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Acta Materialia 56 (2008) 351–357

Simulation of chemically driven inelastic strainin multi-component systems with non-ideal sources

and sinks for vacancies

J. Svoboda a, F.D. Fischer b,c,*, E. Gamsjager b

a Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, CZ-616 62, Brno, Czech Republicb Institute of Mechanics, Montanuniversitat Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria

c Materials Center Leoben Forschung GmbH, Franz-Josef-Strasse 13, A-8700 Leoben, Austria

Received 24 September 2007; accepted 27 September 2007Available online 26 November 2007

Abstract

Recently, a new concept of diffusion and inelastic strain in multi-component systems with non-ideal sources and sinks for vacancieshas been presented, accounting for both chemical and mechanical driving forces. Based on this concept, chemically driven diffusion andinelastic deformation in an Fe–Mn–C-vacancy system, consisting of a bamboo-structured load-free wire, are investigated in the presentpaper. Simulations demonstrate that the evolution of the inelastic deformation depends on the activity of sources and sinks for vacanciesrepresented by grain boundaries and jogs at dislocations. The evolution of the chemical composition is, however, practically not influ-enced by the activity of sources and sinks for vacancies in the studied system.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Bulk diffusion; Simulation; Thermodynamics; Vacancies; Chemically driven inelastic strain

1. Introduction

Diffusion in multi-component systems causes not onlythe evolution of the chemical composition, but may alsobe accompanied by considerable inelastic deformations.A well-documented manifestation of the chemically driveninelastic strain is the Kirkendall effect [1]. It is usuallyexplained by the Darken theory [2], tacitly assuming thatthe deformation of the specimen occurs only in the direc-tion of the diffusive fluxes and stems exclusively from gen-eration and annihilation of vacancies at ideal sources andsinks distributed over the whole specimen. Similar assump-tions can also be found in the very recent literature [3–7]. Adifferent approach was suggested by Ruth and Voigt [8,9],

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.09.038

* Corresponding author. Address: Institute of Mechanics, Mont-anuniversitat Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria.Tel.: +43 3842 402 4001; fax: +43 3842 46048.

E-mail address: mechanik@mu-leoben.at (F.D. Fischer).

who assumed that the vacancies are annihilated at surfacesof pores in the part of the specimen with the faster diffusingcomponent.

A new theoretical concept of diffusion and inelasticdeformation has been published by Svoboda et al. [10],which deals in detail with the influence of non-ideal sourcesand sinks for vacancies on the system evolution. Thedeformation mechanisms acting due to generation andannihilation of vacancies, as well as to diffusion in multi-component systems, namely the diffusion of interstitialcomponents and/or interdiffusion of substitutional compo-nents with different partial molar volumes, are taken intoaccount by this treatment. In addition, this concept alsoaccounts for the influence of the three-axial stress stateon the system kinetics.

The motivation for this paper was to find a typical sam-ple that would enable understanding of the role of vacan-cies, more or less independently of the actual possibilityof observing the vacancies in experiments. Therefore, we

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352 J. Svoboda et al. / Acta Materialia 56 (2008) 351–357

demonstrate the newly developed concept using the exam-ple of a bamboo-structured crystal arrangement, whichmainly serves to avoid the geometrical difficulties, thatwould arise if a general three-dimensional model wereemployed. For this reason, the current paper demonstratesthe applicability of the equations derived in Ref. [10] to rel-atively simple stress-free systems with a thin wire bamboo-structured geometry. The simulations are performed for theFe–Mn–C-vacancy system. The concept [10] enables us toaccount for the actual stress state via generalized chemicalpotentials (see Eqs. (29)–(32) in Ref. [10]). Thus, the cou-pling of already considered processes with further phenom-ena – as may be relevant for the evolution of a systemundergoing a complex thermomechanical treatment – ispossible within this concept. The setup of the numericalmodel for such problems is discussed.

2. Leading equations

Let us repeat in the simplest form the leading equationsfrom Ref. [10] used in the following simulations. Weassume n substitutional components (k = 1, . . .,n), m inter-stitial components (k = n + 1, . . .,n + m) and vacancies(k = 0) in the system. Then the evolution of the site fractionyk of the component k is given by its rate as

_yk ¼ ��Xdiv~jk � ayk ðk ¼ 1; . . . ; nþ mÞ ð1Þ

and for vacancies

_y0 ¼ ��Xdiv~j0 þ að1� y0Þ ð2Þ

The dot symbol denotes the material derivative accordingto time t; the volume �X corresponds to 1 mol of the latticesites (Eq. (6) in Ref. [10]); ~jk is the diffusive flux of thecomponent k (k = 0, . . .,n + m) relative to the lattice (inthe actual configuration being the lattice fixed frame ofreference); and a represents the rate at which vacanciesare generated (a > 0) or annihilated (a < 0) in each repre-sentative volume element (RVE) of the system.

The generalized inelastic strain rate due to diffusion andvacancy generation/annihilation processes, _egc, can be writ-ten as

_egc ¼ a�X0

�X� d

3

Xn

k¼1

ðXk � �X0Þ1� y0

div~jk þXnþm

k¼nþ1

Xkdiv~jk

!ð3Þ

The tensorial product _egc : d (d is the unit tensor) equals justthe shrinking or swelling rate; Xk is the partial molar vol-ume of component k; and �X0 is the partial molar volumeof vacancies approximated by the mean value of the partialmolar volumes of substitutional components (Eq. (3) inRef. [10]). The first term in Eq. (3) describes the strain ratetensor of the volume and shape change due to the genera-tion or annihilation of vacancies. The tensorial product a:dmust be equal to a. The second term in Eq. (3) refers to thevolume misfit developing due to diffusive motion of atomswith different partial molar volumes.

The diffusive fluxes are given by

~jk ¼ �Xnþm

i¼1

Lik grad l�i ðk ¼ 1; . . . ; nþ mÞ ð4Þ

~j0 ¼ �Xn

k¼1

~jk ð5Þ

The generalized chemical potentials, l�k , are expressed for astress-free system by l�k ¼ lk � l0 ðk ¼ 1; . . . ; nÞ, l�k ¼lk ðk ¼ nþ 1; . . . ; nþ mÞ with lk (k = 0, . . .,n + m) beingthe chemical potentials. For calculation of the chemical po-tential for vacancies the ideal solution model can be used,yielding l0 ¼ RT lnðy0=yeq

0 Þ.The Onsager coefficients, Lik, read as (see section 4 of

Ref. [10])

Lik ¼ Akdik � AiAk

Xn

j¼0

Aj ði ¼ 1; . . . ; n and k ¼ 1; . . . ; nÞ,

ð6ÞLik ¼ Akdik ði ¼ nþ 1; . . . ; nþ m or k ¼ nþ 1; . . . ; nþ mÞ

ð7Þwith

Ak ¼y0ykDeq

k

yeq0

�XRTðk ¼ 1; . . . ; nÞ ð8Þ

Ak ¼ykDeq

k�XRT

ðk ¼ nþ 1; . . . ; nþ mÞ ð9Þ

A0 ¼ �1

1� f

Xn

k¼1

Ak ð10Þ

The quantity Deqk is the tracer diffusion coefficient of the

component k for the equilibrium concentration of vacan-cies, yeq

0 ; R is the gas constant; T is the absolute tempera-ture; and f is the geometric correlation factor (f = 0.7815for the vacancy mechanism in fcc alloys and f = 0.7272for bcc alloys). The Onsager coefficients, Lik, coincide withthose derived originally by Manning [11].

We assume that the bulk of a grain contains a disloca-tion structure with randomly oriented dislocation linesand randomly oriented Burgers vectors. In that case, thebulk of the grain with jogs at dislocations acts as an isotro-pic source and sink for vacancies. For H being the jogdensity and a being the lattice spacing, the tensor a fromEq. (3) is given by Eqs. (54) and (62) in Ref. [10] as

a ¼ � 2pl0aHA0�X

3d ð11:1Þ

If we assume the distance between jogs on dislocations tobe (10–100) Æ a, say 30 Æ a, then the product a Æ H can be re-placed by q/30, with q being the dislocation density. ThenEq. (11.1) can be rewritten as

a ¼ � pl0qA0�X

45d ð11:2Þ

As Balluffi has pointed out [12,13], the grain boundaries inpolycrystals act as dominant sources and sinks for vacan-cies and, thus, we address the grain boundaries as ideal

J. Svoboda et al. / Acta Materialia 56 (2008) 351–357 353

sources and sinks for vacancies. The strain rate contribu-tion due to generation/annihilation of vacancies a is givenfor the grain boundary zones normal to the 1-direction byEq. (67) in Ref. [10]:

a ¼�Xdiv~j0

�X0ð1� yeq0 Þ

�X~d� d

3

Xnþm

k¼nþ1

ykXk

!ð12Þ

with ~d being a tensor with the only non-zero component~d11 ¼ 1.

A critical evaluation of the role of the value of the equi-librium vacancy site fraction, yeq

0 , seems to be necessary.Few data on yeq

0 , often much lower than 10�6, have beenreported (see, for example, Ref. [14]). Most usually, thevalue of yeq

0 is unknown. One can assume only small devi-ations of y0 from yeq

0 in many applications. Thus, after acertain transition time period, one can adopt _y0 � 0 inEq. (2) and neglect the values of y0 and yeq

0 with respectto 1. Then it can be shown that in Eqs. (1)–(10), (11.1),(11.2), (12) only the ratio y0=yeq

0 appears and, thus, thekinetics of the system is practically independent of thevalue of yeq

0 itself.

3. Numerical procedure

The evolution Eqs. (1)–(3) for the system, supplementedby Eqs. (4)–(10), (11.1), (11.2) and (12), can be integratedwith respect to time by the Euler method in combinationwith a finite-difference scheme for spatial discretization.The whole treatment is performed in the actual configura-tion (in the lattice fixed frame of reference).

The wire can be considered as a system in whichdiffusion occurs only along the wire axis (the x-axis in1-direction) and, thus, the diffusion fluxes are not treatedas vectors. The deformation occurs in the direction parallelto the wire axis and normal to the wire axis (radial r-direc-tion). The system is divided in the x-direction into s cylin-drical RVEs of the length Dx‘ and the radius r‘(‘ = 1, . . ., s). To each RVE, the mean site fractions, yk,‘,and the corresponding chemical potentials, lk,‘

(k = 0, . . .,n + m, ‘ = 1, . . ., s), are assigned. The values ofdiffusive fluxes, jk,‘ (k = 0, . . .,n + m, ‘ = 1, . . ., s � 1), arecalculated at the boundaries of the RVEs (jk,‘ is the fluxbetween the RVEs ‘ and ‘ + 1). At both ends of the spec-imen, the fluxes as well as the gradients of all chemicalpotentials, lk (k = 0, . . .,n + m), are supposed to be zero;a closed system is assumed. Then the fluxes can be calcu-lated as

jk;‘ ¼ �Xnþm

i¼0

ðLik;‘ þ Lik;‘þ1Þl�i;‘þ1 � l�i;‘Dx‘ þ Dx‘þ1

ðk ¼ 1; . . . ; nþ m; ‘ ¼ 1; . . . ; s� 1Þ ð13Þ

Lik,‘ are the Onsager coefficients, Lik, given by Eqs. (6)–(10), assigned to the RVE ‘. The vacancy flux is given byEq. (5). The divergence of the flux of the component k inthe RVEs 1,‘ and s is given by

ðdiv jkÞ1 ¼jk;1

Dx1

ðk ¼ 0; . . . ; nþ mÞ ð14Þ

ðdiv jkÞ‘ ¼jk;‘ � jk;‘�1

Dx‘ðk ¼ 0; . . . ; nþ m; ‘ ¼ 2; . . . ; s� 1Þ

ð15Þ

ðdiv jkÞs ¼ �jk;s

Dxsðk ¼ 0; . . . ; nþ mÞ ð16Þ

Eqs. (13)–(16), together with Eqs. (11.1) and (11.2) for theRVEs in the interior of the grain, or with Eq. (12) for theRVEs representing the grain boundary, can be inserted intoEqs. (1)–(3). Their integration in time provides the systemevolution.

According to Eq. (3), the individual RVEs expand orshrink during the system evolution. Then the length Dx‘of the RVE ‘ evolves as

D _x‘ ¼ _egc11;‘Dx‘ ð17Þand the radius r‘ of the RVE grows or shrinks according to

_r‘ ¼ _egc22;‘r‘ ¼ _egc33;‘r‘ ð18Þ

4. Results of simulations and discussion

The thermodynamics of the Fe–Mn–C system is calcu-lated using a two-sublattice regular solution model follow-ing the thermodynamic assessments [15]. For austenite, thediffusivity of manganese, DMn, has been given by Kimuraet al. [16], the diffusivity of iron, DFe, can be found in Por-ter and Easterling [17] and that of carbon DC in Agren [18]:

DMn ¼ 96:06� 10�5 exp � 2:983� 105 J mol�1

RT

� �m2 s�1

DFe ¼ 4:9� 10�5 exp � 2:841� 105 J mol�1

RT

� �m2 s�1

DC ¼ 2:343� 10�5 exp � 1:477� 105 J mol�1

RT

� �m2 s�1

ð19ÞThe whole system is in the austenitic state at the chosentemperature T = 1073 K, so DFe = 7.246 · 10�19 m s�2,DMn/DFe = 3.99 and DC/DFe = 2.09 · 106.

The partial molar volumes for iron and manganese aredetermined from [19] as

XFe ¼ 7:205� 10�6 m3 mol�1;

XMn ¼ 7:897� 10�6 m3 mol�1 ð20Þ

and the partial molar volume for carbon is, according to[20],

XC ¼ 4:643� 10�6 m3 mol�1 ð21Þ

Note that the partial molar volume of interstitial carbon isnot, as is often assumed, negligible relative to the molarvolumes of substitutional components.

As the initial state, a wire of the total length 2L ischosen, with two grain boundaries normal to the x-axis

354 J. Svoboda et al. / Acta Materialia 56 (2008) 351–357

positioned at 25% and 75% of the length. Thus, the centralgrain has length L. The wire is divided into 100 cylindricalRVEs of initially identical length. The grain boundaries arepositioned in RVEs No. 25 and 75. The site fractions arechosen as yFe = 0.900, yMn = 0.099, yC = 0.010 in the lefthalf of the wire and yFe = 0.960, yMn = 0.039, yC = 0.030in the right half of the wire. The initial and equilibrium sitefraction of vacancies in the whole wire is assumed to bey0 ¼ yeq

0 ¼ 10�3. This choice of a rather high value of yeq0

drastically increases the stability of the numerical solutionand allows for increasing the time integration step by sev-eral orders of magnitude without any observable influence

Fig. 1. Results of simulations for yeq0 ¼ 10�3, �q ¼ qL2 ¼ 107 and �t ¼ tDFe=L2 ¼

C, yC; (c) site fraction of vacancies, �y0; (d) axial strain, ex (the value of ex reach(f) vacancy flux �j0.

on the accuracy of the results. The sum of the site fractions,yFe + yMn + y0, must be 1 always and everywhere.

The dislocation density, q, in metals ranges typicallyfrom 1011 to 1015 m�2. We assume in our simulations thatthe dislocation density, q, is a free parameter. It is keptconstant during the whole system evolution and does notinfluence the tracer diffusion coefficients in the bulk (negli-gible pipe diffusion is assumed).

The results of simulations are represented by the profilesof site fractions yMn, yC and y0 (yFe = 1 � yMn � y0), by theprofiles of the axial strain ex = Dx/Dxst � 1 and of theradial strain er = r/rst � 1, (Dxst and rst are the starting val-

10�5, 10�4, 10�3 and 10�2. (a) Site fraction of Mn, yMn; (b) site fraction ofes ±0.3 in RVEs with the grain boundary for �t ¼ 10�2); (e) radial strain, er;

J. Svoboda et al. / Acta Materialia 56 (2008) 351–357 355

ues of Dx and the wire radius, r, respectively) and by theprofiles of the diffusive flux of vacancies, j0. It is advanta-geous to eliminate the size, L, for a compact presentationof the results by introducing the normalized quantities�x ¼ x=L ð�x 2 h0; 2iÞ, �t ¼ tDFe=L2, �q ¼ qL2 and �j0 ¼ j0L�X=DFe. Moreover, y0 can be normalized as �y0 ¼ y0=yeq

0 .The results of simulations (profiles of yMn, yC, �y0, ex, er

and �j0) are plotted in Fig. 1 for yeq0 ¼ 10�3, �q ¼ qL2 ¼ 107

and the time instants �t ¼ tDFe=L2 ¼ 10�5, 10�4, 10�3 and10�2. Equivalent simulations were repeated foryeq

0 ¼ 10�5, to show that for yeq0 � 1 the results of simula-

tions are independent of the value of yeq0 , except for a very

short initial transition time period, necessary for the redis-tribution of vacancies. This is demonstrated in Fig. 2,where the profiles of �y0 and �j0 differ for �t ¼ 10�5. All otherprofiles for yeq

0 ¼ 10�5 are identical (within the accuracy ofthe plot) to those for yeq

0 ¼ 10�3 presented in Fig. 1. Fig. 3shows the results of the simulations for yeq

0 ¼ 10�3, the timeinstant �t ¼ 0:03 and different values of �q 6 105,�q ¼ 107; 109 and �q P 1011.

It is evident from the figures that an increased initialvalue of the vacancy site fraction, y0, being equal to yeq

0

has no significant influence on the system kinetics. Thisallows, as already mentioned, a drastic decrease of thecomputational time and avoids the need to use uncertaindata on equilibrium site fractions of vacancies in varioussystems. The results of simulations also clearly demonstratethe influence of sources and sinks for vacancies due to jogsat dislocations in the bulk of the grains on the inelasticdeformation behaviour of the material. The bulk of thegrains can be treated as a region with no sources and sinksfor vacancies for systems with qL2

6 105 and as a regionwith ideal sources and sinks for vacancies for qL2 P 1011.The grain interiors should be treated as objects with non-ideal sources and sinks for vacancies for 105 < qL2 < 1011.

The sources and sinks for vacancies also influence theevolution of the chemical composition due to deformationof the system and deviation of y0 from yeq

0 . However, forthe system chosen, the influence of the sources and sinks

Fig. 2. Results of simulations for �q ¼ qL2 ¼ 107 and �t ¼ tDFe=L2 ¼ 10�5 and twflux, �j0.

for vacancies on the evolution of the chemical compositionis insignificant. It may become significant for systems withmuch larger differences in the starting chemical composi-tion, the diffusivities of substitutional components and/orthe partial molar volumes of the substitutionalcomponents.

5. Outlook

The realistic simulation of a complex thermomechani-cal treatment of an alloy seems to be a very difficult task.One may assume a certain starting configuration of thesystem, given by the distribution of chemical composi-tion, internal stresses and dislocation density (we areaware that the description of the dislocation structureonly by its density, q, need not be satisfactory in manycases) and of the positions of grain boundaries in thewhole specimen. Furthermore, we can assume that thetemperature and loading of the system are prescribedby external conditions. To obtain a proper model ofthe thermomechanical treatment, one must supplementthe concept of Ref. [10] by algorithms meeting dislocationplasticity, grain boundary migration and sliding, grainboundary diffusion, recovery and a proper algorithm forthe stress–strain analysis.

Of course, the thermomechanical treatment must besimulated by an incremental procedure. Each integrationstep starts with the stress–strain analysis providing theactual stress field. Then the algorithm representing thedislocation plasticity, together with the concept of Ref.[10] at hand, yields the increment of the inelastic strainfield. Furthermore, one has to recalculate the dislocationdensities by using algorithms for dislocation multiplica-tion and recovery and to prescribe the motion of thegrain boundaries according to the algorithm for grainboundary migration. It should be taken into account thatthe dislocation density in the region, swept by the migrat-ing grain boundary, is drastically reduced. Moreover, thecontributions from the grain boundary diffusion and

o values of yeq0 ¼ 10�5; 10�3. (a) Site fraction of vacancies, �y0; (b) vacancy

Fig. 3. Results of simulations for yeq0 ¼ 10�3, the time instant �t ¼ 0:03 and the values of �q 6 105, �q ¼ 107; 109 and �q P 1011. (a) Site fraction of Mn, yMn;

(b) site fraction of C, yC; (c) site fraction of vacancies, �y0; (d) axial strain, ex (ex = ±0.30 for �q 6 105, ex = ±0.20 for �q ¼ 107, ex = ±0.024 for �q ¼ 109

and ex = ±0.008 for �q P 1011 in RVEs containing grain boundaries); (e) radial strain, er; (f) vacancy flux, �j0.

356 J. Svoboda et al. / Acta Materialia 56 (2008) 351–357

sliding should be accounted for. If pipe diffusion is con-sidered, the diffusion coefficients in grains must be recal-culated according to the actual local values of thedislocation density. One should also meet the generationof vacancies and corresponding swelling due to disloca-tion motion (see Cuitino and Ortiz [21]) and/or due toirradiation. After all that is done, the next time integra-tion step can be started with the stress–strain analysis.An attempt to couple the diffusion of vacancies withthe elastic stress–strain analysis has already been per-formed by Wu, who applied analytical methods [22].Later, Boettinger et al. [23] presented a semi-analyticalsolution for coupling of the diffusion of vacancies with

a visco-elastic material behaviour for a cylindrical rod.However, the current incremental and iterative frame-work applied to the finite RVEs seems to be inevitablefor the numerical performance of this task.

6. Conclusions

The unified equations for

� diffusion,� generation and annihilation of vacancies at non-ideal

sources and sinks and� diffusional inelastic deformation.

J. Svoboda et al. / Acta Materialia 56 (2008) 351–357 357

In multi-component one-phase systems, including substitu-tional and interstitial components and vacancies, as derivedin Ref. [10], have been solved by numerically simulating thekinetics in the Fe–Mn–C-vacancy system.

To this end, a stress-free bamboo-structured wire at anelevated temperature was assumed. The bulk of grains issupposed to act as a non-ideal isotropic source and sinkfor vacancies. The grain boundaries act as an ideal aniso-tropic source and sink for vacancies. The profiles of straincomponents and of the vacancy fluxes are sensitive to anychange of the grain size and dislocation density in grainscovering a wide range of conditions. On the other hand,site fraction profiles of all atomic species are insensitiveto the grain size and the dislocation density in grains forthe investigated system. The main goal of this work wasthe demonstration of the new theoretical concept outlinedin Ref. [10].

Acknowledgements

Financial support by the Materials Center Leoben Fors-chung GmbH (Project SP19) and by the Research Plan ofthe Institute of Physics of Materials (Project CE-Z:AV0Z20410507) is gratefully acknowledged. This workhas also been supported by the Grant Agency of the Acad-emy of Sciences of the Czech Republic – Grant No.IAA200410601.

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