International Deep Drawing Research GroupIDDRG 2009 International Conference

1-3 June 2009, Golden, CO, USA

NON-PROPORTIONAL HARDENING OF DUAL PHASE STEELS ANDITS CONSTITUTIVE REPRESENTATION

L. Sun I, J.H. Kim 2, and R.H. Wagoner2

1 Department of Mechanical EngineeringThe Ohio State University,

20 I W. 19th AvenueColumbus, OH 43210, USA

email: sun.206@osu.edu

2 Department of Materials Science and EngineeringThe Ohio State University,

2041 N. College RoadColumbus, OH 43210, USA

email: kim.2502({v,osu.edu, wagoner.2@osu.edu

ABSTRACTThe elastic-plastic response of sheet materials during non-proportional paths are seldom

incorporated in constitutive equations used for routine sheet forming simulation, but can have asignificant effect on formability and springback. Monotonic tension and compression, coaxialtension-compression (T-C), coaxial compression-tension (C-T), and two-stage/non-coaxialtensile tests have been perfonned for three grades of dual phase steels: DP590, DP780, andDP980. The reverse or second-stage flow curves have three characteristics: reduced yield stress(Bauschinger effect), rapid transient strain hardening over a few percent strain, and long-term or"permanent" softening. The departure of reverse hardening curves from monotonic ones is largerthan with other typical sheet forming alloys, presumably because of the large second-phasemartensite particles in dual-phase steels.

A Modified constitutive model based on the Chaboche approach (M-C) was developed. Inaddition to one or more standard nonlinear components of the back stress, a linear term wasadded to represent the "permanent" offset of hardening following a stress reversal. Theparameters for the model were fit using the monotonic and reverse tensile test results, and themodel predictions were then compared with large-strain balanced biaxial bulge test results andwith non-coaxial, two-stage tensile tests. All of the effects are captured with reasonable, but notperfect, accuracy.

Draw-bend simulations based on the modified Chaboche (M-C) model reveal thatincorporating non-proportional hardening effects reduces draw-in loads, formability, andspringback as compared with standard isotropic hardening models.

Keywords: Dual-phase Steels; Tension/Compression; Chaboche Model; Non-proportionalLoading; Draw-bend test; Springback; Constitutive Equations; Bauschinger Effect

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L. Sun, J.I-I. Kim, and R.H. Wagoner

1. INTRODUCTION

Rapid developments of advanced materials are creating major opportunities for improvingsociety, such as conserving energy, reducing environmental impact and increasing theperfonnance of transportation vehicles in recent decades. Dual-phase (DP) steels haveoutstanding combinations of strength and ductility. Fundamental technical questions related tofonnability and springback need to be addressed for the efficient use ofDP steels

"Springback" is the elastically-driven change of shapc of a part after a material has becnformed into a useful shape and released from the forming forces or displacement constrains ofthe tooling. The inability to accurately predict springback is a major cost factor in industry. Topredict the formability and springback precisely, the elastic-plastic response of DP steels duringnon-proportional loading paths needs to be incorporated in constitutive equations and thus inapplied simulations [1-5].

In a sheet metal forming process, most material points undergo complex loading modes, forexample, strain path changes. In order to investigate the work hardening sensitivity to strain path,Schmitt et al. [6] introduced the parameter 0 as follows:

where Dj and O2 represent the rate of the deformation tensor during the prestrain andsubsequent loadings. Monotonic, reverse and orthogonal strain paths correspond to 0 = I,-I and0, respectively. Combinations of simple loading paths such as tension, simple shear, torsion andbiaxial tensions have been used to determine the mechanical behavior of sheet metals forcomplex strain/stress paths [7-9]. Special experimentally-observed effects during complexloading, such as hardening stagnation and cross-hardening, have be attributed to the influence ofa developing dislocation microstructure [I, 10-14].

i permane~ softening

'" Transient hardening

", C-T test

DP590.1.4mm

o --, - ._-~--,-o 0.05 0.1 0.15 0.2 0.25

Accumulated Absolute True Strain

....~u;~ 400~ Bauschlnger effect.• /"'5~ ~ AI Tension ••

.c 200<{

••c..~ 600

Figure I. Monotonic and C-T test experimental curves of DP590 that illustrate the three characteristicregions o/reverse hardening: Bauschinger e./lect, rapid transient strain hardening and "permanent ..

.w?ftening.

In present paper, monotonic tension and compression, coaxial tension-compression (T-C),coaxial compression-tension (C-T), and two-stage/non-coaxial tensile tests were performed for

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L. Sun, J.H. Kim, and R.H. Wagoner

three grades of dual phase steels: DP590, DP780, and DP980. Fig.1 illustrates the threecharacteristics in the C-T tests of DP590: (a) Bauschinger effect (early re-yielding after loadreverse) (b) rapid transient strain hardening, and (c) long-term or "permanent" softening. Torepresent these three characteristics, a modified constitutive equation based on the Chabochemodel [15] was developed. A linear term is added to standard nonlinear components of the backstress in order to account for the "permanent" softening effects. The modified Chaboche modelwas compared with two-stage/non-coaxial tensile tests and balanced biaxial bulge test results.Draw-bend simulations based on the modified Chaboche model revealed the importance of non-proportional hardening effects in formability and springback.

2. MATERIALS

DP steels consist of a relatively soft ferrite matrix and a hard martensitic phase. The chemicalcompositions of three grades of Dr steels provided by various suppliers: DP590, DP780, andDP980, are listed in Table I. It is noted that Nb, Ti, Y, and B are all less than 0.003 in thesematerials.

Table I: Chemical composition * olDP steels in weight percent

C Mn P S Si Cr AI Ni Mo Thickness(mm)

DP590 0.08 0.85 0.009 0.007 0.28 0.01 0.02 0.01 <.01 1.4DP780 0.12 2.0 0.020 0.003 0.04 0.25 0.04 <.01 0.17 1.4DP980 0.10 2.2 0.008 0.002 0.05 0.24 0.04 0.02 0.35 1.45

* Chemical composition was analyzed utilizing Baird OneSpark Optical Emission Spectrometer at theGMNA Materials Laboratory following ASTM E415-99a.

3. EXPERIMENTAL PROCEDURE

3.1 Revcrsc path tcsts: C- T and T-C tcsts

An apparatus designed for T-C and C-T tests [16] was installed in a standard tensile testingmachine to perform large-strain reversals. An exaggerated dogbone specimen is subjected tomechanical clamping system using a compressed air cylinder operating on two flat side plates[17]. Teflon adhered to the surface of side plates reduced friction, which was corrected for alongwith the slight biaxial loading. The air cylinder maintained a constant side force of 2.23 or 3.35kN corresponding to side stresses of 0.83 or 1.25 MPa at air pressures (40 or 60psi). AnElectronic Instrument Research LE-05 laser extensometer was used to measure the displacementbetween two fixed points initially 25mm apart on the specimen. The nominal strain rate10-3 s -I was controlled during the reverse path tests.

3.2 Non-coaxial tensilc tests

The large tensile specimens shown in Fig. 2 were used to prestrain the materials along therolling direction (RD) at the nominal strain rate 10-3 S-I . The grips, Fig. 3, were designed forshort plane-strain specimens as described in the literature [18, 19]. ASTM subsize tensilespecimens depicted in Fig. 4 wcre machined along RD, 30°, 45°, 60° and transverse direction(TD) following pre-straining ..

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L. Sun, 1.B. Kim, and R.B. Wagoner

415.1

o

oo RD

TD 100 oo

o

Figure 2. Geomelly of the large tensile specimen (mm)

Figure 3. Grip and serrated gripping wedges/or large tensile specimen

---- JO()---~

----,J J2 0---1_/_1---------1

G,O 'O,Ci I

----------,--iCC C

Figure 4. Geomeliy of the ASTMsllbsize specimen (mm)

3.3 Balanced biaxial bulge tests

Balanced biaxial bulge tests can often achieve a much larger unifonn strain than uniaxialtensile tests. Such tests were performed in cooperation with Alcoa material test laboratory [20].The opening diameter is l50mm and the die profile radius is 25.4mm. The balanced biaxial datawere converted to a tensile effective stress - effective strain bases based on the '79 Hill's yieldcriterion using measured r values (plastic anisotropy ratio) and with a single best-fit m value(yield surface exponent) .

4. RESULTS AND DISCUSSION

4.1 Reverse path tests

Monotonic tensile tests and C-T tests data are compared in Fig 5 for DP590, DP780 andDP980. The three characteristics mentioned earlier are apparent: (a) Bauschinger effect (early re-yielding after load reverse) (b) rapid transient strain hardening, and (c) permanent softening .

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L. Sun, J.I-I. Kim, and R.I-!. Wagoner

The simplest quantification of reverse tests is in tenns of the Bauschinger effect. The scalarparameter fJ [21] represents the normalized difference between the forward and reverse stress, asfollows

lalimvard 1-lareVl'mlfJ = lalim",n/l (2)

where alimvard and arever"" are the flow stresses before and after the reverse loading, respectively.The determination of reverse flow stress depends on the choice of the yield offset. In Fig. 6, twokinds of offset definition, 0.2% yield offset and 0.4% yield offset, were used with no obvioussignificant difference. fJ = 0 when the material shows no Bauschinger effect, whereas fJ = Iwhen the material yields at zero stress. It shows that the Bauschinger effect increases in the orderof DP590, DP780 and DP980, tending to confinn the concept that higher volume of fractions ofhard inclusions increases the departure from isotropic hardening.

The pattern of the C-T test data is revealed in Fig. 7 by shifting the subsequent hardeningdata to a strain origin at the reversal. The stress-strain curves subsequent to the reversal arenearly identical and independent of the prestrain. Furthennore, the subsequent C-T and T-Cstress-strain curves are identical in Fig. 8, as would be expected from classical pressure-independent plasticity theory.

4.2 Modified Chaboche model

In order to incorporate the observed reverse hardening behavior, a constitutive equationbased on a modified Chaboche model was developed. In this M-C model, the kinematicalhardening evolution of backstress a is composed of two parts, a nonlinear tenn al and a linear

tenn a2, as follows

2du, =3C,d~" -ru,dp (3)

2dU2=-C2d~"

3where ~" and p are the plastic strain and equivalent plastic strain, respectively. In the M-C

model, Eq.3, the nonlinear term corresponds to the Bauschinger effect and transient hardening,and the linear term corresponds to the pennanent softening.

The isotropic hardening was taken as an exponential function as follows:R = R,(I-exp(-bp)) (4)

The unifonn elongation domains from one monotonic tension and two stress-strain curvessubsequent to the reversal in C-T tests with 0.04 and 0.08 (0.1 for DP590) prestrain (Fig. 5) wereused to find optimal model coefficients using the least square method. The best-fit values ofcoefficients in the M-C model are summarized in Table 2. It is noted that yield stress Yo also isdetennined by curve fit method so there is a little deviation from the value defined by traditional0.2% yield offset.

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L. Sun, J.H. Kim, and R.I-!. Wagoner

1000 -,- ---r

--O~590-1.4m:-l

1000

'" Monotonic n;-o.. 800 tension 0.. 800~ ' .. ~.... • •• £.£ •.•..•..•. ..~ ..

600 ~u; Iii 600

" ":Jt:-

:Jt:-

" 400 " 400:; :;~ '0.0 Experiment

..q: .0

200q: 200--M-CModel

0 -- --l.___ ~ ---,--_._- 00 0.05 0.1 0.15 0.2 0.25 0

Accumulated Absolute True Strain

OP780.1.4mm

.•. Experi ment

--M-C Model

0.05 0.1 0.15 0.2Accumulated Absolute True Strain

0.25

(a) DP 5901200 -- ,----,----,----,-

Monotonic tension

(b) DP780

.....)000'"0..~~ 800~Iii~ 600t:-"~ 400~.0q:

200

OP980-1.45mm

.•. Experiment

--M-C Model

o - -_. - - '-o 0.05 0.1 0.15 0.2 0.25

Accumulated Absolute True Strain

(c) DP980

Figure 5. Monotonic tensile tests Vs. C- T tests/or three grades 0/DP steels

--.-- --- -.-DP980

~

0.06 0.08

Yield offset: 0.4%

0.1

••

0.08

Yield offset: 0.2%

•

0.06

.~-<I

IDP590

0.04

DP980

"'"DP780 .--'-..._- --

o0.02

0.2

0.4

0.6

0.8

0.1

•- - .i

DP590

.-0.04

0.8

0.6

C!:l..

0.4

0.2

00.02

Prestrain Prestrain

(a) (b)

Figure 6. Parameter fJ = (la-iimI'tJrl/I_Ia-mme I) / Ia-iiwurd I to represent Sal/sehinger effect (a) 0.4% yield

(?[f~et(b) 0.2% yield C?[f~et

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L. Sun, J.H. Kim, and R.H. Wagoner

Tension after Compression800

Monotonic tension

•• 700a.~'" CoT test, T part'"e 600 (0.04 C)en<l>::J>:=<l> 500:;0 C.T test, T part'" (0.10 C).c<{

400DP590.1.4mm

300 ~--,- -' ___ 1_-

0 0.05 0.1 0.15 0.2 0.25Reverse True Strain

Figure 7. Monotonic tension and C-T tests experimental data (?/ierreversal/or DP590

800

700••a.~ 600

'"'"e 500en<l> 400::J>:=.s 300::J0'".c 2004;

100

00

--,-

"-T-C test ( 0.06 T)

DP590-1.4mm

0.05 0.1 0.15 0.2 0.25Accumulated True Strain

Figure 8. T-C Vs. C-T tests experimental data/or DP590

Table 2: Variables in M-C model

Yo C] C2 r R, b StandardDeviation

DP590 312 13501 296 85 246 12 12.28DP780 454 17062 517 72 163 16 15.67DP980 510 31955 828 89 152 17 11.33

The M-C model implemented using the UMA T user subroutine in the commercial finiteelement code ABAQUS Standard 6.7 [22]. Fig. 5 shows that M-C model fits monotonic and C-Ttests well.

4.3 Non-axial tensile tests

In Fig 9, three non-coaxial tensile tests are compared with M-C model predictions using thecoefficients in Table 2 as obtained from fits to independent reverse tensile tests. The labels "XX-YY" denote first the prestrain direction followed by the subsequent strain direction. For the RD-RD (coaxial) case, the reloading curve follows approximately the unloading one and thesubsequent monotonic curve. The subsequent hardening behavior shows greater departure from

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L. Sun, J.B. Kim, and R.B. Wagoner

the monotonic curve as the angle between the prestrain and subsequent axes increases. The M-Cmodel captures much of this mechanical behavior, but underestimates the departure frommonotonic curves for larger angles between the two tensile axes. It is nonetheless clear that theM-C model represents non-coaxial test behavior much better than an isotropic hardening model,which would only reproduce the monotonic curve.

Fig. 10 presents subsequent non-coaxial stress-strain curves using the style of Fig. 7. Thisrepresentation illustrates that the departure from monotonic hardening increases systematically asthe angle between the axes increases. Furthermore, all of the subsequent curves exhibit acommon "permanent" offset at larger subsequent strains.

1200

• ~RD-45°

.•...•...RD-TD

Monotonictension (RD)"~......'.'.'

"".",..• RD-RD...

•• 1000a.~'"'"eii5 800 ..,::l~

600DP980-1.45mm(0.07 prestrain)

400o 0.05 0.1 0.15 0.2

Accumulated True Strain

Figure 9. Non-coaxial tensile tests/or DP980

1200Experimental Data-,-

_1000

'"a.~'"'"e 800ii5.,::l~

600 DP980-1.45mm(0.07 prestrain)

400 - - , ----~o 0.03 0.06 0.09 0.12

Subsequent True Strain

Figure 10. Compare the strain hardening behaviorFom d!llerent second-stage strain path

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L. Sun, 1.B. Kim, and R.B. Wagoner

1400 --,- --,- -,-

_1200 ..•Q.

!.~ 1000i!!iii•.~ 800

~w600

• Tensile Expt

Bulge Expt-M-<:Model

400 - -"--- _.1.......-....._1 - -._-~

o 0.1 0.2 0.3 0.4 0.5 0.6Effective Strain

Figure II. Balanced bulge tests experimental data Vs. M-C model

4.4 Balanced bulge tests

Fig. II compares balanced biaxial effective stress-strain curves with predicted M-C modelones bascd on tensile-range data only. The standard deviations of the predictions andmeasurements over the entire strain ranges shown are 8.06, 15.33, and 11.25, respectively, forDP590, DP 780, and DP 590.

5. SIMULATION OF DRAW-BEND TEST

Preliminary simulations of a few draw-bend tests, including the formability and springback,were carried out using a three-dimensional solid model (ABAQUS element C3D8R) with 5layers through the sheet thickness. The schematic of draw-bend test is shown in Fig 12 [23]. Thespecimen is 71Omm length and 25.4 mm width. A standard isotropic hardening model and theM-C model were used.

For formability simulations, the back end was fixed and the front end was drawn at the rateof 51mmls until failure. For springback, the back force was set to 60% and 80% of the 0.2%offset yield stress and the specimen was drawn to a distance 127mm at the rate of 25.4mm/s. Atthe end of test, the specimen was released and the angle ~() (in Fig. 12) was recorded to indicatethe magnitude of the springback. An R/t ratio (roller radius/ specimen thickness) of 3.4 was usedfor formability and 4.5 for springback. A friction coefficient between the specimen and rollerwas taken as 0.04 in the simulations.

Fig. 13 compares the simulated displacement and engineering stress at the front end for theM-C and isotropic hardening models. The M-C predicts lower front stress than the isotropichardening model: 2.8%, 2.1 % and 2.4% respectively for DP590, DP780 and DP980. (Table 3shows that the springback difference for DP 590 is similar to the reduced tension stress, i.e. areduction of3%.)

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L. Sun, J.B. Kim, and R.B. Wagoner

Fb -Initial

Fb-Loaded

/7777

127mm

Figure J2. Schematic (~ldra\V-bend test

1200R/t=3.4 V1 =50.8mm/s

v,'v,=O 1'=0.04

•• DP980-1.45mmQ.

!. 900: jDP760-1.4mm

~ 'til

~ /DP590~1.4mm

u. 600

-M-C Model~Isotropic Model

300 I -~ --, --, -- ,--o 10 20 30 40 50 60

Front Displacement (mm)

Figure J3. Front displacement-front stress curve in the simulation ql draw-bend test: formabilityobjective

Table 3 Springback angle predictionlor DP590

M-C Model (degree) Isotropic Model (deuree) DifferenceBack end stress (0.6 YS) 27.2 28.0 3.0'YoBack end stress (0.8 YS) 21.5 22.2 3.1 'Yo

6. CONCLUSION

Monotonic tension and compression, T-C, C-T, and two-stage/non-coaxial tensile tests havebeen performed for three grades of dual phase steels: DP590, DP780 and DP980. A modifiedconstitutive equation based on the Chaboche model was developed and fit to the monotonic andreverse test data. The M-C model was probed by simulating non-coaxial tests and comparingextrapolations with balance biaxial bulge test results.

The following conclusions were reached:I. M-C model represents three observed characteristics of the reverse path tests: (a)

Bauschinger effect (early re-yielding after load reverse) (b) rapid transient strain hardening, and(c) permanent softening.

2. Use of the M-C model changes simulated draw-bend springback and tensile forces by 3%.3. Strain-path changes for non-coaxial tensile tests produce complex but systematic

hardening effects. Predictions of M-C model for these paths are much better than isotropichardening models, but not perfect.

4. The M-C model extrapolates accurately to higher strains in balanced bulge tests .

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L. Sun, 1.11. Kim, and R.II. Wagoner

7. ACKNOWLEDGEMENTS

This work was supported cooperatively by the National Science Foundation (Grant CMMT0727641), the Department of Energy (Contract DE-FC26-020R22910), the Auto/SteelPartnership, the Ohio Supercomputer Center (PAS-080), and the Transportation ResearchEndowment Program at the Ohio State University. The authors wish to thank Kun Piao formaking available the improved tension/compression device that is the subject of her MS andPh.D. research, Ji Hyun Sung for providing some of the experimental data and the balancedbiaxial analysis procedure and Jeong- Whan Yoon and John Brem at Alcoa for the balancedbiaxial tests.

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