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

Effect

Stra in

of Volume Fraction of Constituent

Relationship of Dual PhaseSteels

Phases on the Stress-

ThomasHOpER.Shigeru ENDO.Nobuyuki ISHIKAWAand Koichi OSAWAMaterials and Processing Research Center. NKKCorporation, Kokan-Cho. Fukuyama,Hiroshima-ken, 721 -851 OJapan,

(Received on August 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 on a linepi pe steel. Their stress-strain relationshi ps were predicted by micromechanics,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 phases on 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 for

each DP-steel is smaller than that in stage ll and is related to the difference in tensiie strength between theharder 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 phase on 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 FEMcalculations, the stress-strain behavioris related to the microstructure, such as volume fraction and shape of the grains in the DP-steel.

KEYWORDS:micromechanics; FEM;dual phase steel; work hardening rate; ferrite; bainite; martensite.

1. Introduction

It has been reported that critical buckling strain (e) ofpipes 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 critical

buckiing 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 the

buckling resistance. Dual phase materials are reportedto showhigher work hardening ability comparedwithsingle phase materials.2) It was the purpose of this

investigation to clear the effect of the volume fraction

and the properties of the harder phases on stress-strain

relationships of dual phase steels (DP-steels). The final

goal 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 a goodagreementbetweenthe 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-Tanaka meanfield 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, micromechanics could give useful

information to control the stress-strain relationships

and the resulting work hardening behavior of DP-steels.

The stressstrain relationships of ferrite-martensite

and 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 relationships

calculated by using micromechanics and a two dimen-sional FEManalysis have been comparedwith 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 the

harder phases on the stressstrain behavior, such as

@1999 ISIJ 288

ISIJ Internationai, Vol.

Ltiders elongation, of the DP-steel.

2. Introduction of the Prediction Model

2.1. Micromechanics

2,1. 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 elastic

and plastic strain under the mutual constraint of the

other 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. The main 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 meansith phase. i is I for the softer phase andis 2for the harder phase in this paper. Anoutline of the

calculation according to the model is described here-inafter.

2, I .2. The Different Stages

The stress-strain relationship of dual phase materials

can be divided in three stages, which are defined belowand shownin Fig. l.

Stage I: The two constituent phases deform elasti-

cally

Stage II: The softer phase deforms plastically, the

harder phase continues to deform elasti-

cally

Stage 111: Both phases deform plastlcally

It 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 starts

to deform plastically. Themacroscopic yield stress of the

two phase alloy is equal to the (T; of the softer phase. Asthe plastic deformation proceeds in the softer phase, the

discontinuity of the plastic strain at the boundary of the

two 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 phasean

At 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 softer

phase, a flow stress curve of the softer phaseand volumefraction of the harder phase, respectively. A is a functionof the shape of the harder phase and elastic constantsand equals to E(7- 5v)/{lO(1 -v2)} jn case of spherical

grains.

The stage 11 finishes with onset of the plastic flow ofthe harder phase. By using the following simultaneousequations, one can determine epl(II) where plastic flowof the harder phase starts.

39 (1999), No. 3

COCO

O,O

(D

Sh

8t :Total Straln

8p : Plastic StrainHard phase

stage ll '

stage l

stage lll,,"

:

8t2' ,'

8t :

Twophase

internaistress

: Soft phase

:8tl

289

ep2 8 8P P1

True strain

Fig. 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 calculated

by 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

where A8pmeansthe misfit strain and is written:

A8p=(epl(II)+88 -~8p2) """_"(6)pl

2.2. FEMAnalysis

FEMcalculations have been carried out to examinethe influence of the volume fraction of the harder phase

on Ltiders elongation usually occurring in low carbonsteels. According to experimental data achieved by tensile

tests, Ltiders elongation in a ferrite-330/0 bainite DP-steel

was 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 bainite

volume fractions.

Dueto the fact that the stress-strain behavior is related

to the microstructure of DP-steel, SEMpictures of670/0 ferrite-330/0 bainite DP-steel were taken into

account to develop the meshfor the FEManalysis (see

Figs. 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 lying

parallel to the tensile direction. Although each phase is

to be considered to distribute in three dimensions, for

simplification, the problem was treated in a two dimen-sional matter. Each mesh, which stands for a partial

region of the microstructure, consists of 528 elements.It was composedof the eight nodes four-sided and the

O1999 ISIJ

ISIJ International. Vol, 39 (1999), No. 3

six nodes three-sided element and wasanalyzed in planestrain. The Poisson's ratio was taken as 0.3 and theYoungis modu]usas 206OOOMPa.Thecalculations werecarried out in four different time steps which amount-ed to 65 steps. The FEManalysis was carried out byusing the ADlNAprogram.

3. Experimental Procedure

3.1. Microstructure of the DP-steels and the ChemicalComposition of Constituent Phases

In 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. Element meshfor FEMcalculation.

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 steel

was submitted to two different conditions of TMCP(Table 2). In the plate roiling, 1423Kreheat temperature,l 023K CRfinishing temperature and 10 and 30K/secof cooling rates were employed. AcCstarting tempera-ture and cooling rate were altered to obtain different

microstructures. 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 shows microstructure of theDP-steels obtained. The portions which look white andblack in Fig. 3(a) are martensite and ferrite, respectively.

Ferrite-martensite and ferrite-bainite microstructures

are 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 andSEMpictures were taken. Theareas of each phase weregraphically markedand measured. The measuredarea-fraction of the each phase was regarded as representa-tive for the volume fraction of the each phase in the ex-amined DP-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 phase

x=average carbon content of the DP-steel

x.= carbon content of the softer phasexp =carbon content of the harder phase

Microscopic 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 carboncontent of ferrite being para-equilibrium to austenite atthe temperature where transformation terminates. This

Table l. Chemical composition of the examined DP-steel.

(masso/o)

Steel

AB

Table 2. Different heat treatments of the examlnedsteel

Micro-

structure

FerriLe-Martensite

Ferrite-Bainite

Reheating

Temp1423K1423K

Controlled Rolling

Start Finish

l073 K1073K

Accelerated Cool in"

Start Finish Cooling Rate

l023K I023K 573K IOK/s

l023K 973K 573K 30K/s

C 1999 ISIJ 290

ISIJ International, Vol, 39 (1999), No. 3

Table 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 each phaseare 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 TMCPwere 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's Mstemperature model.9) This model is based on the critical

driving force for athermal martensitic transformationand can precisely predict multi-component Fe-base al-

loys' Mstemperature above 300K where thermal con-tribution to the frictional work can be neglected. Inthis calculation, site fraction of substitutional alloy ele-

ments, such as Mn, Si, Nband V is set as the sameasthat of the steel tested. Figure 4 shows those param-eters as a function of carbon content. Figure 5 showsthe CCTdiagram of the tested steel. Transformationbehavior after deformation which simulated the con-trolled rolling of this study wasexamined. It is expect-ed that the martensite (M-A constituent )becomesthe

dominant 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 the

Mstemperature of the martensite whosechemistry wasdetermined by the metallographic examination and listed

in Table 3, is around 600K and higher than the AcCfinishing temperature (573 K), the ferrite-martensite

microstructure can be obtained. In the case of DP-steelB, AcCstarting temperature is less than transformationstarting temperature of ferrite whencooling rate is less

than 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 Phases

Three different single-unit-microstructures as harderspherica] inclusions, such as martensite, bainite and

~-

h~:'

~c5

*o,~E1'

F

Fig.

1ooo

800

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,tlsec

Fig. 5. CCTdiagram of the steel tested.

pearlite, in a softer ferrite matrix were examined. Thetensile tests were carried out with a strain rate of

l .5 mm/minat RTfor the ferrite, bainite and martensitesingle phase steels 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 and were usedto cornpute the macroscopic stress-strain curves of theDP-steels by micromechanics.

For the FEMcalculation, 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-strain

behavior of the DPmaterials,

4. Results and Discussion

4.1. Stress-Strain Curves of Constituent Phases

Theobtained stress-strain curves were computedinto

true stress-strain curves and these weregraphically fitted.

The example is shown in Figs. 6and 7. By fitting the

291 C 1999 ISIJ

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

,11

,L:~

~U,a,5

H

1200

1ooo

800

600

400

200

o

rl-r71'1-- rr,Bainite

/

Calculation

- - Experiment

O 0.02 0.04 0.06 0.08 0.1

True strain c

Stressstrain curves obtained by calculation and ex-periment (bainite).

~~~,,~

,nO

(1)

h

1ooo

800

600

400

200

o

Ferrite-1 20/0Martensite

- - 'f!-tj-

-

Calculation

- - Experiment

Fig. 6. Fig. 8.

O 0.02 0.04 0.06 0.08 0.1

True Strain c

Stress-strain curves obtained by calculation and ex-periment (ferrite120/0 martensite DP-steel).

~~

co

h

2000

1500

1oOo

Martensite

__/?::''l

:l,:

Table 4. Comparisonof the results of the experiment andmicromechanics.

',

Experiment

Micromechanics

TS(MPa)

69l700

Hardening index

1-4% 29~o-uEl

O155 O107

O, 145 O, I18

500Calculation

- - Experiment

OO 0.01 0.02 0.03 0.04 0.05

True 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[eel

Ferrite-Martensite

Ferrite-Bainite

Ferritc-Pearlite

Carbon

content, al,~

Volumefraction

06020.7

O. 120.33

O07

Hardening index

Stage ll Stage lll

O13

0.29

O, 12

0,12

O12

O07

curve graphically, satisfactory agreement between theexperimental curves and the fitted ones was achieved.

The stress-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 CurvesandWorkHardening Behaviorof the DP-steels.

4.2. I .Stress-Strain Curves of the DP-steels

The 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 elongation

were 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)

where K is called strengthening coefficient and n is the

work 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-ferrite

DP-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.

The main purpose of this investigation by meansofmicromechanics was to investigate the influence of the

volume 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 shown in the previous section,

was varied and the work hardening indices of stages II

and 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-

C 1999 ISIJ 292

ISIJ International, Vol. 39 (1999), No. 3

1'a'

,,,

1::

,Da''e

u'

1'

'~

CD=o1:cu,:~'

8~

0.5

0.4

0.3

0.2

0.1

o

V

A,

_:~

~:, •i

Stage lll

~~ 1~, -_

-O- Ferrite-Pearlite

-lh Ferrite-Martensite--*-- Ferrite-Bainite

Stage ll

tA'-_*:':~~

O5 0.6 0.7 0.8 0.9 1Volumefraction of ferrite

The effects of volume fraction of constltuent phases

on work hardening behavior of the DP-steels.

a=,5

CO

o,o=O

0.001 4

0.0012

0.001 o

0.0008

o.o006

0.0004

0,0002

o

! Ferrite-pearlite

Ferrite-Martensite i~

~\~~"~i•••••••---i--]--

-_"It....-.-- Vi

':';'~~~:r-~r, ;:-=---~~~~~~~~~"

,

I ~ I I11~~~~~~~~~~

Illl?'11'? ' IFerrite-Bainite

Fig. 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/pearlite : :/ :l'

' /'! :' : /' '

.__._

__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 ferrite

Theeffects of volumefraction of ferrite on onset strain

of stage 111.

,D:,,,5

,:

G,

,U

O,C,:a)

~,g

=~,*O

~

0.25

0.20

0.15

0.10

0.05300 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.

1ooo

800

600

400

200

o

Fig. 10.

700/0 Fel300/0 Bainite

\.1

,;"'.. tl

_

'-i ~'~' ~"~j

/,,,

d-1;~ ~'

80o/o Fe-20'/o Bainite

90Q/o Fe-1 O'/o Bainite

ference in hardening amongthe different DP-steels is

small. The hardening index is increasing with a highgradient by decreasing ferrite volume fraction. In stageIII, the change of hardening index in relation to the

volume fraction is small, oniy for ferrite-martensite

DP-steel it is higher. Figure 10 showsthat the hardeningin stage 111 is related to the difference in tensile strength

between 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 tensile

strength is, the higher the hardening index is, regardless

of the volume fraction of ferrite.

Taking into account that the evaluation of the data

has been related to a O.080/* Iow carbon TMCPsteel,

the volume fraction of the bainite-phase can be easily

varied. Thus the ferrite-bainite DP-steel seemsto be the

preferable buckling resistant steel.

4.2.3. The Effect of Volume Fraction of the HarderPhaseon Onset Strain of DP-steels

Figure ll showsthe developmentof onset strain wherethe harder phase starts to deform plastically as func-

tions of volume fraction for the 3different investigated

DP-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 0.2

True Strain e

The effects of volume fraction of bainite on yielding

behavior of ferrite-bainite DP-steel.

strain is related to elastic limit of the harder phase. Bycomparing the gradients of the graphs of the 3different

DP-steels, Fig. I I indicates that the more difference in

elastic limit of harder and softer phases is, the greaterthe influence of the volume fraction on the onset strain

is. For the reason pearlite single phaseshowedthe highest

value in elastic limit, the graph of the ferritepearlite

DP-steel has the steepest gradient.

Regarding the buckling resistance of pipes, the onsetstrain is in a range below the buckling strain, which is

about I o/o of nominal strain.1) This fact suggests that

the work hardening behavior of stage 111 could governthe buckling resistance. In this regard the attention for

further investigation should be directed to the beginningof stage 111.

4.3. Influence of the VolumeFraction of the HarderPhaseon the Yielding Behavior of DP-steel

As stated in Sec. 2, the ferrite-bainite DP-steel does

not showLilders elongation. Onthe other hand, ferrite

which is the main constituent phase of the DP-steels has

Luders elongation in its stress-strain curve. Figure 12

showshowthe increase in volume fraction of the harder

phase (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 maximumlocal

C 1999 Is[J

ISIJ International. Vol. 39 (1999). No. 3

c:

Q'

c5

oo~,vE~

0.6

0.5

0.4

0.3

0.2

0.1

o

Fig. 13.

,:

,v

CO,11

oO~,:I

E

li

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

;,'

!-------/:~}~~~~ /30 o/o Bainite '

' ./~~2,20 o/Q ~ainite: /

~-.. 7 -...;--.-

l:i l' r~ .lj

~'f"'~"7.~7,~~~~"""~j10 o/o Bainite

: r ':/ :_.*___1"rFd._ ______~

,7~;

O 0.02 0.04 0.06 0.08 0.1 0.12

Nominal strain

The maximumlocal strain of the softer phases asrunctions of the nominal strain.

0.07

0,06

0,05

0,04

0.03

0,02

0.01

o

Fig. 14.

-f-"" ""'f""" ""'f""'

~//i/

:l:

~j---\i-

10 a/o Bainite I'

~! -~71

' /--'-------:- - --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 strain

The maximumlocal strain of the harder phases asfunctions of the nominal strain.

strain in the softer phase is increasing, while the nominalstrain is rising. By comparlng the DP-steel with different

volume fraction of bainite, the maximumlocal strain

increases with rising nominal strain andshowsthe highestvalue for 300/0 bainite. It becameclear by the FEManalysis, that the peaks of local strain in the softer phase

are located at about 45' of the half circle of the harderphase partial meshat the grain boundary between the

harder and softer phases. This corresponds to the results

of former investigations on this field.1 1) Figure 14 showsthe maxlrnumlocal strain as a function of the nominal

strain for the harder phase. By rising the nominal strain,the maximumlocal strain increases, but is significantly

lower than the maximumlocal 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 local

stress can be considered as apossible reason for reductionIn Lilders e]ongation.

5. Conclusion

Stress-strain relationships of different DP-steels havebeen calculated by using the present method of calcu-lation and examinedin 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 volumefraction 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.

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C 1999 ISIJ 294