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Comput. Methods Appl. Mech. Engrg. 257 (2013) 17–35

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Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier .com/locate /cma

Explicit finite element implementation of an improved threedimensional constitutive model for shape memory alloys

0045-7825/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cma.2012.12.021

⇑ Corresponding author. Present address: Graduate Aerospace Laboratories,California Institute of Technology, Pasadena, CA 91125, USA.

E-mail address: [email protected] (A.P. Stebner).

A.P. Stebner a,⇑, L.C. Brinson a,b

a Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USAb Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 September 2012Received in revised form 20 November 2012Accepted 27 December 2012Available online 16 January 2013

Keywords:Shape memory alloyConstitutive modelNumerical implementationPhase transformation

This article documents a new implementation of a three dimensional constitutive model that describesevolution of elastic and transformation strains during thermo-mechanical shape memory alloy loadingevents assuming a symmetric, isotropic material response. In achieving this implementation, improve-ments were made to the original formulation of the constitutive model. These improvements allow forrobust three-dimensional calculations over a greater range of thermo-mechanical loadings. Furthermore,a new explicit scheme for solving the model equations was derived. This scheme removed the need foruser calibration of the numerical integration parameters and greatly reduced the sensitivity of this expli-cit finite element implementation of a rate independent model to mass scaling. Studies were performedthat quantified both simulation times and convergence of the new scheme along with the original solu-tion scheme of Panico and Brinson for single element and multi-element simulations. The effectiveness ofthe new scheme is apparent in 6 and 30 times reductions in computation expense for selected single andmulti element simulations, respectively.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

1.1. Motivation

The Panico–Brinson constitutive model [1] has become a stan-dard for comparison in modern day development of three-dimen-sional continuum-scale shape memory alloy (SMA) constitutivemodels (e.g., [2–5]), largely because it was the first model at thisscale to phenomenologically consider the ability of a martensitemicrostructure to reorient during non-proportional loading events.In the years since original publication, several numerical imple-mentations for modeling the SMA behaviors described by the ther-modynamic formulation of Panico and Brinson [1] have beendeveloped [1–3,6,7]. However, the original implementationscheme [1,6] is cumbersome to code and limited in numericalrobustness. In this article, we proceed to present a new numericalimplementation, which is comprised of:

� An improved constitutive model that allows for robust perfor-mance under a wider range of thermo-mechanical loadingswhile simplifying the set of model inputs;

� A new explicit integration scheme that is easier to implement,easier to use, and greatly reduces the computational expensewhen compared with the original Panico and Brinson imple-mentation of this model.

This new numerical implementation serves to document robustconvergence of the SMA constitutive law within structural simula-tions using explicit finite elements, and also improve the ability ofother researchers to continue to benchmark new advancements inSMA constitutive modeling. Furthermore, the self-calibratingnumerical scheme we derive provides a simple, physical, robustsolution to solving constitutive equations that describe mechanicsof reversible inelastic deformation governed by a bounded internalvariable. The numerical complications inherent to such a constitu-tive model have been a recent topic of discussion (e.g. [4]).

Expanding upon the first point listed above, while the chosenform of the orientation kinetic function (see Section 1.2 for termi-nology definitions) of the Panico–Brinson model was theoreticallyrobust [1], it was not sufficient under large mechanical loads whenlimited by machine precision (see Section 2.2.1). Thus, a new func-tion has been introduced to describe the evolution of the orientedmartensite volume fraction. Additionally, the number of model in-puts has been reduced while retaining the original material physicscaptured by the Panico–Brinson formulation. Finally, Arghavaniet al. [2] formulated a constitutive model with decoupled sets ofequations describing transformation strain: one for orientation, a

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Fig. 1. A simplified, two-dimensional cartoon of crystal structures and microstruc-tures associated with the reversible, solid-state phase transformation that gives riseto the shape memory behaviors of SMAs. Their significance is discussed in the text.

1 We note that this definition of self-accommodation from a phenomenologicalcontinuum perspective differs from metallurgical definitions that consider micro-structural heterogeneities. It is not necessary to belabor the point in this work, but werefer the reader to [12,13], and references therein for further reading on self-accommodation as well as other microstructural terms used in this work.

18 A.P. Stebner, L.C. Brinson / Comput. Methods Appl. Mech. Engrg. 257 (2013) 17–35

second for reorientation. Inspired by the computational impact of[2], in the newly formulated numerical implementation we arriveat two smaller sets of equations describing the transformationstrain that may be independently solved, even though they arenot fully decoupled in formulation as in [2], and this simplifiesthe numerical implementation.

To the second point, Panico and Brinson originally proposed animplicit integration scheme (Newton–Raphson) for solving thepartial differential equations (PDEs) of their model at integrationpoints of explicit finite elements (FEs) [1]. However, this schemerequires programming the Jacobian of the equations comprisingthe model, which is tedious and error-prone due to the large num-ber of equations to derive and code. This originally proposedscheme and the Jacobian for the improved constitutive model pre-sented in this article are given in Appendix B for consideration inthis light. In creating the first Abaqus Explicit [8] user material sub-routine (VUMAT) of the proposed numerical implementation of thePanico and Brinson model, the Newton–Raphson integration wasaugmented with an explicit constant Lagrange multiplier integra-tion (‘‘Scheme 1’’ in this article, see Section 3.1) that overrodethe implicit calculation in the event Newton–Raphson failed toconverge within a few iterations. The original explicit override VU-MAT was used in the Abaqus calculations published in [6,7].

In this article, complexity, convergence, and computational effi-ciency of two VUMATs that make calculations using unique explicitintegration schemes will be compared: (1) the constant Lagrangemultiplier integration scheme used to override the implicit inte-gration scheme in the original Panico and Brinson VUMAT(Scheme 1, see Section 3.1) and (2) a newly derived explicit inte-gration scheme that allows for more robust computations(Scheme 2, see Section 3.2), which has been implemented into anew VUMAT-the second written by our research group. Compari-sons will be made within the context of non-proportional loadingof a single three-dimensional brick element, as well as loading ofa SMA bar containing a hole, modeled with a two-dimensionalmesh of non-uniform elements.

We proceed with a brief review of the background necessary tounderstand the model features. Subsequently, the improved con-stitutive model is presented in Section 2, and in Section 3 thenumerical schemes used in the two VUMATs are presented, includ-ing the new explicit integration scheme formulated for integratingthe coupled equations of the improved model. Both single andmulti element simulation data are then used to compare and con-trast the convergence and computational efficiency of the VUMATSin Section 4. Finally, implications of these results are discussed inthe conclusion.

1.2. Background

The shape memory behaviors exhibited by SMAs result from areversible solid-state phase transformation between an austenitephase comprised of a high symmetry crystal structure and a lowersymmetry martensite phase. A simple cartoon of crystal structureand microstructure changes during phase transformation is shownin Fig. 1. As indicated in Fig. 1, in this work forward describes aus-tenite to martensite phase transformation while reverse indicatesmartensite to austenite. First let us consider differences in crystalstructure between the phases (Fig. 1a–c). In this example the aus-tenite square (Fig. 1a) transforms to martensite parallelograms(Fig. 1b and c). Note it is possible to impart two unique deforma-tions upon the high symmetry square that result in the same lowsymmetry parallelogram when viewed from the perspective ofeach local coordinate system defined by the axes of Fig. 1b and c.Note that if we restrict the example of Fig. 1 to two dimensionsin the plane of the paper, the two martensite parallelograms arenot frame indifferent [9,10]; that is Fig. 1b cannot be rotated in

the plane of the page to be of the same orientation as Fig. 1c. In thiswork, each unique stretch that may result from transformation of asingle austenite crystal orientation defines a unique variant of mar-tensite. Indeed, in a real material in three-dimensional space thereare more than two unique variants of martensite – in the most pre-valent SMA, NiTi, there are twelve monoclinic variants that maytransform to and from a single body centered cubic austenite ori-entation [11,12].

Next, let us consider microstructures that form from transfor-mation events (Fig. 1d–f). The amount of material that transformsfrom austenite to a given martensite variant is driven by energyminimization [12]. In the absence of an applied load, mixtures ofvariants form in twinned arrangements that self-accommodate1 topreserve the macroscopic shape, since macroscopically there is nodriving force for deformation. This is depicted in transforming fromthe microstructure comprised of repeating units of square austenitecrystals in Fig. 1d to the martensite microstructure that is a mixtureof two parallelogram variants in Fig. 1e. Contrarily, if the materialtransforms in the presence of an applied mechanical load, forminga martensite microstructure of variants whose orientations bestaccommodate the applied load will minimize the energy of the sys-tem. In the simple cartoon, transforming from the austenite micro-structure (Fig. 1d) to a martensite microstructure of a singleparallelogram variant under a tensile load best accommodates themacroscopic driving force for overall elongation (Fig. 1f). Here wehave provided a high-level, over-simplified summary of the conceptsof SMA microstructures during phase transformation to motivate theterminology and features of our model. The reader is referred to[12,13] and references therein for mathematical and crystallographicdiscussions of microstructure and energy minimization in SMAs.

Having provided a high-level description of microstructural dif-ferences between the phases involved in martensitic transforma-tions, we will conceptually connect microstructure changes withthe phenomenological appearance of shape memory behaviors.Toward this purpose, the macroscopic stress–strain–temperature

A.P. Stebner, L.C. Brinson / Comput. Methods Appl. Mech. Engrg. 257 (2013) 17–35 19

responses associated with shape memory behaviors are shown inFig. 2 along with our simplified cartoons of microstructure at theinflection points of the macroscopic responses. Stress-inducedphase transformation (Fig. 2a) may be exhibited by SMAs thatare austenitic at ambient temperature and subjected to changesin mechanical load. Application of the load drives forward transfor-mation, release of the load stimulates reverse phase transforma-tion, and (ideally) recovery of the inelastic strain imparted uponloading is complete, thus stress-induced shape memory behavioris often called pseudo-elasticity or superelasticity. The plateaus inthe stress–strain curve signify inelastic deformation resulting fromthe microstructure changing from the austenite to an orientedmartensite and back again.

Thermally-induced phase transformation is achieved by heat-ing and cooling the alloy above and below its characteristic trans-formation temperatures. Four temperatures typically describe thephase transformation: (1) martensite start and (2) martensite finishtemperatures signify the onset and saturation of the forwardtransformation upon cooling; analogously (3) the austenite startand austenite finish temperatures characterize the reverse trans-formation upon heating. Further described in the caption ofFig. 2 and depicted in Fig. 2b, cooling from austenite to self-accommodated martensite load-free (microstructurally Fig. 1d toe), loading martensite from a self-accommodated state to an ori-ented state (Fig. 1e–f), unloading the oriented martensite (staysFig. 1f), then heating back to austenite (Fig. 1d–f) is called theshape memory effect (SME). Thermally cycling under load betweenaustenite and oriented martensite (Fig. 1d–f and back to d) givesrise to actuation.

Having briefly reviewed shape memory behaviors at a levelrequired to understand the model, this introduction concludes witha brief review of the Panico–Brinson model followed by a definitionof the terminology used to describe the model features, which hasbeen modified from that used in the presentation of the originalmodel [1]. Note that shorthand, operators, and abbreviations as wellas their correlation to that used in [1] are defined in Appendix A.

The constitutive assumptions made in deriving the Panico–Brinson model are [1]:

(1) The total strain may be additively decomposed into elastic(e) and transformation (tr) strains:

Fig. 2.stress-iof martmartenaccommappliedcurves,

e ¼ ee þ etr : ð1Þ

Shape memory behaviors: (a) Pseudo-elasticity is exemplified by large inelastic snduced phase transformation between austenite and martensite. (b) In the shapeensite variants from self-accommodated microstructure (black curve); residualsite into austenite via heating (red curve, bottom path). Subsequent stress-free co

odated microstructure – this process exhibits no macroscopically observable dload is not removed from the alloy as it is thermally cycled through phase transftop path). (For interpretation of the references to colour in this figure legend, th

Furthermore, the rate of transformation strain may be decomposedinto orientation (r) and reorientation (re) parts:

trains inmemorystrain uoling fr

eformatormatioe reader

_e ¼ _ee þ _er þ _ere: ð2Þ

(2) The transformation strain is traceless since there is less than1% volume change exhibited by most SMAs in transformingfrom austenite to martensite crystal structures ([12]):

trðetrÞ ¼ 0: ð3Þ

(3) The martensite volume fraction n may be expressed as a sumof self-accommodated (SA) and oriented (r) martensite:

n ¼ nr þ nSA; ð4Þ

and martensite volume fractions are bounded by zero and unity:

0 6 n; nr; nSA 6 1: ð5Þ

(4) Making the simplifying assumption that the material is iso-tropic and symmetric, the transformation strain is boundedby the magnitude of maximum polycrystalline transforma-tion strain emax

tr .(5) Furthermore, the magnitude of transformation strain is

assumed to only evolve through orientation of martensite,independent of reorientation. Considering this assumptionalong with assumption (4), the evolution equation for nr is:

_nr ¼etr : _erffiffiffiffiffiffiffiffi

3=2p

emaxtr ketrk

: ð6Þ

Another consequence of these assumptions is that reorientationmay only re-direct the transformation strain, not change its magni-tude, thus the tangency condition must always be satisfied betweenthe transformation strain and the reorientation strain rate:

etr : _ere ¼ 0 ð7Þ

(6) Rate and history independent kinetics are also assumed.

The model derivation is achieved by writing an expression forthe Helmholtz free energy incorporating these constitutiveassumptions:

wðee; T; nr; nSAÞ ¼1

2qee : C : ee þ nrhT~g0 � ~u0i þ nSAðT~g0 � ~u0Þ

þ cv T � T0ð Þ � TlnTT0

� �� þ 1

2Hrn2

r: ð8Þ

duced during loading that are recovered during unloading due to reversibleeffect (SME), inelastic strains are induced during loading due to orientation

pon unloading is still recoverable by phase transformation of this orientedom austenite to martensite (blue curve, Temperature axis) results in a self-ion. Actuation may be achieved via a special case of the SME in which then between oriented martensite and austenite microstructures (red and blue

is referred to the web version of this article.)


This free energy is then examined in the Clausius–Duhem inequal-ity to arrive at the following dissipation potential, where X denotesa driving force:

qDp ¼ Xr : _er þ Xre : _ere þ XSA_nSA P 0: ð9Þ

The driving forces are:

Xr ¼ r0 � qðT~g0 � ~u0 þ HrnrÞffiffiffiffiffiffiffiffi3=2

pemax

NðetrÞ

Xre ¼ r0;

XSA ¼ �qðT~g0 � ~u0Þ

ð10Þ

evolution equations for the internal variables:

_er ¼ _krXr

_ere ¼ _krebIðetrÞ : Xre;

_nSA ¼ _kSAXSA

ð11Þ

where bIðetrÞ ¼ I0 � NðetrÞNðetrÞ, and the kinetic relations are intro-duced through limit functions (F):

Fr ¼ kXrk � YrðnrÞFre ¼ kbIðetrÞ : r0k � Yre;

FSA ¼ XSA � YSAðnSAÞð12Þ

which are subject to Kuhn–Tucker conditions:

Fa 6 0; Dka P 0; DkaFa ¼ 0 ð13Þ

where a = r, re, SA.The functions Ya are the kinetic equations; these are docu-

mented in presenting the implementation of the improved modelin Section 2.1.

The updated terminology used in this presentation provides acomplete description of the macroscopic thermodynamic responsesthat are modeled while more accurately correlating with the micro-structural phenomena that give rise to the shape memory behaviors.

� Self-accommodated martensite (‘‘twinned’’ in [1]) transforms toand from austenite in a macroscopically shape-preserving man-ner, corresponding to transformation between austenite andself-accommodated martensite microstructures (Fig. 1d to andfrom Fig. 1e).� Oriented martensite (‘‘detwinned’’ in [1]) forms from self-accom-

modated martensite via an applied load (Fig. 1e and f), and alsoduring phase transformation under load (Fig. 1d to and fromFig. 1f). Formation of oriented martensite gives rise to transfor-mation strain, and in transforming oriented martensite back toaustenite the transformation strain is recovered.� Orientation (‘‘transformation’’ in [1]) encompasses growth or

depletion of oriented martensite. This model feature alwayscontrols the magnitude of transformation strain. In a case ofproportional loading, this feature also controls the transforma-tion strain direction. Note that ‘‘orientation’’ in this sense maymechanistically refer to:

1. Phase transformation under load (austenite to/from ori-ented martensite, Fig. 1d to and from Fig. 1f);

2. Conversion of self-accommodated martensite to orientedmartensite (Fig. 1e to and from f);

3. Furthermore, ‘‘orientation’’ does not include austenite-self-accommodated martensite phase transformation(Fig. 1d to and from Fig. 1e) as this process does not affectthe macroscopic transformation strain.

� Reorientation encompasses changes in orientation, but not themagnitude of transformation strain during a non-proportionalmechanical loading event. This model mechanism reflects theability of an SMA to reorient its martensite variants to form anewly oriented microstructure as proportions of loading

conditions change. This feature was the primary novelty ofthe Panico–Brinson model at the time of publication, thoughmodels published around the same time and since also includethis ability (e.g., [2–4,14,15]).

Relative to the original publication, this terminology is moreprecise since orientation of self-accommodated martensite(Fig. 1e and f) is not a phase transformation process, while trans-formation between austenite and self-accommodated martensite(Fig. 1d to and from Fig. 1e) is a phase transformation process.The total inelastic strain modeled by both the original and im-proved formulations is transformation strain–the ‘‘reorientation’’strain described below is recoverable through phase transforma-tion. Thus it provides better agreement to call the total inelasticstrain ‘‘transformation strain’’ and to allow it to still evolve bytwo mechanisms –‘‘orientation’’ and ‘‘reorientation.’’.

Hence, both the improved model presented in this work and theoriginal Panico–Brinson model simulate three-dimensional evolu-tion of elastic and transformation strains during pseudo-elastic,shape memory effect, and actuation events. They also provide theability for the transformation strain to reorient in a tangentialmanner relative to the transformation surface during non-propor-tional loading events (Eqs. (7) and (10)). Furthermore, the modelsare able to simulate independent energy dissipation events foreach formation of oriented martensite as well as reorientation oforiented martensite (Eqs. (9) and (12)). The current form of themodel still assumes symmetric and isotropic material responsesand neglects plasticity, thermal expansion, finite deformation,minor loop kinetics, cyclic evolution, and rate dependence, all fea-tures that may be added to this core model in future developmentefforts. The reader is referred to [3–5,14–29] and references there-in for examples of advancements in phenomenological modeling ofthese mechanisms. Subsequent to original publication, Panicoincorporated a mechanism to model irreversible strains that resultfrom retained martensite [6,7], a mechanism originally proposedby Malecot et al. [27]. This unrecovered strain mechanism is not re-viewed here nor included in the improved model since the presentfocus is on numerical implementation of the core constitutivemodel (elasticity + transformation), not the mechanics of unrecov-ered inelastic strains during cycling.

Background in hand, we proceed with documenting the im-proved model and the new numerical implementation.

2. The improved model

The model introduced in this section is improved from the ori-ginal formulation [1]. As mentioned in the introduction, it has beenfound that equations, primarily the martensite volume fraction ki-netic equations, must be modified to achieve robust three-dimen-sional computations. In making these modifications, the number ofmodel inputs may also been reduced from 17 to 14. Additionally,since it was assumed that reorientation does not change the mag-nitude of the internal variables (inelastic strains and martensitevolume fractions), but merely reorients the inelastic strain direc-tion with respect to the transformation strain limit surface [1], itis possible to numerically solve for reorientation and orientationindependently [2]. The fundamental mechanics and constitutiveassumptions of the original formulation are not altered in thisnew formulation.

2.1. The model & solution algorithm

Here, the improved model is presented through the rate inde-pendent algorithm used to solve the equations within the VUMATs.We then document the improvements in detail in Section 2.2.


1. Abaqus prescribes increments De and D T to the VUMAT, alongwith values of variables from the previous time step (denotedwith superscript n). Strain and temperature are updated:

T ¼ Tn þ DT ð14Þe ¼ en þ De: ð15Þ

2. The deformation resulting from these increments is assumed tobe purely elastic. A trial stress is calculated using:

rtrial ¼ C : e� entr

� �: ð16Þ

3. A check is made to see if this assumption of elastic deformationviolates the Kuhn–Tucker conditions imposed upon the limitfunctions of orientation and reorientation mechanisms. Thischeck is achieved by first calculating the thermodynamic driv-ing force for orientation:

Xr ¼ r0trial � qðhT~g0 � ~u0i þ HrnrÞffiffiffiffiffiffiffiffi3=2

pemax

tr

N entr

� �: ð17Þ

Next, the limit functions are evaluated:

Fr ¼ kXrk � YrðnrÞ ð18Þ

Fre ¼ bI entr

� �: r0trial

� Yre ð19Þ

The form of the projection tensor bI originally used by Panico andBrinson is bIðetrÞ ¼ I0 � NðetrÞNðetrÞ. The deviatoric identity tensor I0

is defined in full matrix indicial notation as I0ijkl ¼ dikdjl � 13 dijdkl

where d is Kronecker’s delta, or in Voigt notation as I0 ¼ I� 13 II

where I is a 6 � 6 identity tensor and I is the traditional 2nd orderidentity tensor expressed as a 1 � 6 column vector. While the twoforms of the projection tensor are not equivalent, in implementingthis model one may also choose to replace the deviatoric identitysimply with a fourth order identity tensor such thatbI en

tr

� �¼ I� NðetrÞNðetrÞ since in the model bIðetrÞ is always con-

tracted with r0 and ½I0 � NðetrÞNðetrÞ� : r0 � ½I� NðetrÞNðetrÞ� : r0. Thisresult follows from a standard identity of deviatoric tensors, as de-scribed on page 423 of [30].The kinetic scalar Yre represents an internal energy (friction) thatmust be overcome for the SMA microstructure to reorient, and is as-sumed to be constant in its phenomenological average. It may becalibrated via fitting to non-proportional loading data for a givenmaterial. This kinetic dissipation parameter provides hysteresis inreorientation, as a real material would experience. It is assumedto be constant merely because the phenomenological dissipationof a conglomerate of reorienting martensite variants is not quanti-fied well enough to understand the true physical form. However,a physically motivated scalar function may easily be introduced inplace of this parameter once this phenomenon is understood.Analogously, Yr controls the kinetics of nr and in this improvedmodel takes the form:

YrðnrÞ ¼nr tan p nr þ 1

2

� � �þ Cf ; Dnr > 0

ð1� nrÞ Ar þ tan p 12� nr� � ��

þ Cr; Dnr < 0

(ð20Þ

and the direction of Dnr is evaluated through:

Xr : N entr

� �> 0; Dnr > 0

Xr : N entr

� �< 0; Dnr < 0

: ð21Þ

If the conditions

Fr;re 6 0 ð22Þ

are satisfied, then the assumption of the deformation prescribed byAbaqus being completely elastic is valid. r ¼ rtrial; etr ¼ en

tr , and weproceed to Step 6 to solve for self-accommodated martensite. Ifthese conditions are not satisfied, the step is not fully elastic and

the code proceeds sequentially through this algorithm to solve forthe transformation strain.4. If Fre 6 0, skip to Step 5, else the transformation strain is reori-

ented by solving the reorientation residuals for r, Dere, andDkre:

0 ¼ r� C : e� entr � Dere

� �ð23Þ

0 ¼ Dere � DkrebIðetrÞ : r0 ð24Þ

0 ¼ kbIðetrÞ : r0k � Yre: ð25Þ

Note that Eq. (24) could be substituted into Eq. (23) to further re-duce the dimension of the problem, but because we wish to outputDere, we choose to program this less compact system of residuals.5. If Fr 6 0, skip to Step 6, else the transformation strain is evolved

by solving the orientation residuals for r, Der, nr, and Dkr:

0 ¼ r� C : e� entr � Dere � Der

� �ð26Þ

0 ¼ Der � DkrXr ð27Þ0 ¼ kXrk � YrðnrÞ ð28Þ

0 ¼ nr �ketrkffiffiffiffiffiffiffiffi3=2

pemax

tr

ð29Þ

where Xr follows the definition given in Eq. (10). Akin to Step 4, Eq.(27) could be substituted into Eq. (26) if the implementer did notcare to track the evolution of Der, but because we do we choose thisless compact system of residuals. Also note that we include Dere

from Step 4 in Eq. (26), thus if ere evolved in Step 4, r will be calcu-lated in Step 5 for the total transformation strain.In the case of initiation of transformation strain (i.e., while ketr-

k 6 1), N(etr) is numerically singular and the initiation residualsare used (we refer the reader to [1] for their derivation):

0 ¼ r� C : ðe� DerÞ ð30Þ0 ¼ Der � DkrXr ð31Þ

0 ¼ kr0k � qðhT~g0 � ~u0i þ HrnrÞffiffiffiffiffiffiffiffi3=2

pemax

tr

þ YrðnrÞ( )

ð32Þ

0 ¼ nr �kDerkffiffiffiffiffiffiffiffi3=2

pemax

tr

ð33Þ

where the deviatoric stress is used to define the initial transforma-tion strain direction:

Xr ¼ r0 � qðhT~g0 � ~u0i þ HrnrÞffiffiffiffiffiffiffiffi3=2

pemax

tr

Nðr0Þ ð34Þ

6. Finally, the evolution of self-accommodated martensite isdetermined. The self-accommodation driving force isevaluated:

XSA ¼ �qðT~g0 � ~u0Þ ¼ �q~g0 T �M0s

� ð35Þ

The self-accommodation limit function is then calculated:

FSA ¼XSA � Yf

SA nnSA

�; XSADnn

SA > 0�XSA � Yr

SA nnSA

�; XSADnn

SA < 0

(ð36Þ

using the kinetic equations:

YfSA ¼ qcf nn

SA ¼ q~g0 M0s �M0

f

� nn

SA ð37Þ

YrSA ¼ qcr 1� nn

SA

� �þ Yr

T0 þ �remaxtr ð38aÞ

¼ q~g0 A0s �M0

s

� þ A0

f � A0s

� 1� nn

SA

� �þ

�rCM

� �ð38bÞ

If the condition

FSA 6 0 ð39Þ

Table 1Numerical values of the soft constraints imposed upon the kinetic evolution of n by


is satisfied, self-accommodated martensite does not evolve. Other-wise, nSA is incremented through:

the original model (natural logarithm) and the improved model (tangent) for variouschoices of nmin.
nSA ¼ nn
SA þ DnSA ð40Þ
nmin �ln (nmin) tan p nmin þ 1
2

� �� 1.00E�02 4.61E+00 3.18E+01

where to impose the bounds on the volume fractions (Eq. (5)), ifapplicable, the calculation of DnSA is made according to:

1.00E�03 6.91E+00 3.18E+021.00E�04 9.21E+00 3.18E+031.00E�05 1.15E+01 3.18E+041.00E�06 1.38E+01 3.18E+05

DnSA¼�Dnr if nnSAþnr> nmax or T <M0

f and nnþDnr< nmax

ð41Þ
1.00E�07 1.61E+01 3.18E+06 otherwise: 1.00E�08 1.84E+01 3.18E+071.00E�09 2.07E+01 3.18E+081.00E�10 2.30E+01 3.18E+091.00E�11 2.53E+01 3.18E+101.00E�12 2.76E+01 3.18E+111.00E�13 2.99E+01 3.18E+12
DnSA ¼� ~g0

cf DT

� ~g0cr DT þ D�r

CM

(if

FfSA T; nn

SA

�P 0

FrSA T; nn

SA

�P 0

andDT < 0DT > 0

:

ð42Þ
1.00E�14 3.22E+01 3.18E+131.00E�15 3.45E+01 3.18E+14
10−20 10−15 10−10 10−5 1000

200

400

600

800

1000

ξ

Y σ (M

Pa)

Logarithm Constraint

Tangent Constraint

Fig. 3. The logarithm constraint bounding the evolution of nr originally proposed[2] was numerically too soft – as the limits of machine numbers were approached,the value of – ln (nr) did not numerically approach1. The improved law, however,numerically approaches 1 well before machine precision is challenged.

7. The final calculated stress is passed back to Abaqus, along withthe user-defined state variables transformation strain, orienta-tion and reorientation strain increments, and martensite vol-ume fractions, return to Step 1.

2.2. Discussion of improvements

2.2.1. Numerically robust kinetic lawThe most significant modification for robust numerical imple-

mentation is assumption of a new kinetic law for the frictionalforce resisting orientation (Eq. (20)). For discussion purposes, theoriginally proposed kinetic equations governing orientation follow:

YrðnrÞ ¼Af nr � Bf nr lnð1� nrÞ þ Cf ; _nr > 0Arð1� nrÞ � Brð1� nrÞ lnðnrÞ þ Cr ; _nr < 0

(ð43Þ

The value of this function represents a critical effective stress thatmust be overcome for the orientation mechanism of the model toactivate. While in theory the natural logarithm terms drive the va-lue of Yr to1 as the volume fraction saturates (0 6 nr 6 1), in prac-tice this is not the case.

Discontinuities would arise in these equations if nr = 1 or 0, thusnumerically nmin, a small number very close to 0, must be chosen asthe lower bound, and nmax = 1 � nmin is a logical correspondingchoice for an upper bound. Table 1 and Fig. 3 show that in decreas-ing nmin by orders of magnitude toward the limit of machine preci-sion, the logarithm terms approach 50. This does not numericallyreplicate the intended limit of 1. The orientation mechanism iscontrolled by the limit function Eq. (18) and in the state of satu-rated volume fraction, Xr varies proportionally with r0 (see Eqs.(10) and (34)) The result using Eq. (43) in Eq. (18) is that if the ap-plied load is increased a modest amount after the saturation of ori-ented martensite, situation may arise where Fr > 0 even though itis not physically possible for martensite to further orient. Forexample, consider the case in which orientation of martensite dur-ing forward transformation saturates at an effective stress of500 MPa. If we chose nmin = 1e�12, Bf = 5 MPa, Af = 0 MPa, andCf = 70 MPa, all reasonable choices for a polycrystalline SMA suchas NiTi, Yr = 138 MPa when nr = 1. Thus if we continue to loadthe material to the vicinity of 700 MPa, Fr > 0 and the orientationmechanism will attempt to increase the transformation strain,even though physically this is impossible for the material givenour constitutive assumptions (Eqs. (4)–(6)). Thus, we conclude thatthe form of a soft constraint built into this kinetic law is too soft forrobust numerical implementation.

To make this kinetic law more numerically robust and also toreduce the number of calibration constants needed for the model,we have replaced the original assumed form (Eq. (20)) with Eq.(43). In choosing the tangent function instead of the natural loga-rithm we introduce a soft constraint that provides sufficient hard-ening to numerically bound n; the constraint continues to grow by

an order of magnitude in each order of magnitude toward satura-tion of n (Table 1). Additionally, the need for two calibration con-stants Bf and Br is eliminated. In the original model theseconstants were required to impose a constraint to bound n, andcould be adjusted to control the ‘‘roundness’’ of the exit side ofthe hysteresis loop in each forward and reverse transformation.Since the bound is now sufficiently hard, a scaling constant is nolonger required. Since the motivation for the model is to replicatethe mechanics of the overall transformation strain, not smoothingof the hysteresis corners, it is not necessary to reintroduce a tuningparameter for hysteresis smoothing. Furthermore, the constant Af

was redundant considering Hr (see Eqs. (17), (18) and (43)), thusit has also been removed in this improved kinetic law. Ar remainsas a tuning parameter that allows the modeler to calibrate a uniquehardening slope for reverse transformation with respect to forwardtransformation.

2.2.2. Reduced set of model inputs easily interpreted from empiricalmeasurements

In [1], model inputs were defined through 17 scalars: the 6 phe-nomenological material properties E; m;q; ~u0; ~g0; emax

tr combinedwith 3 model material parameters cf ; cr ;Yr

T0 and 8 kinetic tuningparameters Hr,Yre, Af, Bf, Cf, Ar, Br, Cr. However, it is also possibleto instead write the parameters cf ; cr ;Yr

T0 along with the materialproperties ~u0; ~g0 in terms of the four stress-free transformation


temperatures and the Patel–Cohen coefficient. This formulation isdesirable as the specific internal energies and entropies associatedwith the austenite–martensite phase transformation are only occa-sionally reported, but stress-free transformation temperatures arealmost always documented. Additionally, in numerically improv-ing the orientation kinetic law (Eq. (20), discussed further in Sec-tion 2.2.1) the number of kinetic tuning parameters was reducedfrom seven to four. This results in a set of 14 model inputs consist-ing of 9 material properties that are readily published for nearly allSMAs E; m;q; emax

tr ;CM;M0s ;M

0f ;A

0s ;A

0f and 5 kinetic tuning parameters

Hr, Yre, Cf, Ar, Cr. From these inputs, the model-specific parameter scf ; cr ;Yr

T0 and less commonly published material properties ~u0; ~g0

may be directly calculated for convenience of shorthand withinthe numerical implementation:

~g0 ¼CMemax

tr

qð44Þ

~u0 ¼ ~g0M0s ð45Þ

cf ¼ ~g0 M0s �M0

f

� ð46Þ

cr ¼ ~g0 A0f � A0

s

� ð47Þ

YrT0 ¼ q~g0 A0

s �M0s

� : ð48Þ

One final modification may be made to the model inputs to facili-tate direct translation from empirical measurements. Since phasetransformation and orientation are driven by the deviatoric stressin this model, the kinetic parameters Hr, Yre, Cf, Ar, Cr and the Pa-tel–Cohen coefficient CM taken from empirical measurementsshould be scaled by

ffiffiffiffiffiffiffiffi2=3

pprior to calculations. This may either

be done manually, or else written into the numerical code.

2.2.3. New choice of reorientation limit functionThe original proposed reorientation limit function was

Fre ¼ 12 r0 : bIðetrÞ : r0 � Yre [1]. In this improved model, we have in-

stead introduced the form of Eq. (12), as this new form allowsthe dissipation of reorientation Yre to be expressed in units of stressrather than stress squared. Hence, it is now possible to calibratethis parameter through phenomenological observation of an effec-tive onset stress for reorientation, just as an effective onset stress

for orientation is used to calibrate Yr. Note that kbIðetrÞ : r0k andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir0 : bIðetrÞ : r0

qare equivalent expressions in choosing to imple-

ment either form of bIðetrÞ (see Section 2.1, Step 3) and tr(etr) = 0.

2.2.4. Correction to the specific Entropy–Patel–Cohen relationIn the original work, Panico and Brinson defined the Patel–Co-

hen coefficient relationship to the specific entropy as CM = q Dg0

[1]. However, by definition the Patel–Cohen relationship betweenapplied stress and energy of transformation scales as the productof stress and transformation strain [31]. The omission of transfor-mation strain in this relationship resulted in an issue of improperscale. In the improved model, Eq. (44) was adopted from [32] tocorrect the calculations. Note that for purposes of continuum for-mulation, we have oversimplified the original description of thisrelationship, which considered microstructure at the level of habitplanes formed by pairs of variants [31].

3. Numerical schemes for solving orientation and reorientationequations

There are four scenarios in the model that require evolution ofthe internal variables. nSA evolves when Eq. (39) is not satisfied.However, the evolution of self-accommodated martensite may becalculated using the derived explicit solution procedure (Step 6,

Eqs. 35–42), thus it requires no further numerical consideration.Hence there are three scenarios that require sets of coupled equa-tions to be solved:

(i) When Eq. (13) is not satisfied for Fr and ketrk 6 1, nr and etr

will initiate via Eqs. 30–34;(ii) When Eq. (13) is not satisfied for Fre, etr will reorient via Eqs.

23–25;(iii) When Eq. (13) is not satisfied for Fr and ketrk > 1, etr will ori-

ent via Eqs. 26–29.

As mentioned in the introduction, two numerical schemes forsolving these sets of equations are presented in this section. In Sec-tion 4, they are compared through studies of convergence andcomputational efficiency.

3.1. Scheme 1-constant lagrange multiplier increment

This is the simplest of the schemes to implement and thescheme that was used to augment the implicit integration schemeinitially proposed by Panico and Brinson (see Section 1.2). A fixedLagrange multiplier increment is used for both orientation andreorientation Dkr,re = k0. When orientation and/or reorientationKuhn–Tucker conditions require inelastic deformation to be satis-fied (i.e., Eq. (13)), this constant multiplier is used in the strain rateequations (Eq. (11)), and the new transformation strain (Eq. (2)) iscalculated along with the updated stress (Eq. (26)) and martensitevolume fractions (Eq. (29) or (33), depending on Scenario (i) or (iii)stated above). The error from not solving the equations throughintegration propagates into the next step and is compensated forin subsequent calculations. That is, at each step of the finite ele-ment simulation, Eq. (13) will continue to not be satisfied untilthe solution to the equations of the active scenario have converged.While this scheme is simple and the easiest scheme to program,the choice of k0 relative to the parameters of the finite elementsimulations (mesh, time) is sensitive, as demonstrated in Section 4.With this scheme, the algorithm for Steps 4 and 5 is:

� Dkr,re = k0 = constant� Plug Dkr,re into Eqs. (24) and (27) and solve for Der,re

� Use to update r, nr through Eqs. (2), (26), (29)

END

3.2. Scheme 2-explicit integration

A new explicit scheme was developed to relieve sensitivityinherent to the ‘‘Constant Lagrange Multiplier’’ scheme whileavoiding the cumbersome implementation of the Jacobian for the‘‘Implicit Integration’’ scheme (see Appendix B). This new integra-tion scheme is conceptually simple, analogous to the idea of a ra-dial return algorithm [33]. However, for this specific constitutivemodel, robust implementation requires a bit of additional deriva-tion. The gist is to begin with an initial guess of the Lagrange mul-tiplier increments, and then continue to increase Dkr,re until thesolution converges.

First, we introduce a new computation input f, the targetednumber of iterations for each integration. The choice of f is studiedin Section 4. However, generally, choosing smaller f leads to re-duced computation time at the expense of less precise convergencewhile larger f has opposite consequences.

The derivation commences with the observation that eachmechanism will require integration if and only if Eq. (13) is not sat-isfied. Consequently, in solving for the new transformation strain,stress, and martensite volume fractions, the algorithm will loopuntil Eq. (13) is satisfied. This in mind, when Eq. (13) is not


satisfied, we would like each step of the explicit integrationscheme to increment the value of the limit function by

DFr;re ¼ �Fr;refrtrialg

f: ð49Þ

until we converge to a solution that satisfies Eq. (13). Examining theorientation Eqs. (10) and (12) in rate form and noting that _NðetrÞ � 0since the transformation strain is always normalized by its magni-tude, and _Yr � 0 as well since we are explicitly stepping throughtime in small increments:

_Fr ¼_Xr : Xr

kXrk� _Yr �

_r : Xr

kXrk: ð50Þ

To study the order of magnitude of the reorientation limit functionin Eq. (12), it is useful write the expressions bIðetrÞ : r0 andkbIðetrÞ : r0k out in indicial notation and simplify using tr(etr) = 0. Thiscalculation results in2:

bIðetrÞ : r0 ¼ r0 � etretr : retr : etr

ð51Þ

kbIðetrÞ : r0k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir0 : r0 � ðetr : rÞ2

etr : etr

s: ð52Þ

Making the appropriate substitution and taking the time derivativeresults in:

_Fre ¼r : _r� 1

3 trðrÞtrð _rÞ � ðetr : _rþ _etr : rÞ etr :retr :etrþ _etr : etr

etr :retr :etr

� 2

kbIðetrÞ : r0k:

ð53Þ

Next, we recognize that we are solving the reorientation and orien-tation equations individually, and while they are being solved, thetotal strain increment is fixed and we are beginning from theassumption that the total strain increment was elastic (Section 2.1,Steps 1–3). Thus, in isolation the stress rates during the integrationof the equations for each mechanism are (Section 2.1, Steps 4, 5):

_r ¼ �C : _er and _r ¼ �C : _ere: ð54Þ

Substituting these expressions into Eqs. (50) and (53), and also Eq.(27) into Eqs. (50) and (24) into Eq. (53) (all in rate form), we arriveat the expressions:

_Fr ��ðC : _erÞ : Xr

Xr� � _kr

Xr : C : Xr

Xr; ð55Þ

_Fre ¼ � _kre

r� etretr :retr :etr

� : C : ðbIðetrÞ : r0Þ � 1

3 trfC : ðbIðetrÞ : r0ÞgtrðrÞ � kbIðetrÞ : r0k2 etr :retr :etr

kbIðetrÞ : r0k: ð56Þ

Finally, these may be substituted into Eq. (49) in incremental formto derive the initial values of the Lagrange multiplier increments forthis explicit integration scheme:

Dk0r ¼

kXrkFr

fXr : C : Xr; ð57Þ

Dk0re ¼

FrekbIðetrÞ : r0kfðr� etr

etr :retr :etrÞ : C : ðbIðetrÞ : r0Þ � 1

3 trfC : ðbIðetrÞ : r0ÞgtrðrÞ � kbIðetrÞ : r0k2 etr :retr :etr

: ð58Þ

Using these initial values, the equations for orientation and reorien-tation (Steps 4 & 5) may now be independently solved through theexplicit iteration scheme:

2 It may also be desirable to use these substitutions in coding the model as theyallow omission of explicit calculation of the projection tensor.

Dkr;re ¼ Dk0r;re

# iterations = 1While Eq. (13) not satisfied and #iterations 6max iterations� Substitute Dkr,re into Eqs. (24) and (27) and solve for Der,re

� Use to update r, nr,Fr,re through Eqs. (2), (18), (19), (26), (29)� #iterations = #iterations + 1� Dkr;re ¼ #iterations Dk0

r;re

END

These multipliers cannot exactly converge in a single step(f = 1) for all cases since there was some approximation madein their derivations (assumptions that _Yr; _NðetrÞ � 0Þ and be-cause they are calculated by reducing tensors to scalars, andthen multiplying those scalars by components of tensors again.For the same reasons, we cannot expect to always converge inexactly the targeted number of iterations f > 1. In the currentwork we choose max iterations = 10f, and we will documentthe actual number of iterations used for different choices of fin Section 4.

Checks may be implemented to understand if the choices of fand max iterations are appropriate: (1) warning messages maybe printed to the log file if the procedure stopped due to maximum#iterations; (2) in the case of reorientation, numerical implemen-tation of the tangency condition (Eq. (7))

etr : _ere < 1 ð59Þ

may be used to check sensibility of the calculated reorientationstrain increment (the same effective small strain limit 1 as is usedto protect against singularity of N(etr) may be used here). If thereare problems due to poor choice of material parameters, simula-tions parameters, mesh, boundary conditions, or bugs in the code,one of these two conditions will not be met fairly quickly in thesimulation.

4. Results and discussion of simulations using the explicitnumerical schemes

Having formulated two schemes for explicit integration of theimproved constitutive model within an explicit finite elementframework, we proceed to study convergence of each scheme withrespect to the computational parameters k0 for Scheme 1(Section 3.1) and f for the Scheme 2 (Section 3.2). Also, since this

rate independent constitutive model is being implemented intoan explicit FE framework, it is desirable to have a numerical imple-mentation that is insensitive to liberal mass scaling. Mass scaling

artificially inflates the mass of the material to force Abaqus tocalculate a larger stable time increment for the simulation. In aquasi-static simulation using explicit FE, such as one using arate-independent material model, mass scaling may be used to re-duce computation time (larger time increment = fewer simulationsteps = less computation time, provided the order of convergence

Table 2Material properties and kinetic parameters used in the simulations.

Property Value Parameter Value

E 70 GPa Yre 40 MPav 0.4 Hr 200 J/kgq 6.45 g/cm3 Cf 100 MPaMs 258 K Cr 100 MPaMf 248 K Ar 0 MPaAs 278 KAf 288 KCA,M 5.5 MPa/Kemax

tr 0.05

Fig. 4. The brick element and the defined coordinate system.

Table 3Simulated displacement-control loadings applied to the 4 top nodes of the simulatedelement. All displacements were applied with a velocity of 0.16 mm/s.

Step Time Direction

1 10 +y2 10 +x3 20 �y4 20 �x5 20 +y6 10 +x7 10 �y


remains constant). The restriction on mass scaling is that at somepoint the simulation the mass may be inflated too much and theresulting artificially inflated kinetic energy will impart oscillationsupon the quasi-static solution. Thus, one must use caution andcheck carefully for both oscillations and abnormally large valuesof kinetic energy to ensure quasi-static conditions are upheld inthe simulations. See the Abaqus Manual for more details and theequations used for the different mass scaling algorithms availablein Abaqus as well as computation of the stable time increment[8]. In this work, we use the ‘‘Semi-Automatic,’’ ‘‘Whole Model,and ‘‘Fixed’’ options while varying the mass scaling factor. This al-lows us to also examine the robustness of both schemes in the con-text of mass scaling.

The material properties and kinetic tuning parameters used inthese simulations are listed in Table 2. These are not calibratedto a specific shape memory alloy, but are plausible for commer-cially available NiTi [34]. We begin by examining convergence ofthe constitutive law within a single 3D brick element, and thenevaluate the robustness of the most promising schemes when usedto simulate stress-induced phase transformation of a bar with ahole subjected to tension, meshed with a varied element size.

4.1. Simulations of a single 3D brick element

A single 8-node explicit linear brick element (type C3D8R [8])40 mm � 40 mm � 40 mm (see Fig. 4) was subjected to an

axial-shear, non-proportional, displacement-control loading path.These loadings were imposed by fixing the bottom surface of theelement in the 1-and 2-directions and then displacing the nodesof the top surface of the element as indicated in Table 3 andFig. 5. The converged axial and shear components of the calculatedresponses are shown in Figs. 5 and 6, along with the equivalentCauchy stress and Lagrangian strain responses calculated via

req ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

22 þ 3r212

q; eeq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

22 þ43e2

12

r: ð60Þ

The converged oriented martensite volume fraction evolution isalso shown in Fig. 5c. In examining Figs. 5b and 6b, the stress re-sponse may seem counter-intuitive as there are many inflectionpoints that do not correspond to inflection points in the appliedloading path. These inflection points are a result of the drivingforces and kinetics of orientation and reorientation (Eqs. (10),(18), (19), (20)) – because of hysteresis as well as saturation events,the initiation and cessation of activity of these mechanisms doesnot necessarily coincide with inflections in the mechanical loadingof the material. A more detailed discussion of the mechanics of thisconverged response is given in Appendix C.

4.1.1. Scheme 1 convergence sans mass scalingThe responses of simulations using Scheme 1 while varying k0

in order of magnitude increments from 10�5 to 10�12 are shownin stress-space through Fig. 7a and b. In examining the responseson the scale of Fig. 7a it is apparent that for k0 varying from10�10 to 10�12 stresses are grossly over-predicted. Because the va-lue of k0 is not large enough, the explicit finite element continuesto calculate an elastic response using a modulus of 70 GPa in theabsence of the VUMAT calculating a reasonable estimation of theinelastic transformation strain. In examining these same responseson the scale of Fig. 7b, for k0 varying from 10�6 to 10�5 the stresspredictions become non-convergent, over and undershooting thesolution as k0 is now too large. Thus, for this material parameteri-zation, element choice, and prescribed loading, it is demonstratedthat selecting k0 between 10�7 and 10�9 results in convergence ofthe mechanical response. Fig. 7c & d are used to compare Schemes1 & 2, and will be discussed in Section 4.1.3.

These results demonstrate the ability of this scheme to simu-late the thermodynamics of the constitutive model. Furthermore,the explicit finite element calculation is able to compensate for afairly wide range of k0 choices in the VUMAT calculation, span-ning two orders of magnitude. One drawback of this scheme isthat the convergence study must be run to understand if thechoice of k0 is reasonable – there was no way for us to know apriori that k0 between 10�7 and 10�9 would converge; we hadto employ the guess-and-check method. For more complex mod-els, this convergence study becomes quite tedious when checkingmany integration points. Further insights in choice of this schemewill be made in comparing this scheme with Scheme 2 inSection 4.1.3.

4.1.2. Scheme 2 convergence sans mass scalingMechanical response predictions (stress, strain, martensite vol-

ume fraction) converged to those shown in Figs. 5 and 6 for choicesof f ranging from 1 to 500. Insight as to why may be gained consid-ering Fig. 8 in the context of the convergence study of the previoussection. It was demonstrated that the finite element calculationcould accommodate two orders of magnitude difference in the La-grange multiplier increment used in the VUMAT. Fig. 8 shows thatby using Eqs. (57) and (58) to calculate the initial Lagrange multi-plier increments and Scheme 2 to integrate the sets of transforma-tion and/or reorientation equations for Steps 4 & 5, the calculatedinitial Lagrange multiplier increments always fall well within an

−0.05

−0.025

0

0.025

0.05

0.075

ε (m

m/m

m)

ε12

ε22

εeq

−500

−250

0

250

500

σ (M

Pa)

0 10 20 30 40 50 60 70 80 90 1000

0.25

0.5

0.75

1

ξ σ

Time

σ12

σ22

σeq

(c)

(b)

(a)

ii iiiiv

ii iiiiv

vvi i i i

2

1

i i ii

Fig. 5. Converged responses of the simulations in the time domain: (a) the displacement control path represented in axial and shear strain components as well as anequivalent applied strain; (b) the directional and equivalent stress responses; (c) the oriented martensite volume fraction calculation.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

100

200

300

400

εeq

σ eq

−0.05 −0.025 0 0.025 0.05−0.05

−0.025

0

0.025

0.05

ε22

ε 12

−400 −200 0 200 400−200

−100

0

100

200

σ22

σ 12

0 to 1010 to 2020 to 4040 to 6060 to 8080 to 9090 to 100

(c)

(a) (b)

i

i

v

iiiii

iv

v

v

i

i

ii

ii

iii

iii

iv

iv

i

Simulation Time

Fig. 6. Converged responses of the simulations in mechanical domains: (a) the strain control path plotted as axial vs. shear strain; (b) the stress response in the conjugatestress-space; (c) equivalent stress vs. equivalent strain. The small Roman numerals are used to coordinate inflection points in stress-space (b) with minor loop inflectionpoints in equivalent stress–strain space (c).


order of magnitude of the converged solution (the solution con-verged at choices of f = 100, 500). Considering the case f = 1, thelack of convergence of the Lagrange multipliers but convergence

of the stress, strain, and volume fraction calculations in Figs. 5and 6 is significant. It means that using Scheme 2, one may choosenot to integrate the coupled PDEs in Steps 4 & 5 at all (i.e., f = 1)

−4,000 −2,000 0 2,000 4,000−2000

−1000

0

1000

2000

σ22 (MPa)−400 −200 0 200 400

−200

−100

0

100

200

−2000

−1000

0

1000

2000σ 12

(MPa

) −200

−100

0

100

200

10−6

10−7,10−8,10−9

10−10

10−12

1 − 103

104

105

106

108

No Mass Scaling

Δλ = 10−8

(a) (b)

(d)(c)

Δλ0 Value

Mass Scaling Factor

Fig. 7. Stress responses of simulations using the Constant Lagrange Multiplier scheme (see Section 3.1). (a) and (b) depict the responses of varying k0, sans mass scaling; (c)and (d) show the result of varying the mass scaling factor for k0 = 10�8.

10−12

10−11

10−10

10−9

10−8

Δλσ

0 10 20 30 40 50 60 70 80 90 10010−12

10−11

10−10

10−9

10−8

Time

Δλre

1550100500

68 69 70 71 726

6.5

7

7.5

x 10−10

(a)

(b) (c)

ζ Value

Fig. 8. Calculations of (a) the orientation strain Lagrange multiplier increments and (b) the reorientation strain Lagrange multiplier increments during simulations using theExplicit Integration scheme (see Section 3.2) as f was varied. (c) Is a magnified view of the reorientation multiplier calculations about t = 70, showing that while the f = 50simulation is indistinguishable from the converged solution on the scale of figures (a) and (b), it is not fully converged when examined more closely.


and still simulate the correct stress, strain, and volume fraction re-sponses by allowing Abaqus Explicit FE calculations to compensatefor any error caused by the lack of exactness in calculating thetransformation/reorientation Lagrange multipliers. Furthermore,choosing a small number of iterations (i.e., f = 5) quickly allowsthe solution to approach the converged solution for all internaland external variables, and very little difference was seen betweenthe simulated Lagrange multiplier increments and the convergedsolution in choosing f greater than 50 (see Fig. 8c).

Another interesting observation is the number of iterationsused in integrating the equations for Steps 4 & 5, plotted in

Fig. 9. Recall from Section 3.2 that the targeted number of itera-tions f could only be treated as an approximation since tensorswere being contracted to scalars in Eqs. (57) and (58). Thus, wechose to allow for an order of magnitude more iterations in the VU-MAT by setting max iterations = 10f. However, Fig. 9 shows thatwhile f was an approximation, there were never more than f iter-ations required for convergence, regardless of the choice of f. Thus,an implementer of Scheme 2 could restrict the maximum numberof iterations to f + 1 instead of 10f, though we note that this resultdoes not automatically transcend to multiple element meshes sub-jected to non-proportional loading.

10−1

100

101

102

103

# of

iter

atio

nsor

ient

atio

n

0 10 20 30 40 50 60 70 80 90 10010−1

100

101

102

103

Time

# of

iter

atio

nsre

orie

ntat

ion

5001005051

(a)

(b)

ζ Value

Fig. 9. The number of iterations needed to meet the convergence criteria during explicit integration of the systems of equations for (a) orientation and (b) reorientation as fwas varied.

Table 4CPU times (in m:s) of simulations for Scheme 1 using = 10�8 as the mass scaling factorvaried, and for Scheme 2 varying both f and the mass scaling factor.

Factor None 10 102 103 104 105 106 107 108

k0 = 10�8 21:49 7:10 2:19 0:44 0:18 0:11 0:08 0:07 0:06f = 1 22:32 7:23 2:28 0:46 0:20 0:11 0:07 0:06 0:07f = 50 24:51 8:17 2:39 0:50 0:20 0:11 0:07 0:07 0:06f = 100 27:54f = 500 52:10

0 50 100 150 200 250 300 350 4000

50

100

150

σ22

σ 12

Base

ζ=1

ζ=50

Mass Scaling Factor = 108

Fig. 10. The first quadrant of the stress responses of the Explicit Integration schemesimulations for a mass scaling factor of 108, plotted vs. the base (converged)solution.


4.1.3. Comparison of the schemesThrough consideration of the previously presented results (i.e.,

Sections 4.1.1 and 4.1.2), the most significant difference betweenthe two schemes is that Scheme 1 must be manually calibratedand studied for convergence when mesh and/or model parametersare changed while Scheme 2 is self-calibrating, even when explicit

integration of the PDEs for orientation and reorientation is turnedoff (i.e. f set to 1). The question for a user of this constitutive modelthen becomes ‘‘Does the coding complexity and computationalcost of calculating Eqs. (57) and (58) outweigh the benefits ofself-calibration?’’ While the final answer to this question may besubject to both personal opinion and frequency of use of the model,in the remainder of this section we will quantify the computationalcost of each scheme both with and without mass scaling for a sin-gle 3D element, and in Section 4.2 this study of numerical simplic-ity of Scheme 1 vs. benefits of self-calibration of Scheme 2 will beextended through simulations of a more complicated mesh.

To aid in the ensuing discussion, CPU times for the simulationsboth with and without mass scaling are listed in Table 4. Thesesimulations were performed on a single core of a Dell T5500 Linuxworkstation. For the ensuing mass scaling study, k0 = 10�8 waschosen for Scheme 1 since it was in the middle of the observedconvergence interval of 10�7

6 k0 6 10�9 while Scheme 2 wasstudied for both no integration, i.e. f = 1, and f = 50. Mass scalingwas imposed by use of a mass scaling factor [8], which was variedin order of magnitude from 1 (no mass scaling) to 108.

Comparing the results for no mass scaling, Scheme 2 withoutintegration (f = 1) took 3% longer to run. Both schemes providedconverged stress, strain, and volume fraction calculations, but nei-ther scheme provided a converged Lagrange multiplier solution(see the preceding two sections and Figs. 5–8). However, theScheme 2 Lagrange multiplier calculations for f = 1 were of thesame order of magnitude as the converged solution (Fig. 8), whilethe Scheme 1 fixed Lagrange multiplier increment was off by an or-der of magnitude. Integration of the PDEs using 50 iterations in-creased the time by an additional 6%, while providing theadditional accuracy of a converged Lagrange multiplier solution.Further increase of the number of iterations for each integrationscontinued to increase the computational cost with virtually nobenefit, as the Lagrange multiplier calculations did not experiencesubstantial improvement (Fig. 8c).

As mentioned in the introduction to Section 4, since this consti-tutive model is rate independent, it is desirable to have a numericalimplementation that is insensitive to mass scaling so this techniquemay be invoked to reduce the computational cost of simulations.Stress responses of Scheme 1 as the mass scaling factor is varied

1e−6

1e−3

1

1000

ζ=1

0 10 20 30 40 50 60 70 80 90 1001e−6

1e−3

1

1000

Time

ζ=50

1e−6

1e−3

1

1000

Δλ0=1

0−8

1 (none)10102

103

104

105

106

107

108

Kine

tic E

nerg

y (m

J)

1 (none)10102

103

104

105

106

107

108

Mass ScalingFactor

(a)

(b)

(c)

Fig. 11. Total kinetic energy of single 3D element simulations as mass scaling factor was varied: (a) Fixed Lagrange Multiplier scheme with k0 = 10�8 (see Section 3.1); (b)Explicit Integration scheme with f = 1 (see Section 3.2); c) Explicit Integration scheme with f = 50 (see Section 3.2).

Fig. 12. Contours of the evolution of the oriented martensite volume fraction as the specimen is loaded.


Table 5CPU times of the three 2D simulations compared in Section 4.2.

Description CPU Time (h:m:s)

k0 = 10�8, Factor = 102 115:07:24f = 1, Factor = 105 4:35:41f = 50, Factor = 105 4:41:34


are plotted in Fig. 7c and d. Mass scaling for this scheme was robustfor factors less than or equal to 103. Contrarily, stress, strain, andvolume fraction were correctly calculated using Scheme 2 for allmass scaling factors except for 108. For 108, the predictions werestill quite reasonable on the whole, though locally there were someoscillations near load inflection points. The imparting of dynamicresponses upon the quasi-static solution indicates that the thresh-old for increasing the mass scaling factor without affecting the rateindependent calculations had been crossed (see Fig. 10). Fig. 10 alsoshows that these oscillations were more quickly dampened for fullintegration (f = 50) than for self-calibration only (f = 1).

The total kinetic energies of each simulation of the mass scalingstudy are plotted in Fig. 11. Note that for a mass scaling factor of104, this plot would suggest both schemes performed equivalently.However, recall that Scheme 1 calculations converged only for fac-tors 1–103, and in Fig. 11 Scheme 1 kinetic energy oscillates moreseverely than Scheme 2 for this range of mass scaling factors. Fur-thermore, note that while the average kinetic energy of self-cali-brating Scheme 2 does increase for increasing mass scalingfactor, it is relatively oscillation-free for factors of 1–107, all ofwhich provided converged calculations of stress, strain, and vol-ume fraction. Clearly the self-calibrating scheme is more robustwith regard to retaining rate independent, quasi-static conditionsin the presence of mass scaling.

Concurrent consideration of all of these results demonstratesthat the additional computational expense of making the self-

00.050.10

200400

200

400

0 0.02 0.04 0.060

200

400σ eq

0 0.02 ε

0 10 20 30 40 5010

−12

10−10

Δλ

Simulatio

0.25

0.5

0.75

1

ξ σ

(b)

(c)

40 to 60

90 to 100

0 to 10

i

10 to

v

i viii

iii iv

(a)

Fig. C.1. (a) Isolated responses of each loading segment in equivalent stress–strain spacand reorientation Lagrange multiplier evolutions over simulation time. This figure compsuperelastic loading responses of Section 4.1.

calibration calculation for the initial Lagrange multiplier incre-ments using Eqs. (57) and (58) is quickly recovered in the robust-ness of Scheme 2 in the presence of mass scaling. Convergedsimulations using Scheme 2 were obtained in 6–7 s with a massscaling factor of 107 while the least expensive converged simula-tion for Scheme 1 with k0 = 10�8 took 44 s (6–7 times longer) witha mass scaling factor of 103.

4.2. Simulations of a 2D mesh with varying element size

To evaluate the computational efficiency of the schemes in amulti-element simulation, a 3 mm � 25 mm bar with a 0.5 mmdiameter hole placed at the center of the gage area was subjectedto an axial displacement control load of 0.8 mm at a rate of0.1 mm/s. The model was seeded with 28 equidistant nodesaround the diameter of the hole, the seeding was biased alongthe y-axis, top, and bottom edges such that a non-uniform meshof CPS4R [8] 4-node plane-stress elements that was coarse awayfrom the hole and fine near the hole was achieved. This mesh isshown in Figs. 12 and C.1.

The CPU times and computational parameters (k0, f, mass scal-ing factor) of the simulations used for this study are listed in Ta-ble 5. The same k0 and f values used in the mass scaling study ofthe previous section were selected, and the mass scaling factorwas chosen to be one order of magnitude less than the maximummass scaling factor that resulted in a converged solution for thesingle 3D brick element (see Section 4.1.3). Immediately the bene-fit of making the extra calculation to provide self-calibration is evi-denced in noting that the Scheme 1 simulation was on the order of115 h while the Scheme 2 simulations completed in 4.5 h. Fur-thermore, the additional computation of integrating the PDEs inSteps 4 & 5 using 50 explicit iterations does not substantially al-ter the CPU time of the simulation.

Similar to the single element studies, choices of f = 1 and f = 50produced fully converged solutions for stress, strain, and volume

0.04 0.06eq

0 0.02 0.04 0.06

60 70 80 90 100n Time

ΔλσΔλ

re

20 to 40

80 to 9060 to 80

i

iiiii

i

v

iv

20

iiiii

iv i

viiii

iii iv

e are shown along with (b) oriented martensite volume fraction and (c) orientationlements the Appendix C discussion of SMA mechanics during the converged biaxial

Table A.1Superscripts and subscripts.

Notation Definition


fraction calculations. Snapshots of oriented martensite volumefraction contours are shown in Fig. 12. The simulation usingScheme 1 also converged to these solutions.

A,M austenite, martensitee, tr elastic, transformation

(‘‘tr’’ was ‘‘in’’ in [1])r,re,SA orientation, reoriention, self-accommodation

(‘‘r’’ was ‘‘tr’’ and ‘‘SA’’ was ‘‘T’’ in [1])r,SA oriented, self-accommodated

(‘‘SA’’ was ‘‘T’’ in [1])f,r forward, reverses,f start, finish0 indicates a stress-free, reference, or initial condition, as defined in

textn denotes value from previous step

if not used in time-discrete equations, the represented quantity isin the n + 1 (current) condition

Table A.2Variables.

Symbol Description

r Stressr0 Deviatoric stress�r Mises equivalent stress

�r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2r0 : r0

pe Strainw Helmholtz free energyT Temperaturen Martensite volume fraction

(z was used in [1])u Specific energyg Specific entropyk Lagrange multiplierDp DissipationY Dissipative/frictional forceX Thermodynamic driving forceF Limit functionbI Projection tensor (See [41])

5. Conclusions

The numerically improved constitutive model demonstratedrobustness for a complex, non-proportional loading in a single3D brick element, and also in predicting transformation about ageometric defect (i.e., hole) within a rectangular gage section.Though the focus of this work was to evaluate the best methodfor numerical implementation of this constitutive model withinAbaqus Explicit software [8], it may be noted that the phase trans-formation propagation patterns shown in Fig. 12 are qualitativelyconsistent with empirical observations of other researchers [35].The transformation nucleates in lobe-like formations near the de-fect, propagates away from the defect in a crossing pattern, andin the bulk material bands form along shear planes and meld intoregions of uniformly transformed material. This indicates that theimproved model is not only numerically robust, but also able torealistically replicate SMA behaviors.

Two schemes for solving the PDEs of the model were derived. InScheme 1, constant Lagrange multiplier increments were used andthe error in the VUMAT calculation was compensated by the Aba-qus Explicit FE calculation. In Scheme 2, a newly formulated expli-cit integration scheme was used. While this scheme requires extracomputations in the VUMAT, it demonstrated a superior numericalrobustness relative to Scheme 1, highlighted by the followingfindings:

1. Scheme 2 is self-calibrating, thus a convergence study of theVUMAT for the choice of k0 using a guess-and-check methodis not required for Scheme 2 as it is for Scheme 1.

2. Solutions using Scheme 2 converged over 7 orders of magnitudeof mass scaling factor, while Scheme 1 calculations only con-verged for 3.

3. This robustness in sensitivity to mass scaling factor results inScheme 2 being the less expensive of the two schemes inmulti-element simulations, demonstrated by a simulation timeof 4.5 h vs. 115 h. for Scheme 1 in calculating the stress-inducedtransformation response of a bar with a hole.

Furthermore, in studying the choice of f = 1 for Scheme 2 it wasfound that the stress, strain, and martensite volume fraction pre-dictions converged equally as well as choice of f = 50 in all of thesimulations. The only benefit to allowing for multiple iterationsin the integration is if accurate Lagrange multiplier increment val-ues are desired. However, since in this model they are merely anumerical tool for integrating PDEs and have no physical relevance,there is rarely a need to calculate them exactly as the Abaqus Ex-plicit FE solver will compensate for error as long as they are esti-mated within reason (see Section 4.1.2).

If we now consider these two schemes relative to the original,unaugmented scheme proposed by Panico and Brinson in whichNewton’s method was used to implicitly integrate the PDEs,Scheme 2 with f = 1 requires no iteration to achieve a convergedsolution for stress, strain, and volume fraction calculations withinan explicit FE framework. Thus, the true essence of an explicit FEsimulation is recovered with Scheme 2 – in solving the materialmodel, make a single calculation that is close but not necessarilyexact for each time step and let the explicit FE calculation compen-sate for the error. Even though we did not create a VUMAT using anunaugmented version of the originally proposed scheme, clearlyScheme 2 with f = 1 is less computationally expensive than implic-itly integrating the equations at each step of and explicit FE

simulation. Additionally, while Eqs. (57) and (58) are not simple,they are much easier to program than the full Jacobian requiredfor the implicit integration scheme (see Appendix B). Hence, ifimplementing this constitutive model into explicit finite elementsfor studies using mass scaling, Scheme 2 with f = 1 is the bestchoice, and Scheme 2 with f = 50 may be used to ensure a conver-gent solution in all variables.

In considering impacts of this work beyond our model, a similarderivation approach as was used in Section 3.2 may be used to de-rive an integration-free, self-calibrating explicit scheme for othermaterial constitutive models: simply formulate the relationshipbetween driving forces and kinetic equations through Kuhn-Tuckerconditions, then derive the initial Lagrange multiplier incrementfollowing our procedure setting f = 1. Finally, we note that manyother SMA FE modelers chose to use implicit FE solvers (e.g.,[4,29,36,37]), such as Abaqus Standard [8]. In an implicit finite ele-ment framework where implicit integration is an inherent part ofthe solution scheme, the Jacobian in Appendix B will be useful informulating the tangent matrix [8]. While our long-term visionfor developing an SMA constitutive model using this core modelled us to select an explicit FE implementation for model develop-ment, a purely implicit FE implementation may be equally robustas the explicit Scheme 2 presented in this article for this rate andhistory independent improved model.

Acknowledgements

A.S. acknowledges funding through fellowships from the ToshioMura Endowment, Predictive Science and Engineering Design

Table A.3Material properties and model parameters.

Symbol Description

E Effective Young’s modulusv Effective Poisson’s ratioC Isotropic elasticity tensor

(L was used in[1])q density~g0 Specific entropy of phase transformation

~u0 ¼ gA0 � gM

0 (Dg0 was used in [1])~u0 Specific energy of phase transformation

~u0 ¼ uA0 � uM

0 (Du0 was used in [1])cv Specific heat at constant volumeMs Martensite start temperatureMf Martensite finish temperatureAs Austenite start temperatureAf Austenite finish temperatureCA,M Patel–Cohen coefficientY Dissipative/frictional constantemax

tr Maximum effective transformation strain(c in [1])

Hr Hardening coefficient for orientationA,C Kinetic parameters

(a third kinetic parameter B was also used in [1])

Cf ;Cr ;YrT0

Internal parameters that relate transformation temperatureswith specific energy and entropy

Table A.4Numerical parameters.

Symbol Description

1 Onset strain sensitivity parameterk0 Lagrange multiplier increment for Scheme 1f Targeted number of iterations for Scheme 2


Cluster at Northwestern (PSED), Initiative for Sustainability andEnergy at Northwestern (ISEN). A.S. and C.B. acknowledge the sup-port of the Army Research Office, Grant # W911NF-12-1-0013/P00002. Catherine Tupper is thanked for creating the model ofthe beam with the hole as well as many fruitful discussions andtesting development versions of the VUMAT. Pingping Zhu isthanked for continuing validation efforts for the VUMAT.

Appendix A

The following conventions are used in this work:

Scalar-not bold font2nd Order Tensor-bold font4th Order Tensor-capital open-face fontIf A is a tensor then: AA ¼ AijAkl; A � A ¼ AijAjk; A : A ¼ AijAij;

A0 ¼ Aij � 13 dijAkk

The following shorthand is used:

NðAÞ ¼ AkAk

hargi ¼ argþjargj2

kargk is the Euclidean norm (Tables A.1–A.4).

Appendix B

It was originally proposed that Newton–Raphson [38] be usedto implicitly integrate the orientation and reorientation equations[1]. This scheme achieves quadratic convergence, thus less

iterations are needed to solve the coupled equations than in thefull Explicit Integration scheme described in this paper. However,the need to program the Jacobian equations, presented in thisappendix, makes it more tedious to implement.

The algorithm as applied to the improved model is presentedfollowed by the systems of unknowns {x}, equations {a}, and Jaco-bians {D} that must be solved using this algorithm; {Dx} is theincrement of the unknowns calculated during each iteration.

{x0} (initial guesses) = values calculated in the elastic trial step,or from the previous step if they were not changed in Steps 1–4.#iterations = 1While residual of {a} is less than a convergence tolerance and#iterations 6max iterations

� Evaluate {a} and {D} using {xcurrent}
� Solve {Dx} = �{D}�1 � {a}.� Increment {xcurrent} = {xcurrent} + {Dx}.� Calculate residual {a}.� #iterations = #iterations + 1
End

The presentation of the equations to solve ensues, incorporatingthe additional shorthand:

hðT; nrÞ ¼qðhT~g0 � ~u0i þ HrnrÞffiffiffiffiffiffiffiffi

3=2p

emaxtr

ðB:1Þ

dYr

dnr¼

tan p nr þ 12

� �� þ p

cos2 p nrþ12ð Þð Þ ; _zr > 0

� Ar þ tan p 12� nr� ��

þ �pcos2 p 1

2�nrð Þð Þ

� �; _zr > 0

8><>: ðB:2Þ

B.1. Scenario i: Initiation

The number of equations to solve may be reduced from the 20presented in Eqs. (23)–(27) to the 14 stated here by substitutingEq. (25) into Eq. (24) and eliminating Xr.

fxg ¼ fr; etr;Dkr; nrg

fag ¼

r� C : ðe� etrÞ

etr � Dkrr0 � hðT; nrÞNðr0Þ

Fr � kr0k þ hðT; nrÞ þ YrðnrÞ

nr � ketrkffiffiffiffiffiffi3=2p

emaxtr

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

9>>>>>>>=>>>>>>>;

fDg¼

I C 0 0

DkrhðT;nrÞkr0 k

23Iþ r0r0

r0 :r0

� �� 2

3In o

I hðT;nrÞNðr0Þ�r0 DkrhðT;nrÞNðr0Þ

�Nðr0Þ 0 0 qHrffiffiffiffiffiffi3=2p

emaxtr

þ dYrdnr

0 �Nðetr Þffiffiffiffiffiffi3=2p

emaxtr

0 1

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

9>>>>>>>>>=>>>>>>>>>;

B.2. Scenario ii: Reorientation

In solving the reorientation equations, the desired outputs arer,etr. Eqs. (12) and (13) may be consolidated, reducing the numberof equations to integrate to the 13 shown below. Dere may then becalculated post-integration if explicit tracking of its evolution isdesired.

fxg ¼ fr; etr;Dkreg


fag ¼

r� C : ðe� etrÞ

etr � entr � Dkre

bIðetrÞ : r0

Fre � kbIðetrÞ : r0k þ Yre

8>><>>:9>>=>>;

fDg¼

I C 0

Dkreetretretr :etr� I0

� Iþ Dkre

etr :etrIetr : rþetrr�2etretr

etr :retr :etr

� etr

etr :retr :etr�r0

etretr :retr :etr

�r0

kIðetr Þ:r0 kr

etr :retr :etr

�etretr :retr :etr

� �2

kIðetrÞ:r0 k0

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

9>>>>>=>>>>>;
B.3. Scenario iii: Orientation
Eq. (18) may be substituted into Eq. (17) prior to integration.Additionally, since Dere is not evolving during this step, we rede-fine en

tr as entr ¼ en

tr þ Dere in writing the system of equations pre-sented here.

fxg ¼ fr; etr ;Xr;Dkr; nrg

fag ¼

r� C : ðe� etrÞetr � en

tr � DkrXr

Xr � r0 þ hðT; nrÞNðetrÞFr � kXrk þ YrðnrÞ

nr � ketrkffiffiffiffiffiffi3=2p

emaxtr

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

9>>>>>>>>=>>>>>>>>;

fDg ¼

I C O 0 0O I �DkrI �Xr 0

� 23 I

hðT;nrÞketrk Iþ 2 etretr

etr :etr

n oI 0 qHrffiffiffiffiffiffi

3=2p

emaxtr

NðetrÞ

0 0 �NðXrÞ 0 dYrdnr

0 � 1ffiffiffiffiffiffi3=2p

emaxtr

NðetrÞ 0 0 1

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

9>>>>>>>>>=>>>>>>>>>;
Appendix C
Here we provide a more detailed review of the mechanics atplay in the converged response of the non-proportional strain con-trol loading od Fig. 6 imposed upon the brick element in Section4.1. In this discussion, we refer to Fig. C1, which shows (a) theequivalent stress–strain response to each displacement-controlledloading segment documented in Table 3; (b) the oriented martens-ite volume fraction evolution in time; (c) the Lagrange multiplierincrements for each orientation and reorientation. Our review ofthis figure will provide insight into the inflections labeled i–v ofthe stress-space response shown in Figs. 5b and 6b, some of whichdo not correlate with inflections in the loading path.

Observing each orientation and reorientation Lagrange multi-plier increments allows us to examine when orientation, reorienta-tion, or both mechanisms are active. The oriented martensitevolume fraction provides understanding as to whether orientedmartensite is evolving or de-evolving, as this volume fraction iscompletely independent of reorientation (see Section 1.2). Whenconcurrently considered with these responses, the equivalentstress–strain response will allow us to understand axial-shear cou-plings that may not be obvious in examining the response in axial-shear space (i.e. Fig. 6). We now proceed to explain the mechanicsof each loading segment.

0 to 10: An axial displacement of 1.6 mm is applied to thetop surface of the 40 � 40 � 40 mm3 brick elementwhile the bottom surface is fixed (Table 3). Initially,

the loading is modeled as elastic with a 70 GPa effec-tive Young’s modulus. It reaches a critical stress of 325 MPa and beings forward phase transformation(Fig. C1a), which results in forward evolution of ori-ented martensite (Fig. C1b). Only orientation isactive during this segment (Fig. C1c).

10 to 20: Fixing axial displacement, shear displacement of1.6 mm is applied to the nodes of the top surface ofthe brick element (Table 3). This triggers reorienta-tion (Fig. C1c) since the deviatoric stress statechanges directions, but notice there is a slight lag inthe reorientation response. This is because Yre is40 MPa, not 0. Thus dissipation must occur throughEq. (19) before reorientation activates. Here this dissi-pation occurs very quickly due to the sharp change inthe applied load direction. Orientation turns off for avery short time then slowly increases (Fig. C1c). Thus,further orientation of martensite occurs, but it is slowand small in magnitude (Fig. C1b). These mechanismsallow the non-proportional addition of shear to theaxial load state to further increment the material for-ward through phase transformation in equivalentstress–strain space (Fig. C1a).

20 to 40: Path i to ii: Axial displacement of �3.2 mm isapplied to the nodes of the top surface of the brickelement (Table 3). This action initially halts evolu-tion of oriented martensite, which was active atthe end of the previous step; however, reorientationof inelastic strain continues (Fig. C1c). There is alarge drop in equivalent stress (Figs. 5b, 6c,Fig. C1a) as elastic strain is relaxed while the magni-tude of inelastic strain remains constant (again,through Eqs. (6) and (7) reorientation does not affectthe magnitude of transformation strain). However,because the transformation strain is changing direc-tion, the slope of the equivalent stress–strainresponse i-ii is much steeper than that of the initialelastic loading in time segment 0 to 10 ((Figs. 6c andFig. C1a).
Path ii to iii: Very quickly into this step of continuedaxial unloading through zero, reverse transforma-tion of oriented martensite to austenite begins andreorientation turns off (Fig. C1b and c). Thus thematerial begins to proceed through the reverse-transformation portion of a minor hysteresis loop(i.e. a lower plateau) in equivalent stress–strainspace as calculations are made to satisfy Eqs. (26)–(29) (Fig. C1a; see [18,39,40] for further discussionof minor transformation hysteresis loops in SMAs).Simultaneous to reverse transformation withoutreorientation, bIðetrÞ : r0 grows since the non-propor-tional loading is driving the deviatoric stress in anew direction. Eventually, this quantity grows toYre = 40 MPa and through Eqs. (19) and (22) reorien-tation turns on again and redirects the transforma-tion strain until it is aligned with the deviatoricstress. However, reorientation turning on and offalone does not cause sharp inflections in the stressresponse, but instead creates a nonlinear ‘‘rounding’’effect when coupled with the elastic response in theequivalent stress and strain as the tensile loaddiminishes and the axial loading becomescompressive.Path iii to iv: The inflection iii observed in Fig. 6b, cand Fig. C1a is due to cessation of reverse transfor-mation, while inflection iv arises from re-initiation


of forward transformation in the minor loop(Fig. C1). In between, reorientation is the only activeinelastic deformation mechanism, and similar topath i-ii, it only redirects the inelastic strain but doesnot alter its magnitude (Section 1.2, Eq. (7)) and theslope of the equivalent stress and strain loadingagain deviates from the elastic loading response asa result, exhibiting a much harder effective modulus.Path iv to i: Finally, the forward transformation pro-ceeds along the global hysteresis upper transforma-tion plateau along path iv–i until the �3.2 mm axialdisplacement is complete. Note that the plateau isinitially overshot a bit at point iv, Fig. 6c due tothe dissipation Yr that must occur before forwardtransformation re-initiates (Eq. (20)). Reorientationis initially active and decreasing during this path,and it is through this mechanism that the globalequivalent stress–strain response is redirected afterthe overshoot (Fig. C1c). Once the transformationstrain and deviatoric stress directions are realigned,reorientation ceases (Fig. C1c) and this path con-cludes in a forward-transformation to oriented mar-tensite manner (Fig. 6c and Fig. C1).

40 to 60: Shear displacement of �3.2 mm is applied to thenodes of the top surface of the brick element(Table 3). This action immediately triggers reorien-tation and turns off orientation (Fig. C1c), thus theoriented martensite volume fraction does not evolveduring this load path (Fig. C1b). Contrary to the pre-vious load segment, the effective modulus of com-bined elastic and martensite reorientation segmentof the global hysteresis loop in equivalent stress–strain space is more compliant than the elastic-onlyresponse (Fig. 6c and Fig. C1a). The reason it is stifferin Path i–ii and more compliant in Path i-v isbecause of the nature of the reorientation and thecalculation of equivalent strain. Observe from Eq.(60) that in isolation, reorienting the transformationstrain from an axial alignment to a shear alignmentwill increase the effective strain, while the reverseaction will decrease the effective strain as the shearcomponent is weighted by 4/3 relative to the unityweighting of the axial component in this calculation.With this understanding, considering the changes inthe loading path (Fig. 6a): while the elastic stress–strain response is always constant in this model, inSegment 20 to 40 the reorientation event as axialstrain diminished while shear displacement washeld constant pointed the transformation strainfrom an axial-biased toward a more shear-biaseddirection. Even though the magnitude of the inelas-tic strain was constant the calculation of effectiveinelastic strain increased through Eq. (60), whichcombined with the elastic response caused thematerial to appear effectively stiffer. In this segment(40–60), however, the opposite effect is observedsince reorientation is triggered by a change in shearstrain while axial displacement is held constant,biasing the inelastic shear direction from a shearalignment toward an axial alignment, whichdecreases the calculation of effective transformationstrain through Eq. (60).The material requirement of following a constantvolume fraction response is why axial stressdecreases along with shear stress in Fig. 6b. The

inflection v observed in Fig. 6b and c, Fig. C1a duringthis loading segment is a result of crossing the 0 mmshear displacement in unloading to reloading.Reloading occurs along the same constant volumefraction response in equivalent stress–strain spaceas the shear is increased to �1.6 mm displacementin the shear direction (Fig. C1a). Again, the reasonthe inflection point occurs at r12 = �40 MPa insteadof 0 MPa is because of the dissipation necessary toreorient martensite (Eq. (19)).

60 to 80: Axial displacement of 3.2 mm is applied to the nodesof the top surface of the brick element (Table 3),returning this surface to global 1.6 mm axial dis-placement. The mechanisms at play during this loadsegment are identical as from 20 to 40. Their effec-tive response in equivalent stress–strain space issimilar but not identical, again due to dissipativeterms in Eqs. (18)–(20) that provide phenomenolog-ical energy barriers that must be overcome for trans-formation, orientation, and reorientation ofmartensite.

80 to 90: Shear displacement of 1.6 mm is applied to thenodes of the top surface of the brick element, return-ing the surface to 0 mm displacement in the sheardirection (Table 3). Thus, again reorientation of aconstant volume fraction of martensite and elasticunloading concurrently relax the material at a slopethat is more compliant than the elastic-onlyresponse (path i–v, Fig. C1a, Fig. 6c). Again, inFig. 6b the asymmetry about r12 = 0 MPa in thisresponse is due to energy dissipation.

90 to 100: Finally axial displacement of �1.6 mm is applied tothe nodes of the top surface of the brick element,returning the surface to 0 mm displacement in alldirections (Table 3). Through relatively quick energydissipation processes, the shear stress is relaxed(Fig. 6b and c, Fig. C1a). The remainder of this dis-placement allows the material to complete reversephase transformation and finally elastic unloadingin a uniaxial stress state (Fig. C1a, Fig. 6b and c). Thisis done through the orientation mechanism of themodel (Fig. C1b and c). The reason there is a smoothtransition from transformation to elasticity in thereverse transformation as opposed to the sharp tran-sition that was observed in the forward transforma-tion is because of the form of Eq. (20); i.e., thissmoothing is merely a product of the tangent functionused to impose the bounds upon the volume fraction.

References

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