simulations of localized thermo-mechanical behavior in a ... · simulations of localized...

Simulations of localized thermo-mechanicalbehavior in a NiTi shape memory alloy

John A. Shaw *

Department of Aerospace Engineering, The University of Michigan, MI, USA

Received in final revised form 5 October 1999

Abstract

Previous experiments have shown that stress-induced martensitic transformation in certainpolycrystalline NiTi shape memory alloys can lead to strain localization and propagationphenomena when loaded in uniaxial tension. The number of nucleation events and kinetics of

transformation fronts were found to be sensitive to the nature of the ambient media andimposed loading rate due to the release/absorption of latent heat and the material's inherenttemperature sensitivity of the transformation stress. A special plasticity-based constitutivemodel used within a 3-D ®nite element framework has previously been shown to capture the

isothermal, purely mechanical front features seen in experiments of thin uniaxial NiTi strips.This paper extends the approach to include the thermo-mechanical coupling of the materialwith its environment. The simulations successfully capture the nucleation and evolution of

fronts and the corresponding temperature ®elds seen during the experiments. # 2000 ElsevierScience Ltd. All rights reserved.

Keywords: Localization; Phase transformation; Shape memory alloy; Constitutive behavior; Finite elements

1. Introduction

There are a number of remarkable materials, such as shape memory alloys (SMAs),piezoelectric materials, magnetostrictive materials, etc., which exhibit strong cou-pling between their mechanical behavior and other ®elds, such as thermal, electric ormagnetic ®elds, respectively. These materials are often referred to as ``interactive'',or (to stretch the point) ``smart'' materials, in that they sense a change in theirenvironment and respond in a mechanical way. The unique properties of thesematerials are not new, having been discovered 30±60 years ago, yet their use for

International Journal of Plasticity 16 (2000) 541±562

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E-mail address: [email protected] (J.A. Shaw).

application in active structures is relatively new and recently seems to be gainingmomentum.SMAs, such as NiTi (or Nitinol), exhibit two remarkable properties, the shape

memory e�ect and pseudoelasticity (see Fig. 1). The shape memory e�ect ( to inFig. 1) is the material's ability to recover large mechanically-induced strains (up to8%) by moderate increases in temperature (�10±20�C). Pseudoelasticity ( to )refers to the ability of the material in a somewhat higher temperature regime toaccommodate strains of this magnitude during loading and then recover uponunloading (via a hysteresis loop). The underlying mechanism is a reversible marten-sitic transformation between solid-state phases, often occurring near room tem-perature. The transformation can be induced by changes in temperature or bychanges in stress causing a strong thermo-mechanical coupling in the materialbehavior. The material also has very nonlinear mechanical behavior, high internaldamping, and high yield stresses. All of these properties make NiTi a promisingcandidate for novel structural applications (see Perkins, 1975; Funakubo, 1987;Otsuka and Wayman, 1988; Duerig et al, 1990).NiTi's remarkable behavior arises from the interplay of two phases (see lattice

sketches in Fig. 2), a high temperature phase (austenite), having a cubic latticestructure, and a low temperature phase (martensite), having a monoclinic structure(Otsuka et al., 1971). Due to its low degree of symmetry, the martensite phase existseither as a randomly twinned structure (low temperature, low stress state) or astress-induced detwinned structure that can accommodate relatively large strainswithout permanent deformation.

Fig. 1. Thermo-mechanical response of NiTi wire in water: shape memory e�ect ! ; pseudoelastic

response ! (experiment taken from Shaw, 1997).

542 J.A. Shaw / International Journal of Plasticity 16 (2000) 541±562

Despite the fact that the material was discovered nearly 40 years ago (Buehler etal., 1963), constitutive models have developed rather slowly, hampered by the com-plexity of the material behavior and the somewhat limited experimental basis formany years. The materials science literature is rich on the subject, and the under-standing of the micromechanical aspects has reached a mature level. However, theanalytic bridge between microscopic and macroscopic behavior is quite complex andis a recent area of research (see, for example, Ball and James, 1987; Batthacharyaand Kohn, 1996; Siredey et al., 1999). The last decade has seen some notable con-stitutive modeling e�orts (see Tanaka et al., 1986; Brinson, 1993; Levitas, 1994;Boyd and Lagoudas 1994; Patoor et al., 1995) and some new experimental ®ndings(see Leo et al., 1993; Shaw and Kyriakides, 1995; Sittner et al., 1995; Gall et al.,1999), providing new impetus for design and application. Yet, it is fair to say thatreliable constitutive models suitable for many engineering applications are not yetavailable, especially under cyclic loading conditions.In particular, few of the SMA constitutive models acknowledge the material

instabilities which have been observed in pseudoelastic NiTi (Shaw and Kyriakides,1997). [For notable exceptions see James (1983) and the 1D thermodynamic frame-work of Abeyaratne and Knowles (1993) and Knowles (1999)]. A continuum-levelplasticity approach with a special trilinear e�ective stress±strain response wasrecently shown to capture many features of the localized deformation ®elds seenduring unstable stress-induced transformation in uniaxially loaded thin strip NiTi(Shaw and Kyriakides, 1998). The approach was limited to quasi-static, isothermal,irreversible behavior, representative of the material response for very slow loadingrates in a convective medium, such as water. In this paper the approach will be

Fig. 2. Di�erential scanning calorimetry of NiTi strip.

J.A. Shaw / International Journal of Plasticity 16 (2000) 541±562 543

extended to include the thermal interactions between a mechanically loaded NiTispecimen and its environment to capture the nonisothermal response at higherloading rates and in a less convective medium, such as air. Some experimentalobservations from Shaw and Kyriakides (1997) are ®rst reviewed and then themodeling approach is demonstrated through some ®nite element simulations.

2. Experimental observations

This paper focuses on the material instabilities which occur in pseudoelastic NiTiloaded in uniaxial tension. Experimental studies of the phenomena of initiation andkinetics of phase transformation fronts and the associated thermal sensitivities arediscussed in detail in Shaw and Kyriakides (1995, 1997) and Shaw (1997). This sec-tion brie¯y reviews some general experimental ®ndings and discusses in particulartwo experiments that will be simulated in the following section.First, it was shown experimentally that the ``rate e�ect'' in the material is not the

usual viscoelastic phenomena, but rather, a strong interaction between the latentheat of transformation and the material's extreme (rate-independent) sensitivity totemperature. A di�erential scanning calorimetry (DSC) response is given in Fig. 2for the polycrystalline, near-equiatomic NiTi alloy used in Shaw and Kyriakides(1997). The area under each power peak/valley separating the three solid-state pha-ses, austenite (A), martensite (M), and rhombohedral-phase1(R), represents thelatent heat of transformation. A set of displacement controlled, isothermalmechanical responses at various temperatures is provided in Fig. 3a. Notice thenucleation peaks that must be surmounted at the beginning of the stress plateaus(during loading). Fig. 3b shows how the nucleation and transformation stresses(stress plateaus) increase signi®cantly with increasing ambient temperature.The exothermic A!M transition during loading2 tends to cause self-heating,

which in turn raises the material's underlying transformation stress according to Fig.3b. Consequently, the nature of the ambient media, gas versus liquid, plays a sur-prising role in the apparent material response due to the di�erences in the prevailingheat transfer environment (primarily the convective properties). This issue is anunusual, and perhaps counter-intuitive e�ect among typical structural materials. Forexample, a distinct stress plateau (unstable behavior) is observed during transfor-mations if the specimen is kept isothermal, i.e. slow loading rate and a convectivemedium like water. A seemingly stabilized stress±strain behavior (positive tangentmodulus) is observed under adiabatic conditions, i.e. faster loading rate or a rela-tively nonconvective medium, like air. It should be noted that this behavior isopposite to the destabilizing e�ects of temperature rise during adiabatic shearbanding in more conventional materials (Molinari and Clifton, 1987), owing to thedi�erent microstructural deformation mechanisms.

1 Note, the R-phase is not encountered in the remainder of this paper.2 The endothermic transition (M!A) during unloading tends to cause self-cooling, but this transfor-

mation is not the subject of this paper.


Since transitions in untrained NiTi can occur in a mechanically unstable manner,the transformation may occur in a localized way, i.e. through nucleation events andsubsequent propagation of distinct phase fronts along the length of a uniaxiallyloaded specimen. Fig. 4 shows an experiment taken from Shaw and Kyriakides(1997) in which a NiTi thin strip specimen (gage section: t=0.4, w=2.5, L=39mm,and free length: Lf=50.8mm) was stretched at a slow elongation rate(�:f=L � 10ÿ4 sÿ1) in room temperature air (25�C). The specimen was initially auste-

nite, and unstable transformation to martensite began once a critical stress level wasreached. During this transformation, photographs were taken at 40 s time intervals(see Fig. 4a). The large changes in strain caused an observable color change of thespecimen surface due to the disturbance of a naturally occurring brittle oxide layer.In this way the distinctly inhomogeneous evolution of the transformation was

Fig. 3. (a) Pseudoelastic responses of NiTi strip at �:f=L � 10ÿ4 sÿ1; (b) ®t of nucleation and propagation

stresses for A!M transformation (loading).


Fig. 4. Experiment 1. Evolution of A!M transformation (loading) at �:f=L � 10ÿ4 sÿ1 in room air: (a)

photographs, (b) infrared thermal images, [experiment taken from Shaw and Kyriakides (1997)].


tracked optically. Simultaneously, an infrared thermal imaging system providedsynchronized temperature of the specimen (see Fig. 4b where the color temperaturelegend spans 5�C centered around the room temperature).The motion of transformation fronts in Fig. 4a started with a nucleation event3

near the top of the gage section where there was a slight stress concentration at thetaper. A single transformation front, seen as a shear band, moved down the speci-men axis as the transformation progressed from time to time . The exothermictransition is seen in the thermal sequence in Fig. 4b as a yellow spot, then small redspot, moving down the length of the specimen. Once the temperature of the frontrose by about 3.5�C a second nucleation occurred near the lower end of the gagesection at time . At this time there were two converging fronts, they shared theoverall transformation rate (prescribed by the end displacement rate) and travelledhalf as fast as when there was a single front. Consequently, a smaller temperaturepeak existed (green spots) until time . As the fronts neared each another, however,they began to thermally interact, and a temperature rise was observed between timeand time . Once the specimen gage section was completely transformed at time

the specimen temperature returned to ambient.Fig. 5 shows the same type of experiment conducted at a loading rate 10 times

faster (�:f=L � 10ÿ3 sÿ1). In this case there were four nucleation events, two staring at

the ends of the gage length at time and time and two occurring in the interiorbefore time and time , leaving as many as six simultaneously traveling fronts inthe specimen. The thermal sequence showed severe self-heating throughout theexperiment (note the larger 20�C range in the thermal legend). This allowed multiplenucleation barriers to be overcome. In fact, front temperatures exceeded 37�C dur-ing latter parts of the transformation, due to the higher rate of transformation andthe lack of time for convection to occur.During these temperature ¯uctuations, the stress changed according to the pro-

pagating stress±temperature trends of Fig. 3b, interrupted by small stress dipswhenever new fronts were nucleated or when two fronts coalesced. Equilibriumrequired the average axial stress to be constant along the length. Consequently, thenucleation stress of a colder region could be surmounted if the local temperature riseof a propagating front caused its propagation stress level to rise by at least the sizeof the nucleation peak. Therefore, more fronts were observed at higher loading rateswhere local self-heating was more severe.To summarize, the number of propagating fronts was determined by the number

of nucleation events that could occur. This, in turn, was determined by the amountof excess self-heating and nonuniform temperature ®elds necessary to overcome themechanical nucleation barriers. In addition, at these low to moderate strain rates thespeed of all fronts were nearly the same according to the kinematical relationc � �:= n�� , where �: is the elongation rate, n is the number of currently traveling

3 Note: nucleation in this context refers to nucleation of transformation fronts, not nucleation of

martensite. It has been correctly identi®ed that some A!M transformation may procede and follow the

stress plateau (Liu et al., 1998). For example, one can see the homogenous self-heating due to some early

A!M transformation at time in Fig. 4b before the start of localized deformation.


Fig. 5. Experiment 2. Evolution of A!M transformation (loading) at �:f=L � 10ÿ3 sÿ1 in room air: (a)

photographs, (b) infrared thermal images [experiment taken from Shaw and Kyriakides (1997)].


fronts, and �� is the strain jump along the plateau. See, for example, the change infront speed after in Fig. 4a when the second nucleation event occurred.Furthermore, the kinematics of such an inhomogeneous deformation has a sig-

ni®cant e�ect on the aforementioned temperature sensitivity of the material, sincethe actively transforming region is local to the traveling front rather than uniformlydistributed along the length. This induces a hypersensitive ``rate'' e�ect, occurring atglobal strain rates, such as �

:f=L � 10ÿ3 sÿ1, that one might normally consider quasi-

static.

3. Finite element simulations

The ®nite element approach of Shaw and Kyriakides (1998) was a ®rst attempt tocapture the 3-dimensional deformation ®elds during the nucleation and propagationin thin strips under uniaxial tension. It used a conventional J2 plasticity model thatwas calibrated to a special trilinear stress±strain curve. In an isothermal, purelymechanical, setting it successfully captured details of the transformation front fea-tures that have been observed in experiments both on NiTi (martensitic transfor-mation) and ®ne grained mild steel (Luders bands). Since it was a plasticity-basedmodel, it ignored the reversibility of the deformation in NiTi upon unloading. Thislimitation will also be accepted here by focusing only on the A!M transformation.The approach will be extended, however, to include the thermo-mechanical couplingwith the ambient environment. This introduces a time scale into the calculations(that was absent before) which is needed to predict the number of fronts and thestress history for experiments conducted at typical loading rates in air. The numer-ical approach will ®rst be outlined and then comparisons will be made between theexperiments already presented and two corresponding numerical simulations.

3.1. Numerical approach

The existence of distinctly inhomogeneous deformation ®elds has importantimplications on modeling. The global force±displacement (engineering stress±strain)response must lose (momentarily) its positive slope, since during propagation, mul-tiple strain states are possible for a single axial stress state. Several investigators havestudied the behavior of a 1-D continuum solid with a nonconvex strain energy den-sity function (see Erikson, 1975; Falk, 1980; Ericksen, 1991; Abeyaratne andKnowles, 1993). They often treat the ensuing phase front as a mathematical dis-continuity in strain and develop appropriate jump conditions for the movingboundary. Although the problems simulated here are uniaxial in nature (actuallygeneralized plane stress), the response is investigated in a 3-D continuum setting.This precludes abrupt strain discontinuities (the front is a propagating ``neck'' witha ®nite width and pro®le shape) and allows one to follow the evolving deformation®elds without any assumptions on front morphology.The numerical simulations that will be presented were performed with the ABA-

QUS ®nite element software (HKS, 1997) using a fully 3-D, transient, coupled


displacement temperature analysis. From a mechanical point of view, it is a quasi-static analysis since inertial e�ects are neglected, yet it is a transient calculation froma heat transfer standpoint. The constitutive model for the material is based on tra-ditional rate-independent, ®nitely deforming J2 ¯ow theory with isotropic hardening(available as a built-in material type in ABAQUS). In this way the material isassumed to be homogeneous, isotropic, plastically incompressible, and plasticityirreversible. The homogeneity assumption is reasonable, since NiTi is ®ne grained(grain size typically 5m). The isotropic assumption is reasonable if there is no sig-ni®cant crystallographic texture. It is recognized this may not be the case for heavilydrawn or rolled NiTi (see Gall et al., 1999). Since the stress states calculated here arelargely uniaxial tension (with only a small bending and shear component), theassumption does not seem to a�ect the results adversely. The irreversibility of thedeformation does not a�ect the validity of the results provided that unloading doesnot occur below the critical stress for the reverse transformation. Only monotonicloading, �

:> 0, will be analyzed.

The key ingredient in the constitutive model is a trilinear uniaxial true stress±logstrain curve having an up±down±up shape. This is used to calibrate the plasticity¯ow rule. Using the ®t of the measured nucleation stresses and propagation stressesof Fig. 3b, a series of shifted/scaled trilinear curves is constructed to model thetemperature dependence of the isothermal material behavior. The derived tempera-ture dependent stress±strain model is shown in Fig. 6. Each trilinear curve is

Fig. 6. Constitutive model: nominal stress±strain and true stress±log strain.


constructed to model the nucleation stress, �N, propagation stress, �P (i.e. theMaxwell stress of the trilinear model), and transformation strain, �� (see Table 1).The intermediate branch of the trilinear curve has a negative slope in both theengineering (reference) stress and true (Cauchy) stress representations. This isnecessary to simultaneously model the as-measured nucleation peak and transfor-mation strain of the material. (Note, this is distinct from the approach of Hutch-inson and Neale (1983), in which the engineering stress±strain response has anunstable branch but the true stress±strain response is everywhere stable.) However,it poses some theoretical and numerical di�culties in the form of a loss of ellipticityof the underlying incremental boundary value problem and the potential for a mesh-dependent ®nite element calculation. In our case, however, this does not adverselya�ect the results as will be discussed at the end of this section.

The coupled deformation-temperature analysis available in ABAQUS solvessimultaneously for equilibrium and the heat equation. It allows one to choose aportion f� � of the speci®c inelastic work to be converted to an internal heat source asfollows,

q:s � f

1

��:p; �1�

where q:s is the heat generation rate per unit mass, � is the mass density, and ��

:p=� isthe speci®c plastic work rate.This is used here to simulate the release of latent heat during the A!M transfor-

mation. The phase transformation, however, generates latent heat through a speci®centhalpy change �h� �, which is decomposed here into stress independent and stressdependent parts as

�ql � ÿ�hA!M � ÿ �hA!MO ÿ 1

��P��

� �; �2�

where �ql is the speci®c latent heat change and �hA!MO is the stress free speci®c

enthalpy change.Integrating Eq. (1) along the Maxwell stress, neglecting the small change in elastic

energy, and equating with Eq. (2) leads to the temperature dependent factor of Eq. (3).

f � 1ÿ ��hA!MO

�p��3�

Table 1

Trilinear ®t of true stress±log strain

T (�C) E1 (GPa) 1n(1+E1) �true;1 (MPa) 1n(1+E2) �true;2 (MPa) E3 (GPa)

15 52.0 0.006511 338.6 0.05350 322.5 15.025 55.5 0.007621 423.2 0.05399 389.9 15.035 59.1 0.008600 507.9 0.05450 457.6 15.045 62.6 0.009470 592.6 0.05504 525.6 15.055 66.1 0.01025 677.5 0.05559 593.8 15.0


This factor is greater than one (since �hA!MO is negative) and changes with tem-

perature according to the temperature dependent propagation stress. Since ABA-QUS does not permit values greater than one, a value of unity was assigned and thethermal parameters (speci®c heat, thermal conductivity, and convective ®lm coe�-cient) were scaled down by this factor as a function of temperature. This results inthe correct temperature ®eld due to self heating without abandoning the built-incapabilities of ABAQUS.The environment is modeled by assigning a convective ®lm coe�cient to each free

surface of the specimen (a typical value for stagnant air was chosen). Heat conduc-tion and radiation to the environment were judged to be minor heat transfermechanisms and were neglected. As a ®rst approximation all thermal parameters,speci®c heat, thermal conductivity, ®lm coe�cient, and thermal expansion coe�-cient, were modeled to be constant and independent of strain or temperature. (It isknown that thermal conductivity and expansion coe�cient, in particular, maydepend on whether the material is austenite or martensite). Table 2 shows the chosenthermal and physical parameters. The latent heat was a measured DSC value for theNiTi alloy used. The thermal conductivity, k, speci®c heat, C, thermal expansioncoe�cient, �, and mass density, �, are typical values taken from vendor literature. Afew di�erent values of the convective ®lm coe�cient, h, were tried between 2 and 20W/m2K typical of free convection in stagnant air (Incropera and DeWitt, 1996), anda value of 4 seemed to give good results. Table 3 provides the ®tted nucleation andpropagation stresses and the calculated scale factor f as a function of temperature.The temperature of the specimen ends is ®xed at the ambient temperature (25�C),

modeling the grips as perfect heat sinks. This is a reasonable assumption for therather large metallic grips used. One end of the free length of the specimen is ®xed

Table 2

Thermal and physical properties

Parameter Value

Enthalpy change (zero stress) �hA!MO (J/g) ÿ12.3

Thermal conductivity k (W/m K) 18

Convective ®lm coe�cient h (W/m2 K) 4

Speci®c heat C (J/kg K) 837

Thermal expansion coe�cient � (10-6/K) 10

Density � (g/cc) 6.5

Table 3

Mechanical parameters

T (�C) �N (MPa) �P (MPa) �� f

15 336.4 320.6 0.04990 6.00

25 420.0 393.8 0.05018 5.05

35 503.5 467.0 0.05061 4.38

45 587.0 540.2 0.05115 3.89

55 670.6 613.4 0.05178 3.52


while the other end is displaced by �f. Other surfaces are traction free. In addition,only half of the specimen thickness is modeled by enforcing symmetry conditions,zero normal displacement and zero normal heat ¯ux, at the mid-thickness plane.These boundary conditions are speci®ed in Eq. (4).

u 0; y; z� � � 0 u Lf; y; zÿ � � �f; w x; y; 0� � � 0

v 0; 0; z� � � 0 v Lf; 0; zÿ � � 0

T 0; y; z� � � Tamb; T Lf; y; zÿ � � Tamb;

@T

@zx; y; 0� � � 0:

�4�

A few comments should be made regarding possible mesh sensitivity of the ®niteelement calculation when using a true stress±strain model with a negative slope. Ithas been well established that admitting an unstable branch in a true stress±strainresponse leads to discontinuous deformation gradients and ®ne phase structureswhich depend on the numerical mesh (see Tvergaard et al., 1981; Silling, 1988). Thisdi�culty can be resolved by introducing a ``penalty'' in the strain energy density inthe form of strain gradient terms (see for example, Triantafyllidis and Aifantis, 1986;Triantafyllidis and Bardenhagen, 1993; Hutchinson and Fleck, 1996). This e�ec-tively introduces a length scale into the calculation that must be speci®ed a priori.Since the strip thickness is the minimum continuum length scale of interest, itappears that by keeping one ®nite element through the half thickness the same isaccomplished here. Any ®ne phase structures are thereby suppressed in the calcula-tions, and this is acceptable, since that level of detail is not of interest. The homo-genization of the ®ne scale does not seem to adversely a�ect the calculations, and thegood results below seem to con®rm this view. The only practical issue was occa-sional convergence problems at the point where the stress±strain model slope sud-denly turns negative. This was resolved by introducing a small geometricimperfection to initiate the ``transformation'' and by adjusting the time step incre-ment at certain critical points in the calculation.Furthermore, as discussed in Shaw and Kyriakides (1998) only the local neigh-

borhood of a transformation front where material is actively transforming is a�ec-ted, since material elsewhere exists on a stable branch of the stress±strain curve. Forthe isothermal case only minor mesh sensitivity was seen in the details of the neckpro®le when re®ning the in-plane mesh (Shaw, 1997). Adding the temperature cou-pling here seems to improve the situation further, since self-heating has a stabilizinge�ect on the material behavior. A mesh re®nement study was conducted on a sim-pler version of the current problem, and no mesh dependency was observed whenincreasing the planar density of the mesh by nearly two orders of magnitude. Itappears that a very ®ne mesh indeed would be needed to resolve ®ne phase struc-tures within a front pro®le.

3.2. Simulation 1

The ®nite element mesh used in both simulations is shown in Fig. 7. The elementsare eight-node brick continuum elements, with linear strain and temperature


interpolation (ABAQUS type C3D8HT). The gage section of the model consists ofone element through the half-thickness, 12 elements across the width, and 182 ele-ments along the length. The mesh was chosen to give elements which are nearlycubic in shape to avoid introducing any directional bias into the calculation. A slightdent is located along the side near the top of the gage length to control the locationof the ®rst nucleation (shown with the ``<'' in Fig. 7). Otherwise, there are no otherimperfections, and further nucleations are allowed to occur naturally as determinedby the analysis.This ®rst analysis simulates the experiment of Fig. 4 in which the specimen is

loaded at an average strain rate of �:f=L � 10ÿ4 sÿ1 in room temperature (25�C) air.

As stated previously, only the loading portion of the experiment is analyzed, due tothe irreversible nature of the plasticity model used. The results of the calculation areshown in Figs. 8 and 9. Fig. 8 shows the experimentally measured stress history andthe calculated stress history annotated with numeric labels corresponding to thedeformation and temperature contours of Fig. 9. The dotted lines show the localstress-strain model at di�erent temperatures. Fig. 9a shows axial strain contourswhich have been thresholded at a chosen intermediate strain to simulate the beha-vior of the oxide coating in the experiment. Fig. 9b shows the calculated temperaturecontours using a similar temperature legend (centered around room temperatureand spanning 5�C) as the experiment (Fig. 4b).

Fig. 7. Finite element model of NiTi specimen.


The similarities between the strain and temperature contours of Fig. 9 and theexperimental results of Fig. 4 are evident. The ``transformation'' begins just beforetime in the form of a shear band at the top of the gage section where the dent waslocated. The shear band occurs at the well known angle of 55� to the loading axis fora uniaxially loaded thin strip (see Shaw and Kyriakides, 1998). When the transfor-mation is ®rst initiated the calculated stress drops slightly from its critical value(before time in Fig. 8). The nucleation event creates two diverging fronts, yet theupper front is quickly arrested in the tapered region of the specimen, thereby leavingone front traveling down the gage length. As shown, in Fig. 9b the local temperatureof the front rises as it travels and the stress level rises accordingly (Fig. 8), much likein the experiment. Just after the temperature has risen to 28.5�C (time ) the secondnucleation event occurs (time ). A temperature rise of 3.5�C is needed to raise thepropagation stress of the currently traveling front above the nucleation stress of aremote region which is still at ambient temperature (see again Fig. 3). Once thisoccurs, the stress drops again towards the Maxwell stress (Fig. 8). Note the drop inthe stress just after in both the experiment and simulation.Between time and time (Fig. 9a) two traveling fronts exist to complete the

transformation, each traveling at a speed about half as fast as the single travelingfront previously. The calculated thermal sequence (Fig. 9b) shows a general rise intemperature as the fronts travel and then approach each other and thermally inter-act. During this time the stress level rises continuously. Despite the fact that the

Fig. 8. Simulation 1 Ð calculated force-elongation at �:f=L � 10ÿ4 sÿ1.


Fig. 9. Simulation 1. Evolution of A!M transformation at �:f=L � 10ÿ4 sÿ1 in room air. (a) axial defor-

mation at 3% threshold strain; (b) temperature.


temperature rises signi®cantly, there are no untransformed regions at a su�cientlylow temperature to cause any more nucleation events. The stress dips momentarily(just after time ) when the fronts coalesce, a feature also seen in the experiments ofShaw and Kyriakides (1997). When the transformation is complete the temperaturereturns to normal. Except for the nonlinearity which is not modeled before thetransformation starts (Fig. 8), the stress history and the evolution of strain andtemperature ®elds are quite similar to those measured in the experiment. Notice alsothat from time onward in Fig. 9a the front ceases to be a straight angled band.Instead, it becomes a somewhat curved shaped front vaguely resembling (due to therelatively coarse ®nite element mesh) the criss-cross ``®nger'' pattern seen in theexperiment (see times to in Fig. 4a).

3.3. Simulation 2

The second analysis simulates the experiment of Fig. 5 in which the specimen isloaded at a rate ten times faster, an average strain rate of �

:f=L � 10ÿ3 sÿ1, again in

room temperature (25�C) air. Fig. 10 shows the stress history annotated withnumeric labels according to the deformation and temperature contours of Fig. 11.Fig. 11a shows thresholded axial strain contours and Fig. 11b shows the calculatedtemperature contours using the same temperature legend as the experiment (Fig.5b), now spanning 20�C.

Fig. 10. Simulation 2. Calculated force±elongation at �:f=L � 10ÿ3 sÿ1 in room air.


Fig. 11. Simulation 2. Evolution of A!M transformation at �:f=L � 10ÿ3 sÿ1 in room air. (a) axial

deformation at 3% threshold strain; (b) temperature.


The evolution of strain and temperature contours of Fig. 11 are similar to theexperimental results of Fig. 5. Four nucleation events occur, ®rst at the top wherethe dent was located, then at the lower end, and then two in the middle (time to), leading to six simultaneously traveling fronts during the remainder of the trans-

formation. This is the same number of nucleations and fronts seen in the corre-sponding experiment in Fig. 5. The only di�erence is that the third and fourthnucleations are delayed in the experiment (times and ) compared to that of thesimulation (times and ). This can probably be attributed to our imperfectknowledge of the exact convective heat transfer environment. This sequencingcaused some di�erence in the stress histories between the simulation and theexperiment (Fig. 10). During early parts of the transformation the experiment hasonly two moving fronts compared to the six fronts in the simulation. Consequently,more severe self-heating exists in the experiment, and the stress level is higher than inthe simulation (before time ). The two stress histories, however, tend to convergelater when both have six traveling fronts (after time ).At this higher rate the self-heating is more severe (Fig. 11b). The force±displace-

ment response maintains a positive slope except for the minor hiccups duringnucleation and coalescent events. The specimen temperature rises over 13�C aboveroom temperature by time , and locally blooms high o� scale where fronts coalesce(see Fig. 11b). This is quantitatively similar to the evolving temperature ®eld seen inthe latter parts of experiment (see Fig. 5b). As explained before for a displacementcontrolled experiment, the temperature of the transforming fronts drives the currentaxial stress level according to Fig. 3b (propagation stress). Further nucleations occurelsewhere in the specimen only if the nucleation stress is surmounted. Since equili-brium requires a constant average axial stress, this happens only if a colder (by3.5�C), untransformed region exists. The spacing of the nucleation events, therefore,is set by the axial gradient of the temperature ®eld, i.e. determined by di�usivity.Fig. 11a again shows that ``transformation'' occurs in the form of shear bands at

an angle of 55� to the loading axis. The bands align themselves parallel to each otherto minimize the e�ects of kinking and misalignment with the loading axis (see Shawand Kyriakides, 1998). This tendency is also observed during the latter parts (times

to ) of the transformation in the experiment in Fig. 5a. Except for the momen-tary ¯ipping of the band angle to achieve a parallel arrangement (the upper front attime and lower front at time ) the fronts remain straight throughout the trans-formation. Similarly, the bands remain straight for most of the simulation in Fig.11a, except for the topmost and lowermost fronts which ¯ip orientation starting attime to achieve a parallel alignment with the other fronts.

4. Summary and conclusions

Finite element simulations were presented to demonstrate a continuum approachfor the nucleation and propagation phenomena and the severe thermo-mechanicalinteractions seen in a pseudoelastic NiTi shape memory alloy. A side by side com-parison was made between ®nite element simulations and full-®eld measurements of


strain and temperature ®elds taken from experiments of Shaw and Kyriakides (1997)for two loading rates in room temperature air. The overall agreement was quitegood. The number of nucleation events and the calculated evolution of deformationand temperature ®elds (kinetics of transformation fronts) and the force±displace-ment response were strikingly similar to those measured in the experiments.The approach was based on the isothermal method developed in Shaw and Kyr-

iakides (1998), in which an up±down±up trilinear stress±strain response was usedwithin a traditional ®nite-plasticity constitutive model. Here the method was exten-ded to include thermal interactions by constructing a series of trilinear stress±straincurves at di�erent temperatures according to measured NiTi data and then per-forming a 3-dimensional, coupled thermo-mechanical calculation using the ®niteelement program ABAQUS. The latent heat of transformation was treated formallyas a heat release due to inelastic work. This underpredicted the actual latent heatrelease by a known factor, so the material's thermal properties were scaled downaccordingly. It resulted in the same thermal ®elds as if the true parameters had beenused, yet avoided the need to construct specialized user constitutive subroutines. Theirreversible nature of the plasticity model limited applicability of the analysis to theA!M transformation during loading.The simulations served to verify the understanding of the thermo-mechanical

phenomena and con®rmed the previous conclusions of Shaw and Kyriakides (1997);namely:

. Nucleation of transformation fronts within a uniformly stressed austeniteregion requires a higher stress (nucleation stress) than the stress required tosubsequently continue the transformation (propagation stress). For this parti-cular NiTi alloy a 3.5�C temperature di�erence is needed to elevate the pro-pagation stress of the currently transforming region above the nucleation stressof an untransformed region still at room temperature. This temperature dif-ference must exist in the nonuniform temperature ®eld before additionalnucleation events can occur. Therefore, under isothermal, uniform stress con-ditions only one nucleation is possible; whereas, at higher end-displacement ratesthis process can be repeated several times, resulting in multiple transition fronts.

. During unstable transformation, deformation is distinctly inhomogeneous.Nearly uniformly deformed regions (phases) are connected by narrow zoneswith steep strain gradients (transition fronts). In a thin strip specimen thetransformation front takes the form of a propagating shear band at an anglenear 55� to the loading axis. A criss-cross pattern is often observed undernearly isothermal conditions; whereas, bands tend to remain straight duringmore severe self-heating conditions where many fronts exist.

. Each nucleation spawns two transition fronts and active deformation (trans-formation) is limited to the neighborhood of these fronts. As a result latentheat is released in discrete local regions rather than distributed uniformly alongthe specimen length. The transient thermal ®elds depend primarily on specimenthermal properties, geometry, and the convective characteristics of the envir-onment. The localized nature of heat release creates, perhaps surprisingly,


strong thermo-mechanical interactions, and results in signi®cant ``rate'' e�ectsat strain rates one might normally consider quasistatic.

. Coexisting fronts tend to travel at the same speed Ð proportional to the rate ofapplied end-displacement and inversely proportional to the number of activefronts. Thus, more propagating fronts imply lower front speed and a reducedlocal rate of heating.

. A distinct instability and LuÈ ders-like deformation occurs under isothermalconditions (slow loading rate and convective medium). At the other extreme,adiabatic conditions (higher loading rate and/or more insulating medium)cause signi®cant self-heating which leads to a rising force±displacementresponse and multiple transformation fronts. As the number of fronts becomesnumerous the deformation begins to resemble a homogeneous deformation.Therefore, self-heating here acts as a stabilizing mechanism, which is contraryto adiabatic shear banding e�ects in conventional materials where internalheating acts as a destabilizing mechanism.

. While in general an unstable branch in the stress±strain model can lead toundesirable mesh sensitivities in the numerical calculation, it is not a practicalissue here. The relevant continuum length-scale is captured by using one ®niteelement through the half-thickness of the model. Whereas minor mesh sensi-tivity was reported in Shaw (1997) for the isothermal problem, no mesh sensi-tivity is observed here due to the stabilizing e�ects of self-heating.

Acknowledgements

The reported work was performed with the ®nancial support of the University ofMichigan Horace. H. Rackham School. The author also wishes to acknowledgehelpful discussions with N. Triantafyllidis.

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