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 DOI: 10.1177/1045389X12457255

2013 24: 70 originally published online 21 September 2012Journal of Intelligent Material Systems and StructuresAtef F Saleeb, Binod Dhakal, Santo A Padula II and Darrell J Gaydosh

prediction of the cyclic ''attraction'' character in binary NiTi alloysCalibration of a three-dimensional multimechanism shape memory alloy material model for the

  

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Calibration of a three-dimensionalmultimechanism shape memory alloymaterial model for the prediction ofthe cyclic ‘‘attraction’’ character inbinary NiTi alloys

Atef F Saleeb1, Binod Dhakal1, Santo A Padula II2 and Darrell J Gaydosh3

AbstractAs typically utilized in applications, a particular shape memory alloy device or component operates under a large numberof thermomechanical cycles, hence, the importance of accounting for the cyclic behavior characteristics in modeling andcharacterization of these systems. To this end, the present work is focused on the characterization of the evolutionary,cyclic behavior of binary 55NiTi (having a moderately-high transformation temperature range). In this study, an extensiveset of test data from recent cyclic, isobaric, tension tests was used. Furthermore, for the calibration and characterizationof this material, a newly developed, multiaxial, material-modeling framework was implemented. In this framework, multi-ple, inelastic mechanisms are used to regulate the partitioning of energy dissipation and storage governing the evolution-ary thermomechanical response.

KeywordsNiTi, shape memory alloy, thermomechanical, actuation, isobaric, cycling, evolution, transient, attraction state, multi-axial, multimechanism, material modeling

Introduction

In the history of materials, alloys have played an impor-tant role as structural materials. Among the variouscombinations of metals used to form alloys, certaincombinations, such as Ni and Ti, are known to exhibitunique properties, like the ability to remember a shapeafter deformation. Because of these novel and uniqueproperties, they are known as ‘‘shape memory alloys(SMAs)’’ and are desirable for use in various fields ofengineering application; for example, in sensors, actua-tors, biomedical stents and devices, energy absorption,and vibration damping.

The key to this property is the solid-to-solid (diffu-sionless) martensitic phase transformation between ahigh-symmetry stable austenite phase, A(parent), andlow-symmetry, low-temperature, martensite phase,M(daughter), under the effect of temperature and/orstress. From a micromechanic point of view, the trans-formation in binary NiTi is due to the change in crystalstructure that occurs, that is, a high-temperatureordered cubic structure (B2 austenite) to a monoclinic(B19#) structure (Jones and Dye, 2011). This martensi-tic phase transformation enables these alloys to exhibit

thermomechanical behaviors (pseudoelasticity/supere-lasticity, pseudoplasticity, as well as one- and two-wayshape memory effects) that cannot be observed in otherconventional materials (Helm and Haupt, 2003;Lagoudas et al., 2009; Wada and Liu, 2008).

A number of desirable actuator applications requirethe material to operate for large numbers of thermo-mechanical cycles while still exhibiting stable shapememory effect. Hence, a sufficient number of thermo-mechanical cycles, which is known as training, must beapplied to transition the material from unstable beha-vior to stable performance characteristics (Jones andDye, 2011; Matsuzaki and Naito, 2004; Zaki andMoumni, 2007). Some cycling experiments on nearlyequiatomic NiTi SMAs have been reported to exhibit

1Department of Civil Engineering, The University of Akron, Akron, OH,

USA2NASA Glenn Research Center, Cleveland, OH, USA3Ohio Aerospace Institute, Cleveland, OH, USA

Corresponding author:

Atef F Saleeb, Department of Civil Engineering, The University of Akron,

Akron, OH 44325-3905, USA.

Email: saleeb@uakron.edu

poor dimensional stability (Kockar et al., 2008; Wadaand Liu, 2008). Hence, it is important to understandthe evolutionary response of thermomechanical cyclesneeded to achieve a stable configuration producing thedesired usable performance for practical applications.

A variety of constitutive, mathematical models havebeen proposed in the past few decades to characterizethe behavior of shape memory materials. In general,these various models can be classified as either micro-mechanical or phenomenological/macroscopic. Themicromechanical class of models is based upon the fun-damental thermodynamic principles, including changesin the crystallography that occur during transforma-tion, hence accounting for strain differences from thiscrystallographic change for the individual martensitevariants (Gao et al., 2011; Hamilton et al., 2004;Matsuzaki and Naito, 2004). More recently, whenattempts were made to handle polycrystal SMAs, anumber of additional refinements were found necessaryto include in the basic micromechanical formulations.Examples of these refinements include factors such asmutual variant interactions, grain-to-grain level interac-tions, as well as dislocations and other defects (Ikeda,2008; Manchiraju et al., 2011).

On the other hand, there are numerous phenomeno-logical models that were formulated based upon themacroscopic experimental observation of the thermo-mechanical behavior, and they are used for the analysisof engineering applications (Lagoudas et al., 2006,2009; Matsuzaki and Naito, 2004). In particular, vari-ous proposed one-dimensional and three-dimensional(3D) macromechanical models are based upon someassumed forms of the free energy and utilize internalstate variables for various quantities. Some formula-tions use these internal variables to represent quantitiessuch as the martensite volume fraction, martensiteorientation strain tensor, residual strain, transformationstrain, and internal stress induced by repeated phasechange. The corresponding evolution laws in these mod-els are based on classical plasticity theories to accountfor cyclic as well as other effects exhibited by SMAsduring thermomechanical loading. Effects such as self-accommodation, reorientation and detwinning of mar-tensite, superelasticity and one-way shape memoryeffect, effect of training, and the two-way shape mem-ory effect all have been addressed (Helm and Haupt,2003; Lagoudas et al., 2006; Lim and McDowell, 2002;Tanaka et al., 1995; Zaki and Moumni, 2007).

More recently, Saleeb et al. (2011) have formulateda general 3D model that can capture a comprehensiveset of experimental observations for the evolutionaryresponse of SMAs under different thermomechanicalloading conditions. Using this same model, our focusin this article is to calibrate the deformation responseof a binary 55NiTi alloy based upon a set of recent cyc-lic, thermomechanical experiments (courtesy: Dr SantoA. Padula II, NASA GRC).

Experimental observations

The material used in the experiment is a binary NiTialloy; designated as 55NiTi (55% Ni by weight in alloy)with at% stoichiometry Ni49.9Ti50.1, with fully annealedingot austenite start temperature As of 95�C 6 5�C. Anaustenite finish temperature (Af) of 115�C, as foundfrom differential scanning calorimetry, was estimatedunder load-free conditions. The specimen used in theexperiments was fabricated from 10-mm-diameter rodsof varying lengths in the hot-rolled/hot-drawn and hot-straightened condition. Note that these conditions forheat treatments will have significant effects on theresulting thermomechanical behavior of the material(e.g. see Yoon and Yeo, 2008).

Nomenclature for control and response variable inthe experiments

In order to facilitate the presentation and discussion ofresults regarding cyclic behavior of the response, wepresent some nomenclature regarding one thermalcycle, that is, strain versus temperature response in atypical isobaric experiment, as shown in Figure 1. Inthis test, the specimen is loaded to achieve a certain ten-sile load level (hence producing a given constant tensileengineering stress). Subsequently, with the engineeringstress held constant, a number of thermal cycles areperformed between fixed values of lower-cycle tempera-ture (LCT) and upper-cycle temperature (UCT).

In the following, the constant engineering stress andcorresponding absolute true (logarithmic) strainresponse will be represented by symbols s and e,respectively (calculated as in the expressions given inFigure 1). As typically denoted in the literature, wedefine the following set of transformation characteristictemperatures under stress-free conditions, that is, themartensite start (Ms), martensite finish (Mf), austenitestart (As), and austenite finish (Af).

Regarding the observed conjugate strain responseduring thermal cycling, we utilize eM and eA to denotethe measured strain values specific to the two extremesof the temperature range, that is, eM is the correspond-ing strain at the LCT (martensite side) and eA is thecorresponding strain at the UCT (austenite side),respectively. Furthermore, eACT is used to represent theactuation strain calculated as the difference between eMand the succeeding eA for a particular heating branchduring any of the thermal cycles. Note that the signifi-cance of this eACT lies in its usefulness as a measure ofthe actuation capabilities of the SMA component andcan be used to roughly determine the amount of worka material can perform under a given condition.Similarly, we utilize eOLS to represent the open-loopstrain at the LCT (martensite side), also commonlyreferred to as irreversible strain. This is mathematicallydefined as the difference between two consecutive strain

Saleeb et al. 71

eM readings, that is, e(N + 1)M � e(N )

M , where N denotes thethermal cycle number and gives a measure of thedimensional stability of the system in actuation applica-tions (Padula et al., 2011). In all the subsequent figures,SI units are used for engineering stress, s (in MPa),absolute true (logarithmic) strain, e (in %), and tem-perature (in �C).

Observations under cyclic thermal loading

To best demonstrate the evolutionary character of the55NiTi material response, a specific isobaric test caseof constant tensile stress of 50 MPa with 100 thermalcycles, together with all its details, are described here(see Figure 2(a)). To this end, various stages of the evo-lution will be discussed in terms of the nature of theevolutions with continued thermal cycles, during whichthe material progresses from the initial transientresponse, moves into a more regulated strain evolution,and finally approaches a stabilized state.

As can be seen in Figure 2(b), there is a significantchange in strain eM due to the transient that occursduring the first thermal cycle. Subsequent to this transi-ent, the magnitude of open-loop strain is diminishedbut remains significant in the early part of evolution(cycles 2–20 in Figure 2(a) and (b)) in comparison tocycles in the intermediate range of evolution (cycles 20–50 in Figure 2(a) and (b)). Finally, as evident in the testmeasurements depicted in Figure 2(c) to (e), the strainevolution continues with an ever-diminishing rate, sig-naling an approach to a saturated state toward the endof the test (cycles 50–100 in Figure 2(c) and (d)).

Observations under variation of UCT

A test matrix was conducted utilizing different isobaricconditions (i.e. different stress magnitudes with varyingvalues of UCT) in order to investigate the effect of theUCT on the response evolution (courtesy: Dr Santo A.Padula II). In particular, two values of UCT (165�Cand 230�C) and four values of stresses (80, 100, 150 and200 MPa) were used to conduct extended cycle experi-ments. A summary of the results of this test matrix isshown in Figure 3(a) and (b), where cycles 5 and 80were selected to be representative of the early (‘‘soonafter transient’’) and later (‘‘nearing saturation’’) evolu-tion states, respectively.

There are two important observations that can bedrawn from the data mentioned above. First, in the‘‘soon after transient’’ regime, as denoted by e(5)M , highervalues of strain were obtained for lower values of UCT,thus signifying a slower evolution rate in the early stagefor higher values of UCT (see Figure 3(a)). Conversely,in the ‘‘nearing saturation’’ regime, as indicated by e(80)

M ,cycling to the higher UCT leads to higher accumulatedvalues of strain. Second, when the strains in the auste-nite (e(5)A and e(80)

A ) are considered, these same crossing(‘‘non-monotonic’’) patterns with respect to the effectof UCT are observed (see Figure 3(b)).

Observations under multistep isobaric test

In a more elaborate protocol, dubbed the multistep iso-baric test, the stress is varied after a certain number ofthermal cycles. In this case, the test input controls forboth the mechanical tensile load and the temperature

Figure 1. Representation of parameters used and their calculations for the analysis.LCT: lower cycle temperature; UCT: upper cycle temperature.

72 Journal of Intelligent Material Systems and Structures 24(1)

Figure 2. Thermal cycles on bias stress of 50 MPa with UCTof 165�C: (a) 100 thermal cycles (strain vs temperature response); (b)selected strain versus temperature response at initial, intermediate, and later thermal cycles; (c) variation of strains at martensite(eM) and austenite (eA) with cycles; (d) actuation strain (eACT) with cycles; and (d) open-loop strain (eOLS) at martensite with cycles(courtesy: Dr Santo A. Padula II, NASA GRC). (The reader is referred to the web version of this paper to see figures in color.)UCT: upper cycle temperature.

Saleeb et al. 73

changes for heating and cooling are shown in Figure4(a) and (b). The corresponding test output is the mea-sured strain response shown in Figure 4(c) as strain ver-sus temperature. The first step in the test consists ofloading to 75 MPa at room temperature (path 1-2 inFigure 4) followed by 10 thermal (heating/cooling)cycles between 30�C and 200�C at a constant stress of75 MPa (path 2-3-4-.-5, with point 5 indicating theend of the first 10 thermal cycles block). This was thenfollowed by unloading to 50 MPa at 30�C (path 5-6).Subsequently, in the second step, additional 20 thermal-cycles cycles block at a constant stress of 50 MPa wereperformed (see path 6-7-8-.-9, with point 9 indicatingthe end of the second 20 thermal-cycles block). Thisenables the study of the effect of prior history on thematerial evolutionary response (see Figure 4(c)).

Considering the results of this two-step isobaric test,a number of important remarks are in order. First,there are a number of isothermal ‘‘transients’’ thatoccur, such as the loading to 75 MPa at room tempera-ture (path 1-2), which produced a small amount ofstrain increase, and the unloading from 75 to 50 MPa at30�C (path 7-8), which produced a small amount of strainreduction. Second, several non-isothermal ‘‘transients’’were also observed. For example, this includes the firstheating/cooling cycle, in which the heating branch undera stress of 75 MPa (path 2-3) produced a small increase inthe strain, whereas the subsequent cooling branch (path3-4) resulted in a significant amount of strain increase. Incontrast, considering the case involving prior deformationhistory (i.e. after 10 thermal cycles at 75 MPa), the transi-ent behavior during the first thermal cycle at the reduced

Figure 3. Strains at (a) martensite (eM), and (b) austenite (eA) at 5th and 80th cycle of the thermal cycles performed till UCTof165�C and 200�C under different bias stress levels (courtesy: Dr Santo A. Padula II, NASA GRC). (The reader is referred to the webversion of this paper to see figures in color.)UCT: upper cycle temperature.

74 Journal of Intelligent Material Systems and Structures 24(1)

stress level (i.e. path 8-9 for its heating branch and 9-10for its cooling branch) led to a significant amount ofstrain reduction (termed here as ‘‘de-evolution’’), whereas

the subsequent thermal cycles (path 10-11-12-.-13) pro-duced a significant amount of strain increase (termed hereas ‘‘forward-evolution’’).

Figure 4. Two-step isobaric tests: (a) input mechanical load control, (b) input temperature control (between LCT = 30�C andUCT = 200�C), and (c) output strain versus temperature response (courtesy: Dr Santo A. Padula II, NASA GRC). Inset 1. Loadingfrom 0 to 75 MPa at LCT =30�C (path 1-2) followed by the first heating branch (path 2-3). Inset 2. Unloading from 75 to 50 MPa atLCT = 30�C (path 7-8). Insets 3 and 4. Details of response during the second step of the two-step isobaric test corresponding to (c).(The reader is referred to the web version of this paper to see figures in color.)LCT: lower cycle temperature; UCT: upper cycle temperature.

Saleeb et al. 75

In conclusion, there are a number of distinct experi-mental features and response patterns as observed inFigures 2 to 4. These include (1) the existence of signifi-cant transients, both in the first thermal cycle of eachisobaric test for a virgin specimen and during the firstheat following events of unloading at the LCT and (2)an apparent, ever-evolving, response with extendednumber of thermal cycles. These characteristic willprove pivotal in guiding the calibration of the presentSMA material model presented in section ‘‘Model cali-bration and characterization.’’

Brief review of the 3D, multimechanism-based viscoelastoplastic SMA model

As originally formulated, the multimechanism-basedviscoelastoplastic SMA model developed by Saleeb etal. (2011) has targeted the modeling of both ordinaryand high-temperature (where creep/relaxation andother allied time-/rate-dependent response aspectsbecome important) SMA material systems. However,in the present application to 55NiTi (with only a mod-erate range of transformation temperatures, that is, As

of 95�C 6 5�C and Af of 115�C), all high-temperatureeffects are discarded. In particular, it is assumed thatthe SMA material has a weakly rate-dependentresponse, with no viscous effects in the reversibleresponse component and no recovery terms in the evo-lution of the inelastic, kinematic hardening, state vari-ables. Furthermore, the inelastic response is assumed tobe purely deviatoric (i.e. transformation-induced strainis independent of the hydrostatic/volumetric stresscomponents), and the material is taken to be initiallyisotropic.

A Cartesian frame of reference is utilized in the deri-vation of constitutive equations, along with indicialnotation (where the summation is implied for repeated‘‘subscripts’’). Note that ‘‘superscript’’ letters placedbetween parentheses is used as indices to identify sets ofinternal state parameters, and when needed, the sum-mation over these will be indicated explicitly by thesummation symbol. Furthermore, an over dot (_s) sym-bol denotes the rate of change of s with respect to timefor the parenthetical quantity.

In the generalized 3D space, the total strain tensor isdecomposed into elastic and inelastic parts. The straintensor eij (and its rate _eij) is decomposed into reversible(elastic) and irreversible (inelastic) components, ee

ij andeI

ij, respectively. In particular, the tensor eIij is utilized

here to implicitly account for all transformation-induced deformations; that is, forward/reverse transfor-mations between A and M phases at higher tempera-ture, the detwinning of M-phase variants at lowertemperature, as well as reorientations, variant coales-cence, and other allied effects under non-proportionalstates of stress and strain.

The stress tensor, sij, is decomposed into an effective

stress, sij � aij

� �, and internal state tensorial variable,

aij =PN

b= 1

abð Þ

ij , where N indicates the number of inelas-

tic mechanisms whose internal stress-like, and conju-

gate strain-like, internal variables are denoted as a(b)ij

and g(b)ij , respectively, for b = 1, 2, ., N mechanisms.

These are utilized to regulate the energy storage (b =1, 2, 3) and energy dissipations (b = 4, 5, ., N) duringthe evolution of the thermomechanical response of thematerial. Following the discussions in Saleeb et al.(2011), we have assumed here that N = 6 for simplicityin the present initial characterization of 55NiTi, butthis may be expanded on in the future.

There are two fundamental energy potentials, that is,a Gibb’s complementary function, F=FRðsijÞ+FIRðsij,a

bð Þij Þ, and a dissipation function, Oððsij � aijÞ,

aðbÞij Þ, where the subscripts R and IR indicate reversible

(elastic) and irreversible (inelastic) components, respec-tively. Note that the parentheses utilized in F above sig-nify functional dependency in terms of the tensorialarguments inside these parentheses. In all the subsequentequations in this section, the same notations for func-tional dependency will be utilized. Note that based on theassumptions introduced in the beginning of this section,both the internal stress and conjugate strain tensors arepurely deviatoric. With this specific form selected for theabove potential functions, all the governing evolutionary(rate) equations for the inelastic strains and internal vari-ables in the individual mechanisms can be derived follow-ing the procedure detailed in Saleeb et al. (2011). Forconvenience, a summary of the resulting final equation inthe model is given in Tables 1 (basic equations) and 2(transformations and hardening functions).

Finally, note that there are 37 material parameters inthe general forms given in Tables 1 and 2. Among these,there are 14 material parameters (c and d in transforma-tion function and c(b) and d(b) for each of b = 1, 2, .,6) used for the detailed evolutions under different load-ing modes, such as tension, compression, or shear.However, for the present calibration and characteriza-tion of 55NiTi, all the data available are collected undertensile loading (see subsection ‘‘Observations under cyc-lic thermal loading’’).

Therefore, these last 14 material parameters dealingwith tension/compression asymmetry will not be activehere. Furthermore, the elastic modulus, E, andPoisson’s ratio, n, are assumed based on typical dataavailable in the literature for NiTi–based SMA system(see Table 3). In addition, because of the assumed weakrate dependency of the SMA response, we have electedto use a ‘‘large’’ value for ‘‘n’’ and relatively ‘‘smallvalue’’ for ‘‘m’’ to achieve this as recommended inSaleeb et al. (2011) (see Table 3). This leaves a total of21 remaining constants that need to be determined

76 Journal of Intelligent Material Systems and Structures 24(1)

from the calibration procedure described in the follow-ing section.

Model calibration and characterization

Ideally, the calibration procedure of a comprehensivematerial model with an extensive amount of test datashould be automated using sophisticated optimizationtechniques (e.g. see Saleeb et al., 2001, for applicationsto conventional metallic materials). However, such anautomated characterization procedure is not currentlyavailable for the present multimechanism SMAmodel. Instead, we have introduced a number of sim-plifying assumptions in order to facilitate the majoreffort involved in the manual characterization proce-dure utilized here. This is particularly important for acomplex SMA material such as 55NiTi that is knownto be a high evolver and notorious for the stress state-dependency of its thermomechanical response.

Until more detailed information can be obtained, wehave reduced the temperature dependency and stressstate dependency to include only 7 key threshold values(among the available total of 21 material parameters).These are the values k in the transformation functionand k(b) for mechanisms b = 1–6 in the evolution

equations of the internal state variables. Furthermore,we have assumed a multiplicatively-decoupled formfor the temperature dependency and stress state

dependency; that is, k=k(r)(T ) � h(se) and k(b) =

k(r)(b)(T ) � h(b)(se), where T is the temperature and the

multiaxial stress intensity is defined as se =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 sijMijklskl

� ��2

q. In this, k(r) and k

(r)(b) are reference

thresholds (all in stress units) that are a function of tem-perature only, and h (or h(b)) is a nondimensional factor

that is dependent on the stress intensity only.For the purpose of determining the temperature varia-

tion of these aforementioned variables, together with thespecific numerical values of the remaining parametersH(b) and b(b), results from a single isobaric experimentconducted at 100 MPa for 100 thermal cycles between anLCT = 30�C and a UCT = 165�C were utilized for cali-bration. In connection with this ‘‘trial-and-error’’ manualcalibration exercise, several iterations were made toimprove the matching of the model response to the testdata, starting from the initial assumed values of thematerial parameters taken from Saleeb et al. (2011) (spe-cifically Table 2 in that reference). The final outcomefrom this calibration exercise has led to the selected val-ues of the material parameters reported here in Table 3(for H(b) and b(b)) and Table 4 (k(r) and k

(r)(b)). Once the

temperature variations of the reference thresholds wereestablished, estimates for the values of the nondimen-sional, stress-dependent factors h and h(b) were deter-mined from the results of three additional isobaric tests;that is, 10 MPa with 10 thermal cycles to UCT = 165�C,50 MPa with 100 thermal cycles to UCT = 165�C, and80 MPa with 100 thermal cycles to UCT = 165�C. Theobtained values are given in Table 5.

Finally, a number of important remarks are in orderhere regarding the above calibration/characterization pro-cedure. First, from the point of view of qualitative results,an effort was made to capture the two most importantcharacteristics of the evolutionary response exhibited inthe test data; that is, a significant transient change duringthe cooling branch of the first thermal cycle, followed bya gradual evolution with diminishing rates on approach-ing a ‘‘nearly’’ stabilized/attraction state. Second, consid-ering the quantitative comparison, a compromise wasmade with preference given to matching the measuredexperimental response in later cycles of every test (com-pared to the early and intermediate counterparts).

Comparative study

In this section, two different types of information willbe compared. First, the calibrated model response willbe compared to the experimental response used for itscalibration. To this end, two isobaric tests involving vir-gin specimens undergoing 100 thermal cycles underconstant engineering stresses of 50 and 100 MPa were

Table 1. Summary of basic equations of the multimechanismSMA material model.

Equation set 1: decomposition of stress and strain

eij = eeij + eI

ij; aij =PN

b= 1

abð Þ

ij

Equation set 2: specific functional forms for stored energyand dissipation potentials

FR sij

� �= 1

2 sijE�1ijkl skl;

FIR sij,abð Þ

ij

� �=sije

Iij +

XN

b= 1

�H bð ÞGbð Þ

O sij � aij

� �,a

bð Þij

� �=

Zk2Fn

2mdF

Equation set 3: Evolutionary laws

_eij � _eIij =

ddt

∂FR

∂sij

� �=E�1

ijkl _skl; _eIij =

∂O∂sij

_abð Þ

kl = ∂2FIR

∂abð Þ

ij ∂abð Þ

kl

� �1

_gbð Þ

ij ; _gbð Þ

ij = � ∂O∂a

bð Þij

where k, m, and n are material constants, F is thetransformation function (see Table 2), and Eijkl is the isotropicfourth-order tensor of elastic moduli (Young’s modulus, E, andPoisson’s ratio, n).

SMA: shape memory alloy.

Saleeb et al. 77

utilized. Second, the results from a set of multistep iso-baric experiments were compared to the model’s pre-dictions, thus providing a validation test for the presentmultimechanism model.

Cyclic evolutionary response at constant biasengineering stress

The SMA model was used to simulate the experimentalresponse of 55NiTi subjected to 100 thermal cyclesunder the two different isobaric conditions used in themodel calibration. In one case, the system was firstloaded at room temperature (23�C—martensitic load-ing) until a stress of 50 MPa was reached, followed by

100 heating and cooling branches (100 thermal cycles)between LCT of 30�C and UCT of 165�C. In the othercase, a similar loading sequence was followed with theonly difference being that 100 MPa stress value wasused instead.

For the tests conducted at 50 and 100 MPa, respec-tively, the evolutionary responses during thermalcycling are shown in Figures 5(a) and 6(a) and are com-pared to their model prediction counterparts in Figures5(b) and 6(b). Qualitatively, it is noted that the modelwas able to predict the overall evolution of strains rea-sonably well in both tests. However, from the point ofview of quantitative accuracy, the model captured theseevolutions well for the later stage of evolution, but

Table 2. Summary of transformation and hardening functions used in the multimechanism SMA material model.

Equation set 4: transformation and hardening functions

F sij � aij

� �=

1

k2

1

2r2sij � aij

� �Mijkl skl � aklð Þ

� ,

�H bð Þ=k2

bð ÞR

1�h g bð Þð Þ dG bð Þ, for b= 1, 2, 3;

k2bð ÞR

1h G bð Þð Þ dG bð Þ, for b � 4;

8<:

h g bð Þ� �

=

r bð Þk bð ÞH bð Þffiffiffiffiffig bð Þp� � b bð Þ�1ð Þ

k bð Þ+H bð Þffiffiffiffiffig bð Þp� �b bð Þ , for b= 1, 2,

r bð ÞH bð Þ 1+

ffiffiffiffiffig bð Þp

k bð Þ=H bð Þ

�b bð Þ" #

, for b= 3,

8>>>>><>>>>>:

h G bð Þ� �

=H bð Þ 1�ffiffiffiffiffiffiffiffiG bð Þp

r bð Þ

!b bð Þ

h Lð Þ

24

35, for b � 4;

where G bð Þ abð Þ

ij

� �= 1

2k2bð Þ

abð Þ

ij Mijklabð Þ

kl

� �; g bð Þ g

bð Þij

� �=g

bð Þij g

bð Þij ; r= 1+ c

ffiffidp

1+ cffiffiffiffiffiffiffiffiffiffid+ k3

p ; r(b) =1+ c(b)

ffiffiffiffiffid(b)p

1+ c(b)ffiffiffiffiffiffiffiffiffiffiffiffiffiffid(b) + k(b)

3

pk3 = cos 3u; k(b)3 = cos 3u(b); where u and u(b) are Lode’s angle calculated from the invariants of the stress sij and the internal

stress of the bth mechanism, abð Þ

ij (Chen and Saleeb, 1994).

h Lð Þ is the Heaviside function with argument being the loading index L=abð Þ

ij G ij; where .

G ij = ∂F�∂ sij � aij

� �Mijkl =

12 dikdjl + dildjk

� �� 1

3 dijdkl ; with dij is the Kronecker delta.H(b),b(b) and k(b) are the material parameters for the individual hardening mechanism and c, d; c(b), d(b) are the material parametersfor tension/compression asymmetry (Saleeb et al., 2011).

SMA: shape memory alloy.

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

Parameters Units Value

Viscoelastic mechanisms‘‘Deflated’’ elastic stiffness modulus, E MPa 60,000Poisson’s ratio, n — 0.3Viscoplastic mechanismsNumber of viscoplastic mechanisms — 6k=k(r) � h MPa see Tables 4 and 5n — 5m MPa�s 1.00E+ 05

k(b) =k(r)(b) � h MPa see Tables 4 and 5

b(b), b= 1, 2, . . . , 6 — 1, 1, 10, 10, 1, 2.5

H(b), b= 1, 2, . . . , 6 MPa 400 3 103, 300 3 103, 200, 41 3 103, 31 3 103, 600

78 Journal of Intelligent Material Systems and Structures 24(1)

overestimated the amount of strain accumulations dur-ing the early stages of thermal cycling. For a morecomprehensive assessment of these aspects, the data forstrains at the LCT (eM) and at the UCT (eA) wereextracted for each cycle and plotted for both the experi-ment and model predictions (see Figures 5(c) and 6(c)).In addition to this, actuation strains (eACT) and open-loop strains (eOLS) were calculated and plotted for com-parison against those from the experiment (see Figures5(d) and (e) and 6(d) and (e)). The open-loop strainwas observed to decrease with cycles, similar to theexperimental observation (Figures 5(e) and 6(e)). Thisis typical of the response observed during a practical‘‘training’’ protocol of an SMA as it approaches a satu-rated response; that is, the open-loop strain approacheszero and the actuation strain reaches a fixed value(Padula et al., 2011; Jones and Dye, 2011).Furthermore, the magnitude of actuation strain for thecase of 50 MPa was significantly overpredicted in theearly stages of evolution but approached the magnitudeobserved in the experiment at the later stages of cycling(compare summary data in Figure 5(d)). For the testconducted at 100 MPa, both model predictions andexperimental measurements showed comparable actua-tion strains, albeit the details about their evolution dif-fer somewhat (Figure 6(d)). This is in stark contrast towhat was observed for 50 MPa, where the actuationstrains differed in the early part of cycling (comparesummary data in Figures 5(d) and 6(d)). Recall that the

model characterization procedure utilized here (asdescribed in the preceding section) was intentionallydesigned to favor better accuracy of the rate of strainevolution in the later stages of the thermal cycling. Thisis evident by inspecting the summary data in Figures5(c) and 6(c).

Multivariable load-bias, three-step, cyclic thermalresponse

The multivariable load-bias, three-step test is utilizedhere for model validation. This test includes three stepsof loading and thermal cycling, as shown in Figure 7.

In the first step, the virgin 55NiTi material specimenis loaded to a stress of 75 MPa at room temperature(23�C), followed by 10 thermal cycles between an LCTof 30�C and a UCT of 200�C. In the second step, thesame specimen was unloaded to 50 MPa at 30�C (iso-thermal) and 20 thermal cycles were performed at 50MPa to the same UCT as was used for cycling at 75MPa. In the third step, the same specimen was reloadedfrom 50 to 100 MPa at 30�C (isothermal) and 15 ther-mal cycles were performed at 100 MPa to the sameUCT used before.

Both the experiment (Figure 7(a)) and the model pre-dictions (Figure 7(b)) show the usual transient behaviorin the ‘‘early’’ (first 10) thermal cycles of the 75 MPaportion of the response. When unloaded to 50 MPa, thestrain at the LCT was lowered by a small amount. The

Table 4. Temperature dependency of material parameters k(r) and k(r)(b), for b = 1, 2, ., 6.

Parameter(s) Value (MPa) Remarks

T1 = 20�C T2 T3 T4 = 200�C

k(r) 20 (temperature independent)

k(r)(b), b= 1, 2 9.2 0.2 62.2 53.7 For b = 1,2: T2 = 65�C, T3 = 115�C

k(r)(b), b= 3 1.00E+ 21

k(r)(b), b= 4 90 0.001 0.001 400 for b = 4: T2 = 50�C, T3 = 120�C

k(r)(b), b= 5 14 (temperature independent)

k(r)(b), b= 6 52 (temperature independent)

For each parameter, 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 nondimensional factors h and h(b).

Stress (MPa) h h(b), b= 1, 2 h(b), b= 3, 4 h(b), b= 5 h(b), b= 6

0 0.05 0.05 1.00 (stress independent) 0.05 0.0525 0.1 0.18 0.22 0.250 0.15 0.48 0.55 0.575 0.65 0.75 0.75 0.6580 0.7 0.8 0.8 0.7100 1 1 1 1125 1.35 1.2 1.2 1.9

Values are linearly interpolated between shown stress levels.

Saleeb et al. 79

Figure 5. Cyclic response for an isobaric test at bias stress of 50 MPa and a UCTof 165�C: (a) experimental (courtesy: Dr SantoA. Padula II, NASA GRC), (b) 3D multimechanism SMA model, (c) strain at martensite (eM) and austenite (eA) with cycles,(d) actuation strain (eACT) with cycles, and (e) open-loop strain (eOLS) at martensite with cycles. (The reader is referred to the webversion of this paper to see figures in color.)UCT: upper cycle temperature; SMA: shape memory alloy; 3D: three-dimensional.

80 Journal of Intelligent Material Systems and Structures 24(1)

Figure 6. Cyclic response for an isobaric test at bias stress of 100 MPa and a UCTof 165�C: (a) experimental (courtesy: Dr SantoA. Padula II, NASA GRC), (b) 3D multimechanism SMA model, (c) strain at martensite (eM) and austenite (eA) with cycles,(d) actuation strain (eACT) with cycles, and (e) open-loop strain (eOLS) at martensite with cycles. (The reader is referred to the webversion of this paper to see figures in color.)UCT: upper cycle temperature; SMA: shape memory alloy; 3D: three-dimensional.

Saleeb et al. 81

subsequent 20 thermal cycles performed at 50 MPashowed an intermediate (near to saturation) evolutionresponse in both cases. Furthermore, when reloaded to100 MPa, the strain at the LCT was increased by asmall amount and the subsequent 15 thermal cycles per-formed at 100 MPa showed intermediate evolutionarybehavior. This behavior was comparable to the type ofevolution observed in the later cycles of the 100 thermalcycles test performed from the virgin state. The varia-tions of the actuation strain (eACT) and strain at theLCT (eM) and UCT (eA) during this three-part sequenceare depicted by the corresponding time histories inFigure 8(a) and (b) and Figure 9(a) and (b), respec-tively. Note that in Figure 8(a) and (b), the lower peaksconstitute the values of eA as they vary with cycles, andsimilarly, the upper peaks define the changes in eM. It isalso interesting to note the difference in the magnitudesof the actuation strains at the two different stress levels,as exhibited by both the experiment (Figure 8(a)) andthe model predictions (Figure 8(b)). Considering thedetails of strain evolution in the summary plots inFigure 9, there are several discrepancies in the modelpredictions compared to the test results. In particular,the model underestimates the rate of evolution in the 10cycles conducted at 75 MPa and overestimates the tran-sient strain decrease after unloading to 50 MPa. As aresult, the level of strain at the beginning of the 50 MPathermal-cycle block was lower than observed experi-mentally. This, coupled with the observation that boththe experiment and the model attempted to evolve tothe same ‘‘attraction’’ (i.e. stabilized strain state) at theend of the 20 thermal-cycles block, required a higherrate of evolution in the model as compared to theexperiment (see Figure 9). Similarly, the overestimation

in the model of the transient reloading from 50 to 100MPa, together with an unquantified 100 MPa attractionstrain state, produced a slower strain evolution rate inthe model than was observed experimentally. Fromthese observations, it is clear that the richer data contentfrom one or more multi-step tests should be utilized dur-ing the model’s calibration procedure in future efforts.

Finally, considering the combined results presentedin Figures 7 to 9, a number of remarks are in orderhere. Both experimental measurements and model pre-diction indicate significant strain changes in the twomajor transient regimes. In particular, observe the com-parable results for the first heating and coolingbranches, following the unload from 75 to 50 MPa instep 2 and following the reload from 50 to 100 MPa instep 3 (see summary data in Figures 8 and 9). However,as was observed in the previous subsection, there is anoticeable overprediction of the strain evolution in theearly thermal cycling stages after a transient hasoccurred. Despite this overestimation at the beginningof each of the three steps, the model predictionapproached similar values to those of the experiment atthe later stages of thermal cycling in the individualsteps themselves. Once more, this is a byproduct of thebias toward achieving better accuracy during latercycles at the expense of less accuracy in the early partsof the evolution.

Effect of prior history on thethermomechanical response

The additional results presented in this section are allobtained from the model simulation. They are aimed at

Figure 7. Multivariable load-bias, three-step (75-50-100 MPa) with UCTof 200�C, thermal cyclic test, strain versus temperatureresponse: (a) experimental (courtesy: Dr Santo A. Padula II, NASA GRC) and (b) 3D multimechanism SMA model. (The reader isreferred to the web version of this paper to see figures in color.)UCT: upper cycle temperature; SMA: shape memory alloy; 3D: three-dimensional.

82 Journal of Intelligent Material Systems and Structures 24(1)

demonstrating the effect of prior load histories on theevolutionary SMA response. To this end, we considerdifferent loading scenarios in the following subsections.

Transient and evolutionary responses

The first loading scenario is labeled ‘‘A’’ and is utilizedin this section. It involves a loading condition similarto the first two steps of the three-step isobaric testdescribed earlier in subsection ‘‘Multivariable load-bias, three-step, cyclic thermal response,’’ except for thenumber of thermal cycles conducted at the reduced

stress of 50 MPa in the second step, that is, here 100cycles (instead of the 20 cycles performed before). Inaddition to the results shown for scenario A (Figure10(b), (d), and (f)), we have also included the resultsfrom a model simulation for an isobaric test case at 50MPa conducted from the virgin state for comparison(Figure 10(a), (c), and (e)). Note that only part of sce-nario A has an experimental counterpart that was usedduring the model validation presented in the precedingsection.

Although similar in pattern, the strain evolution forscenario A, which corresponds to a test with prior history,

Figure 9. Multivariable load-bias, three-step (75-50-100 MPa) with UCTof 200�C, thermal cyclic, evolution of strain with cycles atmartensite (eM) and austenite (eA): (a) experimental and (b) 3D multimechanism SMA model. (The reader is referred to the webversion of this paper to see figures in color.)UCT: upper cycle temperature; SMA: shape memory alloy; 3D: three-dimensional.

Figure 8. Multivariable load-bias, three-step (75-50-100 MPa) with UCTof 200�C, thermal cyclic test, strain with cycles: (a)experimental and (b) 3D multimechanism SMA model. (The reader is referred to the web version of this paper to see figures in color.)UCT: upper cycle temperature; 3D: three-dimensional.

Saleeb et al. 83

Figure 10. 3D multimechanism SMA model thermal cycles (temperature vs strain response) in isobaric test at 50 MPa up to UCTof200�C: (a) first thermal cycle (on virgin material), (b) first thermal cycle (on material with prior history of 10 thermal cycles at 75 MPaon virgin material), (c) 10 thermal cycles (on virgin material), (d) 10 thermal cycles (on material with prior history of 10 thermal cyclesat 75 MPa on virgin material), (e) 100 thermal cycles (on virgin material), and (f) 100 thermal cycles (on material with prior history of10 thermal cycles at 75 MPa on virgin material). (The reader is referred to the web version of this paper to see figures in color.)UCT: upper cycle temperature; SMA: shape memory alloy; 3D: three-dimensional.

84 Journal of Intelligent Material Systems and Structures 24(1)

Figure 11. 3D multimechanism SMA model thermal cycles (temperature vs strain response) in isobaric test up to UCTof 200�C:(a) 10 thermal cycles at 75 MPa (first loading step in scenario B), (b) 100 cycles at 75 MPa (first loading step in scenario C), (c)subsequent 20 thermal cycles (in red) after unload to 50 MPa (second loading step in scenario B), (d) subsequent 20 thermal cycles(in blue) after unload to 50 MPa (second loading step in scenario C), (e) unload to 50 MPa followed by three thermal cycles showingthe direction of evolution in arrows in scenario B, and (f) unload to 50 MPa followed by three thermal cycles showing the directionof evolution in arrows in scenario C. (The reader is referred to the web version of this paper to see figures in color.)UCT: upper cycle temperature; SMA: shape memory alloy; 3D: three-dimensional.

Saleeb et al. 85

shows some major differences with respect to the virgincase. Considering the transient response during the firstthermal cycle, the response during the heating branch ofthe virgin material at 50 MPa indicates a small rise andsubsequent drop in strain during transformation, but thereis no significant change in strain magnitude at the comple-tion of heating (Figure 10(a)). In contrast to this, thecounterpart response of the prestrained material in sce-nario A exhibited significant strain reduction during heat-ing (Figure 10(b)). On the other hand, the actuation strainresponses of both the virgin and the prestrained materialwere comparable, irrespective of the thermomechanicalhistory differences mentioned above (Figure 10(c) to (f)).

Considering next the short-term evolutionaryresponse during the initial 10 cycles, it is observed thatthe rate of evolution in scenario A is slower relative tothe virgin material case, as shown in Figure 10(d) and(c), respectively. As a result, the total accumulatedchanges in both eM and eA in scenario A are muchsmaller compared to the virgin case as the number ofthermal cycles increased. Remarkably, considering thislong-term response, it is this very difference in strainaccumulation that enabled both conditions (the virginmaterial and the scenario A) to finally approach thesame ‘‘nearly saturated/attracted’’ strain state at theend of the thermal cycles (Figure 10(e) and (f)).

Figure 12. 3D multimechanism SMA model cyclic response (temperature vs strain) in isobaric test up to UCTof 200�C: (a)combined plots of 20 thermal cycles in each of the loading scenarios B and C, showing the corresponding directions of evolution ineach case by arrows (red for scenario B and blue for scenario C), (b) combined plots with continued cycling (100 and 30 thermalcycles in scenarios B and C, respectively), (c) plots of strain at martensite (eM) and austenite (eA) with thermal cycles correspondingto (b), and (d) plots of strain at martensite (eM) and austenite (eA) for scenarios B and C showing the extended cycles necessary toapproach a unique ‘‘attraction’’ deformation state. (The reader is referred to the web version of this paper to see figures in color.)SMA: shape memory alloy; UCT: upper cycle temperature; 3D: three-dimensional.

86 Journal of Intelligent Material Systems and Structures 24(1)

Approach to attraction

To demonstrate the nature of the approach to attraction,we consider two different loading scenarios ‘‘B’’ and ‘‘C.’’The conditions in scenario B are similar to scenario A inthe preceding subsection, except for the number of ther-mal cycles conducted at the reduced stress of 50 MPa inthe second step; that is, here 20 cycles (instead of the 100cycles performed in scenario A) were conducted. In con-trast, scenario C involved 100 thermal cycles at the stressof 75 MPa (instead of the 10 cycles used in scenario B),followed by unloading to 50 MPa and cycling for 20 ther-mal cycles at that reduced stress.

Starting from the virgin state, the first 10 cycles ofscenario B were compared to the first 100 cycles of sce-nario C, and the corresponding differences in strainaccumulation are shown in Figure 11(a) and (b), respec-tively. Figure 11(c) and (d) shows the evolution duringthe second step for the two scenarios. As can be seen,scenario B initially evolved to lower values of strain butquickly reversed and began evolving to higher valueswith diminishing levels of eOLS. This is evident by the‘‘hook-like’’ arrows (see Figure 11(c) and (e)) that indi-cate the reversal in the ‘‘direction’’ of strain accumula-tion. Conversely, scenario C exhibited only decreases instrain accumulation (de-evolution) during all thermalcycles in the second step (Figure 11(d) and (f)).

Note that in Figure 12(a), scenarios B and C showevolution that has not reached a final attraction state.In order to investigate the nature of the attraction char-acter, additional thermal cycles were applied to each ofthe scenarios to determine whether a common attrac-tion point would be observed. For this purpose, Figure12(b) and (c) shows the strain evolutions using a totalof 100 and 30 thermal cycles in scenarios B and C,respectively. Furthermore, thermal cycling was contin-ued by adding 100 and 170 more thermal cycles in sce-narios B and C, respectively (thus giving a total of 200thermal cycles in each of the two scenarios at 50 MPa).Figure 12(d) shows the response corresponding to theseextended cycles for scenarios B and C. Based on theseresults, the cyclic responses for the two different load-ing scenarios are indeed approaching what appears tobe a unique ‘‘attraction’’ deformation state.

Conclusion

This work focused on the investigation of the evolu-tionary aspects of a complex SMA material responseunder cyclic, thermomechanical loading conditions.There are two important contributions here. First, thework provided an initial attempt to use a rather exten-sive set of test data from recent cyclic experiments onthe binary 55NiTi material with moderately high trans-formation temperatures (As of 95�C 6 5�C and Af of115�C at stress-free state). Among other high evolvers,this SMA material system is known to rank at the top.

Second, for the purpose of calibration/characterizationof this material system, the present application utilizesa recently published material model, based on thenotion of multiple mechanisms to regulate the parti-tioning of energy storage and dissipation governing theevolution of the thermomechanical response. In view ofthe results presented, the calibrated model has beenclearly shown to be quite successful in capturing anumber of key, cyclic, response characteristics in bothsingle and multistep isobaric tests. Chief among theseare (1) the ability to account for significant transientchanges (e.g. upon first cooling for the virgin specimenand the first heating following an unloading event for aprestrained material); (2) diminishing rate of evolutionduring intermediate thermal cycles; (3) an approachtoward a ‘‘nearly’’ saturated/attraction state with con-tinued thermal cycling; (4) stress dependency of the dif-ferent amounts of the martensite strain (eM), austenitestrain (eA), open-loop strain (eOLS), and actuationstrain (eACT); and (5) the effect of the prior deforma-tion history on the cyclic response is significant.

Acknowledgements

The authors would like to acknowledge Drs S. M. Arnold andRonald Noebe for their technical guidance and programmaticsupport during the different phases of the project.

Funding

This work was supported by the Fundamental AeronauticsProgram, Subsonic, Fixed-Wing, Project No. NNH10ZEA001N-SFW1, Grant No. NNX11AI57A to the University ofAkron.

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