators under iso-torque thermal cycling. Note that the scale of secondary axial displacement response is one-tenth ofthe scale of the primary tangential displacement response. Also note that the tangential displacement defined in these figures is calculated asðRm � UÞ

� ffiffiffi3p

, where U is the angle of twist (in radians) at the free-end of the tube and Rm is the mean radius of its section

Fig. 14 Comparisons of the evolution of the ‘‘effective’’ displacement responses in (a, b) compression, (c, d) torsion, and (e, f) tension loadingconditions, showing marked intrinsic ATC effect in the Ni50.3Ti29.7Hf20 tube actuator. Note that the ‘‘effective’’ displacement is physically axialdisplacement in case of iso-force (tension/compression) cases whereas it is the tangential displacement for iso-torque loading condition (definedas Rm � U

� ffiffiffi3p

, where U is the angle of twist, in radians, at the free-end of the tube and Rm is the mean radius of its section)


Furthermore, together with the targeted angle of twist,termed here as the primary deformation response, an axialdisplacement was observed in the Ni49.9Ti50.1 and Ni50.3-Ti29.7Hf20 SMA tube actuators under the iso-torque loadingcondition, referred to here as the secondary deformationresponse. However, the magnitudes of the secondary deforma-tion in comparison to the primary deformation were notsignificant under the pure torque loading condition (seeFig. 13). The axial deformation generated under the so-calledpure iso-torque loading condition could be due to the effects ofloading history, path dependence (Ref 32), and/or the �Poyntingeffect� (Ref 15) in the material system caused by large sheardeformation of the SMA materials.

4.3.3 Additional Model Simulations for Ni50.3Ti29.7Hf20Tube. The degrees of asymmetry in the magnitudes of strainsdeveloped under isobaric tensile-verses-compression ex-perimental tests are more acute in the Ni50.3Ti29.7Hf20 materialthan its Ni49.9Ti50.1 counterpart. In order to provide furtherinsight into the effect of the intrinsic ATC parameters used inthe Ni50.3Ti29.7Hf20 material model (as alluded to in sec-tion 4.3.1 above), additional simulations of the tube under puretension, pure compression, and pure torque loading corre-sponding to an effective stress value (i.e., engineering bias-stress) of 100 MPa were carried out. This stress level wasselected following the observed severity in the degree of ATCat such lower stress levels (compare the ratio of strainsmeasured in tension and compression at lower stress of100 MPa to that at a higher stress of 500 MPa in Fig. 3a). Itis worth mentioning that, under this relatively low stressmagnitude, significant portion of the ATC effect is accountedfor, primarily by the intrinsic material parameters, �c� and �d�used in the model. The corresponding results for this supple-mental study are illustrated in Fig. 14, following identicalformats as for the counterpart figures in sections 4.3.1 and4.3.2.

Comparing the results in parts (a), (b), and (c) of Fig. 14,one can clearly see the marked differences in the resultingactuation strokes depending on the mode of the mechanicalloading, i.e., compressive-versus-shear-versus-tension. In par-ticular, when the compression case is taken as a reference, theassociated ‘‘effective’’ displacement in this case is ap-proximately 0.53 mm at the end of the 1st thermal cycle.However, for the counterpart shear (torsion) case, the effectivedisplacement is �0.66 mm; i.e., ratio of nearly 1.24 comparedto the compressive loading mode. Finally, for the tensileloading condition, the corresponding value is approximately0.83 mm, which is about 1.57 times of the compressionreference case. Mathematically, these results can be representedas dðTensionÞACT > dðTorsionÞACT > dðCompressionÞACT in conformity with theoriginal SMA model development (see Fig. 13 and 14 of Ref23). Note the higher ATC effect under the 100 MPa stress levelas compared to that evaluated at the reference value of400 MPa stress level in section 4.3.1.

5. Summary and Conclusion

Comparative studies of the thermomechanical behavior ofNi49.9Ti50.1 and Ni50.3Ti29.7Hf20 SMA thin-walled tube ac-tuators, under iso-force (tension, compression), and iso-torqueloading conditions for extended thermal cycles were performedusing a general 3D SMA modeling framework. In all cases of

loading, effective stresses of 150 and 400 MPa were targetedfor the Ni49.9Ti50.1 and Ni50.3Ti29.7Hf20 tube actuators, respec-tively. Against the background and simulation results obtainedin this investigation, the following conclusions can be made:

(a) The SMA tube actuators under the different loadingmodes demonstrated significant variation in their ther-momechanical response. The amount of actuation strokeand the degree of dimensional stability significantly de-pends on the selected SMA material system as well asthe applied load mode (tension, compression or torque).

(b) The Ni49.9Ti50.1 tube was able to produce more axialand angular actuation with less energy input (i.e., usingless mechanical force and lower upper cycle tem-perature) than its Ni50.3Ti29.7Hf20 counterpart. However,the Ni50.3Ti29.7Hf20 appears to be more dimensionallystable in all cases investigated here. Thus, in actuationsystems where working space is an important designlimitation, the Ni50.3Ti29.7Hf20 tube actuator providessome benefits over the Ni49.9Ti50.1 tube actuator.

(c) Both materials exhibit ATC response characteristics dueto the effect of large deformations. Furthermore, for thecase of the Ni50.3Ti29.7Hf20 tube, the intrinsic ATC ef-fects are accounted for in the model calibration, and thistriggered marked influence on the response of the Ni50.3-Ti29.7Hf20 tube under the different mechanical loadingmodes (compression, shear, tension), especially at lowstress levels (contrast Fig. 7d and 8d versus the resultsof Fig. 14f and b). In particular, considering a valueof 100 MPa of the effective engineering bias-stress,the Ni50.3Ti29.7Hf20 tube showed (see Fig. 14) that theratios of the transformation ‘‘effective’’ displacementsin compression, torsion, and tension are 1:1.24:1.57,respectively. This indicates that the ordering of theresulting actuation strokes under the same loadintensity but different loading modes is as followsdðtensionÞACT > dðtorsionÞACT > dðcompressionÞ

ACT .(d) The model predictions obtained from simulating the two

SMA tubes under the different stress states showedresponse patterns qualitatively similar to that resultingfrom their respective simple homogeneous, isobaric, uni-axial, experimental, and model material-point testsresponses. For instance, compare the results in Fig. 2versus 7(b), and 3 versus 7(d), for the cases ofNi49.9Ti50.1 and Ni50.3Ti29.7Hf20, respectively.

(e) Under the applied iso-torque loading condition, theNi49.9Ti50.1 tube and Ni50.3Ti29.7Hf20 tube produced asecondary (axial displacement) deformation response (inaddition to the primary response). The magnitudes ofthe secondary response, relative to the primary (angle oftwist) deformation response, under the case of pure tor-que loading is insignificant. This, together with the ob-servations made in item (a) above may hint on the moresignificant combined load effects (i.e., axial plus torque)in the case of Ni50.3Ti29.7Hf20 compared to theNi49.9Ti50.1. This will be a topic for our future studies.

Acknowledgments

This work was supported by NASA GRC, the FundamentalAeronautics Program, Subsonic, Fixed-Wing, Project No.


NNH10ZEA001N-SFW1, Grant No: NNX11AI57A to the Univer-sity of Akron. The authors would like to acknowledge Drs. S. M.Arnold and Ronald Noebe for their technical guidance andprogrammatic support during the different phases of the project.

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http://dx.doi.org/10.1117/12.775929

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