multiscale mechanics of trip-assisted multiphase steels: ii. micromechanical modelling

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Acta Materialia 55 (2007) 3695–3705

Multiscale mechanics of TRIP-assisted multiphase steels:II. Micromechanical modelling

F. Lani a,b, Q. Furnemont a, T. Van Rompaey c, F. Delannay a, P.J. Jacques a,*, T. Pardoen a

a Universite catholique de Louvain, Department des Sciences de Materiaux et des Procedes, IMAP, Place Sainte Barbe 2, B-1348 Louvain-la-Neuve, Belgiumb CENAERO, Centre d’Excellence en Aeronautique, Avenue Jean Mermoz 30, B-6041 Gosselies, Belgium

c Department of Metallurgy & Materials Engineering, KU Leuven, Kasteelpark Arenberg 44, 3001 Heverlee, Belgium

Received 14 March 2006; received in revised form 13 February 2007; accepted 14 February 2007Available online 23 April 2007

Abstract

The stress and strain partitioning between the different phases of transformation-induced plasticity (TRIP)-aided multiphase steels isevaluated using a mean field homogenization approach. The change of the austenite volume fraction under straining is predicted using amicromechanics-based criterion for the martensitic transformation adapted to the case of small, isolated, transforming austenite grains.The parameters of the model are identified from the mechanical response and transformation kinetics measured under uniaxial tensionfor two steels differing essentially by the austenite stability. The model is validated by comparing the predictions with tests performedunder different loading conditions: pure shear, intermediate biaxial and equibiaxial. An analysis of the effect of the austenite stabilityon strength and ductility provides guidelines for optimizing properties according to the stress state.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Steels; Mechanical properties; Multiphase microstructure; Martensitic phase transformation; Mean field analysis

1. Introduction

Steel is usually formed by extensive plastic deformation.Improving the strain-hardening capacity is highly beneficialas it contributes to raising both formability and strengthafter forming. One way to increase the strain-hardeningcapacity of steels is to take advantage of a synergy betweensoft and hard phases. This approach can be coupled to othermethods for improving the strength and the ductility of theindividual phases, e.g. using precipitate hardening, solutesolution hardening and grain refinement. In transforma-tion-induced plasticity (TRIP)-assisted multiphase steels,an extra strain-hardening contribution is brought about bythe continuous transformation under loading of one of thephases, austenite, into a harder phase, martensite. As widelydocumented in the literature [1–9] and illustrated in the com-panion paper [10], the extra strain-hardening contribution

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

doi:10.1016/j.actamat.2007.02.015

* Corresponding author.E-mail address: [email protected] (P.J. Jacques).

results from the additional plastic flow induced by the trans-formation strain around the transforming grain as well asfrom the increase of the volume fraction of hard martensite.The TRIP effect has been proven very efficient in a large vari-ety of iron-based alloys such as fully austenitic steels [2,3,6]and ultra-high strength martensitic steels with a fine disper-sion of retained austenite [11]. Progress in both the optimiza-tion of the properties of TRIP-aided multiphase steels andthe design of forming operations requires the developmentof adequate constitutive models. By ‘‘adequate’’, we meana model that accounts for the mechanical and thermo-dynamical first-order effects while maintaining a level ofcomplexity that is sufficiently low to allow further implemen-tation in a finite element code. The first-order effects,described below, involve the influence of (i) the strength mis-match between the different phases (especially the hard mar-tensite), (ii) the retained austenite volume fraction, (iii) thecontinuous character of the martensitic phase transforma-tion and (iv) the stress-state dependence of this martensitictransformation.

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20µm

F+B

A

M

VF VB VA VMσy

F σyB σy

A σyM

a b

dc

Vi

α’

γ

εti

N

σ

σ

Extra hardening

Extra hardening

Fig. 1. The two relevant scales for the material under investigation: (a) and (b) ‘‘RVE’’ scale SEM micrograph showing the four phases (ferrite, bainite,austenite and martensite) and the model of a ferritic matrix with two types of inclusions; (c) and (d) TEM micrograph showing variants formed during thestrain-induced martensitic transformation and the relevant physical parameters at the scale of one grain.

3696 F. Lani et al. / Acta Materialia 55 (2007) 3695–3705

The first scale to be considered is the scale of a represen-tative volume of the microstructure (RVE). Fig. 1a shows atypical micrograph of a multiphase TRIP steel with thefour phases: ferrite, bainite, austenite and martensite. Theload partitioning between the matrix (a mixture of ferriteand bainite) and the inclusions (austenite and martensite)must be properly captured in order to accurately predictthe overall response as well as the kinetics of the transfor-mation, which depends on the current stress state in theretained austenite. A micro–macro scheme allowing theevaluation of the average stress and strain in each phaseis thus the first important ingredient of the model.

The physics at the scale of a microstructural elementcontaining a transforming grain and its environment isillustrated by the transmission electron micrograph ofFig. 1c. Scanning or transmission electron microscopy(SEM or TEM) analyses have shown that the rate of trans-formation results more from the discrete nature of thetransformation than from the continuous progress of atransformation front in each grain: some grains transformat small strains while other transform much later [10,12].Typically, a transformed grain contains 3–8 martensiteplates [12]. In some instances, the transformation of onemartensite plate corresponding to one crystallographic var-iant tends to promote the transformation of other plates ofa possibly different variant leading to a fast transformationof the entire grain. In other instances, the stress relaxationaround the transforming martensite plate hinders the for-mation of the next lath [13]. Other phenomena have tobe considered: (i) each time a variant forms, there is a geo-

metrical constraint on the transformation of the followingvariants due to the small size of the austenitic grains; (ii)increasing straining brings about an increase of the disloca-tion density in retained austenite. There has been, untilnow, no definitive experimental evidence about the positiveor negative effect of the presence of dislocations on thetransformation rate. Finally, the global transformationstrain associated with a fully transformed grain is a dilata-tion of about 4% and a complex combination of sheartransformation strains which depend on the orientationand accommodation between each lath and possibly insidethe laths (by a twinning mechanism).

The model developed in this work is based on (i) amicro–macro model for a three-phase material (one matrixand two types of inclusions) extending the secant meanfield model developed by Tandon and Weng [14]; and (ii)a transformation condition, adapted from the model ofFischer and Reisner [15], for computing the variation ofthe amount of retained austenite with straining. This workfollows earlier efforts by Stringfellow et al. [6], Berveillerand co-workers [8,16,8], Fischer and co-workers [7,13,17–19] and Levitas [20–22], who have mainly focussed on fullyaustenitic TRIP steels. Although the behaviours of fullyaustenitic TRIP steels and multiphase TRIP steels presentobvious resemblances, several differences regarding thestress transfer among the initial phases and the discreteevolution of the transformation turned out to be key inthe modelling strategy (see also Ref. [23]). The objectivesof the present paper are to present, identify and validatethis model, and to provide a parametric study in order to

F. Lani et al. / Acta Materialia 55 (2007) 3695–3705 3697

elucidate the relationships between the phase transforma-tion and the mechanical behaviour, and to help with opti-mization of the TRIP steels.

Section 2 presents the micromechanical model devel-oped for simulating the flow behaviour of TRIP-assistedmultiphase steels. The parameters identification for twohigh-silicon multiphase TRIP steels is briefly described inSection 3. In Section 4, the results of the model are vali-dated by comparison with experimental measurementsunder different stress states and the influence of the austen-ite transformation rate on the strength–ductility balance ofTRIP-assisted steels is studied under different loading con-ditions. The discussion of Section 5 mainly focuses on theoptimal choice of phase stability.

2. Micromechanical model for multiphase TRIP steels

2.1. Micro–macro model for the plasticity of a three-phase

composite with evolving volume fractions of the phases

The material is represented as a composite consisting ofa matrix (ferrite, lumping together the allotriomorphic andbainitic forms) containing two families of homogeneouslydispersed, ellipsoidal inclusions (austenite and martensite)perfectly bonded to the matrix. The response of each phaseis elastoplastic, obeying the J2 deformation theory of plas-ticity, i.e. the simple isotropic von Mises-based non-linearelastic theory, equivalent to the incremental theory for pro-portional loading, a condition which will be respected in allthe applications addressed in this paper. In addition to itssimplicity, the use of an isotropic theory is motivated bythe fact that the investigated steels do not present signifi-cant crystallographic textures. The secant formulation isinspired by Hill’s recognition of the weakening matrix con-straint during plastic deformation [24]. Following Berveil-ler and Zaoui [25], this weakened constraint is taken intoaccount by using secant pseudo-elastic moduli. Thehomogenization is performed using the equivalent inclu-sion method due to Eshelby [26] and the mean fieldapproach of Mori and Tanaka [27,28] to account for inclu-sion–inclusion interactions.

More precisely, the elastic behaviour of phase r isdescribed by the Young’s modulus Er and the Poisson ratiomr. The strain-hardening behaviour of the different phases isdescribed by the Swift representation:

rre ¼ rr

0ð1þ krepr

e Þnr

; ð1Þwhere rr

e is the effective or von Mises stress defined byrr

e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3sijsij=2

p; rr

0 is the yield strength, kr is the Swift coef-ficient, nr is the strain-hardening exponent and epr

e is the

effective plastic deformation defined as epr

e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2eijeij=3

p.

The tensors s and e are the deviatoric parts of r and e.At the onset of yielding, the matrix is replaced by a linearisotropic ‘‘comparison material’’ characterized by twoindependent secant elastic constants EMs and mMs. Thesuperscripts M, I, II and r indicate the matrix, the parentaustenite phase, the transformed phase or any of these

phases, respectively. Under radial loading paths in theframework of the J2 deformation theory and using Eq.(1), the plastic flow rule reduces to:

epr

ij ¼3

2

1

rreh

r

rre

rr0

� 1

� � 1nr

srij: ð2Þ

The average stress rr in each phase is related to the averageelastic strain by:

rM ¼ Cs;Mðe0 þ heimfÞ ð3ÞrI ¼ CIðe0 þ heimf þ eCI � epIÞ ð4ÞrII ¼ CIIðe0 þ heimf þ eCII � epII � eTÞ ð5Þ

where Cs,M, CI and CII are the stiffness tensors of the ma-trix, austenite and transformed phases, respectively; e0 isequal to C s;M�1

�r with �r the overall stress tensor; Æeæmf isthe mean deformation field added in order to account forthe finite concentration of inclusions; eCI and eCII are theperturbation strains; and epI

and epIIare the plastic strains

in the two families of inclusions. The equivalent inclusiontheory consists in replacing the inclusion by an equivalentinclusion made of the matrix material with a stress-freestrain, the eigenstrain e*, so that the stresses in the realand the equivalent inclusions are equal, leading to

C Iðe0 þ heimf þ eCI � epIÞ ¼ C s;Mðe0 þ heimf

þ eCI � epI � e�IÞ; ð6Þ

C IIðe0 þ heimf þ eCII � epII � eTÞ ¼ C s;Mðe0 þ heimf

þ eCII � epII � eT � e�IIÞ:ð7Þ

The perturbation strains eCI and eCII can be related to thetotal eigenstrains ðe�I þ epIÞ and ðe�II þ epII þ eTÞ, respec-tively, using the Eshelby tensors:

eCI ¼ Ss;MI ðe�

I þ epIÞ ð8ÞeCII ¼ Ss;M

II ðe�II þ epII þ eTÞ: ð9Þ

For an isotropic linear comparison material, the Eshelbytensor depends only on the Poisson ratio mM and on the as-pect ratio of the inclusions sr. For identical inclusion aspectratios one has Ss;M

I ¼ Ss;MII ¼ Ss;M. Some tedious but elemen-

tary algebra leads to the following expressions for thestrains in the phases and in the composite:

eM ¼ As;MðcIIÞe0 þ Bs;MðcIIÞepI þ Ts;MðcIIÞðeT þ epIIÞ ð10ÞeI ¼ As;IðcIIÞe0 þ Bs;IðcIIÞepI þ Ts;IðcIIÞðeT þ epIIÞ ð11ÞeII ¼ As;IIðcIIÞe0 þ Bs;IIðcIIÞepI þ Ts;IIðcIIÞðeT þ epIIÞ ð12Þ�e ¼ cMeM þ cIeI þ cIIeII ð13Þ

where the tensors As,M, As,I, As,II, Bs,M, Bs,I, B s,II, T s,M,T s,I, T s,II and S s,M depend on the unknown C s,M (see fullexpressions in the Appendix and see Ref. [29] for more de-tails). The parameters cM, cI and cII are the volume frac-tions of the matrix, austenite and transformed phases,respectively. The stresses are directly calculated using:


rM ¼ C s;MeM;

rI ¼ C IðeI � eIpÞ;

rII ¼ C IIðeII � eIIp Þ:

ð14Þ

The volume fractions cI and cII evolve following the trans-formation criterion described in the next section. The systemof equations is solved using a Newton–Raphson algorithm.

2.2. Micromechanical criterion for martensitic

transformation

The martensitic transformation takes place via the for-mation of favourably oriented variants within the austenitegrains. The criterion proposed by Fischer and Reisner [15]states that the transformation of a micro-volume Vl of aus-tenite occurs when the driving force becomes larger than aresistive barrier. In its detailed form, the criterion can bewritten as:Z

V l

rijjtseTij dV þ V lDuchemðT cÞ

P V lF c þ DCþ W sp þ W s

el: ð15Þ

� The chemical contribution to the driving force is Vl

Duchem(Tc), which is the change of the Helmholtz freeenergy at temperature T for the transformation of a vol-ume Vl.� The mechanical contribution to the driving force writesR

V lrijjtseT

ij dV where rijjts is the ij component of stress inthe variant at the start of the transformation and eT isthe transformation strain tensor. Due to the correspon-dence between the austenite and martensite lattices,martensite can present 24 distinct orientations withrespect to the parent austenite phase [30]. For each ori-entation, the shape and the components of the transfor-mation strain eT,loc in the coordinate system associatedto the variant can be expressed as:

eT;loc ¼0 0 0

0 0 cs=2

0 cs=2 d

0B@

1CA ð16Þ

where d is a dilatational component resulting from astretching along the normal to the habit plane and cs

is the shear component parallel to the habit plane.Knowing the orientation matrix O1 of the austenitegrain in the material and the orientation matrix O2 ofthe variant, the local transformation strain tensor eT,loc,which is the same in each variant, can be expressed inthe global coordinate system using the relationship:

eT ¼ O1O2eT;locOTransp:

2 OTransp:1 : ð17Þ

� The resistive barrier V lF c þ DCþ W sp þ W s

el comprises:(i) a thermodynamic barrier Fc representing the energyneeded to move the atoms to the new equilibrium latticeposition; (ii) the energy DC needed for the creation ofthe new interface between the parent and the trans-

formed phase, which can generally be neglected; and(iii) the plastic and elastic accommodation work W s

p

and W sel, which represent the work done by the stress

fluctuation s caused by the transformation. As shownby Fischer [15], this term is not only dependent on theselected variant but also on the amount of plastic defor-mation prior to transformation. Recent finite elementunit cell simulations of the transformation of a martens-ite lath within an austenite grain have shown that theaccommodation term is not affected by the stress statewhen considering the same level of effective plastic strain[31].

We have adapted the condition (15) to take into accountthe various hypotheses and approximations discussed inSection 1. As only isothermal analysis is considered in thiswork, criterion (15) will be simply written in the form:

rcijjtseT

ij P Gc; ð18Þ

where Gc is a pseudo-dragging force that incorporates all theunknowns regarding the variation of the accommodationterm W s

p þ W sel and the heterogeneities in the microstructure.

The heterogeneity arising from the crystallographic orien-tation texture of the austenite is fully taken into accountwhen calculating r

cijjtseT

ij. A simple expression will be usedfor Gc:

Gc ¼ Gci þ bepI

e ; ð19Þ

where (i) Gci is the initial value of the transformation bar-rier, which will be assumed to follow a Gaussian distribu-tion characterized by a mean value Gc0 and a variance(VarGc0

), taking account of microstructural heterogeneitiessuch as the grain size or the type of phase surrounding thetransforming grain (bainite or ferrite); and (ii) b accountsfor the strain dependence of the accommodation termsW s

p þ W sel (it was shown in Ref. [31] that the mechanical

accommodation work, especially the plastic contribution,changes with the amount of plastic deformation prior tothe transformation) and/or for an effect of an increase ofthe dislocation density on the transformation barrier.

Considering that common multiphase TRIP steels donot exhibit significant crystallographic texture, an isotropicdistribution of orientations will be assumed for the retainedaustenite. The austenite grain population is discretized into2000 orientations, leading to an initial volume fractioncI/2000 for each member of the distribution (for the sakeof simplicity, a ‘‘member’’ will be called a ‘‘grain’’). A dif-ferent initial value of Gci taken randomly according to aGaussian distribution is attributed to each grain. Thisnumber of orientations, i.e. 2000, was required to generateresults independent of the particular random selection ofstability parameters Gci and to give isotropic responses. Itis supposed that the full transformation of a grain involvesthe successive formation of N variants. The transformationcriterion (18) is checked at every step of deformation ineach grain which has not yet completely transformed and


for each of the 24 possible variant orientations. If the crite-rion is satisfied for at least one variant in at least one grain,the volume fraction of martensite cII is increased (and cI

decreased) by an amount cI/(2000 N) multiplied by thetotal number of transforming variants. The value of Gc isupdated at each strain increment according to Eq. (19).

3. Identification of the parameters

The parameters of the model, provided in Table 1, havebeen identified for two high-silicon TRIP-aided multiphasesteels, denoted TRIP1 and TRIP2. These two steels havethe same composition (0.29 wt.% C, 1.42 wt.% Mn,1.41 wt.% Si) but differ in the heat-treatment they haveundergone [10]. After intercritical annealing, steel TRIP1was held for 6 min at 410 �C, resulting in 29% of bainiteand 17% of retained austenite. Steel TRIP2, which was heldfor 4 h at 310 �C, resulting in 33% of bainite and 11% ofretained austenite, is more stable than steel TRIP1. Thein situ stress–strain curves of the phases were obtainedexperimentally by combining the results of neutron diffrac-tion and image correlation measurements, as described inthe companion paper [10]. The evolution of the austenitevolume fraction as a function of plastic strain in uniaxialtension was obtained using Mossbauer spectroscopy [10].The parameters rr

0, kr and nr in Eq. (1) and the parametersGc0, VarGc0

and b characterizing the resistive force in Eq.(19) were identified such as to get the best fitting of thesetwo sets of experimental results. The average total numberof plates in a transforming grain, N, was chosen equal to 4,as suggested by TEM observations (e.g. [12]). The transfor-mation component d is taken equal to the commonly

Table 1Parameters of the model and the corresponding values for steels TRIP1 and T

Parameter Meaning

EM, mM Young’s modulus (GPa) and Poisson ratio of ferrM

0 Yield stress of the ferritic matrix (MPa)kM Swift hardening parameter of the ferritic matrixnM Hardening exponent of the ferritic matrixEI, mI Young’s modulus (GPa) and Poisson ratio of therI

0 Yield stress of the austenite (MPa)kI Swift hardening parameter of the austenite (MPanI Hardening exponent of the austeniteEII, mII Young’s modulus (GPa) and Poisson ratio of therII

0 Yield stress of the martensite (MPa)kII Swift hardening parameter of the martensite (MPnII Hardening exponent of the martensitecI

0 Initial volume fraction of the austenites Aspect ratio of the austenite graingI Orientation distribution function of the austeniteN Number of variants per transforming graind Dilatational transformation strain componentcs Mean shear transformation strain componentGc0 Mean initial value of the stability parameter (MJVarGc0

Standard deviation of the distribution stability (Mb Slope of the linear variation of Gc as a function

accepted value of 0.04 (e.g. [1,18]). Several values weretested for cs, ranging between 0 and 0.2 depending on theamount of accommodation inside the transforming plateand among the different plates appearing in a transforminginclusion. The shear component cs is thus an average valueassociated with a fully transformed inclusion. The choice ofcs influences the identification of the stability parameters,but once cs is fixed (to a value other than 0) and all otherparameters properly identified, the trends predicted bythe model are not significantly affected by the value of cs.Hence, we will only present results obtained withcs = 0.01 (a small value, which leads to a similar impor-tance for the dilatational and shear effects). The yield stressof martensite is taken equal to 2 GPa (see Ref. [10]) and itsstrain-hardening capacity is characterized by n = 0.05 andk = 800.

Fig. 2 compares the experimental results for steelsTRIP1 and TRIP2 and the predictions of the model usingthe identified parameters listed in Table 1. Figs. 2a and 2cpresent the evolution of the phase stresses (obtained exper-imentally by neutron diffraction), whereas Figs. 2b and 2dshow the macroscopic stress–strain curves. The comparisonof the experimental and predicted transformation rates inuniaxial tension is given in Figs. 4a and 4b.

During the identification procedure, it was found thataccounting only for the austenite orientation dispersion doesnot capture the progressive experimental evolution of therate of transformation and that a distribution of stability(i.e. VarGc0

6¼ 0) is absolutely necessary for that purpose. Inan independent study based on the properties of steel TRIP1[31], three-dimensional finite element simulations of a singlevariant transformation showed that the accommodation

RIP2

TRIP1 TRIP2

rite/bainite matrix 220; 0.3 220; 0.3475 720

(MPa) 55 500.27 0.175

austenite 187; 0.3 187; 0.3720 1130

) 62 800.3 0.2

martensite 187, 0.3 187, 0.32000 2000

a) 800 8000.05 0.050.172 0.1151 1Isotropic Isotropic4 40.04 0.040.01 0.01

m�3) 42 55J m�3) 12.5 10.5

of plastic strain 10 60

Fig. 2. Result of the identification procedure for the flow properties of thephases (a stands for ferrite and c for austenite) in terms of a Swiftrepresentation and prediction of the model for the experimental flow curve(a and b) steel TRIP1; (c and d) steel TRIP2. Note that the results are presentedin terms of the overall composite strain and thus do not follow the Swift fit.

Fig. 3. Experimental and predicted variation of the incremental strain-hardening exponent for steels TRIP1 and TRIP2.


term W sp þ W s

el changes with increasing effective plasticstrain. In the present approach, the variation of the accom-modation term is hidden in the parameter b.

4. Results of the micromechanical modelling

4.1. Assessment of the model

One of the objectives in this work is to analyse how theTRIP effect affects ‘‘ductility’’ (‘‘ductility’’ being definedhere as the capacity to resist plastic localization). The incre-mental strain-hardening exponent is defined as nincr =dlnr/d ln e. The onset of diffuse necking in uniaxial tensionoccurs when nincr(e) = e. Fig. 3 compares the experimentaland predicted variations of nincr for steels TRIP1 andTRIP2. The agreement is very good. It is worth noting thatevaluating a constitutive model in terms of the tangentmodulus (i.e. nincr) provides a much more critical assess-ment than simply comparing overall stress–strain curves.

The validation of the model requires a comparisonbetween experiments and predictions for stress states differ-ent to the uniaxial tension state that was used for the iden-tification. Simple shear, Marciniak and equibiaxial loadingconditions were simulated using the identified set of param-eters listed in Table 1 (see Ref. [10]). Fig. 4 compares theexperimental and computed results for the evolution ofthe austenite volume fraction as a function of the effectivestrain for (a) steel TRIP1 and (b) steel TRIP2. The modelsucceeds in reproducing the following experimental fea-tures: (i) the slowest transformation rate occurs in simpleshear; (ii) the rate is larger for uniaxial tension and evenlarger for the Marciniak tests; and (iii) equibiaxial condi-tions lead to a smaller transformation rate than the Marci-niak tests although the stress triaxiality is larger. Planestrain conditions involve the most effective combinationof deviatoric and hydrostatic stress components, leadingto the largest mechanical driving force (see also Ref.[10]). As shown elsewhere [32], the stress-state dependenceof the TRIP effect in multiphase steels is not only related tothe maximum mechanical driving force but also to the dis-tribution of mechanical driving force over the orientationsof the austenite grains. Uniaxial and equibiaxial loadingconditions imply the same maximum mechanical driving

Fig. 4. Comparison of the experimental and predicted evolutions of theaustenite volume fraction as a function of the equivalent strain for variousloading conditions in (a) steel TRIP1 and (b) steel TRIP2.

Fig. 5. Different transformation kinetics used for the optimizationanalysis based on the properties of steel TRIP2. This ‘‘generic material’’is called MAT2. It is exactly similar to steel TRIP2 whenGc0 = 55 MJ m�3. The same methodology is followed for steel TRIP1.


force, leading to similar rates of transformation in the earlystage of transformation. However, the transformation issubsequently accelerated under equibiaxial conditions dueto a larger mechanical driving force on less favourably ori-ented variants. The main difference between the predictionsand the experimental results is for pure shear, for which thepredicted rate is slower than the experimental one. This isbelieved to be due to the experimental difficulty of ensuringa pure shear stress state at large strains.

4.2. Influence of the austenite stability on the

strength–ductility balance

In order to assess the influence of the TRIP effect on thestrength and on the resistance to plastic localization in uni-axial tension, different kinetics of transformation have beenconsidered by varying the mean value Gc0 in the Gaussiandistribution of Gci (Eq. (19)). The flow properties and vol-ume fractions correspond to either steel TRIP 1 or steelTRIP2. These two sets of hypothetic materials, presentinga range of austenite stabilities, will be designated as‘‘MAT1’’ and ‘‘MAT2’’. Fig. 5 presents, for conditionsMAT2, the variation of the austenite volume fraction asa function of straining for various Gc0, varying from the

case where no transformation occurs to the case of aninstantaneous transformation. For each Gc0 value, theoverall true stress–strain curves were computed and thework-hardening evolutions were determined in order topredict the uniform elongation according to the Considerecriterion. Fig. 6 presents the evolution of (a) the uniformelongation, (b) the maximum stress and (c) the productof maximum stress by the uniform strain as a function ofthe austenite stability.

Fig. 6a shows the variation with Gc0 of the equivalentstrain at necking �eu. For both conditions MAT1 andMAT2, uniform elongation reaches an optimum at anintermediate stability Gc0 � 60 MJ m�3. The optimum aus-tenite stability for MAT1 leads to �eu ¼ 0:295, which ismuch larger than the value �eu ¼ 0:16 reached by MAT2.This difference comes from the fact (i) that ferrite and aus-tenite present a lower strain-hardening capacity in MAT2;and (ii) that the volume fraction of austenite is smaller inMAT2 (see Table 1). As shown in Fig. 6b, the ultimate ten-sile stress �ru decreases with increasing Gc0. Indeed, withincreasing stability, the amount of martensite present atthe onset of necking decreases and, for a stability belowthe optimal value, this effect is not compensated by themoderate increase of �eu in MAT1. The plateau at low sta-bility corresponds to the strength of a dual-phase steel andthe plateau at high stability corresponds to the strength ofa duplex steel.

The evolution of the product �ru�eu as a function of Gc0 isgiven in Fig. 6c. An optimal strength–ductility balance isfound again for Gc0 � 60 MJ m�3. It is noteworthy thatthe expected optimal strength–ductility product in MAT1(�340 MJ m�3) is not far from the experimental value(�325 MJ m�3), which corresponds to Gc0 = 42 MJ m�3

(indeed, the processing conditions of steel TRIP1 weredetermined from an experimental campaign aiming at theoptimization of the properties in uniaxial tension [33].)

For sheet-forming applications, two ductility indicesare usually defined: the diffuse necking strain, �eu, and the

Fig. 6. Evolution of (a) the uniform elongation, (b) the maximum stressand (c) the product of the maximum stress and the true strain at necking asa function of the austenite stability for both MAT1 and MAT2.

Fig. 7. Evolution of the major strain at diffuse necking as a function of theaustenite stability for various states of stress for (a) MAT1 and (b) MAT2.Vertical dotted lines indicate the positions of the maxima.

1 For a J2 material under combined plane stress (r3 = 0) and planestrain (e2 = 0) condition, a = 0.5.


localized necking strain, �eloc. The value of the diffuse neck-ing strain �eu corresponding to the maximum principalstrain direction is the solution of [34,35]:

e1 ¼2nincrðe1Þð1þ qþ q2Þð1þ qÞð2q2 � qþ 2Þ ; ð20Þ

where q = e2/e1 is the strain biaxiality ratio. This relation-ship generalizes the Considere criterion. For negative q,the localized necking strain, �eloc, is the solution of [35]:

e1 ¼nincrðe1Þ1þ q

: ð21Þ

For positive q values, the prediction of localization requiresmore advanced analysis such as the Rice and Rudnickibifurcation condition [36], which goes beyond the scope ofthe present paper. The condition (21) for localized neckingis less conservative than the condition (20) for diffuse neck-ing: for uniaxial tension, �eu ¼ nincr whereas �eloc ¼ 2nincr.Fig. 7 presents the variation of �eu as a function of the stabil-ity Gc0 under various loading conditions for (a) MAT1 and(b) MAT2. Different plane stress biaxiality ratios a = r2/r1

were imposed. Depending on the initial stability, the largestnecking strain is found either under uniaxial tension (a = 0)conditions or under plane strain tension conditions(a = 0.5).1 A more stable austenite is thus advantageousfor postponing diffuse necking in plane strain conditions.One also deduces from Fig. 7a that the optimal stabilityslightly shifts to larger Gc0 values with increasing a. Anotherway to represent the results of Fig. 7 is in terms of the recess-ing part of a forming limit diagram expressed in terms ofbiaxial stress ratio a, as shown in Fig. 8 for MAT1 in which

Fig. 9. Variation of the incremental strain-hardening exponent as afunction of the overall tensile strain (in a uniaxial tensile test) for varyingaustenite stability.

Fig. 8. Recessing part of the predicted forming limit diagram of MAT1expressed in terms of the variation of the major strain at necking (bothdiffuse and localized) as a function of the stress ratio.

2 Nevertheless, the role of austenite crystal orientation cannot beneglected because it affects not only the nucleation moment but also thetransformation strain.


both the diffuse necking strain and the localized neckingstrain are presented. It is shown that, under uniaxial tension(a = 0), a large stability (Gc0 = 90 MJ m�3) is beneficial forthe resistance to localized necking but is detrimental for theresistance to diffuse necking.

5. Discussion

The extra strain hardening provided by the TRIP effecthas two sources: (i) the progressive increase of the volumefraction of a hard second phase – like an evolving dual-phase steel; and (ii) the additional plastic deformationdue to the transformation strain. The volume fraction ofaustenite is thus the first essential parameter governingthe properties. Two other parameters mentioned in the pre-vious section deserve some more discussion: (i) the effect ofthe austenite stability and dispersion of stability; and (ii)the effect of the stress state.

5.1. The relationship between strain hardening and austenite

stability

Four different values of austenite stability can be consid-ered: a low value (40 MJ m�3), an intermediate value(60 MJ m�3) and two high values (80 and 100 MJ m�3).The corresponding incremental work-hardening evolutionsare given in Fig. 9. Essentially the same strain at diffusenecking is obtained with both the lowest and the largeststabilities. For the lowest stability, strain-hardening is highat the beginning and reaches a maximum before a truestrain of 0.2. For the highest stability, the work-hardeningkeeps slowly increasing up to the necking condition. Asshown in the previous section, an intermediate stability(60 MJ m�3) is optimal for resisting diffuse necking underuniaxial tension. However, a higher stability is helpful ifone considers resisting localized necking, as also shownin the FLD of Fig. 8. The optimum state for a given local-ization condition is obtained when the transformationreaches its highest rate in the range of strains that precedes

localization. The same conclusion can be drawn for otherstress states.

The identification procedure has shown that the distri-bution of the crystal orientations of the austenite doesnot enable the experimental rate of transformation to bereproduced. Indeed, the spreading of the transformation,which is essential for postponing localization, mostlyresults from the dispersion of austenite stability (character-ized by VarGc0

). Heterogeneities related to the carbon con-tent, size and the nature of the phase surrounding theaustenite grains thus play a major role on the flow proper-ties of TRIP-assisted multiphase steels.2

Experimentally, the austenite stability can easily betuned by changing the testing temperature. Temperaturechanges of about 50–75 �C around room temperature areusually sufficient to cover the range of stability wheremechanical properties change significantly. Experimentalwork showing an optimal uniform strain at intermediatetemperature can be found in Refs. [12,37,38] and showtrends very similar to those predicted in Fig. 6.

5.2. The relationship between strain hardening, stability and

stress state

As shown in Figs. 7 and 8, the fact that the austenite sta-bility depends on the state of stress markedly influences therate of transformation, and hence the hardening rate andductility. This coupling obviously complicates the designof forming operations, which most often involve non-uni-form loading conditions. The strategy to be followed woulddepend on the intended application. If the TRIP effect isused primarily for improving formability, the austenite sta-bility must be chosen in such a way as to be optimum forthe loading conditions in the most critical location of the


formed part. An example of an additional complexitycomes if bending is considered: the outer region deformedin traction will undergo transformation, while fibres undercompression will not transform. The resulting spring backeffect is then quite difficult to predict. An incremental ver-sion of the present model has been developed in order toaddress these issues (see Refs. [39–41]). The componentafter forming will present large gradients of properties withhard (transformed) and soft (untransformed) zones. Alter-natively, if the TRIP effect is used primarily for improvingthe strain-hardening capacity after forming, e.g. for crash-worthiness, forming should be conducted at moderate tem-perature in order to stabilize austenite during deformationand stability must be optimized with respect to the in-service loading conditions. Another way to control the rateof transformation is to modify the hardness of the ferrite,e.g. by changing the grain size, and thus the load transferto the austenite (e.g. see analysis in Ref. [42] for dual-phasesteels).

6. Conclusions and prospects

A micromechanical model for TRIP-assisted multiphasesteels has been developed, validated and used for a para-metric study, leading to the following conclusions:

� The optimum strength–ductility balance is obtained foran austenite presenting an intermediate stability. Toorapid a transformation of austenite leads to a lack ofhardening, which is detrimental to the strength–ductil-ity balance. Too slow a transformation does not bringthe extra strain hardening required to postponelocalization.� Heterogeneities in the microstructure that lead to effec-

tive variations of the austenite stability are favourablefor spreading the effect of the transformation all alongstraining and for postponing localization. There is thusa possibility for microstructural engineering to attemptto control the sources of heterogeneities.� The optimum properties depend on the applied stress

state through both the hydrostatic and deviatoriccomponents.

This work provides guidelines for the development ofoptimal TRIP-assisted multiphase steels. Local controlof temperature should be considered when optimizingforming operations. It is worth noting that optimizationwith respect to fracture resistance should also be care-fully addressed. During forming operations, TRIP-aidedmultiphase steels are known to suffer from a lack ofcrack resistance at notches or surface defects. Indeed,enhancement of austenite stability is commonly broughtabout by carbon enrichment, leading to a very brittlemartensite. Furthermore, the retained austenite is usuallylocated along grain boundaries, which results in the for-mation of a percolating brittle network after transforma-tion [43].

Acknowledgements

Fruitful discussion with L. Delannay, Y. Brechet, R.Kubler, P. Harlet and T. Iung are gratefully acknowledged.The work of Q.F. was supported by a Ph.D. grant from theFRIA (Belgium). The support of the FNRS for F.L. andfor P.J.J. is acknowledged. This research has been carriedout under the interuniversity attraction poles P5/08 andP6/24, funded by the Belgian Science Policy.

Appendix. Homogenization for a three-phase elastoplastic

composite

This appendix provides the expressions for the dummytensors appearing in Eqs. (10)–(12). All bold uppercasesymbols represent fourth-order tensors while lowercasebold Greek symbols represent second-order tensors. Byinvoking that the stress averaged over the composite (cal-culated from Eqs. (3)–(5)) is equal to the macroscopic stress�r ¼ C s;Me0, and using relationships (6) and (7), one gets:

heimf ¼ ðI � Ss;MÞ½cIðe�I þ epIÞ þ cIIðe�

II þ epII þ eTÞ� ðA:1ÞNow introducing (A.1), (8) and (9) into (6) and (7), resultsin the following systems of 12 equations with 12 unknowns:

Pe�I þQe�

II þ DC sIe0 þ RepI þ XðeT þ epIIÞ ¼ 0 ðA:2ÞUe�

I þ Ve�II þ DCsIIe0 þWepI þ YðeT þ epIIÞ ¼ 0 ðA:3Þ

where

P ¼ DCsIðSs;M � cIðSs;M � IÞÞ þ C s;M ðA:4ÞQ ¼W ¼ �cIIDCsIðSs;M � IÞ ðA:5ÞR ¼ DCsIð1� cIÞðSs;M � IÞ ðA:6ÞX ¼ V ¼ �cIDCsIIðSs;M � IÞ ðA:7ÞU ¼ DCsIIðSs;M � cIIðSs;M � IÞÞ þ Cs;M ðA:8ÞY ¼ DCsIIð1� cIIÞðSs;M � IÞ ðA:9Þ

and

DCsI ¼ CI � Cs;M ðA:10ÞDCsII ¼ CII � Cs;M: ðA:11Þ

Let us define the 12 · 18 matrix Ks:

K s ¼�P Q

U V

� ��1DCsI R X

DC sII W Y

!¼

Ks11 Ks

12 Ks13

Ks21 Ks

22 Ks23

� �:

ðA:12ÞThe eigenstrains can then be expressed in terms of the Ks

matrix:

e�I

e�II

!¼

Ks11 Ks

12 Ks13

Ks21 Ks

22 Ks23

� � e0

epI

eT þ epII

0B@

1CA: ðA:13Þ

Introducing the eigenstrains into (4)–(6), leads to Eqs.(10)–(12) with:


As;M¼ I� cIðSs;M� IÞKs11� cIIðSs;M� IÞKs

21 ðA:14ÞBs;M¼�cIðSs;M� IÞðKs

12þ IÞ� cIIðSs;M� IÞKs22 ðA:15Þ

Ts;M¼�cIðSs;M� IÞKs13� cIIðSs;M� IÞðKs

23þ IÞ ðA:16ÞAs;I¼ IþðSs;M� cIðSs;M� IÞÞKs

11� cIIðSs;M� IÞKs21 ðA:17Þ

Bs;I¼ðSs;M� cIðSs;M� IÞÞðKs12þ IÞ

� cIIðSs;M� IÞKs22 ðA:18Þ

Ts;I¼ðSs;M� cIðSs;M� IÞÞKs13

� cIIðSs;M� IÞðKs23þ IÞ ðA:19Þ

As;II¼ I� cMAs;M� cIAs;I ðA:20Þ

Bs;II¼ I� cMBs;M� cIBs;I ðA:21Þ

Ts;II¼ I� cMC s;M� cICs;I ðA:22Þ

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multiscale mechanics of trip-assisted multiphase steels: ii. micromechanical modelling

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