Timescale competition dictates thermo-mechanical responses of NiTi shape memory alloy bars

(1)

Timescale competition dictates thermo-mechanical responses of NiTi shape memory alloy

bars

Zhuo, Mingzhao

DOI

10.1016/j.ijsolstr.2020.02.021

Publication date

2020

Document Version

Final published version

Published in

International Journal of Solids and Structures

Citation (APA)

Zhuo, M. (2020). Timescale competition dictates thermo-mechanical responses of NiTi shape memory alloy

bars. International Journal of Solids and Structures, 193-194, 601-617.

https://doi.org/10.1016/j.ijsolstr.2020.02.021

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ContentslistsavailableatScienceDirect

International

Journal

of

Solids

and

Structures

journalhomepage:www.elsevier.com/locate/ijsolstr

Timescale

competition

dictates

thermo-mechanical

responses

of

NiTi

shape

memory

alloy

bars

Mingzhao

Zhuo

a,b,∗

a Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China b Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 9 September 2019 Revised 21 January 2020 Accepted 12 February 2020 Available online 17 February 2020

Keywords:

Thermo-mechanical coupling Shape memory alloy Phase transition

Thermodynamic driving force Loading-rate effect Timescale ratio

a

b

s

t

r

a

c

t

NiTishapememoryalloys (SMAs)exhibitdistinctthermo-mechanicalbehaviorsaffectedbytheloading frequency,ambient conditions,and thespecimen geometry. Theeffects ofthesefactors areessentially dueto the competition ofdifferent timescales inphase transitions ofNiTi SMAs. However, quantify-ingthetimescale competitionstill remainsachallengefor SMAssubjectedtoforce-or displacement-controlledcyclicloadings.Herewepresentathermo-mechanicallycoupledmodelforone-dimensional SMAbarstoaddresstheeffectsoftimescalecompetitiononthethermo-mechanicalresponses.Scaling themodelgivesadimensionlessnumberλindicatingtheratiooftheloadingtimetothe characteris-tictimeofheattransfer(affectedbyambientconditionsandthespecimengeometry).Themodelshows thatitisthetimescaleratioλthatdictatesthethermo-mechanicalresponses.Comparisonofsimulation resultswithexperimentaldatavalidatesthecoupledmodelandtheeffectsofthetimescaleratioλon thethermo-mechanicalresponses.ThecoupledmodelcanpredicttheresponsesofSMAsunderdifferent combinationsofexternalloadingsandambientconditionsandthusprovideguidelinesforexperimental design.

1. Introduction

Superelastic NiTi shape memory alloys (SMAs) have been widelyusedinpracticalapplicationsfrombiomedicalstentsto vi-bration control devices (Lagoudas, 2008; Jani et al., 2014; Zhuo et al., 2019) and thereby attracted extensive research. The ther-mal and mechanical behaviors of SMAs are intrinsically coupled through phase transitions (Shaw, 2000; Yin et al., 2014; Morin et al., 2011b). In spite of broad constitutive models (Matsuzaki and Naito, 2004; Cisseet al., 2016) available forSMAs, modeling of the timescale competition during SMA phase transitions (He et al., 2010; He and Sun, 2011; Yin et al., 2013, 2014) still re-quires further improvement to consider general external loading modes. For insights into the timescale competition and its im-pact on SMA responses, we present a thermo-mechanically cou-pled model with explicitterms expressing the coupling between the two ﬁelds and the timescale competition, and the model is tailoredspeciﬁcallytoaone-dimensionalSMAbarunderforce-or displacement-controlledcyclicloadings.

∗ _{Corresponding author.}

E-mail address: mzhuo@connect.ust.hk

The mechanical and thermal ﬁelds are fully coupled during phase transitions of NiTi SMAs. On the one hand, the stress-induced phasetransitions causetemperaturevariation. Under ex-ternal loadings, SMA phase transitions are accompanied by re-lease(austenite to martensiteA → M) andabsorption (M→ A) oflatent heat (Auricchio and Sacco,2001; Bernardini and Pence, 2002;Auricchioetal.,2008;Morinetal., 2011a;Yinetal.,2014). Apartfromthelatentheat,theintrinsicdissipationofmechanical energy, manifested as the stress strain hysteresis, is always con-vertedintothermalenergyasanotherheatsource,whichisabout one order of magnitude smaller than the latent heat (Yin etal., 2014). The released/absorbed heat at the domain front of phase transitions will transfer via conduction within the specimen and through convection between the material andthe ambient envi-ronment(Sunetal., 2012), thus leadingtotemperature variation inthe specimen.On the other hand, thetemperature change af-fects the stress strain responses of NiTi SMA bars. According to Clausius-Clapeyronrelation(Yinetal., 2014), thetransitionstress istemperature-dependent:underisothermalconditions,thehigher the ambient temperature, the higher the transition stress (Yin etal., 2013, 2014). It is remarked that for generalthermo-elastic materials, the mechanicaland thermalﬁelds are actually slightly https://doi.org/10.1016/j.ijsolstr.2020.02.021

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coupled (Gough-Joule effect) (Schweizer and Wauer, 2001; Het-narskiandEslami,2009),butthecouplingisnegligiblecompared to that from phase transitions. The thermo-mechanical coupling accountsforexperimentalobservationssuch asthestrong depen-denceofthermo-mechanicalresponsesonloadingrates(Ortíinand Planes, 1989; Leo et al., 1993; Shawand Kyriakides, 1995, 1997; Entemeyeretal., 2000;LimandMcDowell,2002;Auricchioetal., 2008; He andSun, 2010a, 2010b; Zhang et al., 2010; Sun et al., 2012;Morinetal.,2011a,2011b;Grandietal.,2012)(especiallyin cyclicloadinganddeformation(Yinetal.,2014)),onambient con-ditions(Leo etal., 1993;ShawandKyriakides,1995; Iadicolaand Shaw,2004;Shawetal.,2008;Mirzaeifaretal.,2011;Grandietal., 2012),andonthespecimengeometry(Shawetal.,2008; Mirzaei-faretal.,2011).

To understand these observed effects, different timescales in phasetransitionsneedtobeidentified.ForanSMAbarunder ten-sileloading, two timescales exist in the mechanical field:one is the characteristic time for stress wave propagation in the speci-men (Shaw et al., 2008); the other is the characteristic time of loading. Usually, the loading timescale is much longer than that ofthestress wave propagationintensiletests, rendering thebar in mechanicalequilibrium at each time instant. This situation is calledthequasi-staticloading,underwhichthestresscanbesafely assumed uniform throughout the bar. Associated with the ther-malfieldarethecharacteristictimeofheatconductionwithinthe specimenand the characteristic time of convective heat transfer. Thermalfield isintrinsically heterogeneous duetoformation and propagation of localized phase-transition domains (Shaw, 2000; SunandLi,2002; Feng andSun,2006; Churchill etal., 2009;He andSun,2010b;Zhangetal.,2010).Basedonexperimental obser-vations(Yinetal.,2014),transitiondomainsrandomlyforminthe specimen andheat conduction is much faster than the heat re-lease/absorptionat domain fronts. Thus, we ignore the timescale ofheatconductionandthespatialheterogeneityoftemperaturein thespirit ofthelumped analysis(CottaandMikhailov, 1997;Yin etal.,2014).Inconclusion,thetwotimescalesofloadingand con-vectiveheattransferare ofmajor importance,andtheir competi-tionisresponsiblefortheaforementionedeffects.

Anumberofconstitutivemodels(togetherwiththeheat equa-tion and suitable numerical techniques) have been developed to describe the thermo-mechanical responses of SMAs. Liang and Rogers (1990) and Brinson (1993) developed a one-dimensional constitutive model to describe thermo-mechanical behavior of SMAs, reproducing pseudo-elastic and shape memory effects.

Abeyaratne and Kim (1997) presented a one-dimensional model forSMAsunder cyclicloadings. Bernardini andPence (2002) de-rivedmodelsformacroscopicbehaviorofSMAsbasedonafree en-ergyfunction andadissipationfunctionandpresented thermody-namicdrivingforcestoaccount forthehysteresis. Bernardiniand Rega (2017) recently proposed a comprehensive thermomechani-calmodeling framework andcompared theperformancesof vari-ous less-reﬁnedSMA models. In particular, theauthors delivered agermanediscussiononthethermomechanicalcouplingbehavior.

AuricchioandSacco(2001)andAuricchioetal.(2008)established a uniaxial constitutive model based on Helmholtz free energy withaninternalscalarvariable—themartensitefraction;themodel consideredlatentheatandmechanicaldissipationsimultaneously.

ChristandReese(2009)proposed athermo-mechanicallycoupled SMA model in the framework of large strains, with the tension compression asymmetry considered. Morin et al. (2011a) modi-ﬁed the three-dimensional ZM model(Zaki and Moumni, 2007a, 2007b; Moumni et al., 2008) to take into account thermo-mechanical coupling, and then implemented the coupled ZM model into a ﬁnite element code for simulation of a supere-lastic SMA cylinder. The coupled model was also used to com-pare simulations with experimental results of SMA wires under

cyclicloading(Morinetal.,2011b).Lagoudasandcoworkers(Boyd andLagoudas, 1996; Lagoudas etal., 2012) presented a three di-mensional thermo-mechanical model that captured the smooth transition in the thermal and mechanical responses and added stressdependencytotheconceptofcriticalthermodynamic force.

Grandietal.(2012) proposedaone-dimensionalGinzburg–Landau modelforthemacroscopicbehaviorofSMAs.Theinﬂuencesofthe strain rate andambient conditions on the responses were high-lighted.Yuetal.constructedacrystalplasticitybasedconstitutive model(Yuetal., 2013) to describethe cyclicdeformationof NiTi SMAsandthen extended itto describerate-dependent cyclic de-formation (Yu et al., 2014) by considering the internal heat pro-duction andtemperatureevolution. Lateron they proposed a3D thermo-mechanicalmodel(Yuetal., 2015) toconsiderdislocation slippinginaustenitephase.Armattoeetal.(2016)proposeda cou-pled thermo-mechanical model forSMAs focusing on latent heat effectsduringforwardandreversephasetransformations.

The above coupled models can be implemented to simulate theeffectsofthe multiplefactors:the loadingrate,ambient con-ditions, and the specimen geometry (Morin et al., 2011a, 2011b; Grandietal.,2012;Mirzaeifaretal., 2011).However,theseeffects were oftenstudied individually.Typically, thethermo-mechanical responses(e.g.,stressstraincurves,temperatureevolution,andthe hysteresis)wereshownasfunctionsoftheappliedstrainratewith theheat transfercoefficient andspecimenradius fixed atvarious values. For example, the non-monotonic strain-rate dependence curve ofthehysteresis wasshownto movetowards higherstrain rates when the heat transfer coefficient increases (Morin et al., 2011a;Grandietal.,2012)buttowardslowerratesifthespecimen radius increases(Morinetal., 2011a).These observations suggest the potential toincorporate theloading rate, ambientconditions, andthespecimengeometryintoasingleparameterforbetter un-derstandingofthegoverningmechanism.

Actually, recentexperimental and modeling studies (He et al., 2010; He and Sun, 2011; Yin et al., 2013, 2014) have shown that the effects of these factors are attributed to the timescale competition during SMA phase transitions. He et al. (2010) and

Yinetal.(2013)experimentallyshowedthenon-monotonic depen-denceofthehysteresis onthetimescalecompetitionthat reﬂects the strain rate and ambient conditions. He and Sun (2011) pre-sented a model to explain and quantify the strain-rate depen-dence of the hysteresis: they solved the heat transfer equa-tion and calculated the hysteresis from the temperature pro-ﬁle. Yin et al.(2014) reported systematicexperimental results of the thermo-mechanicalresponses at awide rangeof frequencies. These studies found that the frequency-dependent variations in temperature, stress, and hysteresis are determined by the com-petition betweenthetime oftheheatrelease/absorption (i.e.,the phasetransitiontime)andthetimeoftheheattransfertothe am-bient. However, to connect the phase transitiontimescale to the externalloadingtimescale,themodelinginthesestudiesrelieson theassumptionthatthelatentheatrelease/absorptionlinearly de-pendsontheappliedstrainrate.Thisassumptionignoresthetime forelasticdeformationandthusisonlysuitedtothe displacement-controlled loadingbutnot tothe force-controlledcase. Moreover, theevolutionofthestressstraincurvewasnotexplicitlymodeled andpredicted.

The timescale competition is crucial to advancing our un-derstanding of the resultant effect of the previously mentioned three factors, but it still remains a challenge to incorporate all the factors into the timescale competition by developing a cou-pled model that allows for general external loading modes and predicts both the mechanical and thermal responses. This study aims to quantify the timescalecompetition by presenting a one-dimensional model (Section 2) with clearly-formulated coupling terms, followed by scaling of the thermo-mechanically coupled

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Fig. 1. Series model of a one-dimensional SMA bar under tensile loading. Here ξ

represents martensite volume fraction.

model (Section 3.1).Built onmanyof theideasinthe previously cited literature, the one-dimensional modelconsists ofthe stress strain relation (Section 2.2), the heat equation (Section 2.5), and evolution rules (Section 2.4) to describe the progress of phase transitions. Scaling the governing equations permits us to derive a dimensionless number—thetwo-timescaleratio

λ

(Section 3.1). Thenwestudytheeffectsof

λ

onthethermalandmechanical re-sponses, inparticular thedampingcapacity(Section 3.2). Forthe purposeofvalidation,wecomparethemodelpredictionswith ex-perimental results (Section 4). The numerical procedures (return mappingalgorithm(SimoandHughes,1998))forsolvingthe gov-erningequationsareoutlinedinAppendixC.

2. Thermo-mechanicallycoupledmodel

Hereweconsideraone-dimensionalbarunderquasi-static ten-sileloading.Underthissettingwe assume uniformaxialstress

σ

inthebar. Withthelumpedanalysis(CottaandMikhailov.,1997; Yin et al., 2014), we further assume a uniformthermal field and use the average temperature T to describe the whole bar. Spe-cific Gibbs free energy of an SMA bar (Section 2.1) is calcu-latedby the ruleof mixtureswith martensitevolumefraction as the internal variable; the stress strain relation (Section 2.2) and heatequation (Section2.5)arethus derivedaccordingtothefirst andsecond lawsofthermodynamics (Appendix A).To modelthe stress strain hysteresis,we assume positive andnegative thermo-dynamic driving forces (Section 2.3) in the forward and reverse phase transitions, respectively; therelation between the thermo-dynamicdrivingforceandmartensitevolumefraction(Section2.4) is proposed on an empirical basis. Finally, prescribed strain or stress(Section2.6)isalsoincludedtocompletethecoupledmodel.

2.1. GibbsfreeenergyofanSMAbar

Foralinearelasticbar,thespeciﬁcGibbsfreeenergyisreadily availableinreferences(HetnarskiandEslami,2009)andexpressed hereas g =−

σ

₂_E2 −

α

(

T− T0

)

σ

+c σ

T − T0− Tln T T0

+u 0− Ts 0, (1) where E isthe Young’s modulus,

α

is thethermal expansion co-eﬃcient, cσ is the speciﬁc heat capacity (per unit volume) at a constant stress, andu0 ands0 are the initialinternal energy and entropy,respectively,perunit volumeatreferencetemperatureT0 inthestress-freestate.Herethestress

σ

referstotheaxialstress asascalar,notatensorinthethree-dimensionalsetting.

σ

+c Aσ

T − T0− Tln T T 0

+u 0A− Ts 0A. (2) However,martensiteshould besplit intoa linearelasticpartand atransformedpartthataccountsforthetransformationstrain.The workdoneonthetransformationstrainisnotdissipatedbutstored aspotentialenergyinthetransformedpartofmartensite.Itwillbe releasedafterthecompletionofthereversephasetransitionfrom martensiteto austenite. Gibbs free energyof martensiteper unit volumeisthengivenby

g M =−

σ

2 2E M −

α

M

(

ξ

g M+

(

1−

ξ

)

g A =−

σ

2 2E

(

ξ

)

−

α

(

T− T0

)

σ

−

ξ

L

σ

+c σ

T− T0− Tln T T 0

+u 0

(

ξ

)

− Ts 0

1−

ξ

)

s 0A. (7) In Eq.(4), we neglect the smalldifference in thermal expansion coeﬃcient

α

andspeciﬁcheatcapacitycσbetweenmartensiteand austeniteandassumetheyarethesameforbothphases(Lagoudas, 2008;Morinetal.,2011a).

2.2.Constitutiverelations

Substituting Gibbs free energy of the SMA bar Eq. (4) into

Eq.(A.11),weobtainthefollowingconstitutiveequations

=

σ

E

)are deﬁnedinEqs.(5)and(7),respectively. Unlike that stress

σ

and temperature T are assumed uniform, strain

in Eq.(8) and entropys in Eq. (9)are volume-averaged valuesofthewholebar.

2.3.Thermodynamicdrivingforce

Inview ofEq.(A.11),theClausius-Planck inequality (A.10)will bereducedto

D m=−

∂

g

∂ζ

:

ζ

˙≥ 0. (10) For the SMA bar, martensite volume fraction

ξ

can be consid-eredastheinternalvariable(AuricchioandSacco,2001;Zakiand Moumni,2007b;Lagoudasetal.,2012)toreplace

ζ

.InEq.(10),we candeﬁnetheenergyderivativetermasthethermodynamic driv-ingforce

forthe forwardphase transition(A→ M), conjugate totheinternalvariable

ξ

inthesimilarwayasEq.(A.11):

=−

∂

g mix

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Fig. 2. Stress-temperature ( σ-T ) phase diagram (a) and control driving force versus martensite volume fraction ξ(b). In panel (a), M s and M f are martensite start and

finish temperatures for A → M phase transition, while A s and A f are austenite start and finish temperatures for M → A phase transition. In panel (b), = −T ss0 at point C and = −T fs0 at point D. IJ represents the loading/unloading path in the case of incomplete phase transitions ( Appendix B ). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Thus,thephysicalmeaningofthethermodynamicdrivingforce is theGibbsfreeenergydifferencebetweenthetwophases.To calcu-latetheGibbsfreeenergydifference(11),wesubtractEq.(3)from

Eq.(2)andobtain

(

σ

, T

)

=

σ

L− T

s 0+

u 0, (12)

where

s0=s0A− s0M and

u0=u0A− u0M. Here we ignore the smallelastic strain energydifference betweenthe two phasesso thattheycancancelouteachother.

Substituting the thermodynamicdriving force

into Eq.(10), weobtainamoreconciseformoftheClausius-Planckinequality:

D =

_ξ

˙_{≥ 0}_. ₍₁₃₎

)

, (14) where

−

α

(

T − T0

)

σ

−

ξ

L

σ

+c σ

T− T0− Tln T T 0

+u 0

(

ξ

)

− Ts 0

(

ξ

)

−

ξ

(

1−

ξ

)

, (15) andthethermodynamicdrivingforceischangedto

(

σ

, T

)

=

+

u 0−

(

2

ξ

− 1

)

, (16)

where

is deﬁned as the control driving force (Bernardini and Pence,2002)

=

σ

L− T

s 0, (17)

asalinearcombinationofthetwostatevariables

σ

andT. Atequilibrium,theminimizationofthetotalGibbsfreeenergy

gmix requiresthethermodynamicdrivingforcetobezero(

=0), whichgivesthe relation(evolutionrule) betweenmartensite vol-umefractionandcontroldrivingforce:

ξ

=

+

u 0+

2

. (18)

Thenullthermodynamicdriving force,orEq.(18),corresponds to atransitionbandzonedelimitedbytwodashedboundarylinesin the

σ

− T phasediagram(Fig.2a).Inthestress-freestate,the for-wardphase transitionstartsatTs with

ξ

=0andends atTf with

ξ

=1.SubstitutingthesetwoconditionsintoEq.(18)gives

u 0=

Ts+Tf

2

s 0 and

=

T s− Tf

2

s 0. (19)

Thus,theevolutionrule(18)changesto

ξ

=

(

σ

, T

)

+T s

s 0

(

T s− Tf

)

s 0 ,

(20) whichexactlyrepresentslineCDinFig.2b.

The phase transitionlineCD (or

=0in essence)represents a reversible loading-unloading path and no hysteresis is formed duringtheloading-unloading cycle.The nullenergydissipation is howeverincontrasttotheexperimentalobservationofa hystere-sisloopinthestressstraincurveduringphasetransitions.Dueto thefrictional forcesresistingthe motionofinterfacesandenergy barrierbetweenthetwophases,extraenergyisalwaysneededto overcometheseobstaclesandtoproceedthephasetransition.

Tomodel thehysteresis, we choosea positive thermodynamic drivingforcefortheforwardphasetransitionandanegativeforce for the reverse phase transition, in accordance with the non-negativedissipation restriction (13).Hence, in Fig.2a, the transi-tionbanddelimitedbythetwodashedlinesissplitintotwoband zones: the left-hand one with

> 0is for A→ M phase tran-sition andtheother with

< 0for M→ Aphase transition. In thecaseofnullstress,A_→Mphasetransitionstartsatmartensite starttemperatureMsandendsatmartensiteﬁnishtemperatureMf, while M → A phase transition starts ataustenite start tempera-tureAsandendsatausteniteﬁnishtemperatureAf(Brinson,1993; BrinsonandHuang,1996;Auricchioetal.,2008).

AccordinglyinFig.2b,thereversibleloading-unloadingpathCD is split into the forward loading path andthe reverse unloading

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Table 1

Governing equations for an SMA bar under prescribed loadings.

1. prescribed stress or strain: a _σ₌σmax

2 ( 1 − cos ωt) or =

max

2 ( 1 − cos ωt) ; (30)

2. stress strain equation: = σ

E(ξ)+ ξL + α( T − T 0) ; (8) 3. heat equation: b _c σT˙ = −T ασ˙ + T s0ξ ˙ + ξ˙ − hγ( T − T 0) ; (29) 4. evolution rules: ξFT = σ L − T s0 + M ss0 ( M s − M f)s0 , ξRT = σ L − T s0 + A fs0 ( A f − A s)s0 . (21)

a The loading can be applied in other format ( i.e. , the triangular function with a constant loading rate). b_{is given in}_{Eq. (16)}_{and constants therein are given by}_{Eq. (24)}_.

path. The two linear loading andunloading paths indicate linear relationsbetweenthecontroldrivingforce

andmartensite frac-tion

ξ

.Speciﬁcally,analogoustoEq.(20),theevolutionruleforthe forwardphasetransitionis

ξ

FT=

(

σ

,

T

)

+M s

s 0

(

M s− Mf

)

s 0 ,

(21a) andthatforthereversetransitionis

ξ

RT=

(

σ

,

T

)

+A f

s 0

(

A f− As

)

s 0 ,

(21b) where

=

σ

L− T

s0 is restrictedto −Ms

s0<

<−Mf

s0 in Eq.(21a)andto_−A_f

s0<

<−As

s0inEq.(21b).

Now we havefourmaterialparameters to determine:

L,

s0,

u0 and

. The transformation strain

L can be directly mea-sured from experiments, while

s0 is calculated through the slope k of the straight lines (representing constant

values) in the

σ

− T phase diagram (Fig. 2a). The slope k is the coeﬃcient in Clausius-Clapeyron relation and can be obtained from experi-ments(Yinetal.,2014).Thus

s0 iscalculatedas

s 0= k

L. (22)

To determine constants

u0 and

, we assume equal dissipated energy in the forward and reverse phase transitions. This as-sumptionsuggeststhatinFig.2b,thereversibleloading-unloading pathCDcutthehysteresis loopBGEFinhalf,namelythatCisthe middle point of BF and D is the middle point of GE. Therefore, the transformationtemperaturesassociated withCandDare ex-pressedas T s= M s+A f 2 and T f= M f+A s 2 , (23)

respectively.AccordingtoEq.(19),weobtain

u 0=

M s+M f+A s+A f

4

s 0, (24a)

= M s− Mf+A f− As

4

s 0. (24b)

The four transformation temperatures (Ms, Mf, As, and Af) in the stress-free state can be measured from experiments. Con-sidering that phase transitions are determined by the control driving force

(linear combination of

σ

and T, Eq. (17) and

∇

· qrepresentstheoutﬂowof heatthroughtheboundariesofaparticle.Sincethelumped anal-ysis(CottaandMikhailov.,1997)isadopted,weconsidertheSMA barasa whole andusethevolume-averaged heat outﬂowto re-place the local term. To this end, we apply the volume average to

∇

_{· q}anduseGausstheoremtoarriveat

1 V

∇

· qdV = 1 V ∂q · ndS = 1 V hA

(

T − T0

)

=h

γ

(

T − T0

)

, (26) where h is the average coeﬃcient of convective heat transfer,

γ

=A/V is thesurfacearea(A) tovolume(V) ratio,andT0 isthe ambienttemperature.Thevolumeaverageof

∇

· qrepresentsthe heatoutﬂowfromtheSMAbartotheambientenvironment.Since thebarhasnointernalheatsource,theheatsourcetermrcanbe discarded.NowwecanrewriteEq.(25)as

Ts ˙=

_ξ

˙_{− h}

_γ

₍

_T_{− T}₀

₎

_. ₍₂₇₎ Taking the time derivative of the constitutive relation for en-tropy(Eq.(9))andmultiplyingbothsidesbyT,wehave

Ts ˙= T

α

σ

˙ +c σT˙− T

s 0

ξ

˙. (28) Letting the right-hand sides ofEqs. (27) and(28) be equal gives theheatequation:

c σT ˙ =−T

α

σ

˙ +T

s 0

ξ

˙+

ξ

˙− h

γ

(

T− T0

)

. (29) The physical meaning of each term in heat equation (29) is as follows:

1.cσT˙ representsthethermalenergychangerate;

2.−T

α

σ

˙ represents the stress effect on temperature (thermo-elasticeffect)(SchweizerandWauer,2001);

3.T

s0

ξ

˙ denotesthelatentheatcausedbyentropydifference; 4.

ξ

˙ denotes the dissipated mechanical energy manifested by

thehysteresisloop;

5.h

γ

(

T− T0

)

representstheheatoutﬂowtothesurroundings. AsshowninEq.(29),thelatentheatisreleased(T

s0

ξ

˙>0)in theforwardphasetransitionandabsorbed(T

s0

ξ

˙<0)inthe re-versetransition,whilethehysteresisheatisalwaysreleased(

ξ

˙>

0).

2.6.Prescribedloading

Sofarwe havederivedthreeequations:thestressstrain equa-tion(8),heatequation(29),andevolutionrules(21).Themodelis howevernot completedyet.Inexperiments,eithertheaxialforce orenddisplacementwillbeprescribed;accordingly,wecan spec-ifythestressas

σ

)

, (30b)

where

ω

istheangularfrequency,and

σ

maxand

maxarethe max-imumstressandstrainapplied,respectively.Itisremarkedthatwe donotsolvetheequationofequilibriumsinceitisnaturally satis-ﬁedundertheassumptionofuniformstress.Inthe displacement-controlledcase, theenddisplacementisusedtocalculatethe av-eragestrainofthewholebar.

3. Resultsanddiscussion

We first summarizeall thegoverningequations inTable1 for easierreferenceinthefollowingdiscussions.Inthecoupledmodel, there are two types of coupling between the mechanical and thermal fields. The first type is the general thermo-elastic ef-fect(SchweizerandWauer, 2001) thatis manifestedby the ther-malexpansionterm

α

(

T− T0

)

inEq.(8)andthestress rateterm −T

α

σ

˙ inEq.(29).Thesecondtypeiscausedbyphasetransitions: thetransformationstrain term

ξ

L inEq.(8)andthe latentheat termT

s0

ξ

˙ (plusthedissipationterm

ξ

˙) inEq.(29)showthat the two ﬁelds are coupled through the progress of phase tran-sitions. The strong thermo-mechanical coupling phenomena ob-servedinSMAs(Yinetal.,2013,2014)ismainlycausedbyphase transitions, while the general thermo-elastic effect is relatively negligible(SchweizerandWauer,2001).

In the thermo-mechanical model, each ﬁeld involves a timescale. Because of the coupling,the competition betweenthe two timescales results in distinct responses. By scaling the gov-erningequationswecanderiveadimensionlessnumber—the two-timescale ratio (Section 3.1). The effects of the timescale ratio onthermo-mechanicalresponses(Section3.2)andhysteresisloop area(Section3.2.2)arethenstudied.

3.1.Timescaleanalysis

Thetimescalesofmechanicalandthermalﬁeldsaremanifested inEqs.(30)and(29),respectively.Wedenotetheloadingtime,half of a loading-unloading period, astd=

π

/

ω

and regard it as the characteristictimescaleofmechanicalloading.Fortriangular load-ingfunctions,thecharacteristictimescalet_d issimplythetimeto reachthemaximumstress/strain.Letting

τ

=t/tdintheprescribed stress/strain(Eq.(30)),weobtain

σ

=

σ

max

2

(

1− cos

πτ

)

, (31a)

=

max

2

(

1− cos

πτ

u0 and

aregiveninEq.(24),and

λ

isexpressedas

λ

= t d

c σ/

(

h

γ

)

.

To see the physical meaning of the denominator term, we just keeptheheattransferterm−h

γ

(

T− T0

)

intheright-handsideof Eq.(29)anddiscardother terms.ThusthesolutionofEq.(29)has theformof

T= T0+T1e −t/th, (33)

whereT1 isa constant dependingon theinitial condition,andth isexpressedas

t h=

c σ

h

γ

. (34)

Therefore,t_h representsthecharacteristictime ofconvective heat transfer.Thephysicalmeaningof

λ

ishencetheratioofthe load-ingtimetothecharacteristictimeofheattransfer:

λ

=t d

t h.

(35) Speciﬁcally,foranSMAbarofradiusR,thesurfaceareatovolume ratiois

ω

are ﬁxed, changing

γ

also impacts the thermo-mechanical responses, which iscalled the effectof surface area to volume ratio (determined by geometrical shapeandsize).

We can see that the loading rate effect, the effect of ambi-entconditions, andtheeffectof surfacearea to volume ratioare all due to the two-timescale competition through the thermo-mechanicalcoupling.

If h is held close to 0 or _∞, varying

ω

has no effect on the thermo-mechanical responses, i.e., no rateeffect. This is also re-marked by Ivshin and Pence (1994) based on their model. Un-der isothermal condition (h _→ _∞), no strain rateeffect was ob-served in experiments by Grabe and Bruhns (2008). Neverthe-less, the rate-independent responses under nearly adiabatic con-dition(h_→0),aspredictedbyourmodel,stillneedfurther exper-imentalvalidation.

3.2. Effectsofthetimescaleratio

The single parameter

λ

reﬂects the resultant effect of the threeimpacting factors—the loadingrate, theambientconditions, and the surface area to volume ratio. Thus it is possible to ex-haust the thermo-mechanical responses under various conditions by studying the timescale ratio effect. To this end, we simulate the stress strain curves and temperature variation at different

λ

valuesto consider scenarios ranging from the adiabatic to the isothermal conditions. We consider two types ofloading: stress-controlled(Eq.(31a))andstrain-controlled(Eq.(31b)).Material pa-rametersarefromTables2and3andtheyarecalibratedwith ex-perimentsinSection4.1.TheambienttemperatureT0 iskept con-stantatroomtemperature 25°C.

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3.2.1. Onthermo-mechanicalresponses

Fig.3 showsresultsunderthe strain-controlledloading.Three typical

λ

values—1000, 3.2, and 0.01—are chosen to show re-sponses under the isothermal condition, an intermediate heat transfercondition,andtheadiabaticcondition,respectively.In par-ticular,thethree

λ

valuescorrespondtothe threesolid pointsA, B, and C in Fig. 5a. The isothermal case is realized by a large

λ

value (1000) that suggestsvery strongheat exchange withthe ambient environment and thus leads to negligible temperature variation. The bar temperature almost remains constant at the roomtemperatureasshownbytheyellowlineinFig.3b. Accord-ing to Clausius-Clapeyron relation (Yin et al., 2014), no tempera-turechange,nochangeinthephasetransitionstress.Therefore,a stressplateauisobservedinthestressstraincurve(yellowlinein

Fig. 3a). Tobe precise, inthe phase transitionprocesses a small stresschangeoccursduetotheassumptionofthephasetransition band(Fig.2a),whichindicatesdifferentstartandﬁnishstressesof

Table 2

Calibrated model parameters for displacement-controlled (cyclic) loading.

Parameter symbol value unit

Young’s modulus of austenite EA 25 GPa

Young’s modulus of martensite EM 19.4 GPa

temperature dependence of transition stress k 6.8 MPa/K transformation strain L 0.043 -

speciﬁc heat capacity cσ 3.2 × 10 6 J/(m 3 K)

thermal expansion coeﬃcient α 11 × 10 6 _1/K

austenite → martensite start temperature Ms 262.5 K

austenite → martensite ﬁnish temperature M f 261.0 K

martensite → austenite start temperature As 281.0 K

martensite → austenite ﬁnish temperature A f 283.0 K

Table 3

Calibrated model parameters for force-controlled (cyclic) loading. Other model parameters not listed here take the same values as in Table 2 .

Parameter symbol value unit

transformation strain L 0.04 -

austenite → martensite start temperature Ms 259.5 K

austenite → martensite ﬁnish temperature Mf 258.0 K

martensite → austenite start temperature As 273.0 K

martensite → austenite ﬁnish temperature Af 276.0 K

phasetransitionsatthesame temperature.The isothermal condi-tioncanbeachievedinexperimentsbyveryslowloadingandfast ﬂowingairaroundthespecimen(Heetal.,2010;Yinetal.,2014).

The adiabatic condition is approximately achieved by a small

λ

value(0.01),suggestinganegligibleheatexchangewiththe sur-roundings.When the heat exchange isdisregarded, the tempera-ture will change in accordance with the latent heat release and absorption, aswell as thehysteresis heat accumulation(Fig. 3b). Astheforwardphasetransitionstarts,thetemperaturealsostarts to increase (latent heat release) until the end of the transi-tion (II), and then it keeps constant during the unloading pro-cessofmartensite(III).Duringthereversetransition(IV),the tem-perature keepsdecreasing (latent heat absorption) andthen lev-els off (V). At the end of unloading, the latent heat absorption cancels out its release and the accumulated hysteresis heat re-sultsin a smalltemperature rise,which is about5% ofthe tem-perature increase in phase II caused by the latent heat release. This percentage agrees with the direct experimental measure-ment:thevolumetric latentheatl0=7.74× 107J/m3 isabout20 timeslargerthan thesteady-state hysteresisDs=3.61× 106J/m3 inYin etal. (2014). Inparallel withthe temperaturechange, the transitionstress increasessteeply above the stress plateauin the forward phase transition, followed by a parallel stress decrease duringthereverse transition.The stressattheendofthereverse transition is slightly higher than the isothermal transition stress due to the corresponding slight temperature rise. The hysteresis loopisthusslightlysmallerthantheisothermalone.Itisremarked thatweobserveasmalltemperaturedrop—causedbythe thermo-elasticeffect—attheendoftheloadingofaustenite(I). Neverthe-less,thetemperaturedropissosmallthatthethermo-elasticeffect isnegligiblecomparedtothephasetransitioneffect.

When the latent heat release/absorption and heat exchange with the surroundings are comparable (

λ

₌3_.2), the thermo-mechanical responses differ in the following aspects. First, the temperature increases moderately and startsto decreases in the middle of the forward phase transition (the peak in phase II). Astemperatureincreases,thelargertemperaturedifferenceT− T0 enhancesthe heat exchange to the extent that the heat outﬂow outweighs the latent heat release. Second, in the reverse transi-tion(IV),thetemperatureplummetsandisdrivenbelowtheroom temperature, which is due to the fact that the heat released in

Fig. 3. Effects of the timescale ratio λon the stress-strain ( σ-) curve (a) and temperature evolution (b) of an SMA bar under strain-controlled loadings. The loading- unloading cycle is generally divided into ﬁve phases: the loading of austenite (I), the forward A → M phase transition (II), unloading of martensite (III), the reverse M → A phase transition (IV), and unloading of austenite (V).

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Fig. 4. Effects of the timescale ratio λon the stress-strain ( σ-) curve (a and c) and temperature evolution (b and d) of an SMA bar under stress-controlled loadings. The maximum stress ( σmax in Eq. (31a) ) is 520 MPa for (a) and (b), and 312 MPa for (c) and (d). In panel b, the temperature evolution curve for λ= 0 . 001 is divided into ﬁve

phases (in the upper): the loading of austenite (I), the forward A → M phase transition (II), loading and unloading of martensite (III), the reverse M → A phase transition (IV), and unloading of austenite (V). For λ= 2 . 9 , the ﬁve phases I-V are denoted in the bottom. In panel d, the ﬁve phases for λ= 20 and 0.001 are referred to Fig. 3 b.

theforwardtransition isnot completely storedbutlargely trans-ferred out. Third, during the unloading process of austenite (V), thetemperatureincreasesslightlybecauseoftheheatinﬂuxfrom thesurroundings,butthiscompensationissosmallthattheﬁnal temperatureafterafullloading-unloadingcycleisstilllowerthan theroomtemperature. Finally,thetransitionstress isincreasingly higher than the isothermal plateau stress in the forward phase transitionbutlower duringthe reverse phase transition. The en-closedhysteresis loopisthuslargerthantheisothermalone(it is thelargestasshowninFig.5).

valuesareclosetoeachother,duetothesynchronizedincrease ofphasetransitionstressinphaseII. Apparently,thetemperature increase under the adiabatic conditionin Fig. 4d is smaller than thatinFig.4bbecauseoftheincompletephasetransition.

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Fig. 6. Experimental ( Yin et al., 2014 ) and ﬁtted stress-strain ( σ-) curves of an SMA bar under isothermal (25 °C) condition.

3.2.2. Ondampingcapacity

AnimportantquantityofSMAsisthedampingcapacity,which isthehysteresis looparea inthestress strain curveofa loading-unloadingcycle. AsshowninFigs. 3and4,themaximum damp-ingcapacityoccursatanintermediate

λ

value,inagreement with thenon-monotonic variation of hysteresis versus the loading fre-quencyin experimental studies (He et al., 2010; Yin etal., 2013, 2014; Morin et al., 2011a, 2011b; He and Sun, 2011). Here we systematically show the variation of hysteresis as a function of thetimescaleratio

λ

understrain-controlled(Fig.5a)and stress-controlled(Fig.5bandc)loadings.Foreachloadingcase,weshow the hysteresis of the ﬁrst and steady-statecycles. It is remarked that for lower

λ

values, it takes more cycles for the thermo-mechanicalresponsestoreachthesteadystate.

fre-quencyisobserved inexperimentsof Yinetal.(2014)), the tem-peraturegoesbacktotheroomtemperatureaftertheﬁrst loading-unloadingcycle.Therefore,thefollowingcycleswillrepeattheﬁrst one,andallthestressstraincurvescoincidewitheachother (sub-plot3).At

λ

=0.01,thetemperatureishigherthantheroom tem-peratureaftertheﬁrstloading-unloadingcycle(Fig.3b);therefore, theaccumulatedheatwillfurthergrowinthefollowingcyclesand makethestressstraincurvedriftupwardswithreducinghysteresis loopareaasshowninsubplot4.

. Thiscontrasting change pattern is responsible forthe differenthysteresis changetrends inthe two sidesofthe peakpoint.

4. Comparisonwithexperiments

Thissectionappliesthecoupledmodeltosimulateexperiments for the purpose of validation. We ﬁrst calibrate model parame-tersusing thedata ofisothermal testsof NiTiSMAsinYin et al. (2014) andYin(2013),andthencompare modelpredictionswith experimental results (Yin et al., 2014), in terms of stress strain curves,temperatureevolution,andhysteresisloopareaunder var-iousloadingfrequencies.

4.1. Modelparametercalibration

InFig.6,redcirclesshowtheisothermal(atroomtemperature 25°C)stressstraincurveofanSMAbarsubjectedtouniaxial ten-sion(Yin etal., 2014),whilethesolid bluelineistheﬁttedcurve bythecoupledmodel.Heretheisothermalconditionismaintained byaverylowloadingfrequency.Itisshownthatexternalloading ﬁrstcausestheelastictensionofaustenite(AB),andfurther load-ingleadstotheforwardphasetransitionfromausteniteto marten-site(BC), followedby elastic loading of martensite(CD). Reverse loadingcauseselasticunloadingofmartensite(DE),andcontinued unloadingisaccompaniedbythereversephasetransition(EF)and subsequentelasticunloadingofaustenite(FA).

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Fig. 7. Stress-strain ( σ-) curves of the ﬁrst and last cycles (a) and temperature evolution (b) at a low frequency 0.0 0 07 Hz for displacement-controlled cyclic loading. In panel (a), the red solid line is covered by the blue line. Experimental data are from Yin et al. (2014) .

From the experimental stress strain curve, we directly obtain thefollowing parameters:Young’s modulusofausteniteEA (slope of AB), Young’s modulus of martensite EM (slope of DCE), and the transformation strain

isfrom Table5.1ofLagoudas(2008).Theseparametersareusedtoﬁtour modeltotheexperimental isothermalstress straincurve, andthe ﬁttedcurvewellcapturesthetrendoftheexperimentalcurve.All calibratedparametersaresummarizedinTable2.

4.2. Displacement-controlledcyclicloading

Pronouncedcouplingphenomenahavebeenobservedespecially in cyclic loading-unloading of SMA bars at various loading fre-quencies (Yin et al., 2014). In the following, we compare pre-dicted thermo-mechanical responses by our model with experi-mental data of displacement-controlled cyclic tests in Yin et al. (2014).Speciﬁcally,thecyclicstressstrain curvesandtemperature evolutionareofinterest.Inexperiments,thedisplacementuis pre-scribedas

u =u max

2

(

1− cos2

π

f t

)

, (36)

whereumaxisthemaximumdisplacementandfistheloading fre-quency, relatedto the angular frequencyby

ω

₌2

π

f . Formodel prediction,theparameters usedarethoseinTable2.Theambient temperatureT0 iskeptconstantatroomtemperature25°C.Next, resultsatthreedifferentloadingfrequenciesfarecompared.

Fig. 7 shows the stress strain curvesof the ﬁrst and last cy-cles (3 cycles in total) and temperature evolution at a low fre-quency 0.0007 Hz. For the numerical simulation, the character-istic time of heat transfer takes t_h=31s according to Fig. 3 in

Yin etal.(2014); thetimescaleratioisthus calculatedas

λ

=23. Sincetheloading-unloadingismuchslowerthantheheattransfer,

thetemperaturevariesslightlyandgoesbacktotheroom temper-atureafterthe forwardandreverse phasetransitions inthe sim-ulation.Thetemperaturevariationsinallloading-unloadingcycles arethe same,andhencethe stress straincurves ofthethree cy-cles coincide witheach other. A discrepancybetween the model prediction and experimental results in Fig. 7a is that, the stress duringtheforwardphasetransitionmonotonicallyincreasesin ex-perimentsbutdeceasesattheendoftheforwardphasetransition accordingtothesimulation.Thisstressdecreaseinmodel predic-tionis causedby thetemperaturedecrease (Fig. 7b)in thelatter halfoftheforwardtransition:accordingtoClausius-Clapeyron re-lation,thetransformationstress decreaseswithatemperature de-crease.

Fig. 8 shows the stress strain curves and temperature evolu-tionatanintermediatefrequency0.04Hz.Forthenumerical sim-ulation, we take t_h=30s (Yin et al., 2014) and hence

λ

=0.42. ThetemperaturevariationinFig.8bshowsan evolutionfromthe transient stage to the steady state with decreasing mean tem-perature. After the ﬁrst cycle the temperature is lower than the room temperature because the heat transfer is neither as strong astocompensate thelatentheat absorption inthereversephase transition (as occurs in Fig. 7) nor as weak as to save the la-tent heat released in the forward phase transition (as occurs in

Fig. 9). With increasing loading cycles, the temperature decrease accumulatesand ﬁnally reaches an equilibrium. Due to the tem-perature decrease, the stress strain curve drifts downwards in

Fig.8abecauseofthetemperaturedependenceofthe transforma-tionstress(Clausius-Clapeyronrelation).

Fig. 9 shows the thermo-mechanical responses at a high frequency 1 Hz. In numerical simulation we use th=26s (Yin et al., 2014) and hence

λ

=0.02. The

λ

value is so small that theheat transferwith theambientenvironment can be dis-regarded(i.e.,adiabaticcondition).ThusthetemperatureinFig.9b oscillatesdueto thelatentheat releaseandabsorption inthe re-versiblephasetransitions.Themeantemperaturekeepsincreasing becauseoftheaccumulationofhysteresisheat.Thethermaleffect on the mechanical responses reﬂects in the upward shift of the stressstraincurvefromFig.9atob.Itcanbeseenthatourmodel cancaptureessentialfeaturesofthetemperatureevolutionandthe trendofthestressstraincurvechange.

4.3.Force-controlledcyclicloading

Besidesthe displacement-controlled tests,we further compare ourmodelpredictionwithexperimentaldatafromforce-controlled

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Fig. 8. Stress-strain ( σ-) curves of the ﬁrst and last cycles (a) and temperature evolution (b) at an intermediate frequency 0.04 Hz for displacement-controlled cyclic loading. Experiment data are from Yin et al. (2014) .

Fig. 9. Stress-strain ( σ-) curves of the ﬁrst and last cycles (a) and temperature evolution (b) at a high frequency 1 Hz for displacement-controlled cyclic loading. Experiment data are from Yin et al. (2014) .

cyclictestsbyYin(2013).TheaxialforceFisspeciﬁedas

F=F max

2

(

1− cos2

π

f t

)

, (37)

whereFmax isthe maximumforce applied.Here we considerthe loadingfrequencyof1Hz.Thestressstraincurvesoftheﬁrstand

last(steady-state)cyclesandthetemperatureevolutionareshown in Fig. 10. Considering that the specimens for force-controlled tests in Yin (2013) might not be the same batch as those for displacement-controlled tests in Yin et al. (2014), we recalibrate some parameters by ﬁttingthe ﬁrst-cyclestress strain curve;the updated (comparedto Table 2) model parameters are separately

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Fig. 10. Stress-strain ( σ-) curves of the ﬁrst and last cycles (a) and temperature evolution (b) at loading frequency 1 Hz for force-controlled cyclic loading. Experiment data are from Yin (2013) .

Fig. 11. Comparison of our model prediction with experimental data in terms of the damping capacity H (stress strain hysteresis) of the steady-state cycle versus the dimensionless timescale ratio λ. Panel (a) is for displacement-controlled loading while panel (b) is for force-controlled loading. The three stress values (520 MPa, 312 MPa, and 104 MPa) correspond to the maximum forces F max = 50 0 0 N , 30 0 0 N, and 10 0 0 N applied in experiments ( Eq. (37) ), respectively.

listed in Table 3 but those not changed are omitted. We then use the updated parameters to predict the stress strain curve of the lastcycle(Fig.10b) andtemperatureevolution(Fig.10c).The modelisabletocapturethedriftofstressstraincurvetothe left-handside(Fig.10b)duetotheincrease oftemperatureandhence thetransitionstress.

4.4.Dampingcapacity

In Fig. 11, we compare the hysteresis of the steady-state cycle between our model prediction and experimental data (Yin et al., 2014; Yin, 2013). Fig. 11a shows results for the displacement-controlledcyclicloading.Ourmodel prediction(red