Constitutive Modeling of Proportional and Non-Proportional Hardening …li.mit.edu/Stuff/RHW/Upload/15.pdf · 2011. 9. 15. · modified Chaboche (M-C) hardening model incorporating

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Constitutive Modeling of Proportional and Non-Proportional

Hardening of Dual-Phase Steels

Li Sun and R. H. Wagoner*

Department of Materials Science and Engineering

The Ohio State University

2041 N. College Road,

Columbus, OH, 43210, USA

ABSTRACT

The elastic-plastic response of sheet materials during non-proportional paths is seldom

incorporated in constitutive equations used for routine sheet forming simulation, but can

have a significant effect on formability and springback. Monotonic tension and

compression, coaxial tension-compression (T-C), coaxial compression-tension (C-T), and

two-stage/non-coaxial tensile tests were performed for three grades of dual phase steels:

DP590, DP780, and DP980. The reverse 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 the reverse hardening

curves from monotonic ones is larger than with other typical sheet forming alloys,

presumably because of the effects of large second-phase martensite particles in dual-

phase steels.

A Modified constitutive model based on the Chaboche approach (M-C) was developed.

In addition to one or more standard nonlinear components of the back stress, a linear term

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was added to represent the “permanent” offset of hardening following a stress reversal.

The parameters for the model were fit using the monotonic and reverse tensile test results

only, and the model predictions were then compared with large-strain balanced biaxial

bulge test, non-coaxial, two-stage tensile tests, and draw-bend springback tests. The M-C

model captured the response following a path change much better than isotropic

hardening models, but there is a fundamental difference between large-strain hardening

after a path change that occurs between path reversals and changes of principal strain axis.

The M-C model can reproduce either behaviors, but not both simultaneously.

Keywords: Dual-phase steels, Chaboche model, non-proportional loading, draw-bend test,

springback, constitutive equations

To be submitted to the Journal of Material Processing Technology Manuscript date: February 15, 2011 * Corresponding author at: Department of Materials Science and Engineering, The Ohio State

University, 2041 N. College Road, Columbus, OH, 43210, USA, Tel.: +01 6142922079 E-mail address: [email protected] (R.H.Wagoner).

mailto:[email protected]

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1. Introduction

Rapid developments of advanced materials are creating major opportunities for

improving society by conserving energy, reducing environmental impact and increasing

the performance of transportation vehicles. Dual phase steels, which are composed of

numerous hard martensite islands in a soft ferrite matrix, have outstanding combinations

of strength and ductility. However, their widespread application has been limited by

complex, unknown constitutive behavior that can affect formability and springback after

forming.

The inability to accurately predict springback is a major cost factor in industry.

Contrary to implementations intended for many formability simulations, plastic

constitutive equations must be known accurately in order to evaluate accurately the stress

and moment before unloading and thus to predict springback. This is a particular problem

when the plastic deformation path includes reversals, as is typically the case in sheet

forming as the sheet is bent and unbent while being drawn over a die radius (Gau and

Kinzel, 2001; Chun et al., 2002b; Geng and Wagoner, 2002; Li et al., 2002b; Yoshida et

al., 2002; Yoshida and Uemori, 2003a; Chung et al., 2005). The Bauschinger effect (i.e.

early reverse yielding) and the elastic-plastic response during non-coaxial loading paths

must thus be incorporated in constitutive equations (Geng and Wagoner, 2002; Li et al.,

2002a; Tarigopula et al., 2008; Uemori et al., 2008; Wang et al., 2008).

Linear kinematic hardening was proposed (Prager, 1949, 1956; Ziegler, 1959) to

reproduce the Bauschinger effect. Later developments were aimed at also treating the

subsequent hardening behavior after a stress reversal building on linear kinematic

hardening. Those were based on multi-surface/piecewise linear plastic moduli (Mroz,

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1967, 1969), and two-surface representations with continuously varying plastic moduli

(Krieg, 1975; Dafalias and Popov, 1976; Tseng and Lee, 1983; Lee et al., 2007). The

other approach is based on nonlinear kinematic hardening (Armstrong and Frederick,

1966; Chaboche, 1986), which has been adopted and refined widely (Chaboche, 1986;

Ohno and Wang, 1991, 1993; Jiang and Sehitoglu, 1996; Jiang and Kurath, 1996;

Basuroychowdhury and Voyiadjis, 1998; Ohno, 1998; Voyiadjis and Basuroychowdhury,

1998; Chun et al., 2002a; Chun et al., 2002b; Yoshida and Uemori, 2002; Yoshida et al.,

2002; Yoshida and Uemori, 2003b; Lee et al., 2005).

In a sheet metal forming process, most material points experience complex loading

modes, for example, abrupt strain path changes that may occur during bending and

unbending over a die radius, or upon subsequent, discrete forming operations. In order to

investigate the work hardening sensitivity to strain path, the parameter θ (Schmitt et al.,

1985) was introduced as follows:

1 2

1 2

:θ =⋅

D DD D

(1)

where 1D and 2D represent the rate of the deformation tensor during the prestrain and

subsequent loading. Monotonic, reverse and orthogonal strain paths correspond to

1, 1θ = − and 0, respectively. Combinations of simple loading paths such as tension,

simple shear, torsion and biaxial tensions have been used to determine the mechanical

behavior of sheet metals for complex strain/stress paths (Khan and Liang, 2000; Bouvier

et al., 2006; Khan et al., 2007; Tarigopula et al., 2008; Verma et al., 2011). Special

experimentally-observed effects during complex loading, such as hardening stagnation

and cross-hardening, have been attributed to the influence of a developing dislocation

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microstructure (Hiwatashi et al., 1997; Hoc and Forest, 2001; Li et al., 2003; Haddadi et

al., 2006; Wang et al., 2006; Wang et al., 2008) .

In the present work, three dual-phase (“D-P”) steels are subjected to non-proportional

strains paths. D-P steels are of particular interest, not only because of their growing

practical importance, but also because their hard-martensite-island/soft-ferrite-matrix

microstructure promotes dramatic departures from isotropic hardening following stress

reversals (Sun and Wagoner, 2011). Figure 1 illustrates the 3 main features of this

response:

(a) Bauschinger effect (early re-yielding after load reverse)

(b) rapid transient strain hardening, and

(c) long-term or “permanent” softening.

Results such as those shown in Figure 1 are used in the current work to construct a

modified Chaboche (M-C) hardening model incorporating a linear term to reproduce the

“permanent” softening. The M-C model is then used to simulate various tests for

comparison with corresponding test results.

2. Experimental and Simulation Procedures

Several kinds of experiments were conducted, some to obtain a set of constitutive

equations taking into account non-proportional hardening, the remainder to test the

constitutive equations by comparison with simulation and experiments.

2.1. Materials

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Three grades of D-P steels were used: DP 590, DP 780, and DP 980. The numbers

refer to the nominal or target ultimate tensile strengths in MPa. The chemical

compositions for each steel are presented in Table 1, along with sheet thicknesses (all are

near 1.4mm). The principal difference in the various grades is the fraction of martensite

in the microstructure, the stronger grades having more martensite. These differences are

typically accomplished by changes in hardenability, as can be seen by the higher Mn, Cr,

and Mo contents for the stronger grades.

2.2. Co-Axial Tension-Compression Testing

Tension-compression (T-C) and compression-tension (C-T) tests were performed using

a standard tensile testing machine (MTS 810) and a special set of fixtures (Boger et al.,

2005; Piao et al., 2011). An exaggerated dogbone specimen was subjected to a constant-

force mechanical clamping system normal to the sheet plane using a compressed air

cylinder operating on two flat side plates. Unless otherwise stated, a nominal strain rate

of 10-3/s was imposed. Teflon was adhesively bonded to the surface of side plates to

reduce friction, which was corrected for along with the slight biaxial loading. The air

cylinder maintained a constant side force of 2.23 or 3.35 kN corresponding to side

stresses of 0.83 or 1.25 MPa. An Electronic Instrument Research LE-05 laser

extensometer was used to measure the displacement between two fixed points initially

25mm apart on the specimen.

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Compensation for the biaxial stress and friction caused by the side plate forces is

required for proper comparison with standard tensile tests. For the biaxial correction, the

von Mises yield stress was assumed effective with stress follows

( )2 2 212 a t a tσ σ σ σ σ= − + + (2)

where aσ and tσ are the axial and thickness direction stress, respectively. The biaxial

stress effect is small because the side stresses are small relative to the yield stress.

The friction between the specimen and side plates is assumed to obey a Coulomb

friction law:

friction sideF Fµ= (3)

where µ is the Coulomb friction coefficient. frictionF is the additional axial force

attributed to friction, and sideF is the applied side force. The friction coefficients, Table 2,

were determined from separate tension or compression tests carried out with 3 values of

side force, sideF =0 (only for tension), 1.12 KN (only for compression), 2.23 KN and 3.35

KN. At a given strain, say at 1ε ε= , a least squares line was passed through a plot of

.friction sideF vs F , with a slope of 1µ . A similar procedure was carried out at strain as follows:

iε = 0.03, 0.05, 0.07, 0.10 for tension, and iε = 0.01, 0.02, 0.03, 0.035 for compression.

And the value of iµ were averaged to obtain the final values of µ presented in Table 2.

Details of the test procedures, fixtures, and correction procedures have been presented

elsewhere (Boger et al., 2005).

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2.3. Non-Coaxial Tensile Testing

Large tensile specimens Fig. 2(a), were machined to allow application of a prestrain in

the rolling direction (RD) using a hydraulic Instron 1322 testing machine. The details of

the grips and serrated gripping wedges, designed for plane-strain tesnion testing, have

been described in the literature (Wagoner and Wang, 1979; Wagoner, 1980, 1981;

Wagoner and Laukonis, 1983; Laukonis and Wagoner, 1984). The width of the large

tensile specimen was limited by the load capacity of the testing machine and the load cell

(150 kN).

The large specimens were pre-strained in uniaxial tension to 7% plastic strain at a

constant nominal strain rate 3 110 s− − . Photo gridding revealed that the strain was uniform

and homogenous in a central region. ASTM subsize tensile specimens, depicted in Fig. 2

(b), were cut along RD, 030 , 045 , 060 and transverse directions (TD) in the uniform

strain region. Uniaxial tensile tests were then performed to evaluate the strain hardening

behavior after the change of tensile axis orientation. An Electronic Instrument Research

LE-05 laser extensometer was used to measure extension during pre-straining and

subsequent tension for strain measure.

The comparison of stress-strain curves between standard T-C tensile samples and large

tensile samples was shown in Fig. 3. The result indicates that the difference is small

enough to be omitted and any significant differences in non-coaxial tests are the result on

non-coaxiality rather than from machining procedure or specimen characteristics.

In order to distinguish the different experimental results, the label “XX-YY” was

introduced here to specify the prestrain direction “XX” followed by the subsequent strain

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direction “YY”. There is no significant difference between the results obtained from RD

and TD directions in the uniaxial tensile tests without pre-strain, Fig. 4. The plot implies

that the initial in-plane plastic anisotropy is insignificant.

2.4. Hydraulic Bulge Testing

Balanced biaxial bulge tests can typically achieve a much larger uniform strain than

uniaxial tensile tests. Such tests were performed in cooperation with Alcoa using

procedures established there (Young et al., 1981). The bulge specimens were thinned by

machining from one side to a thickness of 0.5mm, and tensile tests of the thinned

specimens confirmed that ultimate tensile strength was the same before and after

machining, within 10 MPa (Sung et al., 2010). The opening diameter was 150mm and the

die profile radius was 25.4mm. The balanced biaxial data were converted to a tensile

effective stress – effective strain bases based on the ‘79 Hill’s yield criterion using

measured r values (plastic anisotropy ratio) and with a single best-fit m value (yield

surface exponent) obtained from tensile and balanced biaxial tests.

2.5 Draw-Bend Springback (DBF) Testing and Simulation

The draw-bend springback test (Wagoner et al., 1997; Carden et al., 2002; Wang et al.,

2005), shown schematically in Fig. 5, reproduces the mechanics of deformation of sheet

metal as it is drawn, stretched, bent, and straightened over a die radius entering a typical

die cavity. The draw-bend test system has two hydraulic actuators set on perpendicular

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axes and controlled by standard mechanical testing controllers. A 25mm-wide strip cut in

the rolling direction was lubricated with a typical stamping lubricant, Parco Prelube MP-

404, and wrapped around the fixed tool of radius 6.4mm (R/t of 4.3). The front actuator

applied a constant pulling velocity of 25.4 mm/s to a displacement of 127mm while the

back actuator enforced a pre-set constant back force, bF , set to 0.3 to 0.9 of the force to

yield the strip in tension (based on the 0.2% offset yield stress). After forming, the sheet

metal was released from the grips and the springback angle θ∆ (shown in Fig. 5) was

recorded.

Simulations of draw-bend springback tests were performed for DP 980 with a three-

dimensional finite element model having 5 layers of solid elements (ABAQUS element

C3D8R) through the sheet thickness, 215 elements in length and 5 elements in half width

(as reduced by symmetry). For other tests, the M-C model was used in equation form (i.e.

for a single finite element) for comparison with results.

3. Reverse Tension-Compression and Results

Monotonic tensile tests and C-T tests (after biaxial and frictional corrections) described

in the experimental and simulation procedure sector are compared in Fig. 6. Compresive

prestrains with the magnitudes of 0.04, 0.06 and 0.08 (0.1 for DP590) were applied. The

three characteristics introduced by Fig. 1 are apparent: (a) Bauschinger effect (early re-

yielding after load reverse) (b) rapid transient strain hardening, and (c) permanent

softening.

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Other patterns can be discerned more readily by shifting the subsequent hardening data

to a strain origin at the point of each reversal, as in Fig. 7. The stress-strain curves

subsequent to the reversal are very similar, independent of the pre-strain, particularly for

the higher strength alloys, DP780 and DP980. (For DP980 an initially high hardening rate

and higher flow stress is followed by a “stagnation” strain range for larger pre-strains.)

To first order, the monotonic initial hardening curves are even similar to the reverse

curves at various pre-strains. Figures 8 compare C-T tests with corresponding T-C tests.

The results are identical to within normal experimental scatter. They tend to confirm two

things: (1) the friction-correction procedure was adequate for these cases, and (2) the

deformation mechanisms involved are symmetric (i.e. forward and reverse deformation

are equivalent) and pressure-independent.

The simplest quantification of reverse tests is in terms of the Bauschinger effect. The

scalar parameter β (Abel, 1987) represents the normalized difference between the

forward and reverse stress, as follows

forward reverse

forward

σ σβ

σ

−= (4)

where forwardσ and reverseσ 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. 9, two kinds of offset definition, 0.2% yield offset and 0.4% yield offset,

were used with no obvious significant difference. 0β = when the material shows no

Bauschinger effect, whereas 1β = when the material yields at zero stress. It shows that

the Bauschinger effect increases in the order of DP590, DP780 and DP980, tending to

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confirm the concept that higher volume of fractions of hard inclusions increases the

departure from isotropic hardening.

4. Modified Chaboche (M-C) Model and Its Implementation

In order to incorporate the observed reverse hardening behavior as shown in Fig. 6, a

constitutive equation based on a modified Chaboche (M-C) model was used. In this M-C

model, the kinematical hardening evolution of backstress α is composed of two parts, a

nonlinear term 1α and a linear term 2α , as follows

1 2

1 1 1

2 2

2323

d C d dp

d C d

γ

= +

= −

=

p

p

α α α

α ε α

α ε

(5)

where d pε and p are the plastic strain increment and the von Mises equivalent plastic

strain, respectively. In the M-C model, Eq. 5, the nonlinear term corresponds to the

Bauschinger effect and transient hardening, and the linear term corresponds to the

permanent softening.

The isotropic hardening was taken as an exponential function as follows:

0 (1 exp( ))sR R R bp= + − − (6)

where 0R , sR and b are constants.

The uniform elongation domains from one monotonic tension and two stress-strain

curves subsequent to the reversal in C-T tests at 0.04 and 0.08 (0.1 for DP590) prestrain

were used to find optimal model coefficients using the least squares method. The best-fit

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values of coefficients in the M-C model are summarized in Table 3. The yield stress 0R

is determined by the curve fit method so there is some deviation from the value defined

by experimental 0.2% yield offset.

The M-C model was implemented using the UMAT user subroutine in the commercial

finite element code Abaqus Standard 6.7.

5. Comparison of M-C Model Predictions with Experiments

5.1 C-T Simulations and Tests

The comparison of M-C model predictions with C-T test experimental data was shown

in Fig. 6, with reveals that M-C model reproduces the monotonic and C-T tests with good

fidelity for the three grades of DP steels.

5.2 Non-Coaxial Tests and M-C Model

Figures 10 presents subsequent non-coaxial stress-strain curves after 0.07 prestrain in

the RD direction. Fig 10a, the experimental results, illustrate that the departure from

“RD-RD” stress-strain curve increases systematically as the angles to the rolling direction

increases. For the RD-RD (coaxial) case, the transition from the elastic zone to the plastic

zone was sharp, but the non-coaxial curves exhibit “transient hardening” typical of low

yield stress and rapid plastic hardening rate (Tarigopula et al., 2008). The transient

hardening behavior vanishes after a few percent of subsequent plastic strain, with all of

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the subsequent curves exhibiting a convergence. The M-C model predictions, Fig. 10b,

are based on the model parameters fit only to monotonic tension and compression-tension

data. They predict the features seen in the experiments: progressively gentler yielding at

lower stress as the angle between prestrain and past strain directions increase. The

tendency of the subsequent flow curves to converge is not reproduced because of the

linear term in the M-C model, which is required to reproduce the reverse hardening

adequately.

In Fig.11, three non-coaxial tensile tests are compared directly with M-C model

predictions using the coefficients in Table 3 as obtained from fits to independent reverse

tensile tests. For the RD-RD (coaxial) case, the reloading curve follows approximately

the unloading one and the subsequent monotonic curve. The subsequent hardening

behavior shows greater departure from the monotonic curve as the angle between the

prestrain and subsequent axes increases. The M-C model captures much of this

mechanical behavior, as the transient yield and hardening behavior dependent on

subsequent strain paths, but underestimates the departure from monotonic curves for

larger angles between the two tensile axes. It is nonetheless clear that the M-C model

represents the non-coaxial test behavior much better than an isotropic hardening model,

which would only reproduce the monotonic curve.

Comparison of Fig. 6 with Fig. 10 and 11 reveals that the M-C model predicts

adequately the Bauchinger effect (i.e. initial yield) and the transient hardening (i.e.

subsequent 2-3% strain) following an abrupt path changes. It does so whether the path

changes is a reversal (i.e. compression to tension, Fig. 6) or is a change of tensile axis

(Fig. 11). However, the long-term subsequent hardening behavior for the two kinds of

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path changes seems to be fundamentally different. For reverse tests, a “permanent”

change of flow stress is apparent, equivalent to a change of effective strain that persists

for subsequent strains of at least 0.1-0.15 (Fig. 6). Conversely, for non-coaxial two-stage

strain paths, Fig. 10a, the second-stage hardening curves rejoin the monotonic curve at a

subsequent strain of 0.08-0.10. The M-C model can be fit to reproduce either behavior,

but not both simultaneously. Nonetheless, the major part of the subsequent hardening

behaviors is remarkably consistent with and well represented by the M-C model.

5.3 Balanced Biaxial Tests and M-C Model

Fig. 12 compares balanced biaxial effective stress-strain curves with predicted M-C

model ones based on tensile-range data only. The large-strain monotonic behavior is well

represented by the M-C model: the standard deviations of the predictions and

measurements over the strain ranges shown are 8, 15, and 11 MPa, respectively, for

DP590, DP 780, and DP 980.

5.4 Draw-Bend Springback Simulations and Tests

The nonlinear unloading behavior of Young’s modulus flowing plastic deformation

has been widely observed (Morestin and Boivin, 1996; Augereau et al., 1999; Cleveland

and Ghosh, 2002; Caceres et al., 2003; Luo and Ghosh, 2003; Yeh and Cheng, 2003;

Yang et al., 2004; Perez et al., 2005; Pavlina et al., 2009; Yu, 2009; Zavattieri et al., 2009;

Andar et al., 2010) and should be considered in the springback simulation. In the present

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work, the “chord modulus” (i.e. the slope of a straight line drawn between stress-strain

points just before unloading and after unloading to zero applied stress) was utilized to

represent the changed Young’s modulus as follows

0 (1 exp( ))E E K Dp= − − − (7)

where 0E is the traditional Young’s modulus, 208GPa, and K and D are material

parameters to be determined from measured tensile unloading and loading behavior (Sun

and Wagoner, 2011). The modulus depends on the overall plastic strain at unloading, but

otherwise is constant during unloading and reloading. For DP980, 562K GPa= and

175D = .

Three constitutive laws were used to predict the springback for draw-bend test. Each

constitutive model was fit to tension and compression-tension tests as already presented.

Shorthand labels refer to constitutive approaches as follows:

Chord/Chaboche: Chord modulus (Eq. 7), M-C model

C0/Chaboche: Traditional elastic constants Co, M-C model

Chord/Iso: Chord modulus (Eq. 7), isotropic plastic hardening model

Table 4 compares the springback angles for the draw-bend tests and simulations results.

The standard deviations were calculated as follows

2model expt

1( )

N

i

N

θ θσ =

−=

∑ (8)

where modelθ and exptθ are the simulated and experimental springback angles, respectively.

N is the number of results compared. The results show that the Chord/Chaboche model

gives an improved result for the draw-bend test simulation. The plastic (M-C) model

accounts for a reduction of prediction error by a factor of three. The use of the chord

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modulus reduces the prediction error by ~30% corresponding to the magnitude of the

change of modulus.

A novel general model for general 3-D nonlinear unloading (Sun and Wagoner, 2011)

was also applied to draw-bend springback problem, as shown in Table 4. When used in

conjunction with the M-C plastic model, the springback prediction error was further

reduced to 02.6 .

6. Conclusions

1. Following a strain path reversal, the subsequent hardening shows three

characteristics: a) yield at a lower stress than before the path change (Bauchinger

effect), b) a strain range (~0.02-0.03) of rapid, transient strain hardening, and c) a

long term ( ε ~0.15) flow stress lower than for monotonic deformation

(“permanent” hardening).

2. Non-coaxial tensile tests show very similar behavior in terms of the Bauchinger

effect and transient hardening, but do not show the permanent hardening.

3. The magnitude of the Bauchinger effect and transient hardening in non-coaxial

tensile tests very systematically and particularly with angle between the tensile

axes in pre-straning and past-straining.

4. The M-C model, as fit to monotonic and reverse tension-compression data,

reproduces non-coaxial tensile tests adequately, much better than isotropic

hardening ones. There is, however, a systematic error in the predictions at large

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strains following a path change because of the differences of material behavior, i.e.

the existence vs. absence of “permanent” hardening.

5. The large strain monotonic behavior of dual-phase steels is reproduced well by

M-C model.

6. Prediction of draw-bend springback (which involves path reversals) is improved

by a factor 3 when using the M-C model as compared with isotropic hardening.

Acknowledgements

This work was supported cooperatively by the National Science Foundation (Grant

CMMI 0727641), the Department of Energy (Contract DE-FC26-02OR22910), the

Auto/Steel Partnership (which provided materials), the Ohio Supercomputer Center

(PAS-080) (which provided computer resources), General Motor (which provided tensile

testing and chemical analysis) and the Transportation Research Endowment Program at

the Ohio State University.

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Table.1: Chemical composition* of DP steels in weight percent

C Mn P S Si Cr Al Ni Mo Thickness

(mm)

DP590 0.08 0.85 0.009 0.007 0.28 0.01 0.02 0.01 <.01 1.4

DP780 0.12 2.0 0.020 0.003 0.04 0.25 0.04 <.01 0.17 1.4

DP980 0.10 2.2 0.008 0.002 0.05 0.24 0.04 0.02 0.35 1.43

* Chemical composition was analyzed utilizing Baird OneSpark Optical Emission

Spectrometer at the GMNA Materials Laboratory following ASTM E415-99a.

26

0

200

400

600

800

0 0.05 0.1 0.15 0.2 0.25

Monotonic tension

C-T test DP590-1.4mm

Accumulated Absolute True Strain

Abs

olut

e Tr

ue S

tres

s (M

Pa)

Bauschinger effect

Transient hardening

Permanent softening

Compression Tension

Fig.1. Monotonic and reverse compression-tension (C-T) test experimental curves that illustrate the three characteristic regions of reverse hardening: Bauschinger effect, rapid

transient strain hardening and “permanent” softening.

27

Table 2: Friction coefficients for DP590, DP780 and DP980 Tension Compression DP590 0.040 0.092 DP780 0.067 0.166 DP980 0.105 0.165

28

Fig. 2b

Fig.2 (a) Geometry of tensile specimens used for non-coaxial tests: the large tensile specimen (mm), (b) ASTM subsize specimen (mm)

RD 45° TD

Fig.2a

29

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08

True

str

ess

(MPa

)

True strain

Large sample tension

Standard T-C sample monotonic tension

DP980-1.43mm

Fig. 3 Comparison of measured strain hardening from standard C-T sample and large tensile sample

30

0

200

400

600

800

1000

1200

0 0.03 0.06 0.09 0.12 0.15

RDTD

True

Str

ess

(MPa

)

True Strain

DP980-1.43 mm

euUTS (MPa)

0.111990.9RD0.106989.0TD

-4.53%-0.09%TD Vs. RD

Fig. 4 Uniaxial tensile tests for RD and TD directions of DP980

31

F b

θ V=25.4 mm/sStroke=127 mm

Start

Final shape

grip

Start

Finish

Finish

grip

Bending, unbending and friction

Fig. 5 Schematic of draw-bend test

32

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

ExperimentM-C Model


Monotonic tension

DP590-1.4mm

Abs

olut

e Tr

ue S

tres

s (M

Pa)

C-T tests

Fig. 6a Fig. 6 Compression of results of compression-tension (C-T) and monotonic tensile tests with modified Chaboche (M-C) model simulations: (a) DP 590 (b) DP 780 (c) DP980

33

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

ExperimentM-C Model

Monotonic tension

DP780-1.4mm


Abs

olut

e Tr

ue S

tres

s (M

Pa)

C-T tests

Fig. 6b

34

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

Abs

olut

e Tr

ue S

tres

s (M

Pa)


Monotonic tension

DP980-1.43 mm

C-T tests

Fig. 6c

35

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

Abs

olut

e Tr

ue S

tres

s (M

Pa)

Reverse True Strain

DP590-1.4mm

C-T test, T part (0.04 C)


C-T test, T part ( 0.10 C)

Monotonic tension

Fig.7a

Fig. 7 Monotonic tension and C-T tests experimental data after reversal (a) DP590 (b) DP780 (c) DP980

36

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

Monotonic tension

Accumulated true strain

True

str

ess

(MPa

)

DP780-1.4mm




Fig.7b

37

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

Monotonic tension Tr

ue s

tres

s (M

Pa)


DP980-1.43mm




Fig.7c

38

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

Monotonic tension

C-T test ( 0.06 C)T-C test ( 0.06 T)

DP590-1.4mm

Accumulated True Strain

Abs

olut

e Tr

ue S

tres

s (M

Pa)

Fig.8a

Fig. 8 Compression of tension-compression (T-C), compression-tension (C-T), and

monotonic test results: (a) DP590 (b) DP780 (c) DP980

39

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

Monotonic tension

DP780-1.4mm

C-T test ( 0.04C)

T-C test ( 0.04T)


True

str

ess

(MPa

)

Fig.8b

40

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

True

str

ess

(MPa

)


Monotonic tension

C-T test ( 0.04C)

T-C test ( 0.04 T)

DP980-1.43mm

Fig.8c

41

0

0.2

0.4

0.6

0.8

1

0.02 0.04 0.06 0.08 0.1

β

Prestrain

DP980

DP780

DP590

Yield offset : 0.4%

Fig. 9a

0

0.2

0.4

0.6

0.8

1

0.02 0.04 0.06 0.08 0.1

β

Prestrain

DP980

DP780

DP590

Yield offset : 0.2%

Fig. 9b

Fig.9 Comparison of Bauchinger parameter ( ) /forward reverse forwardβ σ σ σ= − from C-T

tests: (a) 0.4% yield offset (b) 0.2% yield offset

42

Table 3: Best-fit coefficients in the M-C model

0R (MPa)

1C (MPa)

2C (MPa)

γ sR

(MPa) b Standard

Deviation (MPa)

DP590 312 13501 296 85 246 12 12 DP780 454 17062 517 72 163 16 15.67 DP980 511 38000 1193 100 135 11.9 14.55

43

400

600

800

1000

1200

0 0.03 0.06 0.09 0.12

Experimental DataTr

ue S

tres

s (M

Pa)

Subsequent True Strain

RD-RDRD-30o

RD-45o

RD-60o

RD-TD

DP980-1.43mm (0.07 prestrain)

Fig.10a Fig. 10 Comparison of subsequent hardening curves following a prestrain of 0.07 in the

RD: (a) experimental results, (b) M-C model predictions.

44

400

500

600

700

800

900

1000

1100

1200

0 0.02 0.04 0.06 0.08 0.1 0.12

RD-RDRD-30o

RD-45o

RD-60o

RD-TD

DP980 (7% prestrain) M-C Model Simulation

Subsequent Strain

True

Str

ess

(MPa

)

Fig.10b

45

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2

True

Str

ess

(MPa

)

Accumulated True Strain

Monotonic tension (RD)

RD-RD RD-45o

RD-TD

DP980-1.43mm(0.07 prestrain)

ExperimentM-C Model

Fig. 11 Detailed comparisons of non-coaxial two-stage tensile tests with M-C model predictions.

46

400

600

800

1000

1200

1400

0 0.1 0.2 0.3 0.4 0.5 0.6

Tensile ExptBulge ExptM-C Model

Effe

ctiv

e St

ress

(MPa

)

Effective Strain

DP980-1.43mm

DP780-1.4mm

DP590-1.4mm

Fig 12 Balanced bulge tests experimental data vs. M-C model

47

Table 4: Measured and simulated springback predictions (in degree)

Fb 0.3 0.6 0.8 0.9 <σ> ∆θ ∆θ-∆θExp ∆θ ∆θ-∆θExp ∆θ ∆θ-∆θExp ∆θ ∆θ-∆θExp

Experiment 63.5 53.9 45.9 37.5 0 Chord/Chaboche 74.1 10.6 56.1 2.19 48.4 2.51 43.6 6.07 6.3

C0/Chaboche 53.7 -9.83 43.9 -10 36.5 -9.4 32.9 -4.59 8.8 Chord/ISO 90 26.5 71.8 17.9 58.4 12.5 52.4 14.9 18.7

Documents

Constitutive Modeling of Proportional and Non-Proportional Hardening …li.mit.edu/Stuff/RHW/Upload/15.pdf · 2011. 9. 15. · modified Chaboche (M-C) hardening model incorporating