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1
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
2
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).
3
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,
4
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
5
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
6
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.
7
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).
8
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
9
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
10
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.
11
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
12
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
13
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
14
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
15
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
16
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
17
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
18
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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25
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
Accumulated Absolute True Strain
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
Accumulated Absolute True Strain
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)
Accumulated Absolute True Strain
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 (0.06 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
C-T test, T part (0.04 C)
C-T test, T part (0.06 C)
C-T test, T part (0.08 C)
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)
Accumulated true strain
DP980-1.43mm
C-T test, T part (0.04 C)
C-T test, T part (0.06 C)
C-T test, T part (0.08 C)
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)
Accumulated true strain
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
)
Accumulated true strain
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