constitutive modeling and ﬁnite element approximation of b2-r … · transformations in nitinol...

Constitutive modeling and finite element

approximation of B2-R-B19′ phase

transformations in Nitinol polycrystals

Arkaprabha SENGUPTA† and Panayiotis PAPADOPOULOS†

†Department of Mechanical Engineering

University of California, Berkeley, CA 94720-1740, USA

Table of Contents

1. Introduction

2. Crystallography of B2-R and R-B19′ transformations

3. Balance laws and constitutive modeling of

B2-R-B19′ transformations in single crystal

3.1 Balance laws3.2 Constitutive assumptions3.3 Transformation conditions3.4 A simple constitutive model

4. Algorithmic implementation

5. Numerical simulations

5.1 Deformation of a single-crystal cube5.2 Tension of thin-walled tube

6. Conclusions

Appendix A: Numerical procedure for minimizingthe forward and reverse transformation functionalsAppendix B: Consistent tangent modulusReferences

Abstract

A path-dependent constitutive model is proposed for the stress-induced phase trans-formation of Nitinol single crystals between austenite, rhombohedral and martensitephases. A multi-directional active search strategy is employed to resolve the path andstate of the transformation. The resulting algorithm is implemented using a multi-scalefinite element-based method which extends the applicability of the model to texturedNitinol polycrystals. Representative numerical simulations are included and show goodagreement with experimental results.

Keywords: Superelasticity; Nitinol; R-phase; constitutive modeling; finite element method.

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Modeling of multi-step phase transformations

1 Introduction

Nitinol is a Ni-Ti alloy used in several engineering systems, including biomedical devices,

such as endovascular stents, angioplasty guidewires, vena-cava filters, etc.. One of the main

reasons for Nitinol’s commercial success is its pronounced superelastic behavior. This is

the capacity to sustain large deformations that can be completely reversed upon removal

of the stress, and is due to a stress-induced solid-solid phase transformation at constant

temperature. Most constitutive models of phase transformations in Nitinol account only

for the austenite-martensite transformation and neglect the formation of an intermediate

rhombohedral phase (R-phase). The formation of R-phase in superelastic Nitinol can be

regulated by the heat treatment of the specimen and is critically dependent on the spec-

imen temperature, see, e.g., [1], [2] and [3]. Experiments in [4] indicate that R-phase is

stress-induced from austenite, since there is no initial R-phase and the deviation from lin-

earity in the apparent elastic (pre-martensite) range could not have been a result of mere

reorientations of pre-existing R-phase variants. This conclusion is also supported by the

work in [5], where it is argued that the austenite-R-phase transformation in polycrystals is

(at least in part) responsible for the stress-strain non-linearity in this pre-martensitic zone,

as well as for tension/compression asymmetry. Generally, the plateau of the austenite-

R-phase transformation is much narrower and exhibits much smaller hysteresis compared

to the austenite-martensite transformation [4]. However, the presence of R-phase influ-

ences the transformation along the R-phase-martensite path, thus rendering the formation

of different martensite phases dependent on the load path. Further experimental evidence

in [2,3] and discussions on R-phase in [6] suggest that formation of R-phase and martensite

phases from austenite are two competing phase transformations. As a result, phase trans-

formation during a load increment takes place along the path that is most energetically

favorable at the current state. This path-dependence introduces considerable theoretical

and numerical challenges for the modeling of phase transformations in superelastic Niti-

nol. Similar challenges are also envisioned in Cu-based shape-memory alloys that typically

involve more than one type of martensite forming from the austenite phase.

To address the constitutive modeling of the combined austenite-R-phase-martensite

transformation in single crystals, a novel approach is proposed for determining the current

transformation state for given history and loading conditions. This yields the current state

by minimization of appropriate functionals for forward or reverse transformation along

each of the potential transformation paths, thus extending the procedure used in [7] for

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the (single) austenite-martensite path. However, given the history-dependence, the mini-

mization here is taken to be incremental in nature, as in the case of plasticity and texture

evolution in polycrystals, see [8] and [9]. On the implementation front, the algorithmic

problem is rendered tractable by allowing phase transformation in each time step only

along the single most feasible path. In fact, it is demonstrated that this procedure leads to

a well-posed optimization problem. The proposed methodology is subsequently extended

to textured polycrystals through homogenization of the response of periodic Representa-

tive Volume Elements (RVE) defined at Gauss points of each macro-scale finite element,

following earlier work in [10].

The outline of this article is as follows: Section 2 discusses the procedure for determin-

ing the habit-plane variants associated with the austenite-R-phase (B2-R) and R-phase-

martensite (R-B19′) transformations. This is followed by the introduction of the main

ingredients of the plasticity-like approach to modeling the multi-component stress-induced

phase transformation in Section 3. Next, Section 3.4 introduces the specific model, and is

followed by the presentation of the algorithmic approach in Section 4. Computer imple-

mentation aspects and results of numerical simulations are provided in Section 5. Closing

remarks are offered in Section 6.

2 Crystallography of B2-R and R-B19′ transformations

It has been observed that a rhombohedral phase (R-phase) may be formed from the parent

austenite phase during the mechanical loading of Nitinol, see, e.g., the experiments in

[11, 12]. The crystallography of this phase transformation is briefly reviewed here by way

of background to the proposed modeling of the complete transformation process.

The rhombohedral unit cell is obtained by stretching the austenite unit cell along the

[111] direction. The corresponding Bain strain depends on the rhombohedral angle α,

defined as the angle between the deformed [100] and [010] axes of the austenite unit cell

in the R-phase. It has been found out that α varies continuously with temperature, with

the angle usually decreasing with decreasing temperature [6, 13, 14]. In [14], an empirical

formula is proposed for the temperature dependence of α. Based on this formula, the

angle α is shown to be approximately equal to 89.5 at room temperature. Given that the

present work concerns the quasi-static loading of Nitinol polycrystals, it is reasonable to

assume that nearly isothermal conditions apply and that the rhombohedral angle α can

be taken to be constant and equal to the value predicted in [14]. This is consistent with

3


experimental evidence that shows a narrow R-phase plateau, thus suggesting only a small

deviation from the original cubic lattice of the austenite phase [4].

The habit-plane variants of the B2-R transformation can be deduced by appealing to

a procedure established in [15,16]. To this end, start by noting that the stretch tensor U1

corresponding to the deformation of the unit austenite cell with α = 89.5 is combined

with the rotations of the cubic Laue group Lc to yield a set of four unique stretches

U = RU1RT : R ∈ Lc. This set is consistent with the results in [12, 16]. Subsequently,

the twinning equation (a special form of the well-known Hadamard compatibility equation)

is solved for every lattice correspondence variant pair (i : j). This equation takes the form

RijUi − Uj = a ⊗ n , (1)

where Ui,Uj ∈ U and the twin rotation Rij , the twin shear a, and the unit normal to the

twin plane n are to be determined. Necessary and sufficient conditions for the existence

of solutions to (1) are given in [15]. These conditions are satisfied by all six possible pairs

of lattice-correspondence variants of the B2-R transformation, as explained in [16], and

the general solution provided in [15] is employed here. Lastly, the habit-plane equation is

solved for each twin pair in the form

Rij(λRijUi + (1 − λ)Uj) = I + b ⊗ m , (2)

where I is the second-order identity tensor, and the habit plane rotation Rij , the twin vol-

ume fraction λ, the shape strain b and the habit-plane normal m are to be determined. The

above equation gives the compatibility between averaged deformation of twinned marten-

site and austenite. Appealing to the twinning equation (1), equation (2) can be written

as

Rij(Ui + λa ⊗ n) = I + b ⊗ m . (3)

A general algorithm for obtaining solutions to (3) is outlined in [15] and the resulting

twelve B2-R habit-plane variants are listed in Table 1. The number of B2-R habit-plane

variants is in agreement with the crystallography of B2-R-B19′ transformations in [17],

where twelve R-phase variants with 110 < 001 > compound twins were found to form

for α < 90.

In contrast to the B2-R case, the variants of the R-B19′ transformation are not well-

characterized through experiments. Applying the analytical approach discussed earlier, the

possibility of formation of compound and Type-I martensitic twins has been explored in [17]

for certain values of rhombohedral angles. Here, the habit-plane solutions appear to depend

4

strongly on the parent R-phase variants, which introduces added modeling complexity. At

the same time, experimental observation for Nitinol under superelastic loading appears

to be limited to only Type-II martensite twins. Further, it is possible that even though

other martensitic variants are initially formed to preserve compatibility with the R-phase

variants, they are eventually reoriented to the well-known B19′ variants with continued

loading. In the presence of insufficient experimental evidence of the occurrence of such

vastly different martensitic variants, the twenty-four habit-plane variants that are usually

observed in B2-B19′ transformation [7, 18] are considered in this work. The habit-plane

solutions are based on the following lattice parameters: a0 = 3.015A for austenite, and

a = 2.889A, b = 4.120A, c = 4.622A and monoclinic angle β = 96.8 for martensite.

The preceding choice of martensitic variants implies that there is no one-to-one corre-

spondence between the R-phase variants and the B19′ variants formed along the R-B19′

path. The implications of this important assumption will be examined in the ensuing

constitutive development.

3 Balance laws and constitutive modeling of B2-R-B19′ trans-

formations in single crystal

This section gives a brief recount of the continuum balance laws and discusses the con-

stitutive modeling of B2-R-B19′ transformation in Nitinol single crystal. It extends the

framework for B2-B19′ transformations in single crystal pursued in [7] to account for the

formation of the intermediate R-phase.

3.1 Balance laws

The linear and angular momentum balance equations are written in local referential form

as

Div(FS) + ρ0f = ρ0a ,

S = ST ,(4)

where S is the second Piola-Kirchhoff stress tensor, F is the deformation gradient, ρ0 is

the mass density per unit referential volume, f is the body force per unit mass, and a is

the acceleration. Likewise, the energy equation takes the form

ρ0ǫ = ρ0r − Divq0 + S · E , (5)

5


in terms of the internal energy per unit mass ǫ, the heat supply per unit mass r, the heat

flux vector q0 resolved over the geometry of the reference configuration, and the Lagrangian

strain E.

In this work, the existence of an entropy function is assumed and Clausius-Duhem

inequality holds. A local referential statement of this inequality can be expressed as

ρ0ηθ ≥ ρ0r − Div q0 +q0 · Grad θ

θ, (6)

where η denotes the entropy per unit mass and θ the absolute temperature.

3.2 Constitutive assumptions

Following earlier work on constitutive modeling of single superelastic crystals [7, 19], and

motivated by the theory of rate-type, rate-independent plasticity, the stress response is

viewed as one of a parametrized thermoelastic material for given values of the austenite-

martensite, austenite-R-phase and R-phase-martensite volume fractions. These are denoted

by ξAM1α , ξAR

β and ξRM2γ , with α = 1, . . . , nam, β = 1, . . . , nr, and γ = 1, . . . , nrm, where

nam, nr and nrm are the total numbers of potential M1-martensite, R-phase and M2-

martensite variants. A distinction is made between martensite produced from austenite

(M1) and R-phase (M2), because the two classes of martensites typically involve different

variants with different constraints on their evolution.

The R-phase variant volume fractions ξRα , α = 1, . . . , nr, are determined by trans-

formation along the austenite-R-phase and the R-phase-martensite paths. These can be

additively decomposed into the volume fractions ξARα , α = 1, . . . , nr, due to the forward

austenite-R-phase transformation and the changes ΞM2Rα , α = 1, . . . , nr, in the R-phase

volume fractions due to forward and reverse transformations between the R-phase and the

M2-martensite phase. This decomposition is mathematically expressed as

ξRα = ξAR

α + ΞM2Rα . (7)

The R-phase formed along the austenite-R-phase path is obtained by time-integrating the

corresponding rate, which can be defined as

ξARα = ξR

α

∣∣ξ

RM2β

=const., (8)

where ξRα are assumed to depend continuously on time. While the volume fractions ξAR

α are

always positive, note that ΞM2Rα are positive only when the net variant-wise transformation

between R-phase and M2-martensite phase favors the former. Here, ΞM2Rα are assumed to

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be functions of the history of the R-phase and RM2 transformation and the current volume

fraction of M2-martensite, and can be generally expressed as

ΞM2Rα (t) = ΞM2R

α

(Hτ<t

(ξRβ (τ)

, ξRM2

γ (τ))

, ξRM2

γ (t))

, (9)

where Hτ<t

(·) denotes the history of the variable (·). The set F of volume fractions of all

martensite- and austenite-induced R-phase variants is now defined as

F =ξAM1

α , ξARβ , ξRM2

γ

=

(ξAM1

1 , ξAM1

2 , · · · , ξAM1

nam , ξAR1 , ξAR

2 , · · · , ξARnr , ξRM2

1 , ξRM2

2 , · · · , ξRM2

nrm )

. (10)

These are subject to the constraints

ξAM1

α ≥ 0 , ξRβ ≥ 0 , ξRM2

γ ≥ 0 (11)

andnam∑

α=1

ξAM1

α +nr∑

β=1

ξRβ +

nrm∑

γ=1

ξRM2

γ ≤ 1 , (12)

where ξRβ is given by (7).

The case of forward R-phase-martensite transformation requires special treatment, be-

cause it involves transformation between multiple variants of two phases, in contrast to the

austenite-R-phase and the austenite-martensite transformations, where the parent phase is

the same single variant phase. As discussed in Section 2, the R-phase variants do not have

one-to-one correspondence with the M2-martensitic variants formed from R-phase. The

rates of change of volume fractions of R-phase and M2-martensitic variants are subject to

the conservation of volume fraction, namely

nr∑

α=1

ΞM2Rα +

nrm∑

β=1

ξRM2

β = 0 . (13)

An additional constraint stems from the fact that the net production of M2-martensite

cannot exceed the net supply of R-phase. This leads to the condition

nrm∑

β=1

ξRM2

β dt ≤

nr∑

α=1

ξRα +

nr∑

α=1

ξARα dt . (14)

A specific rate-type form of the constitutive law (9) is now stipulated to characterize

the transformation between individual M2-martensite variants and R-phase variants, con-

sistently with the preceding constraints. In fact, separate rules are imposed on forward

7


and reverse R-phase-martensite transformations. This is due to the different nature of

these two classes of transformations within the proposed model. Indeed, in the forward

R-phase-martensite transformation, the rate of ΞM2Rα can be generally taken to depend

on the rate of the state variables ξRM2

β and the current volume fractions of the R-phase,

as suggested in (9). However, in the reverse R-phase-martensite transformation, the same

rate can depend only on the rate of the state variables ξRM2

β , since R-phase need not be

present, at least at the onset of this transformation.

Based on the preceding observations, it is assumed that when forward R-phase-martensite

transformation occurs, the volume fractions of R-phase variants are reduced in proportion

to their existing volume fractions. Therefore, the rate of reduction in the volume fraction

of R-phase variant α is

ΞM2Rα = −

ξRα∑nr

α=1 ξRα

nrm∑

β=1

ξRM2

β . (15)

Clearly, (15) is consistent with the constraint in (13). It is also consistent with (11)2 in

that, when only forward R-phase-martensite transformation occurs, then ΞM2Rα = ξR

α and

the R-phase volume fraction at time t + dt is

ξRα + ξR

α dt = ξRα −

ξRα∑nr

α=1 ξRα

nrm∑

β=1

ξRM2

β dt ≥ 0 , (16)

where (14) is used.

The reverse R-phase-martensite transformation is only subject to the constraint condi-

tion (13), since (14) holds trivially. In the absence of variant correspondences, one needs to

identify the R-phase variants produced from M2-martensite using alternative means. The

approach adopted here is to admit a limited set of nrv potentially active R-phase variants

that are equally probable, hence would have equal rates of production. These variants may

be chosen using history information (e.g., they may coincide with the R-phase variants

present in the latest stage of R-phase generation) or using a kinetics-based criterion (e.g.,

the magnitude of the current thermodynamic force needed to generate them are equal). In

either case, the rate of growth of R-phase variants takes the form

ΞM2Rα = −

1

nrv

nrm∑

β=1

ξRM2

β . (17)

The Helmholtz free energy is defined by

Ψ = ρ0(ǫ − ηθ) , (18)

8

and its constitutive dependencies are expressed here as

Ψ = Ψ(E,F , θ;H) . (19)

While explicit representation of time dependencies is omitted in (19), Ψ is assumed to

depend on the values of E, F and θ at the current time t, as well as on the history

term H = Hτ<t

(ξRα , ξ

RM2γ ) of the R-phase and RM2 transformation. The fact that the

history of R-phase and RM2 transformation does not affect the instantaneous rate Ψ of the

Helmholtz free energy, is emphasized by setting H aside in the argument list of (19). The

choice of independent variables in (19) underlines the fact that ξRα cannot be viewed as

a state variable in addition to F , because its rate is uniquely determined from the rates of

F and from H, as seen from ((7)-(9)).

It follows from equations (5), (6) and (18) that, during a homothermal (Grad θ = 0)

and isothermal (θ = 0) superelastic process, the condition

S · E − Ψ ≥ 0 (20)

holds true. Using (19) one finds that

(S−

∂Ψ

∂E

)· E−

nam∑

α=1

∂Ψ

∂ξAM1α

ξAM1

α −

nr∑

β=1

∂Ψ

∂ξARβ

ξARβ −

nrm∑

γ=1

∂Ψ

∂ξRM2γ

ξRM2

γ ≥ 0 . (21)

Given that E can be varied independently from ξAM1α , ξAR

β and ξRM2γ due to the existence

of elastic processes at all transformation states, the standard Coleman and Noll argument

can be applied to equation (21), leading to

S =∂Ψ

∂E(22)

and

D = −

nam∑

α=1

∂Ψ

∂ξAM1α

ξAM1

α −

nr∑

β=1

∂Ψ

∂ξARβ

ξARβ −

nrm∑

γ=1

∂Ψ

∂ξRM2γ

ξRM2

γ ≥ 0 . (23)

Equation (23) shows that the dissipation rate D, defined as the rate of work done by the

thermodynamic forces −∂Ψ

∂ξAM1α

, −∂Ψ

∂ξARβ

, and −∂Ψ

∂ξRM2γ

, is non-negative.

The forward and reverse phase transformation of a given variant α are characterized

by transformation functions Y f,Kα and Y r,K

α , K = AM1, AR,RM2, identifies any one of the

three potential transformation paths discussed in Section 3.2 and

Y f,Kα = Y f,K

α (E,F , θ;H) , Y r,Kα = Y r,K

α (E,F , θ;H) . (24)

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The condition Y f,Kα = 0 (resp. Y r,K

α = 0) is equivalent to the variant α being in a state

of forward (resp. reverse) transformation relative to the K-transformation path. These

conditions define the sets of forward (resp. reverse) active variants as

J f,K(E,F , θ;H) =α | Y f,K

α (E,F , θ;H) = 0

(25)

and

J r,K(E,F , θ;H) =α | Y r,K

α (E,F , θ;H) = 0

. (26)

During forward or reverse transformation, one may write

Y f,Kα ξK

α = 0 (27)

or

Y r,Kα ξK

α = 0 , (28)

for each variant α of the K-transformation path. Therefore, for forward and reverse trans-

formation along K-th transformation path, (23), (27) and (28) imply that

nv(K)∑

α=1

(−Y f,Kα −

∂Ψ

∂ξKα

)ξK

α ≥ 0 (29)

andnv(K)∑

α=1

(−Y r,Kα −

∂Ψ

∂ξKα

)ξK

α ≥ 0 . (30)

Here, nv(K) is the total number of variants of the K-th transformation path, namely

nv(AM1) = nam, nv(AR) = nr and nv(RM2) = nrm.

Specific forms of the transformation functions in (24) are derived by noting that trans-

formation of a typical variant α occurs when the thermodynamic force −∂Ψ

∂ξKα

acting on

the variant reaches a critical value Fc,Kα > 0, as explained in [20]. Due to the crystal sym-

metry of the variants, Fc,Kα is the same for all variants undergoing transformation along

a given transformation path K. Furthermore, the critical force for reverse transformation

is assumed to be the negative of that for forward transformation. Under these assump-

tions, critical force-based forward and reverse transformation functions are postulated by

generalizing the development in [7] to

Y f,Kα = −

∂Ψ

∂ξKα

−Fc,Kα ,

Y r,Kα = −

∂Ψ

∂ξKα

+ Fc,Kα .

(31)

10

Taking into account equations (27), (28) and (31), it follows that during forward and

reverse transformationnv(K)∑

α=1

(−

∂Ψ

∂ξKα

−Fc,K

)ξK

α = 0 (32)

andnv(K)∑

α=1

(−

∂Ψ

∂ξKα

+ Fc,K

)ξK

α = 0 . (33)

Note that in equations (32) and (33) all the variants along the K-transformation path have

been included, although it suffices to consider only the variants in the active sets J f,K and

J r,K. Equation (32) implies that along each of the three forward transformation paths, the

functional

Φf,K = Φf,K(ξK

α) = Ψ(ξK

α) +∑

α∈J f,K

Fc,KξK

α (34)

is minimized during forward transformation for given strain, temperature and transfor-

mation history, where ξKα are the corresponding volume fractions of active variants and

Ψ(ξKα) = Ψ

∣∣(E,θ,F\ξK

α;H)=const. A unique minimum exists provided that the Hessian

matrix∂2Φf,K

∂ξKα∂ξK

β

is positive-definite. A general functional that is minimized along any given

forward transformation path can be constructed by including the dissipation potential for

all three such paths, and takes the form

Φf = Φf (F) = Ψ(F) +∑

α∈J f,AM1

Fc,AM1ξAM1

α +∑

β∈J f,AR

Fc,ARξARβ +

∑

γ∈J f,RM2

Fc,RM2ξRM2

γ ,

(35)

where Ψ(F) = Ψ∣∣(E,θ;H)=const

. A minimum of the above functional may, in general, involve

transformation along all three transformation paths.

An analogous procedure may be followed to construct a functional Φr for the case of

reverse transformation as

Φr = Φr(F) = Ψ(F) −∑

α∈J r,AM1

Fc,AM1ξAM1

α −∑

β∈J r,AR

Fc,ARξARβ −

∑

γ∈J r,RM2

Fc,RM2ξRM2

γ .

(36)

Clearly, owing to the history-dependence of Ψ, the minimization properties of the

functionals Φf (F) and Φr(F) in (35) and (36) are incremental in nature. This, in ef-

fect, implies that the minima of Φf and Φr at time t are determined for a given history

H = Hτ<t

(ξRα , ξ

RM2γ ) of the R-phase transformation. Restricting attention to the forward

transformation case and taking into account (25) and (31)1, the incremental minimizers of

11


(35) coincide with the stationary points of the linear functional Φf,Kt , defined as

Φf,Kt (ξK

α) = ˙Ψ(ξK

α) +∑

α∈J f

FcξK

α . (37)

Therefore, in the general case, a forward transformation leads to the stationarity of the

linear functional

Φft (F) = ˙Ψ(F)+

∑

α∈J f,AM1

Fc,AM1 ξAM1

α +∑

β∈J f,AR

Fc,ARξARβ +

∑

γ∈J f,RM2

Fc,RM2 ξRM2

γ , (38)

where F = ξAM1α , ξAR

β , ξRM2γ . An analogous derivation readily follows for the

reverse transformation, and leads to a corresponding linear functional Φrt (F), written as

Φrt (F) = ˙Ψ(F)−

∑

α∈J r,AM1

Fc,AM1 ξAM1

α −∑

β∈J r,AR

Fc,ARξARβ −

∑

γ∈J r,RM2

Fc,RM2 ξRM2

γ . (39)

3.3 Transformation conditions

Recalling (23), it follows that phase transformation along the K-path entails the satisfaction

of the reduced dissipation inequality

−

nv(K)∑

α=1

∂Ψ

∂ξKα

ξK

α ≥ 0 . (40)

Taking into account (25) and (31)1, this becomes

∑

α∈J f,K

Fc,KξK

α ≥ 0 . (41)

Likewise, for reverse transformation, equations (26) and (31)2 imply that

∑

α∈J r,K

Fc,KξK

α ≤ 0 . (42)

Since Fc,K is assumed positive, equations (41) and (42) imply that

∑

α∈J f,K

ξK

α ≥ 0 , (43)

during forward transformation, and

∑

α∈J r,K

ξK

α ≤ 0 , (44)

during reverse transformation. Equations (43) and (44) are the transformation (loading)

conditions that will be used in the algorithmic development to distinguish between the

forward and reverse transformation states of the K-path.

12

3.4 A simple constitutive model

In the case of pure B2-B19′ transformation (identified here as austenite-M1 martensite

transformation), the Helmholtz free energy was assumed in [7] to take the form

Ψ = Ψ(E, ξAM1

α , θ) =1

2(E − Et) · C(E − Et) + BM (θ − θ0

M )

nam∑

α=1

ξAM1

α . (45)

Here, BM is a constant chemical energy coefficient and θ0M is the thermodynamic equilib-

rium temperature between the two phases. These constants generally attain different values

for austenite-R-phase pairs in the case of a two-step B2-R-B19′ transformation. Hence, the

Helmholtz free energy is generalized to

Ψ = Ψ(E,F , θ;H) =1

2(E − Et) · C(E− Et)+

BM (θ − θ0M )

nam∑

α=1

ξAM1

α +nrm∑

γ=1

ξRM2

γ

+ BR(θ − θ0

R)nr∑

β=1

ξRβ . (46)

Here, BM and BR are the chemical energy coefficients of the martensite phases and the

R-phase, respectively. Note that the AM1 and RM2 transformations are associated with

the same chemical energy coefficient BM and equilibrium temperature θ0M due to the earlier

assumption that both transformations involve the same set of martensite variants. Also,

Et is the total macroscopic Lagrangian transformation strain. This is taken to depend

linearly on the martensitic and R-phase volume fractions and is given by

Et =

nam∑

α=1

ξAM1

α Et,AM1

α +

nr∑

β=1

ξRβ E

t,Rβ +

nrm∑

γ=1

ξRM2

γ Et,RM2

γ . (47)

Setting L ∈ AM1, R,RM2, the variant transformation strains Et,Lα relative to the (com-

mon) austenite phase are defined as

Et,Lα =

1

2

(bL

α ⊗ mL

α + mL

α ⊗ bL

α + (bL

α · bL

α)mL

α ⊗mL

α

), (48)

in terms of the phase-dependent unit outward normal mLα to the habit plane and the

transformation displacement bLα. It should be noted that the R-phase variants generated

from austenite or M2-martensite have the same transformation strains, denoted by Et,Rα .

Also, given that the R-phase is obtained by a small distortion of cubic austenite lattice,

the elastic constants of the R-phase are assumed to be coincident to those of the austenite

phase. Adopting the rule of mixtures, this implies that the fourth-order elasticity modulus

13


C in (46) can be approximately written as

C =

1 −

nam∑

α=1

ξAM1

α −

nrm∑

β=1

ξRM2

β

Ca +

nam∑

α=1

ξAM1

α +

nrm∑

β=1

ξRM2

β

Cm , (49)

in terms of the respective elasticity moduli Ca and Cm of the austenite and martensite

phases.

The interaction energy among the different martensite or R-phase variants is neglected

in the Helmholtz free energy (46), since it was seen in our own experiments that it does

not contribute substantially to the overall stress response [4]. Likewise, no pure thermal

energy term is included, because interest is focused on slow loading processes, which result

in small variations of the ambient temperature. Lastly, no thermomechanical coupling

term is included, because its effect is secondary for the given temperature and deformation

ranges.

Using equations (22) and (46), the second Piola-Kirchhoff stress for this model is given

by

S = C(E −Et

). (50)

Likewise, equation (46) implies that the functions defined in equation (31)1 for the forward

austenite-martensite and austenite-R-phase transformations are given by

Y f,AM1

α = Y fα (E,F , θ;H) = C(E − Et) ·Et,AM1

α −(BM (θ − θ0

M) + Fc,AM1)

, (51)

and

Y f,ARβ = Y f

β (E,F , θ;H) = C(E − Et) ·Et,Rβ −

(BR(θ − θ0

R) + Fc,AR)

, (52)

respectively. In the derivation of (52), use is also made of (7) and (9).

The critical driving force Fc,RM2 on the martensite variants during transformation

along the RM2 path is taken to be

Fc,RM2 = Fc,AM1 −Fc,AR , (53)

reflecting the relative magnitude of the resistance to RM2 transformation, as compared to

the AM1 and AR transformations. Taking, again, into account (46) and (31)1 it follows

that

Y f,RM2 =[Et,RM2

α +

nr∑

β=1

Et,Rβ

∂ΞM2Rβ

∂ξRM2α

]· C(E −Et)

−(BM (θ − θ0

M ) + Fc,AM1 + BR(θ − θ0R)

nr∑

β=1

∂ΞM2Rβ

∂ξRM2α

−Fc,AR)

. (54)

14

Appealing to (13) and assuming continuity of the functions ΞM2Rα of equation (9) both in

time and in the volume fractions ξRM2α , it is immediately seen that

nr∑

β=1

∂ΞM2Rβ

∂ξRM2α

= −1 , (55)

therefore (54) reduces to

Y f,RM2 =[Et,RM2

α +

nr∑

β=1

Et,Rβ

∂ΞM2Rβ

∂ξRM2α

]· C(E −Et)

−(BM (θ − θ0

M ) + Fc,AM1 − BR(θ − θ0R) −Fc,AR

), (56)

where a time-discretized form of∂Ξ

M2R

β

∂ξRM2α

will be derived in Section 4. Analogous expressions

hold for the reverse transformation functions Y r,K, for each of the K-transformation paths,

with Fc,K being replaced by −Fc,K.

4 Algorithmic implementation

As argued in Section 3.2, the forward and reverse transformations are governed by the

incremental constrained minimization of functionals Φf and Φr, defined in equations (35)

and (36), respectively. The minimization is incremental in nature, due to the fact that

it minimizes the change to the values of the functionals along each of AM1, AR and

RM2 transformation paths. The constraints involved in this minimization are due to the

magnitude of the volume fractions, as in (11) and (12), as well as due to limits in the

production of M2-martensite placed by the supply of R-phase, as in (14). The latter,

together with the fact that the particular M2-martensite variants grown during a forward

transformation process are dependent on the extant R-phase variants, underline the history-

dependence of the transformation and justify the incremental nature of the minimization

advocated in this model.

During a load increment, it is possible for the functionals Φf and Φr to attain minima

along a linear combination of two or three transformation directions relative to the previous

equilibrium state. However, given the large separation in activation levels between R-phase

on the one end and M1- and M2-martensitic phases on the other end, it is reasonable to as-

sume that such overlap of multiple transformation paths during a small load increment can

be neglected. In fact, experiments conducted on Nitinol under nearly isothermal conditions

at room temperature,, see, e.g., [4], support this observation, where the martensitic plateau

15


is seen at significantly higher stress levels than the R-phase plateau. Consequently, in the

proposed algorithmic development the transformation between different phases is taken to

occur only along one of the three possible paths during each increment of loading. The ac-

tual path is determined at each loading step to be the one that minimizes the corresponding

functional Φf or Φr among all three possible paths.

An additional advantage of reducing the minimization problem into three directional

minimization sub-problems is that this gives rise to a computationally tractable constrained

convex optimization statement. As discussed in [7], the number of active variants in each of

the three sub-problems needs to be restricted to six in order to maintain a positive-definite

Hessian. By minimizing along three different transformation paths, the restriction in the

number of variants along the active path can be easily enforced. In fact, the procedure of

selecting the six variants based on minimum thermodynamic force, suggested in [7] for the

case of a unidirectional AM1 transformation, is carried out here along each of the three

candidate transformation paths. Furthermore, the constraints on the volume fractions are

easily enforced, since the directional minimization affects only one of the three volume

fractions, while keeping the other two constant.

The constraints on volume fractions for all phases can be put in the form

−ξL

α ≤ 0 , ξL

l −

nv(L)∑

α=1

ξL

α ≤ 0 ,

nv(L)∑

α=1

ξL

α − ξL

u ≤ 0 , (57)

where L ∈ AM1, R,RM2 and ξL

l , ξLu are lower and upper limits of the total variant

volume fractions, such that 0 ≤ ξL

l < ξLu ≤ 1. These limits attain different values for

forward and reverse transformation along each of the three transformation paths. The

values of ξL

l and ξLu for the case of forward transformation are given in Table 2. For reverse

transformation, ξL

l = 0 for all paths and ξLu is the same as the corresponding ξL

l for forward

transformation.

Minimization of Φf and Φr, defined in (35) and (36), subject to the inequality con-

straints (57), leads to

−∂Φf

∂ξKα

+ λα + λl − λu = 0 , −∂Φr

∂ξKα

+ λα + λl − λu = 0 , (58)

The preceding equations are deduced by direct differentiation with respect to the vari-

ables F in (10) using the relation (7). Here, λα, α = 1, 2, . . . , nv(L), λl and λu are

nonnegative-valued Lagrange multipliers intended to enforce the constraints in (57), sub-

16

ject to the complementary slackness condition

nv(L)∑

α=1

λα(−ξL

α) + λl(ξL

l −

nv(L)∑

α=1

ξL

α) + λu(

nv(L)∑

α=1

ξL

α − ξL

u ) = 0 . (59)

In (59), nv(L) denotes the number of variants for phase L. Now, taking into account (35),

(36) and (46), the constraint minimization of Φf and Φr at time t = tn+1 leads to the

forward and reverse conditions

Cn+1

(En+1 − Et

n+1) · Et,Kα,n+1 −

(FK

ch + Fc,K)

+ λα,n+1 + λl,n+1 − λu,n+1 = 0 ,

Cn+1

(En+1 − Et

n+1) · Et,Kα,n+1 −

(FK

ch −Fc,K)

+ λα,n+1 + λl,n+1 − λu,n+1 = 0 ,(60)

subject to

nv(L)∑

α=1

λα,n+1(−ξL

α,n+1) + λl,n+1(ξL

l,n+1 −

nv(L)∑

α=1

ξL

α,n+1) + λu,n+1(

nv(L)∑

α=1

ξL

α,n+1 − ξL

u,n+1) = 0 ,

(61)

where the notation [·]n+1 = [·](tn+1) is adopted. In the preceding equations, the elastic

modulus Cn+1, defined in (49), is assumed to be incrementally constant, namely

Cn+1 =

1 −

nam∑

α=1

ξAM1

α,n −

nrm∑

γ=1

ξRM2

γ,n

Ca +

nam∑

α=1

ξAM1

α,n +

nrm∑

γ=1

ξRM2

γ,n

Cm . (62)

This explicit update of the elastic modulus preserves its positive-definiteness and guarantees

the uniqueness of the minimization problem, as can be readily concluded from (60). Also,

the total transformation strain at time tn+1 is given by

Etn+1 =

nam∑

α=1

ξAM1

α,n+1Et,AM1

α +

nr∑

β=1

ξRβ,n+1E

t,Rβ +

nrm∑

γ=1

ξRM2

γ,n+1Et,RM2

γ , (63)

consistently with the definition in (47). In equations (60), the transformation strain vari-

ables Et,Kα and E

t,Kα are defined as

Et,Kα,n+1 =

∂Etn+1

∂ξKα

=

Et,AM1α for K = AM1

Et,Rα for K = AR

Et,RM2α −

Pnrβ=1

ξRβ,n

Et,RβPnr

β=1ξRβ,n

for K = RM2

(64)

and

Et,Kα,n+1 =

∂Etn+1

∂ξKα

=

Et,AM1α for K = AM1

Et,Rα for K = AR

Et,RM2α −

Pnrvβ=1

Et,Rβ

nrvfor K = RM2

(65)

17


for the forward and reverse transformation cases, respectively. The expression for Et,Kα,n+1

for the RM2 transformation, where, as argued in Section 3.2, no one-to-one relation exists

between the R-phase and the M2-phase variants, is derived by writing the explicit time-

discretized form of (15) as

ΞM2Rα,n+1 = ΞM2R

α,n −ξRα,n∑nr

β=1 ξRα,n

nrm∑

β=1

(ξRM2

β,n+1 − ξRM2

β,n ) . (66)

Taking into account (66), it follows that

∂ΞM2Rα,n+1

∂ξRM2

β,n+1

= −ξRα,n∑nr

α=1 ξRα,n

. (67)

Therefore, the gradient of Φf along the RM2 variants in (58)1 can be expressed as

−∂Φf

∂ξRM2

β,n+1

= [Et,RM2

β −

∑nrα=1 ξR

α,nEt,Rα∑nr

α=1 ξRα,n︸︷︷︸

eEtβ,n+1

] · Cn+1(En+1 − Etn+1)−

[BM (θ − θ0M ) − BR(θ − θ0

R)]︸︷︷︸F

RM2ch

−Fc,RM2 (68)

where use is made of the discrete counterparts of (35), (46), (55), (63) and (67). Likewise,

for reverse RM2 transformation, the time-discretized form of (17) is

ΞM2Rα,n+1 = ΞM2R

α,n −1

nrv

nrm∑

β=1

(ξRM2

β,n+1 − ξRM2

β,n ) . (69)

This implies that

∂ΞM2Rα,n+1

∂ξRM2

β,n+1

= −1

nrv, (70)

hence, the gradient of Φr along the RM2 variants in (58)2 is given by

−∂Φr

∂ξRM2

β,n+1

= [Et,RM2

β −

∑nrvα=1 E

t,Rα

nrv︸︷︷︸bEt,RM2

β,n+1

] · Cn+1(En+1 − Etn+1)−

[BM (θ − θ0M) − BR(θ − θ0

R)]︸︷︷︸F

RM2ch

+Fc,RM2 , (71)

where, again, use is made of (36), (46), (55), (63) and (70). Note that the time-dependence

of Et,Kα,n+1 stems from the time-dependent choice of the set nrv, as discussed in Section 3.2.

18

Taking into account (68) or (71), the chemical forces FK

ch in (60) are given by

FK

ch =

BM (θ − θ0M) for K = AM1

BR(θ − θ0R) for K = AR

BM (θ − θ0M ) − BR(θ − θ0

R) for K = RM2

. (72)

The numerical procedure for minimizing Φf and Φr is outlined in Appendix A.

The discrete counterparts of (43) and (44) for forward and reverse transformation can

be written as∑

α∈J f,Kn+1

(ξK

α,n+1 − ξK

α,n) ≥ 0 (73)

and∑

α∈J r,Kn+1

(ξK

α,n+1 − ξK

α,n) ≤ 0 , (74)

respectively. Both conditions are checked at each time step for the minimizing paths K

of both the forward and the reverse transformation functionals. As in [7], a flag is used

to keep track of the state at time tn and resolve the state at time tn+1 based on the

inequality conditions (73) and (74). By way of initialization, the flag is set to ‘forward’

and is switched to ‘reverse’ upon satisfaction of (74). Subsequently, the flag is switched

back to ‘forward’ upon satisfaction of (73). The algorithm for selecting the direction and

path of transformation is outlined in Box 1.

The algorithmic tangent modulus employed for Newton-like solution of the equilib-

rium equations consistently with the preceding algorithmic development is included in

Appendix B.

5 Numerical simulations

In this section, the predictive capacity of the proposed model of B2-R-B19’ transforma-

tions is illustrated by simulating tension loading/unloading response for single crystal and

polycrystal structures. In the latter case, the model is employed for a polycrystalline

NiTi thin-walled tube in tension and the results are compared to those obtained in pre-

viously conducted experiments [4]. Texture data for the tube were collected from several

tube specimens and processed using the Berkeley Texture Package (BEARTEX) [21]. For

the polycrystalline case, the numerical implementation involved a multi-scale approach, in

which both the coarse and the fine scale were spatially resolved using the finite element

19


method, see [10] for details. All numerical simulations were conducted using FEAP, a

general-purpose finite element program partially documented in [22].

5.1 Deformation of a single-crystal cube

The main purpose of this simulation is to demonstrate the capability of the proposed

model in simulating all possible phase transitions in a NiTi single crystal. As explained in

Section 3.2, the possible phase transitions in NiTi crystals are B2-R (AR), B2-B19′ (AM1)

and R-B19′ (RM2). Earlier work has concentrated on the AM1 path, see, e.g., [7,19]. The

mechanical response during the other two possible transformations, is highlighted here.

A 10mm cubic block is fixed along the X1-, X2- and X3-directions on faces with out-

ward normals −E1, −E2 and −E3, respectively, and subjected to a homogeneous normal

displacement of up to 0.75E1mm at the face with outward normal E1, see Figure 1. The

austenite and martensite Young’s moduli are taken to be Ea=55.0 GPa, Em=20.0 GPa,

while Poisson’s ratio is set to ν = 0.3 for both phases, namely the elasticity tensors Ca

and Cm are assumed isotropic. In addition, BR=0.35 MPa/0C, BM=0.607 MPa/0C, θ0R=

20.85 0C, θ0M= -0.5 0C and θ = 22.00C are used in the simulations. The values of the pre-

ceding parameters are chosen based on thin-tube experiments on NiTi in [4]. Further, the

critical driving forces are set to Fc,R =0.11 MPa and Fc,M =6.5 MPa. The response of the

cube undergoing AR and RM2 transformations is depicted by the equivalent stress-strain

curves in Figure 2 for applied displacements of 0.11mm and 0.75 mm. The plot shows

how the plateau for AR transformation occurs at relatively low stress and is substantially

narrower than the one for RM2 transformation. Also, the hysteresis of AR transformation

is shown to be much smaller than that of the RM2 transformation.

5.2 Tension of thin-walled tube

The proposed model is employed to simulate the response of a thin-walled tube in tension.

Details of the supporting experiments can be found in [4] and comparisons with a single

crystal-based model are contained in [7]. The tube has constant thickness in the gauge

(middle) area and is thicker at the ends to accommodate gripping and inhibit end effects due

to phase transformation. Quasi-static loading at a rate of 10−4/sec is employed to ensure

nearly isothermal conditions. Initially, a single-scale solution is obtained by assuming

that each finite element represents a single crystal and that all crystals have the same

orientation according to the dominant sheet texture of the parent austenite phase found

20

in the experiments, see [7]. Here, similar simulations are first carried out for B2-R-B19′

transformation and compared with the experimental results. The tube mesh is constructed

with 630 eight-node brick elements and full integration (eight-point) is employed at every

element. The tube is fixed in the longitudinal and circumferential directions at both ends,

see Figure 3. The predicted stress response shown in Figure 4 includes a flat plateau when

the tube undergoes R-B19′ transformation that matches the experimental measurements.

However, the single-scale model over-accentuates the deviation from the linearly elastic

response due to B2-R transformation. Also, the predicted transition from the B2 (or R) to

martensite phase is smoother than that measured in the experiments along with much lower

stress plateau during unloading. Figure 5 shows the normal stress in the axial direction of

the tube.

To fully resolve the actual texture, a multi-scale formulation of the proposed model is

implemented. This formulation employs finite element discretizations for both the coarse

(continuum) and fine (microstructure) scales with periodic boundary conditions for the lat-

ter. The size of the fine-scale volume is determined so as to guarantee statistical represen-

tation of all dominant crystal orientations, see [10] for a detailed discussion. In the tension

problem, this leads to a representative volume element of 43(=64) crystals at each Gauss

point of the coarse scale. As in [10], a higher value of the critical driving force is chosen

for the martensitic nucleation regime opposed to the propagation regime. Specifically, the

critical driving force during nucleation of martensite was chosen as Fc,M =9.0 MPa, while

that during propagation of martensite phase boundary was taken to be Fc,M=6.2 MPa.

This differentiation is not needed for R-phase nucleation, since the R-phase effects only a

minor distortion to the austenite lattice and its accommodation does not have a high en-

ergy barrier. This choice is also supported by experiments, where no increase in the stress

during R-phase transformation is observed. All other parameters are chosen as in Sec-

tion 5.1. To further reduce the computational expense, and adopting the approach taken

in [10], a relatively coarse mesh with 220 eight-node brick elements is employed to model

the tube in the continuum scale and a one-point integration method with stabilization is

adopted [23].

The numerical predictions for stress response are compared to the experimental mea-

surements in Figure 6. Both the deviation from linearity due to the R-phase and the jump

in the stress due to the initiation of the martensite transition are well represented by the

model. The normal stress along the longitudinal axis of the tube at the onset and comple-

tion of martensitic transformation are shown in Figure 7. In Figure 8, the R-phase volume

21


fractions are shown at the onset and completion of martensitic transformation. This illus-

trates how the R-phase volume fraction in the gauge section is decreased as the R-phase

is annihilated in favor of martensite phase with increased loading. The martensite volume

fraction at these two stages is displayed in Figure 9.

6 Conclusions

A path-dependent incremental minimization-based constitutive model is proposed for austenite-

R-phase-martensite phase transformation. This model is formulated in a manner that

circumvents the need for crystallography characterization of the R-B19′ transformation.

The corresponding algorithmic formulation resolves the complexity of the potential multi-

component transformation by stipulating that only the most energetically favorable trans-

formation path is activated during any quasi-static load increment. This renders the numer-

ical implementation computationally tractable and reproduces the austenite-R-phase and

R-phase-martensite phase transformation plateaus at the single-crystal level. An extension

to textured polycrystals is readily effected by adopting a finite element-based multi-scale

methodology. This yields results that are in good agreement with experimental measure-

ments both for the R-phase and martensitic transformation plateaus, as well as also gives an

estimate of the relative proportion of the different phases at any load level. The proposed

formulation can be extended to include detailed habit-plane variant kinetics for the R-B19′

transformation, provided that quantitative experimental evidence becomes available on the

underlying microstructure.

Acknowledgments

This work was supported by the National Science Foundation under contract CMMI-

0409294 with the University of California, Berkeley. The authors would like to thank

Professor R.L. Taylor of the University of California, Berkeley for useful discussions on the

implementation of the multi-scale finite element method.

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24

APPENDIX A: Numerical procedure for minimizing the for-

ward and reverse transformation functionals

The forward and reverse transformation paths are characterized by functionals Φf and

Φr defined in equations (35) and (36), respectively. For the forward transformation case,

recalling that only one of the three transformation paths K is assumed active, the total

transformation strain at time tn+1 is given by

Etn+1 =

nam∑

α=1

ξAM1

α,n+1Et,AM1

α +

nr∑

β=1

ξRβ,n+1E

t,Rβ +

nrm∑

γ=1

ξRM2

γ,n+1Et,RM2

γ

=

∑namα=1 ξAM1

α,n+1Et,AM1α +

∑nrβ=1 ξR

β,nEt,Rβ +

∑nrmγ=1 ξRM2

γ,n Et,RM2γ for K = AM1

∑nrα=1 ξR

α,n+1Et,Rα +

∑namβ=1 ξAM1

β,n Et,AM1

β +∑nrm

γ=1 ξRM2γ,n E

t,RM2γ for K = AR

∑nrmα=1 ξRM2

α,n+1Et,RM2

α,n+1 +∑nr

β=1 ξRβ,nE

t,Rβ (1 +

ξRM2n,tot

ξRn,tot

) +∑nam

γ=1 ξAMγ,n E

t,AMγ for K = RM2

,

(A.1)

where ξRM2

n,tot and ξRn,tot are the total RM2 martensite and R-phase volume fractions re-

spectively at time tn. Taking into account (35), (64) and (A.1), the general form of the

derivative of Φf with respect to ξK

β,n+1 is

−∂Φf

∂ξK

β,n+1

= Et,Kβ,n+1 · Cn+1(En+1 − Et,in

n )−

Et,Kβ,n+1 · Cn+1

nv(K)∑

γ=1

ξK

γ,n+1Et,Kγ,n+1 −

(FK

ch + Fc,K)

, (A.2)

where Et,inn is the net transformation strain of the two inactive paths during the time

interval (tn, tn+1].

25


Likewise, for the reverse transformation case,

Etn+1 =

nam∑

α=1

ξAM1

α,n+1Et,AM1

α +

nr∑

β=1

ξRβ,n+1E

t,Rβ +

nrm∑

γ=1

ξRM2

γ,n+1Et,RM2

γ

=

∑namα=1 ξAM1

α,n+1Et,AM1α +

∑nrβ=1 ξR

β,nEt,Rβ +

∑nrmγ=1 ξRM2

γ,n Et,RM2γ for K = AM1

∑nrα=1 ξR

α,n+1Et,Rα +

∑namβ=1 ξAM1

β,n Et,AM1

β +∑nrm

γ=1 ξRM2γ,n E

t,RM2γ for K = AR

∑nrmα=1 ξRM2

α,n+1Et,RM2

α,n+1 +∑nr

β=1 Et,Rβ (ξR

β,n +ξ

RM2n,tot

nrv) +

∑namγ=1 ξAM

γ,n Et,AMγ for K = RM2

.

(A.3)

Again, using (36) and (65), it follows that the general form of the derivative of Φr with

respect to ξK

β,n+1 is

−∂Φr

∂ξK

β,n+1

= Et,Kβ,n+1 · Cn+1(En+1 − Et,in

n )−

Et,Kβ,n+1 · Cn+1

nv(K)∑

γ=1

ξK

γ,n+1Et,Kγ,n+1 −

(FK

ch −Fc,K)

. (A.4)

The expressions (A.2) and (A.4) are substituted in the constrained minimization equa-

tions (58). The latter are solved using an active set strategy, as in [7] for the austenite-

martensite case. To this end, a feasible descent direction for each of the functionals Φf and

Φr is obtained by gradient projection on the active set of linear constraints rather than on

the entire constraint set, see, e.g., [24, Chapter 14]. Preliminary to formulating an active

set strategy, define the extended set of Lagrange multipliers as an 8-dimensional column

vector

[λen+1] =

[λ1 · · · λ6 λl λu

]Tn+1

(A.5)

and the set of the corresponding volume fractions for L-phase as a 6-dimensional column

vector

[ξL

n+1] =[ξL

1 · · · ξL

6

]Tn+1

. (A.6)

Then, the Kuhn-Tucker condition (61) can be expressed in matrix form as

[λen+1]

T ([A][ξL

n+1] − [hL

n+1]) = [0] , (A.7)

26

where

[A] =

−1 0 0 0 0 0

0 −1 0 0 0 0

. . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 0 0 0 −1

−1 −1 −1 −1 −1 −1

1 1 1 1 1 1

=

[a1]T

[a2]T

. . .

[a6]T

[a7]T

[a8]T

(A.8)

and

[hL

n+1] =[0 0 · · · 0 − ξL

l,n+1 ξL

u,n+1

]T. (A.9)

Also, let [Πn+1] be the na × 8-dimensional projection matrix from the 8-dimensional space

of all constraints to the na-dimensional space of all active constraints, where, as argued

earlier, na ≤ 8. It follows that [λn+1] = [Πn+1][λen+1], where [λn+1] is the na-dimensional

column vector of the multipliers associated with active constraints. Now, equations (60)

and (61) are written respectively in matrix form as

[QK

n+1][ξL

n+1] +([Πn+1][A]

)T[λn+1] = [cK

n+1] ,

[Πn+1][A][ξL

n+1] = [Πn+1][hL

n+1] .(A.10)

In (A.10), the matrix [QKn+1] has components

QK

αβ,n+1 =

Et,Kα,n+1 · Cn+1E

t,Kβ,n+1 (forward)

Et,Kα,n+1 · Cn+1E

t,Kβ,n+1 (reverse)

, (A.11)

along path K, where α, β = 1, . . . , 6, while the 6-dimensional column vector [cKn+1] has

components

cK

α,n+1 =

Et,Kα,n+1 · Cn+1

(En+1 − E

t,inn ) −

(FK

ch + Fc,K)

(forward)

Et,Kα,n+1 · Cn+1

(En+1 − E

t,inn ) −

(FK

ch −Fc,K)

(reverse). (A.12)

Since [Qn+1] is positive-definite and [Πn+1][A] is a na×6-dimensional matrix of rank na,

the system (A.10) can be solved uniquely for [ξKn+1] and [λn+1] in the form

[ξL

n+1] = [Ωn+1][cK

n+1] + [∆n+1]T [Πn+1][h

L

n+1]

[λn+1] = [∆n+1][cK

n+1] −[[Πn+1][A][QK

n+1]−1([Πn+1][A]

)T ]−1[Πn+1][h

L

n+1] ,(A.13)

where Ωn+1 and ∆n+1 are obtained by inverting the system of equations in (A.10).

The active constraints are determined iteratively using a standard active set strategy,

see, e.g., [24, Chapter 14]. Once the active constraints are specified at an iteration k (hence,

27


the matrix [Π(k)n+1] is known), they are enforced in equality form as in (A.10)2. To reduce

the cost of identifying the active constraints, the initial guess for the active set at time tn+1

is taken to coincide with the active set determined at the previous solution time tn.

APPENDIX B: Consistent tangent modulus

The consistent tangent modulus for transformation along path AM1 was derived in [7] and

is given by

∂Sn+1

∂En+1= Cn+1 −

nrm∑

β=1

nrm∑

γ=1

Ωβγ,n+1

(Cn+1E

t,AM1

β

)⊗(Cn+1E

t,AM1

γ

). (A.14)

The consistent tangent modulus for AR transformation is entirely analogous to that of AM1.

Therefore, here only the modulus for the RM2 transformation path is derived. The second

Piola-Kirchhoff stress in terms of R-phase and the two martensitic phase transformation

strains at time n + 1 can be expressed as

Sn+1 = Cn+1

(En+1 −

nam∑

α=1

ξAMα,n Et,AM

α −

nr∑

β=1

ξRβ,n+1E

t,Rβ −

nrm∑

γ=1

ξRM2

γ,n+1Et,RM2

γ

). (A.15)

The algorithmic tangent along the RM2 transformation path is given by

∂Sn+1

∂En+1= Cn+1 − Cn+1

[ nr∑

α=1

Et,Rα ⊗

∂ΞM2Rα,n+1

∂En+1+

nrm∑

β=1

Et,RM2

β ⊗∂ξRM2

β,n+1

∂En+1

], (A.16)

where (7) is exploited. Now, using (66), one may write for the forward RM2 transformation

that∂ΞM2R

α,n+1

∂En+1= −

ξRα,n∑nr

α=1 ξRα,n

nrm∑

β=1

∂ξRM2

β,n+1

∂En+1. (A.17)

Similarly, for the reverse RM2 transformation, equation (69) leads to

∂ΞM2Rα,n+1

∂En+1= −

1

nrv

nrm∑

β=1

∂ξRM2

β,n+1

∂En+1. (A.18)

Also, recalling the constrained minimization equations (A.12) and (A.13)1, it follows that

∂ξRM2

β,n+1

∂En+1=

nrm∑

γ=1

Ωβγ,n+1Cn+1Et,RM2

γ,n+1 (A.19)

28

for forward RM2 transformation, and

∂ξRM2

β,n+1

∂En+1=

nrm∑

γ=1

Ωβγ,n+1Cn+1Et,RM2

γ,n+1 (A.20)

for reverse RM2 transformation. Substituting (A.17) in (A.16), one finds that

∂Sn+1

∂En+1= Cn+1 −Cn+1

[−

∑nrα=1 ξR

α,nEt,Rα∑nr

α=1 ξRα,n

⊗

nrm∑

β=1

∂ξRM2

β,n+1

∂En+1+

nrm∑

β=1

Et,RM2

β ⊗∂ξRM2

β,n+1

∂En+1

](A.21)

or, upon rearranging terms and invoking (64)3,

∂Sn+1

∂En+1= Cn+1 − Cn+1

[nrm∑

β=1

(E

t,RM2

β −

∑nrα=1 ξR

α,nEt,Rα∑nr

α=1 ξRα,n

)⊗

∂ξRM2

β,n+1

∂En+1

](A.22)

= Cn+1 − Cn+1

[nrm∑

β=1

Et,RM2

β,n+1 ⊗∂ξRM2

β,n+1

∂En+1

]. (A.23)

Substituting (A.19) into (A.22)2, the consistent tangent modulus for forward transforma-

tion takes the final form

∂Sn+1

∂En+1= Cn+1 −

nrm∑

β=1

nrm∑

γ=1

Ωβγ,n+1

(Cn+1E

t,RM2

β,n+1

)⊗(Cn+1E

t,RM2

γ,n+1

). (A.24)

Similarly, for reverse RM2 transformation, one may start from (A.16) and use (A.18) and

(A.20) to deduce the consistent tangent modulus as

∂Sn+1

∂En+1= Cn+1 −

nrm∑

β=1

nrm∑

γ=1

Ωβγ,n+1

(Cn+1E

t,RM2

β,n+1

)⊗(Cn+1E

t,RM2

γ,n+1

). (A.25)

It is noted that the consistent tangent modulus for the RM2 path is similar in math-

ematical structure to that of the AM1 and AR paths, with the variant transformation

strains being replaced by generalized strain terms Et,RM2

β,n+1 and Et,RM2

β,n+1 , respectively, for the

forward and reverse transformation case.

29


1. Copy the active constraints and volume fractions for all transformation paths from

time tn to tn+1.

2. Copy the transformation flag from time tn to tn+1. (flag = ‘f’ for ‘forward’ or flag=‘r’

for ‘reverse’).

3. Loop over the transformation paths K ∈ AM1, AR,RM2.

Minimize the forward and reverse transformation functionals Φf and Φr, re-

spectively, along path K following the active set strategy algorithm given in

Appendix A of [7]. Let the three sets of minima be Φf,Km and Φr,K

m , respectively,

for K-th path.

4. End loop

5. Determine Kf = arg minK

Φf,K

m | K ∈ AM1, AR,RM2

and

Kr = arg minK

Φr,K

m | K ∈ AM1, AR,RM2.

6. If flag = ‘f’ and∑

α∈J f,Kf

n+1

(ξKf

α,n+1 − ξKf

α,n) > tol or

flag = ‘r’ and∑

α∈J r,Kr

n+1

(ξKr

α,n+1 − ξKr

α,n) < −tol, then

Retain flag and compute stress and consistent tangent moduli based on forward

or reverse transformation along Kf or K

r, respectively.

7. If flag = ‘f’ and∑

α∈J r,Kr

n+1

(ξKr

α,n+1 − ξKr

α,n) < −tol,

Set flag = ‘r’ and compute stresses and consistent tangent moduli based on

reverse transformation along Kr.

8. If flag = ‘r’ and∑

α∈J f,Kf

n+1

(ξKf

α,n+1 − ξKf

α,n) > tol,

Set flag = ‘f’ and compute stresses and consistent tangent moduli based on

forward transformation along Kf .

Box 1: A summary of the algorithmic procedure for determining the transformation state

at time tn+1.

30

# b m

1 b2 b1 −b3 m2 m1 −m3

2 b3 b1 −b2 m3 m1 −m2

3 b3 −b2 b1 m3 −m2 m1

4 b2 −b3 b1 m2 −m3 m1

5 b1 b2 −b3 m1 m2 −m3

6 b1 b3 −b2 m1 m3 −m2

7 b1 b3 b2 m1 m3 m2

8 b1 b2 b3 m1 m2 m3

9 b2 b3 b1 m2 m3 m1

10 b3 b2 b1 m3 m2 m1

11 b3 b1 b2 m3 m1 m2

12 b2 b1 b3 m2 m1 m3

Table 1: Components of the twelve habit plane vector pairs (bα, mα) of R-phase generated

from the following vectors by appropriate orthogonal transformations:

[b] = [0.00000 0.00865 0.00008], [m] = [0.00000 -0.00440 -0.99999]

31


XX

XX

XX

XX

XX

Path/LLimits

ξL

l ξLu

AM1/AM1∑nam

α=1 ξAM1α 1 −

∑nrβ=1 ξR

β −∑nrm

γ=1 ξRM2γ

AR/R∑nr

β=1 ξRβ 1 −

∑namα=1 ξAM1

α −∑nrm

γ=1 ξγRM2

RM2/RM2∑nrm

γ=1 ξRM2γ

∑nrβ=1 ξR

β +∑nrm

γ=1 ξRM2γ

Table 2: Values of ξL

l and ξLu for the case of forward transformation

32

Figure 1: Deformation of single crystal cube: Boundary conditions

33


0 0.02 0.04 0.06 0.08 0.1 0.120

100

200

300

400

500

600

700

Equivalent strain

Mis

eseq

uiv

ale

nt

stre

ss

0.75mm0.11mm

Figure 2: Deformation of single crystal cube: Equivalent Cauchy stress vs. equivalent

Lagrangian strain in tension with end displacements of 0.11 mm and 0.75 mm

34

Figure 3: Thin-walled tube in tension: Finite element mesh and boundary conditions

35


0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

50

100

150

200

250

300

350

400

450

Equivalent strain

Equiv

ale

nt

stre

ss

Experiment 1

Experiment 2

Simulation

Figure 4: Thin-walled tube in tension: Equivalent Cauchy stress vs. equivalent Lagrangian

strain from experiments and from a single-scale sheet-texture simulation

36

2.05E+02

2.28E+02

2.50E+02

2.73E+02

2.96E+02

3.18E+02

3.41E+02

3.64E+02

3.86E+02

4.09E+02

4.32E+02

1.82E+02

4.55E+02

_________________ S T R E S S 3

Time = 3.00E+01

Figure 5: Thin-walled tube in tension: Normal stress distribution along axis of tube at 6%

tensile strain predicted from a single-scale sheet-texture simulation

37


0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

50

100

150

200

250

300

350

400

450

Equivalent strain

Equiv

alen

tst

ress

(MPa)

Experiment 1

Experiment 2

Simulation

Figure 6: Thin-walled tube in tension: Equivalent Cauchy stress vs. equivalent Lagrangian

strain response from experiments and multi-scale simulation with full texture resolution

38

2.22E+08

2.40E+08

2.58E+08

2.76E+08

2.94E+08

3.12E+08

3.30E+08

3.48E+08

3.66E+08

3.83E+08

4.01E+08

2.04E+08

4.19E+08

_________________ S T R E S S 3

Time = 9.00E+00

2.04E+08

2.23E+08

2.42E+08

2.60E+08

2.79E+08

2.97E+08

3.16E+08

3.35E+08

3.53E+08

3.72E+08

3.91E+08

1.86E+08

4.09E+08

_________________ S T R E S S 3

Time = 3.20E+01

Figure 7: Thin-walled tube in tension: Normal stress component along the longitudinal

direction at 1% and 6% equivalent strains for textured tube

39


4.75E-01

4.98E-01

5.21E-01

5.44E-01

5.67E-01

5.90E-01

6.13E-01

6.36E-01

6.59E-01

6.82E-01

7.05E-01

4.52E-01

7.28E-01

____________________ R-PHASE VOL. FRCN.

Time = 9.00E+00

1.01E-01

1.57E-01

2.14E-01

2.71E-01

3.28E-01

3.85E-01

4.41E-01

4.98E-01

5.55E-01

6.12E-01

6.68E-01

4.39E-02

7.25E-01

____________________ R-PHASE VOL. FRCN.

Time = 3.20E+01

Figure 8: Thin-walled tube in tension: R-phase volume fractions at 1% and 6% equivalent

strains for textured tube

40

2.88E-03

5.76E-03

8.64E-03

1.15E-02

1.44E-02

1.73E-02

2.01E-02

2.30E-02

2.59E-02

2.88E-02

3.17E-02

0.00E+00

3.45E-02

______________________ MARTENSITE VOL. FRCN.

Time = 9.00E+00

5.79E-02

1.16E-01

1.74E-01

2.32E-01

2.89E-01

3.47E-01

4.05E-01

4.63E-01

5.21E-01

5.79E-01

6.37E-01

0.00E+00

6.95E-01

______________________ MARTENSITE VOL. FRCN.

Time = 3.20E+01

Figure 9: Thin-walled tube in tension: Martensite volume fractions at 1% and 6% equiv-

alent strains for textured tube

41

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