Effect of Volume Fraction of Constituent Phases on the Stress-Strain Relationship of Dual Phase...

8/22/2019 Effect of Volume Fraction of Constituent Phases on the Stress-Strain Relationship of Dual Phase Steels

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ISIJ International, Vol. 39 (1999), No. 3, pp. 288-294

EffectStra in of VolumeFraction of ConstituentRelationship of Dual PhaseSteels Phases on the Stress-ThomasHOpER.Shigeru ENDO.Nobuyuki ISHIKAWAnd Koichi OSAWAMaterials and Processing Research Center. NKKCorporation, Kokan-Cho.Fukuyama,Hiroshima-ken, 721 -851OJapan,

(Received onAugust 20. 1998. accepted in final form on November18. 1998)

Ferrite-martensite and ferrite-bainite dual phasesteels (DP-steels) were prepared by applying acceleratedcooli ng (AcC) process ona linepi pe steel. Their stress-strain relationshi pswere predicted bymicromechanics,ln the pred ictions, the stress-stra in relationsh ips of the constituent phaseswhosechemistries weredeterminedby microscopic examinations and somethermodynamicdata were used. The effect of volume fraction ofthe constituent phaseson the stress-strain relationships of the DP-steels wasalso examined. According tothe applied model, asimpie stress-strain curve can bedivided into three stages. Asa resu It of this investigation,work hardening takes piace in stage ll and at the beginning of stage lll. Further, in stage ll, the hardeningrate is strongly dependent on the volume fraction of the harder phase, In stage lll, the hardening rate foreach DP-steel is smaller than that in stage ll and is related to the difference in tensiie strength betweentheharder and the softer phases.Furthermore the second investigation by meansof FEManalysis wascarried out in order to evaluate theinfluence of variation of the volume fraction of the harder phaseon the stress-strain behavior of a DP-steel.Tensile tests showedthat by increasing the amountof the harder phase (bainite) in the DP-steel, LOderselongation disappears. According to the results obtained by the FEMalculations, the stress-strain behavioris related to the microstructure, such as volume fraction and shape of the grains in the DP-steel.KEYWORDS:icromechanics; FEM;dual phase steel; work hardening rate; ferrite; bainite; martensite.

1. IntroductionIt has been reported that critical buckling strain (e) of

pipes forced by uni-axial compression is related to workhardening index (n) and pipe dirnension (t/D), and wel]expressed by an equationi) :e=4/3 n0.5 . t/D .......... ..........(1)

Further, recent investigations reveal that the criticalbuckiing strain is governed by the work hardening be-havior of relatively low strain range and Ltiders elonga-tion in stress-strain relations also has influence on thebuckling resistance. Dual phase materials are reportedto showhigher work hardening ability comparedwithsingle phase materials.2) It was the purpose of thisinvestigation to clear the effect of the volume fractionand the properties of the harder phases on stress-strainrelationships of dual phase steels (DP-steels). The finalgoal is to develop materials having high work hardeningability and the resulting buckling resistance by optlmizingtheir microstructure. For this purpose, it must be im-portant to makeprecise predictions of stress-strain re-lationships of dual phase steels.Onthe other hand, to reach agoodagreementbetweenthe calculations and the experiments of flow stress ofthe DP-steels, different approaches have been applied.Micromechanics is said to be a promising methodpre-

dicting the flow stress of the DP-materials.3~5) Thismethod is based on the equal strain model and a satis-factory agreement between experiments and calcula-tions has been found. The model is based on con-siderations of Eshelby's inclusion theory,6) the Mori-Tanakameanfield concept7) and the von Mises typeplastic flow rule. Wenghas also proposed the secantmethod to calculate stress-strain curves of hot-rolledsteels.8) In micromechanics, a simple stress-strain curveof a DP-steel can be divided into three stages. It suggeststhat the effect of the volume fraction and strength of theharder phaseson stress-strain curves of each stage couldbe clear. In this sense, micromechanicscould give usefulinformation to control the stress-strain relationshipsand the resulting work hardening behavior of DP-steels.The stressstrain relationships of ferrite-martensiteand ferrite-bainite DP-steels produced by acceleratedcooling (AcC) process have been examined in terms ofthe hardening behavior and Ltiders elongation in rela-tion to volume fraction. The stress-strain relationshipscalculated by using micromechanics and a two dimen-sional FEMnalysis have beencomparedwith those byexperiments. As Luders elongation of constituent phasesin DP-steels causes somedifficulties in calculation bymicromechanlcs, the FEManalysls was carried out toevaluate the influence of the volume fraction of theharder phases on the stressstrain behavior, such as

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Ltiders elongation, of the DP-steel.2. Introduction of the Prediction Model2.1. Micromechanics2,1

. I . Basic Concept of the ModelEvery DP-steel consists of two single phases or unit-microstructures, which are combined to a mixed mi-crostructure. Each constituent phase undergoes elasticand plastic strain under the mutual constraint of theother constituent during the macroscopic deformationof the DP-steel. Tomota has presented a continuummodel for deformation of a DP-steel.3) In order tocalculate the macroscopic stressstrain curves of DP-steels, each composite should be given in form of a nu-merical equation. Themain stress (a)-strain (8p) equa-tion used in the continuum plastic theory is the follow-ing Swiftis equation:

N= ..........(2)cri =ai ' (bi +ep) ...where i means th phase. i is I for the softer phase andis 2for the harder phase in this paper. Anoutline of thecalculation according to the model is described here-inafter.2, I .2. TheDifferent StagesThe stress-strain relationship of dual phasematerialscan be divided in three stages, which are defined belowand shownin Fig. l.Stage I: The two constituent phases deform elasti-callyStage II: The softer phase deforms plastically, theharder phase continues to deform elasti-

callyStage 111: Both phases deform plastlcallyIt is important to state that the elastic constants ofboth constituents are assumedto be equal. Whenanapplied uni-axial tensile stress cr~3 reaches the yield

strength of the softer phase, (T;, the softer phase startsto deform plastically. Themacroscopic yield stress of thetwo phasealloy is equal to the (T; of the softer phase. Asthe plastic deformation proceeds in the softer phase, thediscontinuity of the plastic strain at the boundary of thetwo phases increases. This leads to the internal stress,which hinders the further plastic flow in the softer phaseand aids the onset of the plastic flow in the harder phasewith a~3 Iess than the yield stress of the harder phaseanAt stage II, a stress-strain curve of a two phasemateri-al can be written as:

(T~3=al[8pl]+.fA8pl ........ ..........(3)where epl, (T1[epl] and ,fare plastic strain of the softerphase, a flow stress curve of the softer phaseandvolumefraction of the harder phase, respectively. Ais a functionof the shape of the harder phase and elastic constantsand equals to E(7- 5v)/{lO(1 -v2)} jn case of sphericalgrains.The stage 11 finishes with onset of the plastic flow ofthe harder phase. By using the following simultaneous

equations, one can determine epl(II) where plastic flowof the harder phase starts.

39 (1999), No. 3

COCOO,O(DSh

8t :Total Straln8p : Plastic Strain Hard phase

stage ll'

stage l

stage lll ,," :

8t2 ' ,' 8t :

Twophaseinternaistress

: Soft phase:8tl

289

ep2 8 8P1True strainFig. l. Stressstrain curves of two phase material, hard andsoft phases showing the 3stages of the model.

cr~3 = (Ti [8pl(II)] +fA8pl(II) ,....,......... (4a)(T~3=a2b~- (1 - f)Aepl(II) ............... (4b)At stage 111, both constituent phases deform plas-

tical]y, so that the stress-strain curve can be calculatedby soiving the following simultaneous equations accord-ing to a given small plastic strain increment of eachphase (~8pi and 6ep2) at step by step.

(T~3= (T1[epl(II)+ 8epl] +.fAA8p .,...,......(5a)(T~3=(T2[88 J-(1-/)AAe ..........(5b)p2

whereA8pmeansthe misfit strain and is written:A8p=(epl(II)+88 -~8p2) """_"(6)

pl2.2. FEMnalysisFEMalculations have been carried out to examinethe influence of the volume fraction of the harder phaseon Ltiders elongation usually occurring in low carbonsteels. According to experimental data achieved by tensiletests, Ltiders elongation in a ferrite-330/0 bainite DP-steelwas not observed. This gives a reason for the assump-tion that the distribution of local strain in the DPmi-crostructure prevents the occurrence of Ltiders elonga-tion. In order to verify the infiuence of the microstruc-ture on this phenomenon,FEMcalculation has beencarried out. Thevolume fraction of bainite in softer fer-rite was gradually increased in steps of 100/0. Threecalculations were conducted with l0-300/0 bainitevolume fractions.Dueto the fact that the stress-strain behavior is relatedto the microstructure of DP-steel, SEMpictures of670/0 ferrite-330/0 bainite DP-steel were taken intoaccount to develop the meshfor the FEMnalysis (seeFigs. 2and 3(b)). Themicrostructure of the ferrite-bain-ite DP-steel is considered to be long stretched harderparticles surrounded by softer ferrite as matrix and lyingparallel to the tensile direction. Although each phase isto be considered to distribute in three dimensions, forsimplification, the problem was treated in a two dimen-sional matter. Each mesh, which stands for a partialregion of the microstructure, consists of 528 elements.It wascomposedof the eight nodes four-sided and the

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ISIJ International. Vol, 39 (1999), No. 3six nodes three-sided element andwasanalyzed in planestrain. The Poisson's ratio was taken as 0.3 and theYoungismodu]usas 206OOOMPa.Thecalculations werecarried out in four different time steps which amount-ed to 65 steps. TheFEManalysis was carried out byusing the ADlNAprogram.3. Experimental Procedure3.1. Microstructure of the DP-steels and the ChemicalComposition of Constituent PhasesIn this paper, stress-strain curves of the DP-steelsobtained by experiment were compared with thosepredicted. Then, the influence of the volume fraction andstrength of constituent phases, such as martensite, bain-

ite and pearlite, on stress-strain relationships of the

(1~. HF

Fig.Ferrite Bainite

2. Elementmeshfor FEMalculation.

HF(~

Fig. 3. SEMmicrostructure of steels tcsted.(a): ferrite and martensite, (b): ferrite and bainite

DP-steels was investigated. In the predictions of thestressstrain relationships of the DP-steels, the stressstrain relationships of the constituent phases, whosechemlstries were determined by volume fraction of eachphase and somethermodynamicdate, were used. Table1shows the chemlcal composition of the used steel. AO.080/0 Ccontaining line pipe steel was used. This steelwas submitted to two different conditions of TMCP(Table 2). In the plate roiling, 1423Kreheat temperature,l 023KCRinishing temperature and 10 and 30K/secof cooling rates were employed. AcCstarting tempera-ture and cooling rate were altered to obtain differentmicrostructures. In the case of DP-steel A, the aimedmicrostructure consists of bainite, whereas DP-steel Bhas ferrite-bainite microstructure. Thebainite is regardedas a DP-steel composedof ferrite and martensite (M-A constituent). Figure 3 showsmicrostructure of theDP-steels obtained. Theportions which look white andblack in Fig. 3(a) are martensite and ferrite, respectively.Ferrite-martensite and ferrite-bainite microstructuresare observed for DP-steel Aand Steel B, respectively.The chemistries of the constituent phases were deter-mined by microscopic examinations and somethermo-dynamic data. In order to determine the volume frac-tions of the harder phases in these DP-steels, samplesof the DP-steels were metallographically prepared andSEMictures were taken. Theareas of each phaseweregraphically markedand measured. Themeasuredarea-fraction of the each phasewas regarded as representa-tive for the volume fraction of the each phase in the ex-aminedDP-steel. By a given average content of carbonof the DP-steel and the determined volume fractions ofthe consisting two single phases, the carbon content ofeach second phase can be calculated by the followingequation:

,f=(x-x.)/(xp-x.) .......... ...,,.....(7)where,f =volumefraction of the harder phasex=average carbon content of the DP-steelx.= carbon content of the softer phasexp=carbon content of the harder phaseMicroscopic examination of the DP-steels showsthatthe volume fractions of the martensite and bainite are12 and 330/0, respectively. The carbon content of theferrite is assumedto be 0.0150/0, which is the carbon

content of ferrite being para-equilibrium to austenite atthe temperature where transformation terminates. ThisTable l. Chemical composition of the examinedDP-steel.

(masso/o)

SteelAB

Table 2. Different heat treatments of the examlnedsteelMicro-structure

FerriLe-MartensiteFerrite-Bainite

ReheatingTemp1423K1423K

Controlled RollingStart Finishl073K1073K

Accelerated Cool in"Start Finish Cooling Ratel023K I023K 573K IOK/sl023K 973K 573K 30K/s

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ISIJ International, Vol, 39 (1999), No. 3Table 3. Chemical composition of the different phases.

(masso/o)1200

CFerrite OO15

Martensite O6Bainite O2

carbon content is calculated by Thermocalc. Transforma-tion termination temperature during accelerated cooling(AcC) which is determined by CCTdiagram is around750K (see Fig. 5). As transformation temperature rangeduring AcCis relatively low, such as less than 973K,para-equilibrium is assumedto take place. Therefore,contents of substitutional elements, such as Si. Mn NbandV, in eachphaseare fixed at the samevalues as thosein the base material. The chemical compositions deter-mined are shownin Table 3.In order to evaluate the development of the micro-structure, the parameters of TMCPere assessed bycomparing with the thermodynamic data and trans-formation behavior after deformation of the steel used.As for the thermodynamic data, such as the To, Mstemperature and the extrapolated para-equilibrium Acmcurve were calculated using the Thermocalc software.TheMstemperature is calculated using the Ghosh'sMstemperature model.9) This model is based on the criticaldriving force for athermal martensitic transformationand can precisely predict multi-component Fe-base al-loys' Mstemperature above 300Kwhere thermal con-tribution to the frictional work can be neglected. Inthis calculation, site fraction of substitutional alloy ele-ments, such as Mn, Si, NbandV is set as the sameasthat of the steel tested. Figure 4 shows those param-eters as a function of carbon content. Figure 5showsthe CCTdiagram of the tested steel. Transformationbehavior after deformation which simulated the con-t rolled rolling of this study wasexamined. It is expect-ed that the martensite (M-A constituent )becomesthedominant second phase, when the AcCfinishing tem-perature is lower than Mstemperature of residual aus-tenite at the transformation terminate temperature.10)Cementite could be observed if the AcCfinishing tem-perature is higher than Ms temperature and less thanthe extrapolated para-equilibrium Acmcurves. As theMstemperature of the martensite whosechemistry wasdetermined by the metallographic examination and listedin Table 3, is around 600K and higher than the AcCfinishing temperature (573 K), the ferrite-martensitemicrostructure can be obtained. In the case of DP-steelB, AcCstarting temperature is less than transformationstarting temperature of ferrite whencooling rate is lessthan 5K/s being equivalent to cooling rate of air cool.Furthermore, as residual austenite was cooled with30K/s, the ferrite-bainite microstructure can be ob-tained.3.2. The AssumedStress-Strain Curves of the Con-stituent PhasesThree different single-unit-microstructures as harderspherica] inclusions, such as martensite, bainite and

~-h~'~5*o,~E1'F

Fig.

1ooo800

600

'Tb"""""'"' """~::;"':?7'(~'~)~'a/(o;+~")~ '\

/ Ms~AcCFinish

4000.001 O.O1 0.1 1.0 10

Carboncontent, C/wt'/.4. Calculated To, Ms and phase boundaries and thecooling finishing temperature employed.11oo

~e 1000FJG,*:, 900~~*o~E 800oH 700

60030k/s IOk/sj

1 10 1OO IOOOCooling time from I073K,tlsecFig. 5. CCTdiagram of the steel tested.

pearlite, in a softer ferrite matrix were examined. Thetensile tests were carried out with a strain rate ofl .5 mm/minat RTfor the ferrite, bainite andmartensitesingle phasesteels whosechemistries were listed in Table3. In the case of pearlite, data based on a pearlite-Swiftequation5) wasapplied and combinedwith ferrite singlemicrostructure and calculated in the samemanner. Ascarbon content of pearlite was assumedas 0.70/0, thevolume fraction of pearlite was determined as 7olo.Tensile tests with 3o/o Pre-strained ferrite samples havebeen carried out in order to avoid Lilders elongation,which makesthe graphical determination of the Swiftequation for the ferrite be impossible. The fitted curveswere approximated by the Swift equation andwere usedto cornpute the macroscopic stress-strain curves of theDP-steels by micromechanics.For the FEMalculation, the stress-strain curves ofthe constituent phases, ferrite and bainite respectively,were also taken from tensile test data. In case of theferrite single phase, a discontinuous stress-strain curvewasused in order to examineits effect on the stress-strainbehavior of the DPmaterials,4. Results and Discussion4.1. Stress-Strain Curves of Constituent PhasesTheobtained stress-strain curves werecomputedintotrue stress-strain curves and these weregraphically fitted.The example is shown in Figs. 6and 7. By fitting the

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ISIJ Internationa[, Vol. 39 (1 999), No. 3

,11,L:~~U,a,5

H

12001ooo800600400200o

rl-r71'1-- rr,Bainite

/Calculation- - Experiment

O 0.02 0.04 0.06 0.08 0.1True strain cStressstrain curves obtained by calculation and ex-periment (bainite).

~~~,,~,nO(1)

h

1ooo800600400200o

Ferrite-1 20/0Martensite

- - ' f!-tj- -

Calculation- - Experiment

Fig. 6. Fig. 8.

O 0.02 0.04 0.06 0.08 0.1True Strain cStress-strain curves obtained by calculation and ex-periment (ferrite120/0 martensite DP-steel).

~~co

h

2000

15001oOo

Martensite__/?::''l

:l,:

Table 4. Comparisonof the results of the experiment andmicromechanics.',

ExperimentMicromechanics

TS(MPa)69l700

Hardening index1-4% 29~o-uElO155 O107O,15 O, I18

500 Calculation- - ExperimentOO 0.01 0.02 0.03 0.04 0.05True Strain e

Stress-strain curves obtained by calculation and ex-periment (martensite).

Table

Fig. 7.

5. Hardening index ,11~ and ,In] of calculated volumefraction.

S[eelFerrite-MartensiteFerrite-BainiteFerritc-Pearlite

Carboncontent, al,~

Volumefraction

06020.7

O.120.33O07

Hardening indexStage l Stage lO130.29O,12

0,12O12O07

curve graphically, satisfactory agreement between theexperimental curves and the fitted ones was achieved.Thestress-strain curves, which were used in the presentcalculation, were assumedas:Ferrite : o,0568(T=675 (0.002 +8p) [MPa]Martensite : a=3560 (O.OOOI+8p)0.21 [MPa]Bainite : a=1470 (0.0005 +8p)o,142 [MPa]Pearlite : cr =1298 (O.002+8p)0.108 [MPa]4.2. Stress-Strain CurvesandWorkHardeningBehaviorof the DP-steels.4.2. I . Stress-Strain Curves of the DP-steelsThe stress-strain curve of the ferrite-120/0 martensiteDP-steel (Steel A) wascalcu]ated by micromechanicsandcomparedwith the experiment (see Fig. 8). Table 4sum-marizes the stress-strain relationships obtained by bothmethods. The tensile strength and uniform elongationwere calculated by using plastic instability condition de-scribed as:

(T-da/ds=0 ........ ..........(8)Thework hardening index is measuredby fitting flow

curves in somestrain ranges with:a=K8" .... ..........(9)

whereK is called strengthening coefficient and n is thework hardening index.

According to Fig. 8and Table 4, sufficient agreementwas found between the calculation and the experiment.Comparingstrain hardening behavior of the experimentand micromechanics for the 120/0 martensite DP-steelfor strain in ranges of I to 4o/o and 2o/o to uniformelongation, sufficient agreementbetween the experimentand the calculations was reached.Thehardening behavior in stage 11 and 111 of DP-steelswlth determined volume fractions of pearlite, martensiteand bainite will be comparedaccording to Table 5. Bycomparing the 3dlfferent DP-steels, the bainite-ferriteDP-steel shows the best performance, concerning thehardening indlces in stages 11 and 111. This maycomefrom the higher volume fraction of bainite comparedwith those of martensite and pearlite used.4.2.2. The Effect of Volume Fraction of the HarderPhaseand the Difference in Strength on Stress-Strain Behavior of DP-stee]s.Themain purpose of this investigation by meansofmicromechanicswas to investigate the influence of thevolume fraction on the hardening behavior and macro-scopic stress-strain behavior of the DP-material in gen-eral. The volume fraction of the harder phases whosestress-strain curves are shownin the previous section,wasvaried and the work hardening indices of stages IIand 111 were calculated. Figure 9explains the effect ofthe volume fraction of each constituent on the hard-ening index of the examined steel. In stage II, the dif-

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1'a',,,1::,Da''eu'

1''~CD=o1:cu,:~'8~

0.5

0.4

0.3

0.20.1

o

VA,

_ :~ ~:, i

Stage lll~~ 1~, -_

-O- Ferrite-Pearlite-lh Ferrite-Martensite--*-- Ferrite-BainiteStage l

tA'-_*:':~~

O5 0.6 0.7 0.8 0.9 1Volumefraction of ferriteThe effects of volume fraction of constltuent phaseson work hardening behavior of the DP-steels.

a=,5COo,o=O

0.00140.00120.001o0.0008o.o0060.00040,0002

o

! Ferrite-pearlite

Ferrite-Martensite i~~\~~"~i---i--]---_"It....-.-- Vi':';'~~~:r-~r, ;:-=---~~~~~~~~~",

I ~ I I 11~~~~~~~~~~Illl?'11'? ' I Ferrite-BainiteFig. 9.

0.30 -1- 50 o/o Ferrite-'-' 60 a/o Ferrite iMartensite ~j- - 70 a/a Ferrite- ~~ ' 80 a/o Ferrite : IJ":~" 90 a/a Ferrite )'l~_ : __._ .:J~::i__ :--Bainite"'~-'- : ';~ : ;f:/JL/earlite : :/ :' ' /'! :: /' '

.__. ___1 '

: /!' : L' 4A;/ :;~//-::~j-~1(;---- *;'rJ'-~ ..r j ..i -'/ ,~ /: _:' ' . _ _A-_._:i_L.L_~~'1;'~~~~: " f~ '

Fig. Il.

,:,,L:~,b

Cl),D:,H

04 05 06 07 08 09 1Volumefraction of ferriteTheeffects of volumefraction of ferrite ononset strainof stage 111.

,D:,,,5

,:G,,UO,C,:a)~,g=~,*O~

0.250.200.150.100.05300 400 500 600 700 800 900 1000

Difference in Tensiie Strength, ATS[MPa]The effects of difference in the strength on workhardening behavior of the DP-steels.

1ooo800600400200o

Fig. 10.

700/0 Fel300/0 Bainite\.1

,; "'..tl

_ '-i~ '~' ~"~j/,,d-1;~ ~'

80o/o Fe-20'/o Bainite90Q/oFe-1 O'/o Bainite

ference in hardening amongthe different DP-steels issmall. The hardening index is increasing with a highgradient by decreasing ferrite volume fraction. In stageIII, the change of hardening index in relation to thevolume fraction is small, oniy for ferrite-martensiteDP-steel it is higher. Figure 10 showsthat the hardeningin stage 111 is related to the difference in tensile strengthbetween the harder and the softer phase. The volumefraction of ferrite wasaltered from 50 to 90 o/, for eachof the DP-steels. The greater the difference in tensilestrength is, the higher the hardening index is, regardlessof the volume fraction of ferrite.Taking into account that the evaluation of the datahas been related to a O.080/* Iow carbon TMCPteel,the volume fraction of the bainite-phase can be easilyvaried. Thus the ferrite-bainite DP-steel seemsto be thepreferable buckling resistant steel.4.2.3. The Effect of Volume Fraction of the HarderPhaseon Onset Strain of DP-steelsFigure ll showsthe developmentof onset strain wherethe harder phase starts to deform plastically as func-tions of volume fraction for the 3different investigatedDP-steels. It can be stated that an increase in volumefraction of harder phase reduces the onset strain.According to the applied calcu]ation model, the onset

293

Fig. 12.

O 0.05 0.1 0.1 5 0.2True Strain e

The effects of volume fraction of bainite on yieldingbehavior of ferrite-bainite DP-steel.

strain is related to elastic limit of the harder phase. Bycomparing the gradients of the graphs of the 3differentDP-steels, Fig. II indicates that the moredifference inelastic limit of harder and softer phases is, the greaterthe influence of the volume fraction on the onset strainis. For the reason pearlite single phaseshowedhe highestvalue in elastic limit, the graph of the ferritepearliteDP-steel has the steepest gradient.Regarding the buckling resistance of pipes, the onsetstrain is in a range below the buckling strain, which isabout I o/o of nominal strain.1) This fact suggests thatthe work hardening behavior of stage 111 could governthe buckling resistance. In this regard the attention forfurther investigation should be directed to the beginningof stage 111.4.3. Influence of the VolumeFraction of the HarderPhaseon the Yielding Behavior of DP-steelAs stated in Sec. 2, the ferrite-bainite DP-steel doesnot showLilders elongation. Onthe other hand, ferritewhich is the main constituent phase of the DP-steels hasLuders elongation in its stress-strain curve. Figure 12showshowthe increase in volume fraction of the harderphase (bainite) reduces the extension of Luders elon-gation of the DP-steel. Ltiders elongation is hardly seenin stress-strain curve of the 30 ~/* bainite containing steel.As shownin Fig. 13, the value of the maximumocal

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c:

Q'c5oo~vE~

0.60.50.40.30.20.1

oFig. 13.

,:,vCO,11oO~:IE

li------":---~--

;,'!-------/:~}~~~~ /

30 o/o Bainite ' ' ./~~2,20 o/Q ~ainite/~.. 7 -...;--.-l:i l' r~ .lj~'f"'~"7.~7,~~~~"""~j 10 o/o Bainite: r ':/ :.*___1"rFd._ ______~,7~;O 0.02 0.04 0.06 0.08 0.1 0.12

Nominal strainThemaximumocal strain of the softer phases asrunctions of the nominal strain.

0.070,060,050,040.030,020.01o

Fig. 14.

-f-"" ""'f""" ""'f""'~/i/ :l:~

j---\i-10 a/o Bainite I '~! -~1' /

--'-------:- - --F;' ------;eO o/a Bainite'-

-----_ ,

-:\--------i-------i /.f~-1~~~ldll:

I_//'_'_'1

~~bj/ i"'-'20 e/o Bainite';.

O 002 004 006 008 O1 012Nominal strainThemaximumocal stra in of the harder phases asfunctions of the nominal strain.

strain in the softer phase is increasing, while the nominalstrain is rising. Bycomparlng the DP-steel with differentvolume fraction of bainite, the maximumocal strainincreases with rising nominal strain andshowsthe highestvalue for 300/0 bainite. It becameclear by the FEManalysis, that the peaks of local strain in the softer phaseare located at about 45' of the half circle of the harderphase partial meshat the grain boundary between theharder and softer phases. This corresponds to the resultsof former investigations on this field.1 1) Figure 14 showsthe maxlrnum local strain as a function of the nominal

strain for the harder phase. By rising the nominal strain,the maximumocal strain increases, but is significantlylower than the maximumocal strain of the softer phase.Increase in local strain difference betweenthe harder andsofter phases leads to an increase in local stress, eventhough nominal strain is small. The increase in localstress canbeconsidered as apossible reason for reductionIn Lilders e]ongation.5. ConclusionStress-strain relationships of different DP-steels havebeen calculated by using the present method of calcu-lation andexaminedin terms of hardening index in rela-tion to volume fraction of the harder phase. It can bestated that:(1) The method of micromechanics is a usefulinstrument to predict stress-strain behavior of Dualphase steels.(2) It becameobvious that in stage 11 the volume

fraction of the harder phase is the predominant factorfor the hardening index. In stage 111 the hardening indexis strongly influenced by the difference in tensile strengthbetween the harder and the softer phases.

(3) The onset strain where the harder phase beginsto deform plastically, is smaller than the buckling strain.

(4) The conducted FEManalysis gives proof thatthe volume fraction of the harder phase and the dis-tribution of the local strain are linked to the reductionof Ltiders elongation.REFERENCES

l) B, Katoh: Nihon Ke'lchiku Gaklcai Ronbunhokokusyu,No. 204(1972), 9.2) S. EndoandMNagae: ISIJln/.,36 (1996), 95.3) Y.Tomota, K. Kuroki.T. Mori and I.Tamura: Mate,'. Sci. Eng.,24 (1976), 85.4) K. Tanakaand T. Mori: Ac!a Mc'tal!., 18 (1970), 941,5) Y. Tomota: ISIJ In!., 32 (1992), 343.6) J. D. Eshelby: P,'oc. R. Soc. (L0,7d0,1). A241 (1957), 376,7) T. Mori and K. Tanaka: Acta Meta!!., 21 (1973), 571,8) G. J. Weng:J. Mech. Phys. Solids, 38 (1990), 419.9) G. Ghoshand G. B. Olson: Acta Meta!!.,42 (1994), 3361.lO) S. Endo and M, Nagae: 35th Mechanical Working and SteelProcessing, (1 990). 453.ll) N. Shikanai: ISIJlnt.,32 (1992), 335.

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Effect of Volume Fraction of Constituent Phases on the Stress-Strain Relationship of Dual Phase...

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