strain hardening behavior of dual-phase steelsengconfintl.org/pastconf/11avmettransexample.pdf ·...

Strain Hardening Behavior of Dual-Phase Steels

V. COLLA, M. DE SANCTIS, A. DIMATTEO, G. LOVICU, A. SOLINA,and R. VALENTINI

A detailed qualitative and quantitative examination of the microstructure and mechanicalproperties of three di!erent classes of DP600 and DP450 dual-phase (DP) steels was carried out.The tested DP steels are characterized by di!erent alloying elements: aluminum, boron, andphosphorus. Among them, aluminum DP steels showed the lowest percentages of hard phases,while phosphorus DP steels exhibited the highest resistance values. The Hollomon, Pickering,Crussard–Jaoul (CJ), and Bergstrom models were used to reproduce the strain hardeningbehavior of DP steels. Relationships that correlate the fitting parameters with the chemicalcomposition and the thermal cycle parameters were found, and the predictive abilities of dif-ferent models were evaluated. The Pickering equation, among the tested models, is the best onein the reproduction of the experimental stress-strain data.

DOI: 10.1007/s11661-009-9975-1! The Minerals, Metals & Materials Society and ASM International 2009

I. INTRODUCTION

CAR-REDUCED weight and elevated safety param-eters are the modern design criteria for car manufac-tures, being essential to fulfill customer’s expectations,legal requirements, and standards. Innovative steelgrades with increased formability, high strength level,and high strain hardening index have been developednot only for weight reduction purposes, but also toimprove vehicles’ crash safety.[1] Advanced high strengthsteels, especially dual-phase (DP), transformationinduced plasticity (TRIP), and twinning induced plas-ticity (TWIP) steels, exhibit promising results in terms ofsuperior mechanical properties, also considering thattheir extraordinary mechanical properties can be tai-lored and adjusted by a correct choice of alloyingelements and processing routes.

Dual-phase steels constitute a family of high strengthstrip grades, with a characteristic multiphase structure,consisting of hard second-phase islands (usually mar-tensite at around 20 pct) dispersed in a ferrite matrix.Main characteristics of DP steels are continuous yield-ing behavior, lower yield/tensile strength ratios, higherwork hardening rates at low strain, and higher levels ofuniform and total elongation[2] than HSLA steels withsimilar yield strength.

Also, TRIP steels consist of a ferrite matrix with auniform dispersion of hard second phases (martensite orbainite). These steels also contain retained austenite involume fractions greater than 5 pct, which progressively

transforms to martensite during plastic deformation,thus increasing the work hardening rate at higher strainlevels.Finally, TWIP steels are austenitic alloys in which

mechanical twinning is the prominent deformationmode. These steels exhibit exceptional combinations ofductility, work hardening, and ultimate strength. Thestrain-hardening rate of TWIP steel is lower than that ofDP and TRIP steels, but their local hardening capacityis superior, resulting in increased uniform elongation. Inparticular, the excellent capacity of local hardening ofTWIP steel is mainly related to its single-phase struc-ture, compared with DP and TRIP steel.[3]

The present work is focused on better characterizingstrain hardening behavior of conventional DP steels. Inorder to produce the DP microstructure, cold rolling offerrite/pearlite microstructure, followed by intercriticalannealing in the (c+ a) region between A1 and A3 andfinal quenching at su"ciently high cooling rate, isusually imposed. The cooling path is crucial to allowthe transformation of austenite to martensite, takinginto account that other low temperature transformationproducts and retained austenite might also form.[2,4]

There have been numerous attempts to describe thestress strain or strain hardening behavior of DP steelsand to correlate the strain hardening ability withmicrostructure.[4–7] The usual approach to describe thestress-strain curves and strain hardening behavior ofmetals is the use of analytical expressions, because thisprocess allows the plastic part of the curve to be treatedby certain parameters that can be applied to the study offormability and deformation mechanisms. This is par-ticularly important in the automotive industry, whereeasy formability information is very useful for the designof panels and associated dies.[8]

The strain hardening behavior of DP steel is com-monly analyzed using equations based on variousconstitutive equations. The most common methodsare the Hollomon analysis, the Crussard–Jaoul (CJ)

V. COLLA, Technical Research Manager, and A. DIMATTEO,Young Researcher, are with the Scuola Superiore di Studi Universitarie di Perfezionamento Sant’Anna – Pisa, Italy. Contact e-mail:[email protected] M. DE SANCTIS and A. SOLINA, AssociateProfessors, G. LOVICU, Research Assistant, and R. VALENTINI,Assistant Professor, are with the Dipartimento di Ingegneria Chimica,Chimica Industriale e Scienza dei materiali, Universita di Pisa – Pisa,Italy.

Manuscript submitted March 18, 2008.

METALLURGICAL AND MATERIALS TRANSACTIONS A

analysis, and the modified CJ analysis, all based on theHollomon, the Ludwik, and the Swift equations. Theseanalytical methods indicated that in most cases thestress-strain curves of DP steels cannot be described by asimple parabolic function over a uniform strain rangeand, moreover, that DP steel deform primarily with twoor three distinct stages.[9,10]

Finally, a model that considers the dislocation gen-eration during plastic deformation has been developedby Bergstrom.

The aforementioned stress-strain relationships aresummarized in Table I. The aim of the present articleis to compare the fitting ability of these strain hardeningmodels and to find formulas to correlate the fittingoutput parameters to the chemical composition and theprocess parameters of DP 450 and DP 600 steels.

II. MATERIALS AND METHODS

The present study was performed considering threedi!erent classes of DP600 and DP450 DP steels char-acterized by di!erent alloying elements: boron, alumi-num and phosphorus. The chemical compositions ofalloys are reported in Table II.

The specimens for the tensile tests were obtained by aGleeble 3800 thermomechanical simulator. The thermaltreatment consisted of a typical galvannealing treat-ment. Specimens were heated within the intercriticalrange at temperatures of 780 "C, 800 "C, and 840 "Cusing a heating rate of 3 "C/s. They were held at thesetemperatures for 60 seconds before cooling. The coolingrates ranged between 20 "C/s and 30 "C/s depending onthe specimen thickness.

Samples were cut along the rolling plane and thenmounted, mechanically ground, and polished. Then theywere etched using Le Pera’s solution for 30 seconds. Themicrostructure of samples was examined using opticalmicroscopy (OM) and scanning electron microscopy(SEM) techniques. The volume fractions and relativedimension of hard phases (martensite and bainite)

were measured using image analyzer software, ImagePro-Plus.* True stress–true strain curves were obtained

using a Galdabini Sun** 1000 electronic tester.

Once experimental data on the plastic behavior ofsteels were obtained, the plastic regions were fitted usingdi!erent reported models (Hollomon, Pickering, CJ, andBergstrom). The obtained fitting parameters were cor-related to the chemical composition of steels (C, Mn, Si,B, Al, and P) and to process parameters (annealingtemperature, line speed, and steel sheet thickness) bylinear regressions. The calculated parameters are used topredict tensile curves.In the present work, due to the fact that only a small

amount of data were available, parameters in each setare calculated by a linear combination of chemicalelements and process parameters, where the multiplyingcoe"cients are chosen by means of linear regression.The base formula for parameter prediction is a linear

combination of chemical elements and of functions ofprocess variables that have the best correlation:

Xj ! x0j "X

i

aij#Ei "X

i

bij#f$Pi% &1'

where Xj is the tested model parameter; Ei is the weightpercentage of the chemical element analyzed; f(Pi) is the‘‘simple’’ function of process parameters; and x0j, aij,and bij are the optimized coe"cients.The root-mean-square of residual percentage (rmsrp)

was chosen as a performance index for both fittingcurves and curves obtained by regression formulas. Thermsrp was calculated as follows:

Table I. Strain Hardening Models

Model Stress-Strain Expression Equation References

Hollomon r ! ken 1 5, 11, 12Pickering r ! A" Be" Clne 2 6, 13Crussard–Jaoul r ! r0 " Ken 3 14 through 17Bergstrom r ! r0 " k#

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1( exp$(0:5#e%

p4 18 through 23

Table II. Chemical Composition of DP Steels

Steel C (Wt Pct) Mn (Wt Pct) Al (Wt Pct) P (Wt Pct) B (ppm)

DP450B 0.04 to 0.07 1.1 to 1.4 0.03 to 0.05 0.01 to 0.03 >20DP600B 0.10 to 0.13 1.4 to 1.7 0.03 to 0.05 0.01 to 0.03 >20DP450P 0.10 to 0.13 1.1 to 1.4 0.03 to 0.05 0.06 to 0.1 >20DP600P 0.13 to 0.16 1.4 to 1.7 0.03 to 0.05 0.06 to 0.1 >20DP450Al 0.04 to 0.07 1.1 to 1.4 0.8 to 1.0 0.01 to 0.03 —DP600Al 0.10 to 0.13 1.4 to 1.7 0.8 to 1.0 0.01 to 0.03 —

*Image Pro-Plus is a trademark of Media Cybernetics, Bethesda,MD.

**Galdabini Sun is a trademark of Galdabini, Varese, Italy.


rmsrpfit !

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Prex

rfit(rexrex

) 100" #2

s

N&2'

rmsrpregr !

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Prex

rregr(rexrex

) 100" #2

s

N&3'

where rex is the experimentally measured true stress; rfitand rregr are the true stresses calculated by fit and byregression formulas, respectively; and N is the number

of experimental points of each true stress–true straincurve. The average value of rmsrp for the whole DP dataset and for each DP class was calculated in order tocompare the predictive ability of every single model.

III. RESULTS AND DISCUSSION

A. Microstructural and Compositional Features andTheir Relation with Main Tensile Properties

For the sake of brevity, results from the quantitativemetallography and SEM analysis of materials will be

Fig. 1—Optical micrographs of a sample of each DP tested class intercritically annealed at 800 "C and cooled at 26 "C/s.


shown and reported only for DP steels annealed at800 "C and cooled at a rate of 26 "C/s.

Figures 1 and 2 show steel microstructures asobserved using OM and SEM. After etching with theLe Pera reagent, the ferrite usually appears, in opticalmicroscopy, gray or light brown, while martensite phaseappears white and bainite dark brown (Figure 1).

In many cases, OM was not powerful enough toclearly distinguish martensite from bainite islands. Thisis because SEM imaging was also used, since at highermagnification obtained by SEM, it was possible tounivocally identify the typical lamellar structure ofbainite (Figure 2).

Table II summarized the results from quantitativemetallography on DP steels annealed at 800 "C cooledat a rate of 26 "C/s. As expected, the volume fraction ofhard phases increases with increasing steel grades,ranging from about 10 to 15 pct for DP 450 to 20 to25 pct for DP 600.Figure 3 shows engineering stress-strain curves from

DP450 and DP600 specimens annealed at 800 "C cooledat a rate of 26 "C/s. It can be observed that DP600P andDP450P steels exhibit the highest values of tensile andyield strengths, possibly because of the solid solutionstrengthening e!ect of phosphorus[24] additions. It isalso apparent that phosphorus-rich steels do not exhibit

Fig. 2—SEM micrographs of a sample of each DP tested class intercritically annealed at 800 "C and cooled at 26 "C/s.


continuous yielding behavior. The addition of alumi-num, because of its ability to reduce the existence field ofaustenite,[25] results in lower percentages of hard phases.In the tested steels, an addition of about 0.9 wt pct ofaluminum also reduced the amount of austenite formedat di!erent intercritical temperatures, as shown inFigure 4. Consequently, the carbon content of theintercritical austenite increased, thus improving steelhardenability, as evident from the comparison of micro-graphs of DP450Al reported in Figure 1. Table IIIindicates the lower volume fractions of the second phaseof DP450Al in respect to other steels. As opposed toDP450B and DP450P, no bainite is visible in themicrostructure of DP450Al. This is because at the sameintercritical temperature a lower quantity of austenite isformed, which, consequently, will result in more con-centration of carbon, thus increasing significantly thehardenability during quenching.

Furthermore, for all considered chemistries, themartensite islands exhibit an irregular shape even

though they are roughly equiaxed. The size of themartensite island ranged from approximately 1 to 4 lm,and they are inhomogeneously distributed. The islandsare predominately formed along ferrite grain bound-aries, as shown in Figure 2.It was observed that the size, morphology, and

distribution of martensite islands were dependent onboth the intercritical temperature and the adoptedheating rate.[26] In many cases, it was not possible toestablish the dimensions and morphology of martensiteislands with a su"cient precision, because of theirreduced size and edge e!ects, which made the observa-tion of island interiors di"cult. The annealing temper-ature did not have strong influence on average islanddiameters, although a slight increase in dimension forincreasing annealing temperatures was observed. On thecontrary, the postannealing cooling rate had a pro-nounced e!ect, since slower cooling rates inducedsignificant reduction of island diameters. Obviously,increasing the steel grades increases the dimensions ofmartensite islands.

B. Strain Hardening Modeling

1. Hollomon analysisThe Hollomon parameters were determined by

fitting the experimental true stress–true strain datawith Eq. [1] in Table I. In Figure 5, the comparisonbetween the experimental data for the plastic region

Fig. 3—Stress-strain curves for DP450 and DP600 DP steel intercritically annealed at 800 "C and cooled at 26 "C/s.

Fig. 4—Austenite volume fraction as a function of temperature fortwo DP 450 DP steels with di!erent aluminum contents. Equilibriumcalculation.

Table III. Volume Fraction of Hard Phasesand Their Dimensions (Intercritically Annealed

at 800 !C and Cooled at 26 !C/s)

SteelPct HardPhases

Ferrite GrainSize (lm)

Size of MartensiteIslands (lm)

DP450B 14 ± 3 7.9 ± 2.5 2.1 ± 0.5DP600B 22 ± 4 7.5 ± 2.5 3.1 ± 0.7DP450P 15 ± 4 6.9 ± 1.5 1.6 ± 0.4DP600P 24 ± 5 5.8 ± 1.0 1.7 ± 0.5DP450Al 9 ± 3 7.2 ± 2.0 1.7 ± 0.4DP600Al 20 ± 4 7.0 ± 2.0 1.8 ± 0.6


and both the fit curve and that obtained by regressionanalysis for some DP steels are reported. Regressioncurves are also shown. They are shifted from theinterpolating curves in order to clearly identify andcompare both of them.

The averages of the parameter values found for all thesamples analyzed in each class are reported in Figure 6.

As expected, it is evident that both K and n vary with thesteel grade, being higher for DP600 steels.

2. Pickering analysisThe Pickering model is usually indicated as the ‘‘STS

model’’ (STS represents strength, tilt, and shape). Theregion of the plastic tensile curve is simply considered to

Fig. 5—Examples of the fitting ability of the Hollomon model. The regression curve was shifted in order to see both the interpolating curves.

Fig. 6—Average of fit parameter values calculated for each DP class for all the tested models.


result from three main terms: the constant A (roughlythe yield strength), a logarithmic term (C) that roughlydescribes the shape of the curve, and a linear term (B) toconsider the variation from a simple logarithmic behav-ior (Eq. [2] in Table I).

In this work, this model was proposed to describe thestress-strain curve of DP steels, even though no specificrelationship was used to take into account the e!ects ofmicrostructural features. In order to increase its fittingability, the Pickering equation (Eq. [2] in Table I) wasmodified as follows:

r ! A" B$e( e0% " C ln$e( e0% &4'

where e0 is the yield point strain.Relevant parameters were determined by fitting the

equations to the experimental true stress–true straindata. Figure 7 shows the comparison between theexperimental data of the plastic region and both thefitting curve and that obtained by regression analysis forsome DP steels. Regression curves were shifted in orderto observe more clearly both interpolating curves. Theaverage values of parameters for di!erent samples arereported in Figure 6, where it can be observed that A, B,and C values vary according to the steel composition.

From the present work, the prediction ability of thePickering model is quite surprising, also considering itssimplicity and the absence of any relationship tomicrostructural features of steels. However, the predic-tion ability of the curve obtained by regression analysisis less e!ective than that of the fitting curve, as evident inthe example reported in Figure 7. This is possibly due tothe rather high number of parameters used in thePickering model. Each model parameter was analyzedseparately to find a linear correlation with chemical andprocess parameters. Thus, the more parameters there arethe more errors are introduced drawing the curve.Although the Hollomon analysis resulted in less e!ectivefitting experimental data, the presence of only twoparameters allowed the fit and regression curves to becloser to each other than Pickering’s curves.

3. CJ analysisThe CJ model is an extension of the Hollomon model,

where the use of an additional parameter (r0 in Eq. [3])takes into account the yield strength of steel, thusallowing better modeling of the plastic region.It was suggested that the CJ analysis exhibits several

advantages with respect to other analytical methods,being also more suitable for analyzing the strainhardening behavior of DP steel.[9] Moreover, both mand K can be easily obtained from the experimentalln(dr/de) vs ln e curves.Steels with a microstructure composed of two phases

very di!erent in strength, such as ferrite and martensitein DP steels, are expected to deform into two distinctstages.[15] It has been suggested that the two-stagedeformation behavior of DP steels is related primarily tothe development of dislocation structures in the ferritematrix and the deformation state of martensite (elasticor plastic). In the first low-strain stage, only ferritedeforms plastically, while martensite remains elastic; inthe second high-strain stage, both ferrite and martensitedeform plastically. Therefore, within a certain straininterval around the transition strain et, the deformationstate of martensite changes from elastic to plastic.[9]

The consequence for CJ analysis would be thepresence of two linear regions in the ln(dr/de) vs ln ecurves, with slopes in the first (low strain) stage largerthan those in the second (high strain) stage, as shown,for example, by Jiang et al.[15]

The DP steels analyzed in the present work do notexhibit these characteristic two-stage curves. Figure 8shows representative ln(dr/de) vs ln e curves of DP450Aland DP600B steels, and no linearity or marked slopechange can be observed. Taking into account thisexperimental evidence, using the standard CJ model,some errors are introduced, i.e., the poorness of predic-tion ability of curves and the arbitrary choice of thepoint where the slope changes.Under these circumstances, a parabolic expression

was chosen to approximate the ln(dr/de) vs ln e curve inthe CJ equation:

Fig. 7—Examples of the fitting ability of the Pickering model. The regression curve was shifted in order to see both the interpolating curves.


lndrde

$ %! A" B ln e" C ln e$ %2 &5'

By the way, the possibility to easily calculate m and kparameters can be lost, but, as shown in Figure 9, theexperimental data result is much better interpolated.

The absence of regions with di!erent slopes in theplastic region is probably due to the reduced dimensionsof martensite islands and to the presence of third phaseswith intermediate strength. Jiang et al.[15] used CJanalysis for several DP steels that showed a well-definedtwo-stage deformation behavior. In their study, ferriteand martensite islands had comparable dimensions, thusmaking the martensite deformation an essential part ofthe entire deformation mechanism. In the present study,the dimension of martensite islands is much smaller thanthat for ferrite grains (for instance, Figure 1 and thedata in Table III). Moreover, the presence of bainite inalmost all DP steels introduced the deformation of athird phase with an associated dislocation structure,which contributed to lose evidence for an abrupt changeof slopes. Figure 10 shows the 600 B DP steel ln(dr/de)

vs ln r compared with the results from Jiang et al.[15]

The A6 DP steel of Reference 15 was chosen for thecomparison, because its martensite phase percentage

Fig. 8—Representative experimental ln(dr/de) vs ln e curves for DP450 and DP600 DP steels.

Fig. 9—Examples of the fitting ability of the Crussard–Jaoul model. The regression curve was shifted in order to see both the interpolatingcurves.

Fig. 10—600 B DP steel ln(dr/de) vs ln r compared with thatpublished by Jiang et al.[15]


and dimension of ferrite grain size are very similar tothose of the 600 B DP steel. The two steels exhibit amarkedly di!erent behavior (Figure 10).

Moreover, as reported by other authors,[27] it could betoo simplistic to believe that at a particular value of theplastic strain all martensite islands would start to deformplastically. Because of the inhomogeneities in martensitedistribution and the heterogeneous deformation of theferrite matrix, which may exist during some stage ofdeformation of the composite, martensite particles can-not be assumed to deform plastically all the same time.An acceptable model may be that at some stage ofdeformation the martensite particles with appropriateorientations or located in regions where the local stress islarger begin to deform plastically and the others stay intheir elastic state. As the deformation of the compositecontinues, the rest of the martensite particles will deformgradually from elastically to plastically. As in the presentwork, the true stress–true strain curves reported by Lianet al.[27] do not show a two-slope behavior, but a smoothchange of the curve slope with true stress increasing, asthat of the present work.

By integrating expression [5], the following formula ofr vs e was found:

r( r0 !eA"

B(1$ %24C

!!!p

perf (1"C"2C log e

2!!!C

p" #

2!!!!C

p

2

64

3

75

e

e0

&6'

From an analysis of the influence of the three param-eters, A, B, and C, on the resulting stress-strain curve, itis evident that all have an influence on strength.Increasing all parameters, the stress-strain curve rises,exploiting a higher tensile strength.

The comparison between the experimental data of theplastic region and both the fit curve and that obtainedby regression analysis for some DP steels reported inFigure 8. Regression curves were shifted in order to seeboth interpolating curves.

The averages of the parameter values found for all thesamples analyzed in each class are reported in Figure 6.

From Figure 6, it is evident that A and B parametersvary with the steel grade: B is higher for DP600 steelsthan DP450, and A shows the opposite behavior.The values of the parameters A, B, and C are strongly

a!ected by the presence of phosphorous.

4. Bergstrom modelIn this work, to improve the fitting ability of the

Bergstrom model, it was modified, adding two moreparameters, according to the following formula:

r ! r0 " A# 1( exp$(B#ep%& 'n &7'

where ep is the plastic strain; r0 is the strengthparameter; and a, b, and n are material constants. Theaverages of the parameter values found for all thesamples analyzed in each class are reported in Figure 6.The comparison between the experimental data of the

plastic region and both the fit curve and that obtainedby regression analysis for some DP steels are reported inFigure 11. Regression curves were shifted in order to seeboth interpolating curves.From Figure 6, it is evident that a and b parameters

vary with the steel grade, being higher for DP600 steels.

Fig. 11—Examples of the fitting ability of the Bergstrom model. The regression curve was shifted in order to see both the interpolating curves.

Fig. 12—Fitting and predictive ability of all the tested models.


The parameter n is influenced both by the steel gradeand the main alloying element.

C. Models Comparison

The average rsmrp was calculated for the fit curvesand for the curves obtained by regression formulas.Figure 12 shows the average rsmrp for all of the testedDP steels.

The fitted curves are in very good agreement with theexperimental ones, the rmsrp always being less than 2 pct.Also, the curves drawn using the parameters obtained by

regression formulas are in good agreement with theexperimental data, rmsrp always being less than 3.5 pct.From the analysis of the average rmsrp calculated for

each model, reported in Figure 13, it is evident that thePickering model is the ablest to reproduce the experi-mental tensile curves for DP steels.The Bergstrom model is the worst one. This model is

usually used to reproduce the tensile curves of single-phase materials. The Bergstrom formula used in thisresearch does not take into account the ferrite-martens-ite structure of DP steels, and in particular the defor-mation gradient present between the soft and the hard

Fig. 13—Fitting and predictive ability of all the tested models for each DP class.


phases when stress is applied. The CJ model reproduceswell the experimental data, rmsrp always being less than4.5 pct. The complexity of the CJ model that uses fiveparameters fit well with the experimental data, butpenalized the obtained regression curves. On the con-trary, the simplicity of the Hollomon model, with onlytwo parameters, makes the curves obtained by regres-sion formulas very close to the fitting ones.

The analysis shows that there is not a marked e!ect ofthe steel grade and of the main alloying elements on thepredictive ability of the di!erent methods.

IV. CONCLUSIONS

On the basis of the analysis of the true stress–truestrain curves using the main strain hardening modelspresent in the literature (Hollomon, Pickering, Crus-sard–Jaoul, and Bergstrom) that was carried out andpresented in this article, the following conclusions canbe drawn.

1. For all the tested DP steels, no evident transitionbetween the strain hardening behavior at low strainsdue to the plastic ferrite deformation and that athigh strains due to plastic martensite deformationwas revealed. This evident remark let the authorsprefer a parabolic expression of the ln(dr/de) vs ln efor the Crussard–Jaoul analysis.

2. The fitting curves well reproduce the experimentalones for all the analyzed models. Among them, thePickering model shows the best fitting ability, whilethe Bergstrom model shows the worst fitting ability.The Bergstrom model lack of fitting ability is prob-ably due to the fact that it was originally thoughtof for single-phase materials.

3. Expressions were found to calculate all the testedmodel parameters by the chemical composition andthermal cycle parameters. Also, the curves drawnusing parameters calculated by these formulas wellreproduce the experimental ones, although, obvi-ously, they are less accurate than the fitting curves.

4. Among the tested models, although the Hollomonfit is not very good, its simplicity let the predictedcurves be very close to the fitting ones.

5. Among the tested models, the Pickering analysisexhibited the best fit and prediction ability.

6. No influence of steel grades and main alloyingelements was revealed on the reproductive andpredictive ability of all models.

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