IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 16 (2007) 2102–2115 doi:10.1088/0964-1726/16/6/013

Phase diagram kinetics for shape memoryalloys: a robust finite elementimplementationXiujie Gao1,3, Rui Qiao1,4 and L Catherine Brinson1,2,5

1 Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA2 Department of Materials Science and Engineering, Northwestern University, Evanston,IL 60208, USA

E-mail: cbrinson@northwestern.edu

Received 12 March 2007, in final form 23 July 2007Published 8 October 2007Online at stacks.iop.org/SMS/16/2102

AbstractA physically based one-dimensional shape memory alloy (SMA) model isimplemented into the finite element software ABAQUS via a user interface.Linearization of the SMA constitutive law together with completetransformation kinetics is performed and tabulated for implementation.Robust rules for transformation zones of the phase diagram are implementedand a new strategy for overlapping transformation zones is developed. Theiteration algorithm, switching point updates and solution strategies aredeveloped and are presented in detail. The code is validated via baselinesimulations including the shape memory effect and pseudoelasticity and thenfurther tested with complex loading paths. A hybrid composite withself-healing function is then simulated using the developed code. Theexample demonstrates the usefulness of the methods for the design andsimulation of materials or structures actuated by SMA wires or ribbons.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Shape memory alloys (SMAs) have been widely used inmany applications and they are being increasingly investigatedfor advanced active control materials and systems. SMAsare a class of metals named for their ability to recover aparticular shape in two unique ways. The ‘shape memoryeffect’ allows the recovery of an inelastic deformation byheating. Meanwhile, the ‘superelastic effect’ allows immediaterecovery upon load removal for deformations that are largerthan possible with normal elasticity. This behavior isenabled by a reversible thermoelastic phase transformationin the material that is controlled by temperature and stresslevels [1, 2]. Austenite (A), the high-symmetry parentphase which exists at high temperature, transforms to self-accommodated (sometimes called ‘twinned’) martensite (Mt),

3 Present address: General Motors Research and Development, Warren,MI 48090, USA.4 Contributed equally as first author.5 Author to whom any correspondence should be addressed.

a low-symmetry product phase, upon cooling with no appliedload. Loading applied to either austenite or self-accommodatedmartensite at the corresponding temperature range resultsin ‘oriented’ (or ‘detwinned’) martensite (Mo) with amacroscopic strain in the direction determined by the appliedload. Heating martensite causes a reverse transformation toaustenite, resulting in the recovery of any orientation strain.The temperatures at which the transformations occur at zeroload are generally referred to as Mf, Ms, As, Af, where ‘M’and ‘A’ are the phase and ‘f’ and ‘s’ refer to the ‘finish’ and‘start’ temperatures for the given transformation.

Since shape memory alloys have many unique properties,a number of constitutive laws have been developed to capturethe mechanical behavior of the SMA response. Mostearly models were one-dimensional (1D), phenomenologicalapproaches, including the models by Tanaka et al [3], Boyd andLagoudas [4] and Brinson [5, 6]. Subsequently, researchershave extended the 1D models in various ways; for example,Sittner et al [7] and Briggs and Ostrowski [8] have paidspecial attention to hysteresis in thermal cycling. Additionally,

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Phase diagram kinetics for shape memory alloys

many three-dimensional (3D) models have been developed forshape memory materials, including the models by Qidwai andLagoudas [9], Juhasz et al [10], Gao et al [11], Boyd andLagoudas [12], Brocca et al [13], Bouvet et al [14], and Panicoand Brinson [15]. However, most current 3D models are eithernot accurate enough to model all aspects of complicated shapememory behavior or too computationally intensive to be usedeasily for large-scale applications. At the same time, a 1Dmodel is sufficient to characterize the mechanical response ofcertain structures such as wires and ribbons which composethe majority of SMAs used in applications. In addition, sinceit is not necessary for a 1D model to consider all possiblemartensite variants explicitly, these models have much simpleranalytic forms and are much more efficient to use in numericalsimulations. Therefore, using a unified 1D model to representSMA wires in structures, many useful applications can beanalyzed and optimized with a finite element technique.

Currently, most of the existing numeric implementationsof 1D shape memory alloy material models focus on specificstructure or mechanical response. Birman et al applied theTanaka model in a study of local stresses in composite materialsystems reinforced with NiTi wires [16]. Lee and Lee usedthe Liang and Rogers model to examine the thermal bucklingand post-buckling behavior of laminated composite shells withSMA wires [17]. Shu et al implemented the Boyd andLagoudas model in their study of a flexible beam with SMAwires [18]. Sun et al investigated the thermomechanicaldeformation of an SMA wire reinforced elastic beam basedon the Brinson model [19]. In each of these cases, the SMAconstitutive law was implemented in an ad hoc manner onlyfor the specific application of study. Very few works haveillustrated the implementation of an SMA model into the finiteelement method for general analysis. Brinson and Lammeringdescribed a nonlinear 1D finite element procedure for trusselements to analyze the behavior of shape memory alloys [20].In another study [21], Auricchio and Sacco presented the finiteelement formulation of a 1D shape memory alloy constitutivemodel, which was implemented as a small-deformation beamfinite element. While both of these can be considered generalanalysis methods, in both cases the transformation kineticsincorporated into the implementations was not rigorous andapplies only for simple loading cases.

Additionally, in most of the popular commercial finiteelement analysis software, there is no rigorous shape memoryalloy material type integrated in the packages. While a fewdo have a material type entitled ‘shape memory alloy’, theseare very simplistic implementations which do not have thecapability to fully characterize the complex material behaviorof shape memory alloys under general thermomechanicalloading conditions. For example, MAT 030 of LS-DYNAcan only simulate superelastic response of shape memoryalloys [22]. ABAQUS also provides a UMAT subroutine(user material) to reproduce simple superelastic behaviorof alloys such as nitinol, Similarly, ANSYS includes asuperelastic nitinol model. Although Terriault et al [23]have implemented a bilinear model as a user programmedmaterial model into ANSYS which allows representation ofboth mechanical and thermal hysteresis in one dimension, thebilinear model is a coarse approximation, accurate only forsimple pseudoelasticity without the ability to handle more

complex loading paths. Since many powerful applicationsof SMAs include reorientation effects, partial transformationsand cyclic loading, such simplistic models will not serveadequately for material representation or design. Therefore,in this paper we utilize the kinetics developed by Bekker andBrinson [24], extend them to cover the entire phase diagram,and implement the SMA constitutive law and kinetics intofinite elements. The kinetic algorithms developed by Bekkerand Brinson [24] are mathematically rigorous and proved tobe robust under considerably complex loading paths. Weimplement the modified model into the general purpose finiteelement analysis software ABAQUS via a user interface tocombine its powerful finite element analysis capability withthe rigorous representation of SMA phase transformationand reorientation. With the resulting code we can conductnumerical analysis of smart structures using SMA wires orribbons as actuation elements.

This paper is organized as follows. First, the 1D SMAconstitutive law and transformation kinetics are reviewed anda consistent solution strategy for overlapping transformationzones is proposed. Then the linearization of the SMAconstitutive law together with the transformation kinetics isperformed and tabulated, including the residual vector. Theimplementation of the model into ABAQUS is described.The iteration algorithm, switching point updates and solutionacceptance strategies of the subroutine itself are all given indetail. Finally, the developed code is used to duplicate baselinesimulation results as well as to simulate the complex behaviorof composites embedded with SMA wires or ribbons.

2. SMA constitutive law and transformation kinetics

SMA wires and ribbons in smart structure applicationsare subjected to a variety of thermal and mechanicalload profiles, including cyclic loading, partial forward andreverse transformation and simultaneous changes in load andtemperature. Changes in temperature and/or load are intendedto actuate the SMA material to provide a crystallographicstate change accompanied by strain and stress changes. Inorder to efficiently model and design the optimal use ofSMA capability in such structures, the constitutive law andkinetics chosen must be robust and able to capture complexthermomechanical cycles. With this in mind, in this sectionwe briefly review the constitutive model and kinetics to beimplemented into ABAQUS. We also present an extension tothe previously developed kinetic description to allow consistentsolutions when loading paths proceed through an overlappingtransformation zone.

2.1. SMA constitutive law

For this work, the 1D constitutive relations developed byBrinson [5, 6], based on previous work by Liang and Tanaka,are utilized [3, 25]. This constitutive description is derivedfrom a thermodynamics basis and has a relatively simplemathematical expression. An important aspect is that itincludes only easily quantifiable engineering variables andmaterial properties.

Based upon energy balance equations and work byTanaka [3], the constitutive law can be written as [5, 6]

S = D(ξ )(E − εLξo) + �(T − T0). (1)

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Here, the second Piola–Kirchhoff stress, S, is related to theGreen strain, E , temperature, T , and martensite fraction, ξ .D(ξ ) is the Young’s modulus of the material, � is related tothe thermal coefficient of expansion (� = −Dα, where α isthe thermal coefficient of expansion and D is the modulus ofthe material) and εL is the maximum residual strain for thegiven SMA material. T0 is the reference temperature at whichthe thermal strain is defined to be zero. It is worth noting thatan important feature of this model is to distinguish betweenthe self-accommodated (twinned) and oriented (detwinned)martensite in the material. Self-accommodated martensite,ξt, is temperature induced with no associated macroscopicstrain, while oriented martensite, ξo, has a crystal structurethat has been preferentially oriented by the applied load andis accompanied by a macroscopic strain6. The total martensitevolume fraction is composed of these two components:

ξ = ξt + ξo. (2)

The use of the two martensite internal variables allows thismodel to capture the full range of SMA behavior, includingshape memory effect and pseudoelasticity. In general, theYoung’s modulus of an SMA is taken to be a function ofthe martensite fraction of the material. Here a simple linearfunction is chosen to account for this dependence.

D(ξ ) = ξ Dm + (1 − ξ)Da (3)

where Dm is the modulus value for the SMA as 100%martensite and Da is the modulus value for the SMA as 100%austenite. Although recent work [26] demonstrates that theapparent difference in martensite and austenite modulus maybe caused by limited early transformation (prior to the stressplateaus), nevertheless a simple dependence of the moduluson phase fraction is a reasonable method to incorporate theeffective material response. If the thermal strain componentis neglected (since it is orders of magnitude smaller than thetransformation strain under moderate temperature changes) theconstitutive law is simplified further to

S = (ξ Dm + (1 − ξ)Da)(E − εLξo). (4)

2.2. Transformation kinetics

The transformation kinetic equations relate the evolution ofthe martensite volume fraction with stress and temperature.The choice of transformation kinetics is critical to ensurerobust performance of the overall constitutive response of theSMA material due to the history dependence of the material’stransformation behavior. In this paper, a phase diagrambased kinetic description which was rigorously derived byBekker and Brinson is adopted [24]. This kinetic algorithmhas mathematical simplicity and at the same time accuratelyaccommodates complex thermomechanical loading paths byuse of switching points to account for the material history. Wedescribe the kinetics in some detail here and also propose anextension of the formulation to properly account for loadingpaths which pass through an overlapping transformation regionin the phase diagram.

6 Note that we deliberately depart from earlier notation in which the orientedmartensite was denoted by an ‘s’ in favor of the subscript ‘o’ for orientedmartensite.

Figure 1. A typical one-dimensional phase diagram of shapememory alloys. The shaded regions are the transformation stripswhile other regions are ‘dead zones’ where no change in martensitefractions occurs. � is an arbitrary loading path with points B–Gdenoted on it, where C and G are the entrance and exit pointsrespectively; F is the ‘current point’, D is a switching point wheretransformation stops and E is the most recent switching point, wheretransformation restarts and which carries the material history for thecurrent point.

During SMA phase transformation, one importantphysical quantity is the volume fraction of martensite, ξ , whichis the sum of self-accommodated martensite ξt and orientedmartensite ξo. To assist with the description of 1D SMAbehavior, Cauchy stress and temperature phase diagrams havebeen widely used. Figure 1 is the phase diagram used inthis paper. It is composed of transformation/reorientationzones [A] (martensite to austenite), [t] (austenite to self-accommodated (twinned) martensite), [M] (austenite and/orself-accommodated martensite to oriented martensite) and[o] (self-accommodated martensite to oriented (detwinned)martensite). The overlapping zone of [o] and [t] is denotedas [o, t]. The other regions are either dead zones in whichno transformation or orientation occurs7 or the plastic region,which is specifically avoided for most SMA applications and isnot discussed further here. In the transformation/reorientationzones, martensite volume fraction change occurs only when theloading path has a positive projection on the normal directionvector of the transformation zones (for example nA) such that

τ · ni > 0 i = M, A, o, t (5)

where τ is the tangent direction of a loading path. Also shownin figure 1 is an arbitrary loading path �, with a switching pointat D in the [M] zone at which the transformation8 stops beforecompletion due to the change in direction of the loading path.The entering point C, the ending point G and the restart point

7 It is noted that very small amounts of transformation occur in specificallyoriented grains even in the dead zones, as has been demonstrated in microscopyand x-ray studies [7, 27]. However, such transformations do not addmeasurably to the strain and are neglected in the transformation kinetics.8 Note that in the remainder of the paper, in general phrasing we often usethe word ‘transformation’ to refer to either transformation or reorientation forsimplicity. However, the algorithms are necessarily precise in distinguishingthese two situations. Transformation occurs with a phase change from Ato M (or vice versa) and is associated with an increase (decrease) in totalξ by increasing (decreasing) the ξo value. Reorientation occurs when self-accommodated martensite is oriented by stress and is associated with anincrease in the ξo value at the expense of the ξt value, with no change in thetotal ξ value. Either reorientation or transformation can occur in zones [o],[o, t] and [M] and it is important that the algorithms appropriately reflect thedistinction to allow robust material response prediction.

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E are also counted as switching points. These switching pointsare used in the kinetic algorithm to turn the transformation onand off and to incorporate the history of the material. Onlythe values of the variables at the most recent switching pointare retained and used in the mathematical formulation, whichleads to its compact form and rapid computation.

The essential aspects of the kinetic mathematicalformulation are outlined here, and further details can be foundin the previous paper [24]. However, the original workonly explicitly addressed the transformation kinetics in theregions where the temperature is higher than Ms. Thus, here,we extend the formulation to the all regions, with carefulattention to the overlapping zones [o, t]. The region [o, t]requires a new formulation since the kinetics described in [24]produce unexpected results if one simply applies the kineticsof [o], [M] or [t] zones to this region. Although Chunget al independently proposed a implementation strategy for theoverlapped region (while this manuscript was in review) [28],the transformation kinetics used here are more mathematicallyrigorous and robust under all loading conditions. Detaileddiscussion of the formulation for [o, t] is reserved for the nextsection. In this section, we first introduce the basic kineticsfor all transformation/reorientation regions except for [o, t].The martensite fraction evolution in the zones for forward andreverse transformation can be written as

ξ = FA = ξ j f A(ZA)

ξ = FM = ξ j + (1 − ξ j) f M(ZM)(6)

where subscript j represents the switching point and f A andf M are transformation functions with values between 0 and1. In the next section, equation (6) will be specified forξo and/or ξt as appropriate for each of the transformationregions individually. A variety of transformation functionshave been used to interpolate the value of ξ betweenzero and one, including linear, cosine and exponentialfunctions [12, 25, 29–32]. In this paper the cosine functionis chosen for its mathematical simplicity in differentiation andintegration and its slow and symmetric beginning and end oftransformation. Thus, the transformation functions are definedas

f A(Zi ) = 1 − 12 [1 − cos(π Zi )] (7)

f M(Zi) = 12 [1 − cos(π Zi)] (8)

where Zi is a distance ratio in zone i with reference to the lasttransformation switching point, j . Zi has values between 0and 1 and is defined as

Zi(T , σ ) = ρ i − ρ ij

ρ i0 − ρ i

j

i = A, t, M or o (9)

where ρ i is the distance from the current point to the entranceboundary of zone i . Note that by this definition ρ i = 0 at theentrance and ρ i = 1 at the exit point. ρ i

j is the distance fromthe last transformation switching point to the boundary and ρ i

0is the width of transformation strip. The geometric meaningsof these distances are illustrated on the loading path in figure 1.

The distance ρ i is calculated based on its definition via thedot product of the normal direction to zone i and the difference

Table 1. Entrance and exit points and normal directions for differentzones.

Zone (T iin, σ

iin) (ni

1, ni2) (T i

out, σiout)

[A] (As, 0) CA√1+C2

A

, −1√1+C2

A

(Af, 0)

[t] (Ms, 0) −1, 0 (Mf, 0)

[M] (Ms, σ crs ) − CM√

1+C2M

, 1√1+C2

M

(Ms, σcrf )

[o] (Ms, σ crs ) − Cd√

1+C2d

, 1√1+C2

d

(Ms, σcrf )

vector between two points:

ρ ij = ni

1(Tj − T iin) + ni

2(σ j − σ iin)

ρ i0 = ni

1(Ti

out − T iin) + ni

2(σiout − σ i

in)

ρ i = ni1(T − T i

in) + ni2(σ − σ i

in)

(10)

where (ni1, ni

2) is the normal direction of zone i , (T iin, σ

iin) an

entrance point of transformation zone i , (T iout, σ

iout) an exit

point and (T ij , σ

ij ) the last transformation switching point. In

this formulation, (T iin, σ

iin) is used as the switching point if the

material just starts transformation. Table 1 shows the entranceand exit points used in this paper and the normal directionsgiven the slope Ci in zone i . Note that the entrance and exitpoints of the strip used in equation (10) can be any point on thestart or finish boundary, but for convenience in implementationthe points in table 1 are used.

The Z -distance in equation (9) can then be calculated andwritten as

Zi = ρ i − ρ ij

ρ i0 − ρ i

j

= ni1(T −T i

in)+ni2(σ−σ i

in)−ni1(Tj−T i

in)−ni2(σ j−σ i

in)

ni1(T

iout−T i

in)+ni2(σ

iout−σ i

in)−ni1(Tj−T i

in)−ni2(σ j −σ i

in)

= ni1(T − Tj ) + ni

2(σ − σ j )

ni1(T

iout − Tj) + ni

2(σiout − σ j )

, i = A, t, M or o.

(11)

This Z -distance is then used in equation (6) to determine themartensite fraction evolution according to the loading path.

3. Analytical and numerical formulation

Brinson and Lammering [20] have provided a detailedderivation of the linearization of the weak form of momentumbalance and the SMA constitutive law and its finite elementformulation. It was shown the formulation differs fromthe standard finite element formulation for nonlinear trussmembers only by a factor dependent on the transformationzone, termed the H factor. However, Brinson and Lammeringused a much simpler and less robust kinetic description in theirwork. Thus, in this section, we derive and tabulate the Hfactor for all the transformation zones of the kinetic model justdescribed and give the residual vector for the finite elementformulation.

To assist in implementation, three special points (groupsof variables defined to store the solutions) are used: the lastswitching point denoted by ‘swit’, the last converged pointdenoted by ‘conv’ and the current estimated point denoted by

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‘esti’. Herein, the estimated point refers to the trial solutionsin current iteration. If not specified, ε denotes the currentGreen strain in the rest of the paper. Also note that the Cauchystress–temperature phase diagram is treated as a second Piola–Kirchhoff stress–temperature phase diagram for simplicity;since only small displacements are involved, the error involvedis negligible.

In the transformation zone [A], the material transforms tothe parent phase, and consequently both ξo and ξt decrease. Ifthe loading path has a projection along the normal direction nA

then the estimated martensite fraction can be calculated fromthe fraction at the last switching point and the Z -distance ofthe current point (equations (6) and (7)) as

ξ estio = ξ swit

o f A(ZA) = ξ swito [ 1

2 + 12 cos(π ZA)]

ξ estit = ξ swit

t f A(ZA) = ξ switt [ 1

2 + 12 cos(π ZA)].

(12)

The linearization of the stress-induced martensite volumefraction becomesd

dε{ξo(x + εu)}|ε=0 = d

dε

{ξ swit

o

[1

2+ 1

2cos(π Zi )

]}∣∣∣∣ε=0

= ξ swito [sin(π Zi )] − 1

2πni2 × d

dε{S(x + εu)}|ε=0

ni1(T

iout − Tj) + ni

2(σiout − σ j )

(13)

where x is the current position and u is the displacement. Thelinearization of the temperature-induced martensite volumefraction is obtained by replacing ξ swit

o with ξ switt .

Similarly the linearization of the Young’s modulusbecomesd

dε{D(ξ(x + εu))}|ε=0 = (Dm − Da)(ξ

switt + ξ swit

o )

× [sin(π Zi)] − 12πni

2 × ddε

{S(x + εu)}|ε=0

ni1(T

iout − Tj) + ni

2(σiout − σ j )

. (14)

Therefore the linearization of the second Piola–Kirchhoffstress becomes

d

dε{S(x + εu)}|ε=0 = H DFT · grad u

= 1

(1 − H1 H2)DFT · grad u (15)

where D is the estimated Young’s modulus, u is thedisplacement vector and F is the estimated deformationgradient for a 1D truss element given below:

F ={ 1 + u,x

v,x

w,x

}. (16)

Here u, v and w are the three estimated Cartesian componentsof the displacement vector where the ‘esti’ superscripts areomitted to be consistent with the notation in the literature [20].

The H1 and H2 factors in equation (15) are given by

H1 = sin(π Zi )− 1

2πni2

ni1(T

iout − Tj) + ni

2(σiout − σ j )

H2 = (E − εLξ estio )(Dm − Da)(ξ

swito + ξ swit

t ) − εL D · ξ swito .

(17)In the transformation zone [t] the stress-induced martensite ξo

remains unchanged, while the temperature-induced martensiteξt increases with decreasing temperature to the maximum value

Table 2. Parameters Ci1 and Ci

1s in the H2 factor for different zonesallowing a unified H factor (equation (22)) to be used inequation (15) in all cases excluding the overlapping [o, t] zone.

Zone Ci1 Ci

1s

[A] ξ swito + ξ swit

t ξ swito

[M][o] −(1 − ξ swito − ξ swit

t ) −(1 − ξ swito )

[t] −(1 − ξ swito − ξ swit

t ) 0[o, t] see equation (24)

1 − ξo. Thus, in the [t] transformation zone, the estimatedmartensite fractions can be written as

ξ estio = ξ swit

o

ξ estit = 1 − ξ swit

o − (1 − ξ swito − ξ swit

t ) f A(Z t).(18)

Note that, for ease of programming, here f A is used inall estimated martensite calculations through the relationshipf M = (1 − f A) from equations (7) and (8). In this case, theZ -distance in zone [t] is used directly in the expression for f A

such that f A = 1 at Z t = 0 and f A = 0 at Z t = 1. H1 is thesame as that in equation (17) and H2 is given by

H2 = (E − εLξ estio )(Dm − Da)(−1)(1 − ξ swit

o − ξ switt ). (19)

For transformation zones [M] and [o], the self-accommodated martensite reorients and the austenite trans-forms to oriented martensite when the loading path has positiveprojection along the normal direction of each zone. Thus, theestimated martensite fraction can be written as

ξ estio = ξ swit

o + (1 − ξ swito )(1 − f A(Zi ))

ξ estit = ξ swit

t f A(Zi ).(20)

H1 is the same as that in equation (17) and H2 is given by

H2 = (E − εLξ estio )(Dm − Da)[ξ swit

t − (1 − ξ swito )]

− εL D(−1)(1 − ξ swito ). (21)

Upon examination, the H2 factor can be rewritten into auniform expression with the introduction of two parametersCi

1 and Ci1s, which are functions of the martensite fraction

at the switching points and the active transformation zone asshown in table 2. Using these new parameters, the H factorin equation (15) can be written for all transformation zones(excluding the [o, t] zone) as

H = 1

1 − H1 H2

= 1

1 − H1[(E − εLξ estio )(Dm − Da)Ci

1 − εL DCi1s]

. (22)

A significant issue that has not been addressed in previousimplementations of phase diagram kinetics is the behavior inthe overlapping region [o, t], which can be seen in the phasediagram in figure 1. In this zone, physical restrictions areviolated if we choose to simply apply the kinetics of one ofthe different regions discussed above. For example, if weassume that the overlapping region is only part of region [o],only stress-induced transformation (orientation, ξo � 0) isallowed; consequently, a loading history like path 1 shownin figure 2 will cause unrealistic results. Since only stress-induced transformation could occur, the decreasing stress will

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Figure 2. A close look at the overlapping region, where three loadingpaths are shown.

not change the martensite fraction of the material. Thus,the total martensite fraction of SMA would be less than oneeven as the temperature decreases below Mf, which obviouslycontradicts the physical response of SMAs. Similarly, it is alsoproblematic to treat the overlapping region as part of region[t]. Under this assumption, in path 2 in figure 2, the materialmay enter the region [Mo] with retained self-accommodatedmartensite (ξt > 0), because the oriented martensite fraction inthe material will not change if we apply the kinetics in region[t] as shown in equation (18).

Therefore, we must allow both stress-induced andtemperature-induced transformations to occur in the overlap-ping region simultaneously. Since only kinetics for a singletransformation were discussed in the earlier work [24], it isnecessary for us to extend the kinetics to address the phasetransformation in the overlapping region [o, t]. The followingstatements must be satisfied by the extended kinetics:

(1) The total martensite fraction must be equal to or less thanone.

(2) The stress-induced martensite fraction must be equal toone upon reaching region Mo, where σ � σ cr

f .(3) The total martensite fraction must equal one when T �

Mf.(4) The transformation kinetics must be continuous with other

transformation regions.

Abiding by these restrictions, the new kinetics proposed inthe [o, t] overlapping region are as follows:

(a) If the loading path has a positive projection only on onenormal direction of [t] or [o], then the kinetics in the region [t]or [o] respectively are followed. For instance, path 3 in figure 2has only a positive projection on the normal direction of [o](no in figure 2); thus, we apply the kinetics of region [o] asequation (20) to calculate the martensite fraction.

(b) If the loading path has a positive projection on thenormal directions of both [o] and [t], we split the procedureinto two parts to calculate the martensite fractions. Sincestress can alter the self-accommodated martensite, ξt, whiletemperature decrease does not affect the oriented martensite,ξo, we first calculate the martensite fraction induced bytemperature based on equation (18), so we have a pair ofupdated martensite fractions; then we use equation (20) withthe updated martensite fractions as ξ swit

s , ξ swito to calculate

the martensite fraction induced by stress change. Themathematical formula of the kinetics can be rewritten as

ξ estio = ξ swit

o + (1 − ξ swito )(1 − f A(Zo))

ξ estit = [1 − ξ swit

o − (1 − ξ swito − ξ swit

t ) f A(Z t)] f A(Zo).(23)

These equations can be utilized to calculate the martensitefraction when a loading path has components in both no and nt

directions, such as path 2 in figure 2. The linearization of theabove equation can be performed and incorporated into the Hfactor:

H = 1

1 − H t1 H2 − Ho

1 H3(24)

where

H t1 = sin(π Z t)

− 12πnt

2

nt1(T

tout − T t

j) + nt2(σ

tout − σ t

j )

H2 = (E − εLξ estio )(Dm − Da)[−(1 − ξ swit

t − ξ swito ) f A(Zo)]

= (E − εLξ estio )(Dm − Da)C

t1mix − εL DC t

1smix

C t1mix = −(1 − ξ swit

t − ξ swito ) f A(Zo)

C t1smix = 0

Ho1 = sin(π Zo)

− 12πno

2

no1(T

oout − T o

j ) + no2(σ

oout − σ o

j )

H3 = (E − εLξ estio )(Dm − Da)[−(1 − ξ swit

t − ξ swito ) f A(Z t)

− εL D(ξ swito − 1)]

= (E − εLξ estio )(Dm − Da)C

o1mix − εL DCo

1smix

Co1mix = −(1 − ξ swit

t − ξ swito ) f A(Z t)

Co1smix = ξ swit

o − 1.

In the implementation, the projection of the loading path isestimated before computation to determine which kinetics ((a)or (b)) to apply in the overlapping region [o, t].

In the finite element formulation of Brinson andLammering’s work [20], the nonlinear element stiffness matrixfor shape memory alloy material behavior was derived.However, no formulation for the residual vector or theright-hand side vector was given. As a complement, theresidual vector for a 1D truss element is given below, usingequation (16).

R = −S AlBTF (25)

where S is the 1D second Piola–Kirchhoff stress and A and lare the cross-sectional area and the length of the truss memberrespectively. The matrix B is a 3 × 6 matrix containing thederivatives of the shape functions, and readers are referred tothe original paper for the exact form.

4. Model implementation using ABAQUS’ userelement interface

The constitutive and kinetic laws described above wereimplemented in the general finite element software, ABAQUS.ABAQUS was chosen due to the existence and flexibility of itsuser element interface. To have a user subroutine which can beversatile and robust to apply in composites/structures analysis,several important issues were addressed in the implementation:

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Figure 3. In order to handle small excursions into the compressivestress regime, the transformation zones [t] and [A] are extendedvertically below the T axis, and are shown in hashed regions in thefigure. The curved line is an arbitrary loading path with six points onit to explain the iteration algorithm.

(a) Since even uniaxial embedded SMA wires intended tobe used only in tension only may experience smallcompressive loading due to load transfer from thematrix, an extended phase diagram to consistently allowcompressive stresses in the SMA wires is implemented toenable robust application of this user subroutine.

(b) ABAQUS (as most other finite element packages)passes displacement and temperature increments tothe subroutine and requires element stiffness matrixand residual vector. This leads to iteration of theSMA subroutine since the stress is unknown. Dueto the nonlinearity of the SMA constitutive law andtransformation kinetics this iteration to convergence istime-consuming. Thus, a proper iteration algorithm isnecessary to improve the convergence rate.

(c) SMA constitutive and kinetic laws are history dependent;therefore, the subroutine was programmed to maintain andtrack the switching points to enable proper calculation ofthe phase transformation kinetics.

(d) Since the subroutine solution may be rejected byABAQUS, the subroutine was programmed to maintain aset of converged values (ξ conv

t , Sconv, . . .) while estimatinga new set of values (ξ esti

t , Sesti, . . .).

In the following sections, the points above are furtherdiscussed to explain our implementation.

4.1. Extension to limited compressive phase diagram

Uniaxial SMA wires embedded in composites or uniaxialSMA elements with relatively larger cross-sectional area canundergo compression and even experience transformationunder compressive loading. Different researchers haveproposed fully extended phase diagrams and constitutive lawsfor the compressive behavior of SMAs [21, 33, 34]. Althoughthe constitutive law used here is intended and validated onlyfor tensile stress, it is extended for limited compressive stresstogether with the extended phase diagram shown in figure 3.In order to handle such small excursions into the compressiveregime that may occur in more complex structures even whenthe SMA elements are intended to be loaded only in tension,this extension is essential in the numerical implementation.

In figure 3, the vertical boundaries for the [t] zoneare extended naturally into the compressive region. Insteadof a mirrored [A] zone in the compressive region [34],the boundaries of the [A] zone are extended with verticalboundaries. This vertical extension of [A] offers simplicity inimplementation and easier convergence for the limited rangeof compressive stresses intended. Additionally, a reducedYoung’s modulus can also be used during compression toaccount for buckling. These measures allow simulationsto experience small excursions into compression of theSMA component. For applications in which compression issignificant, more detailed extension as in [21, 33, 34] wouldneed to be implemented.

4.2. Iteration algorithm

Given the temperature and strain by ABAQUS, the usersubroutine needs to iterate to a solution for the stress. Dueto strongly nonlinear constitutive and transformation laws andgeometry, a binary search algorithm is used to iterate tosolution. The following steps are implemented into the usersubroutine for this iteration:

(i) First, a linear prediction is performed based on theincrements of stress and temperature which are passedfrom ABAQUS. Herein, the linear prediction meansthat the material is treated as elastic and no phasetransformation is considered.

(ii) If the predicted path does not involve an activetransformation zone, i.e., elastic movement only on thephase diagram, the predicted solution is accepted andthe subroutine returns to ABAQUS after appropriatehousekeeping of the switching points. For example, seethe paths Y → W or V → W shown in figure 3. Althoughthe path Y → W crosses the zone [M], it does not involvephase transformation since the normal direction of theloading path has a negative projection on nM.

(iii) If the predicted path involves an active transformationzone like R → S or W → Y in figure 3, iterative analysisis needed to calculate the martensite fractions and thestress. In this implementation, a binary search isperformed to solve the nonlinear equations (15) and (22).Although some other algorithms like a Newton algorithmhave a faster convergence rate, the linear convergence rateof the binary search is sufficient and convenient for thecurrent computation. Once a solution is found withinthe tolerance based on the accuracy requirement, thesolution is accepted and returned to ABAQUS. Anotherpossible situation is that the subroutine is unable to find aconvergent solution because the current increment is toolarge. Under such circumstances, the subroutine informsABAQUS to reduce the increment and redo this procedurefrom step (i).

It is worth noting that the solution even within thetolerance is not accepted by the subroutine if we find theloading path may not be unique. For example, if the loadingpath involves an active transformation but crosses the entirezone within one step, like R → U in figure 3, the solution willbe rejected and the increment will be reduced.

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Time Step(b)

Figure 4. (a) Stress–strain curve calculated by the finite element code illustrating the pseudoelastic effect. (b) The associated martensitefraction change during loading and unloading.

Table 3. Material properties for the nitinol alloy used in the examples (Burton et al [37], original data from Dye [38]; Liang [39]; Liang andRoger [25]).

Moduli, density

Transformationtemperature(◦C)

Transformationconstants

Maximumresidual strain

Da = 67 × 103 MPa Mf = 9 CM = 8 MPa ◦C−1 εL = 0.067DM = 26.3 × 103 MPa Ma = 18.4 CA = 13.8 MPa ◦C−1

� = 0.55 MPa As = 34.5 σ crs = 100 MPa

ρ = 6448.1 kg m−1 Af = 49 σ crf = 170 MPa

4.3. Switching point updates and information tracking

The switching point is introduced to incorporate the history ofthe SMA material in the transformation kinetics; thus, trackingthose points is essential for the proper calculation of martensitefractions. In this implementation, the switching pointinformation is recorded as variables and carefully updated afteran estimated solution is accepted by ABAQUS. The updatingstrategy follows the definition of switching points in [24]: onlyboundary points and the points where the projection of loadingpath on transformation zone vectors changes sign are recordedas switching points. In this extended implementation it is alsoimportant to note that, in the overlapping zone [o, t], there aretwo normal direction vectors of transformation zone, nt andno. Thus, the switching point is updated when either vectorproduct, τ · nt or τ · no, changes sign.

Since the user subroutine is a small part of ABAQUSand only accounts for the mechanical behavior of the SMAmaterial during the calculation, much of the finite elementanalysis procedure is invisible to the subroutine. Therefore,the solution proposed by the subroutine may be rejected byABAQUS due to reasons other than the SMA elements. Inthe implementation, a set of variables was introduced tostore the converged values computed by the subroutine so thecalculation can step back when it is necessary.

5. Results and discussion

In this study, the user subroutine discussed above was usedwith ABAQUS to simulate the baseline behavior of SMAelements as well as applied to composites embedded with SMAwires or ribbons. The baseline simulations verify the ability

of the code to capture the necessary fundamental SMA effectsas well as to capture cyclic or partial transformation loadingpaths. The application examples provide an indication of howthe code can be used in design-based simulations and illustrateimportant features such as the ability to start with pre-strainedmartensite or accommodate complex loading paths. Theexamples given here were calculated using ABAQUS Standard6.4 and the parameters used are from existing experimentaldata (table 3). It should be noted that the constitutive modeldoes not attempt to capture the few initial cycles commonto stabilize SMA parameters [35, 36] and thus the materialproperties used should be stabilized values.

5.1. Baseline stress–strain curves for both shape memory andsuperelastic effects

In order to perform the most basic test on the finite elementcode developed and in order to present a basis for thesubsequent applications of SMAs, two simple uniaxial tensiletests illustrating the thermomechanical response of SMAs wereperformed: the pseudoelastic effect at high temperature (T =50 ◦C > Af) and the shape memory effect at low temperature(T = −10 ◦C < Mf). The results are also compared withexperimental data from Liang [39].

The stress–strain response and martensite fractionevolution for high temperature (above Af) are shown infigure 4. Initially, the stress-induced martensite variable iszero and the material loads elastically; then the austenitetransforms to oriented martensite with stress increasing slowlyuntil all the austenite is converted to oriented martensite (ξo =1); the inverse transformation happens during the unloadingprocedure.

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(a)

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Time Step

(b)

Heat to recover

[o] [A] [t] ξo

ξt

1 2 3 4 5

Figure 5. (a) SMA stress–strain curve illustrating the shape memory effect. (b) Associated martensite fraction versus time step.

The shape memory effect is demonstrated in figure 5:the material shows elastic behavior while the load remainsbelow the critical stress, after which the self-accommodatedmartensite transforms to oriented martensite. Unloadingresults in only elastic recovery leaving a residual strain. Tocomplete the shape memory effect, the material is heatedto a temperature above the austenite finish temperatureat zero stress; during this stage, the residual strain isrecovered. Figure 5(b) shows the change of martensite fractionduring the complete procedure: the initial material is fullyself-accommodated martensite, with conversion to orientedmartensite during loading (the [o] shaded region). As thetemperature is raised above the austenite start temperature, theoriented martensite begins to transform to austenite (the [A]shaded region). During the cooling stage, the material returnsto self-accommodated martensite (the [t] shaded region). Fromthis example, we can see that the thermomechanical behaviorof shape memory alloys is clearly and accurately described bymaintaining two martensite internal variables: oriented, ξo, andself-accommodated, ξt, martensite.

5.2. SMA behavior under cyclic loading

Cyclic loading was applied to this finite element procedureto show the robustness of this code under the types of cyclicloading experienced by many SMA elements in actuationapplications. The loading paths of two examples9 are shown infigure 6. In the first example, the temperature is cycled underconstant stress (σ = 2.3 × 108 Pa) between two fixed points:A2i in the [M] and A2i+1 in the [A] strip (i = 0, 1, 2 . . .). Notethat the material was initially 100% oriented martensite beforebeginning the simulation. The result is depicted in figure 7,where transformation regions are indicated by the shaded areas.For the first few cycles, the inner loops drift slightly, but theyconverge to a limit cycle after several cycles. The outmost loopshows the envelope function for complete forward and reversetransformation.

The second example shows the SMA behavior subjectedto isothermal stress cycling with gradually diminishing rangeof stress, shown as Bi · · · B2n+1 in figure 6 (T = 56.5 ◦C). As

9 More simulations and comparisons with experimental results were shownin [24] to demonstrate the capabilities and the robustness of this constitutivelaw and the transformation kinetics.

Figure 6. The loading paths on the phase diagram. Two examplesare shown here: the first one (denoted by A) is a cyclic heating andcooling procedure under constant stress; the second one (denoted byB) is a cyclic loading under constant temperature.

the number of cycle increases, the stress range decreases to thestress levels at the entrances to the [M] and [A] strips. Figure 8shows the stress-induced martensite fraction response versusthe stress, which again converges to a limit cycle. Similarcycling to limit cycles has been observed experimentally andin kinetic modeling [24, 30, 35], and the ability to capturethis response demonstrates the capabilities of the kineticmathematical description and its numeric implementation.

5.3. Restrained recovery

In this case, a material with prestrain is constrained to maintainthe initial constant strain level as the temperature increased. Asshown in figure 9, a material with initial self-accommodatedmartensite is first loaded to 6.5% strain (A to C), and then isconstrained to maintain the deformation as the temperature israised through As (C to D). Because the material is restrainedas the inverse transformation to austenite occurs (D to E), andif unrestrained the material would recover the residual strain,extremely large internal stresses are incurred. It is clearlyshown in figure 9 that the internal stress increases quicklyduring transformation to austenite upon heating. When thismaterial is cooled, the stress decreases rapidly in the region ofthe austenite to martensite transformation (F to G). Figure 9also shows the change of martensite fraction corresponding

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Phase diagram kinetics for shape memory alloys

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(a)

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ξo

(b)

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Temperature

20 30 40 50 60 70 80

0.08

0.06

0.07

0.04

0.05

0.02

0.03

0.00

0.01

Str

ain

Figure 7. (a) Martensite fraction loops and (b) strain loops formed by cycling between [M] and [A] strips with loading path A from figure 6.The loading path starts at A0 and follows the sequence A0 A1 A2 A3 A4; then it oscillates between A3 and A4. In this example ξ = ξo at alltimes.

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0 100 200 300 400 500 600Strain

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Str

ess

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500

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100

0

Figure 8. (a) Martensite fraction response and (b) stress–strain response to isothermal cycling of the SMA material with graduallydiminishing range of stress. The loading path is B0, B1, B2, . . . , B2n−1, B2n . In this example ξ = ξo at all times.

to the temperature. It is worth noting here that when thetemperature decreases below Ms through the [o, t] zone, theoriented martensite fraction does not change since the vectorproduct of the loading path direction with no is less thanzero (equation (5)); however, since the vector product of theloading path direction with nt is greater than zero, the self-accommodated martensite increases until the total martensitefraction reaches one. Comparing the upper and lower plots infigure 9(a), the corresponding relation between the switchingpoints can be clearly observed. From this plot, we can seethat the stress level upon cooling is not identical to the initialstress, though subsequent cycling results in an attractor loopsimilar to those in figure 7. This stress difference betweenpoints C and H illustrates the history dependence of the SMAresponse and arises in this first cycle due to the longer traversalof the [M] strip during cooling. While the strain is constantin the specimen after point C and the oriented martensitefraction nearly identical before and after the temperature cycle,the stress at low temperature associated with that orientedphase fraction is not unique and depends explicitly upon thethermomechanical loading history.

5.4. Self-healing SMA composite

One of the established uses of this user element (UEL) codeis to simulate the behavior of shape memory alloy wiresembedded in a composite. SMA wires in a compositecan be used for actuation such as shape control, alteringeigenmodes by creating internal stresses, or even for self-healing by crack closing. Here we show a brief demonstrationof the UEL employed in the latter example. The motivationof this simulation is to assist the design of self-healingcomposites using shape memory alloys. A prototype of thiskind of material has been demonstrated by the Olson group atNorthwestern University [40]. In the prototype, the matrix isbrittle and several SMA wires are embedded unidirectionallyin the matrix. During the loading process, the matrix fractures,and the wires undergo a martensitic phase transformation andbridge the crack as it propagates. When the composite is heatedabove the reversion temperature of SMA wires, a clampingforce applied by the wires due to the phase transformation pullsthe crack faces together and provides crack closure. Uponfurther heating, the matrix material partially melts, providingcrack welding, and the crack is healed.

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A

B

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(MP

a)

HC

G

D

FE

ξo

ξt

Figure 9. (a) Stress and martensite fraction response due to thermalcycle under constrained conditions (inset). Switching points B, D, F,and G are denoted by x. (b) Stress response upon subsequent cyclicthermal loops. Note that the stress levels are identical after the firstloop since the history is consistent for each subsequent loop.

Similar to an earlier work [37], a finite element modelwas constructed consisting of a brittle metal matrix and five

(a) (b)

Figure 10. (a) Diagram of the finite element model at initialconfiguration for the SMA wire reinforced composite. (b) Schematicof the composite model with interface elements between the wire andthe matrix.

parallel and evenly spaced SMA wires (figure 10). The matrixis modeled with linear plane strain elements and the materialproperties are similar to those used in a prototype composite(see [37]). The NiTi wires are modeled with user-definedtwo-node truss elements (properties from table 3) with theUEL code developed here applied to capture the behavior. Toguarantee full crack closure, the embedded wires need to bepre-strained to contain some oriented martensite. Thus, inthis simulation the wires have an initial oriented martensitefraction of 0.40. As the bonding between SMA wire and matrixmaterial was very weak, wire elements were not bonded inthe simulation and were fixed to the matrix only at the outsideedges. This model is loaded in the direction of the fibers anda crack is allowed to propagate along a line perpendicular towires.

During the loading process, the crack propagates whenthe stress at the crack tip reaches the prescribed failurestress. The SMA wires accommodate the elongation by thephase transformation from austenite or self-accommodatedmartensite to oriented martensite and bridge the crack. Whenthe entire model is heated, the reverse transformation frommartensite to austenite occurs in the wires, which causes thewires to shorten and pull the fractured matrix halves backtogether. Figure 11 shows contour plots of von Mises stressesnear the crack tip at different stages after crack propagationbegins in the composite. Figure 12 illustrates the loading pathfor the composite overlaid on the phase diagram for the SMAwires. The corresponding martensite volume fraction in wiresis also shown below in the figure. Similar to the restrainedrecovery example, the temperature must be raised to well above

(a) (b) (c) (d)

Figure 11. Crack propagation simulation: (a) crack halfway through the matrix; (b) crack fully opened, matrix halves held together by wires;(c) heat causes the wires to shorten, which closes the crack; (d) crack fully closed and plasticity reversed. The SMA wire locations are shownby horizontal lines.

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400

700

600

500

300

200

100

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ess

(MP

a)

ξo

Figure 12. Loading path for cracked composite overlaid on the phasediagram for the SMA wires (upper plot), where switching points aredenoted by x. Martensite volume fraction versus temperaturecorresponding to loading procedure (lower plot).

the austenite finish temperature to significantly transform theSMA wires to austenite so that the wires can generate largeenough internal stress to fully close the two matrix halves.Upon cooling, although the martensite fraction increases againmodestly, a tensile closure stress remains in the wires tomaintain crack closure. Such a finite element implementationof the SMA constitutive law in an application enables acritical design process, including examining the effects oftransformation temperatures, matrix plasticity, interface effectsand reloading.

The finite element of the self-healing composite can befurther extended to examine the effect of interfacial bondingbetween the SMA and the matrix. As illustrated in figure 10(b),two layers of interfacial elements are used surrounding the

wire, and the SMA wire is also discretized into elements.The material properties of the interface elements are assignedproperties identical to the matrix material initially, but decreaseto a low value as the shear stress reaches a critical threshold.With this configuration, we can investigate the progress of theinterfacial debonding depending on the critical shear stress andexamine the stress distribution in the wires.

Figure 13 is an example that shows the comparison ofthe stress distribution in the wire for different interfacialbonding strengths when the crack propagates through thematrix (configuration in figure 11(b)). For this illustration, themiddle wire in the matrix is selected as representative here.From this figure, we can see that tensile stress is distributedalong a large region of the wire for the weak interface sample,which corresponds to a large amount of interface debondingduring the loading stage and thus wire transformation. In thecase of strong bonding between the fiber and the matrix, onlya small part of the SMA wire sustains most of the stress asthe crack passes. In fact, the large stresses in this latter casemay induce plastic deformation in the SMA wire which cannotbe recovered by simple heating, and thus, has no contributionto the composite healing. This simulation provides an insightinto the variation of stress along the wire during the crackpropagation in such composites and can help to design optimalinterfacial conditions to promote crack healing. From these andother simulations, it was observed that weak interfaces wereessential to provide sufficient transformation for the healingprocess.

6. Conclusion

Shape memory alloys have been widely used in manyapplications and they are being increasingly investigatedfor use in advanced active control materials and systems.Due to the prevalence of the wire or ribbon form, a 1Dfinite element analysis tool is an essential tool for realisticdesign and development of these applications. Currentlyonly very rudimentary models are available for SMAs inexisting commercial finite element software. In this paper,we have developed a user element subroutine for ABAQUSimplementing a rigorous 1D shape memory alloy constitutive

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Figure 13. The upper plots are the stress distribution in the wire for (a) a weak interface and (b) a strong interface. The lower plots areschematics of wire showing an intact interface (grey) and a failed interface (black) for each case.

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model. As part of the development, the kinetic law wasextended to properly account for an overlapping region in thetransformation phase diagram. The validity of the code istested via baseline simulations such as shape memory effectand pseudoelasticity. Importantly, the code is based on arigorous SMA constitutive and kinetic law and can thus handlethermomechanical loading cases that other models cannot. Inparticular, the appropriate tracking of self-accommodated andoriented martensite and the capture of the complete phasediagram with a consistent kinetic algorithm allows complexcyclic loading paths, partial transformations and constrainedcases to be accurately modeled. The examples illustrate thevalidity of the code and its potential use in the design ofSMA applications. For future improvements, transformationzones with non-parallel boundaries and a full compressionzone should be considered. Additionally, modifications ofthe model to account for the establishment of two-way shapememory through thermomechanical cycling would be valuable.

Acknowledgments

Support is gratefully acknowledged from NSF-CMS-0089977,NASA-NCC-1-02007/S4 and from Los Alamos NationalLaboratory under award No. 11923-001-00-3P-Mod #6/W-7405-ENG-36.

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