international journal of plasticity complex unloading behavior:...

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Complex unloading behavior: Nature of the deformationand its consistent constitutive representation

Li Sun, R.H. Wagoner ⇑Department of Materials Science and Engineering, The Ohio State University, United States

a r t i c l e i n f o

Article history:Received 29 September 2010Received in final revised form 16 December2010Available online 24 December 2010

Keywords:Unloading behaviorYoung’s modulusSpringback predictionDual phase steelsConstitutive model

a b s t r a c t

Complex (nonlinear) unloading behavior following plastic straining has been reported as asignificant challenge to accurate springback prediction. More fundamentally, the nature ofthe unloading deformation has not been resolved, being variously attributed to nonlinear/reduced modulus elasticity or to inelastic/‘‘microplastic’’ effects. Unloading-and-reloadingexperiments following tensile deformation showed that a special component of strain,deemed here ‘‘Quasi-Plastic-Elastic’’ (‘‘QPE’’) strain, has four characteristics. (1) It is recov-erable, like elastic deformation. (2) It dissipates work, like plastic deformation. (3) It is rate-independent, in the strain rate range 10�4–10�2/s, contrary to some models of anelasticityto which the unloading modulus effect has been attributed. (4) To first order, the evolutionof plastic properties occurs during QPE deformation. These characteristics are as expectedfor a mechanism of dislocation pile-up and relaxation. A consistent, general, continuumconstitutive model was derived incorporating elastic, plastic, and QPE deformation. Usingsome aspects of two-yield-function approaches with unique modifications to incorporateQPE, the model was implemented in a finite element program with parameters determinedfor dual-phase steel and applied to draw-bend springback. Significant differences werefound compared with standard simulations or ones incorporating modulus reduction.The proposed constitutive approach can be used with a variety of elastic and plastic modelsto treat the nonlinear unloading and reloading of metals consistently for general three-dimensional problems.

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1. Introduction

Highly nonlinear unloading following plastic deformation has been widely observed (Morestin and Boivin, 1996; Auge-reau 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), with the apparent unloadingmodulus reduced by up to 22% for high strength steel (Cleveland and Ghosh, 2002) and 70% for magnesium relative to thebond-stretching value (Caceres et al., 2003). The magnitude of the reduction depends on the plastic strain and alloy. In addi-tion, the effect can differ with rest time after deformation, heat treatment and strain path (Yang et al., 2004; Perez et al.,2005; Pavlina et al., 2009).

Nonlinear unloading behavior has been variously attributed to residual stress (Hill, 1956), time-dependent anelasticity(Zener, 1948; Lubahn and Felgar, 1961), damage evolution (Yeh and Cheng, 2003; Halilovic et al., 2009), twinning or kinkbands in HCP alloys (Caceres et al., 2003; Zhou et al., 2008; Zhou and Barsoum, 2009, 2010), and piling up and relaxationof dislocation arrays (Morestin and Boivin, 1996; Cleveland and Ghosh, 2002; Luo and Ghosh, 2003; Yang et al., 2004).

0749-6419/$ - see front matter � 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijplas.2010.12.003

⇑ Corresponding author.E-mail address: [email protected] (R.H. Wagoner).

International Journal of Plasticity 27 (2011) 1126–1144

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International Journal of Plasticity

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The conceptually simplest idea to account for nonlinear unloading is second-order effects in elasticity. A simple calcula-tion shows that such effects are far too small to be realistic for typical structural metals. The elastic response of metals re-lated to atomic bond stretching is very nearly linear because the elastic strain normally attainable before dislocations moveis very small. Second-order elasticity can be expressed as follows (Powell and Skove, 1982; Wong and Johnson, 1988):

e ¼ rE0þ d

rE0

� �2

ð1Þ

where r, e are the uniaxial stress and strain, respectively, E0 is the initial (handbook value of) Young’s modulus, and d is anonlinearity parameter with a value 5.6 reported for ‘‘Helca 138A’’ steel (Powell and Skove, 1982; Wong and Johnson, 1988).Eq. (1) predicts a change of modulus by approximately 3% for a dual phase (DP) steel having an ultimate tensile strength of980 MPa, which makes it one of the strongest steels considered for room-temperature forming applications (and thus withthe largest anticipated second-order effects).

For the more physically plausible dislocation pile-up and release mechanisms, mobile dislocations move on slip planesuntil stopped by grain boundaries or other obstacles, thus forming dislocation pile-ups (or similar structures such as polar-ized cell walls). When the applied stress is reduced, the repelling dislocations move away from each other, providing addi-tional unloading strains concurrent with elastic unloading strains from atomic bond relaxation. Moving dislocationsdissipate work by exciting lattice phonons (Hirth and Lothe, 1982). Thus, while such pile-up-and-release strains are expectedto be at least partly recoverable, they cannot be energy preserving.

One practical consequence of the changed unloading modulus is the challenge of simulating springback accurately. Thegeneral rule is that the magnitude of springback is proportional to the flow stress and inversely proportional to Young’s mod-ulus (Wagoner et al., 2006). Simulations of springback are improved markedly by taking the observed unloading behaviorinto account (Morestin and Boivin, 1996; Pourboghrat et al., 1998; Li et al., 2002b; Fei and Hodgson, 2006; Zang et al.,2007; Vrh et al., 2008; Halilovic et al., 2009; Yu, 2009; Eggertsen and Mattiasson, 2010).

Nearly all of the proposed practical approaches to incorporating complex unloading behavior rely on adopting a ‘‘chordmodulus’’ (i.e. the slope of a straight line drawn between stress–strain points just before unloading and after unloading tozero applied stress) (Morestin and Boivin, 1996; Li et al., 2002b; Luo and Ghosh, 2003; Fei and Hodgson, 2006; Zang et al.,2007; Ghaei et al., 2008; Kubli et al., 2008; Yu, 2009). The chord modulus model has conceptual and practical advantages:incorporating it in existing software is no more difficult than altering Young’s modulus in the input parameters, and option-ally treating it as a function of the strain before the unloading begins.

The nonlinearity of unloading1 unfortunately creates significant accuracy limitations of any chord modulus treatment ofspringback. Even for 1-D loading and unloading, it can be understood that unloading to any internal stress other than zero (i.e withany non-zero residual stress for a given element) will have inherent errors. For the standard sheet-metal case of bending undertension, the residual stresses can be of the same general magnitude as the stresses in the loaded condition, and can be both tensileand compressive (Li et al., 2002a). More general 3-D unloading with any material model other than linear elasticity introduces thepossibility of unloading path effects on springback. For standard elastic-plastic laws, these occur by the path-dependency of plasticdeformation. Such differences have been reported even for simple isotropic hardening (Li and Wagoner, 1998; Li et al., 2002b) andthere are suggestive measurements of such effects possibly related to kinematic hardening or to two-yield surface models (Khanet al., 2009). More fundamentally, as will be shown, the physical phenomenon of nonlinear unloading is not truly elastic (i.e. it isnot energy preserving), nor is it truly plastic (i.e. irrecoverable), and thus loading and unloading excursions may follow consider-ably different stress–strain trajectories than expected, either from elastic-only or elastic-plastic constitutive models.

The plastic constitutive equation during loading (and possibly unloading) must also be known accurately for springbackapplications in order to evaluate the stress and moment, particularly before unloading. This is particularly true when theplastic deformation path includes strain increment reversals, as for example being bent and unbent while being drawn overa die radius (Gau and Kinzel, 2001; Chun et al., 2002; Geng and Wagoner, 2002; Li et al., 2002b; Yoshida et al., 2002; Yoshidaand Uemori, 2003a; Chung et al., 2005). Nonlinear kinematic hardening (Chaboche, 1986, 1989) has shown to be an effectivemethod for prediction of springback under such conditions (Morestin et al., 1996; Gau and Kinzel, 2001; Zang et al., 2007;Eggertsen and Mattiasson, 2009; Taherizadeh et al., 2009; Tang et al., 2010).

In view of the state of understanding of the unloading modulus effect, simple experiments were performed to reveal thenature of the phenomenon using a high-strength steel, DP 980, chosen to accentuate the deviations from bond-stretchingelasticity. DP 980 is a dual-phase steel with nominal ultimate tensile strength of 980 MPa. Based on inferences drawn fromthese results, a consistent, general (3-D) constitutive model was developed to represent the observed variation of Young’smodulus, and the required parameters were determined. Dubbed the QPE model (Quasi-Plastic-Elastic), it was implementedin Abaqus/Standard (ABAQUS) and compared with tensile tests with unloading/loading cycles at various pre-strains, withreverse tension/compression tests, and with draw-bend springback tests. QPE introduces a third component of strain inaddition to traditional elastic and plastic strains, here called ‘‘QPE strain.’’ The QPE strain is similar to one envisionedelsewhere in 1-D form (Cleveland and Ghosh, 2002) that is recoverable (elastic-like) but energy dissipative (plastic-like).

1 We ignore here geometric nonlinearities arising from large rotations and displacements that can occur even for very small strains, as well as nonlinearitiesassociated with differences between true and engineering strain, for example. Such nonlinearities can be, and are, readily handled by standard finite elementapproaches, whereas nonlinearities arising from the material’s constitutive response are typically not considered.

L. Sun, R.H. Wagoner / International Journal of Plasticity 27 (2011) 1126–1144 1127


2. Experimental procedures

2.1. Materials

DP 980 steel was selected for testing because dual-phase steels have large, numerous islands of hard martensite phase ina much softer ferrite matrix. (A few experiments were also performed for DP780, but it was not tested extensively to fit thenew model.) The islands serve to strengthen the composite-like material at large strain, but also to lower the yield stress byproviding stress concentrators initially. DP 980 is the strongest alloy typically considered for cold forming applications,where springback is likely to be a significant issue. Because of the high strength and large, numerous obstacles, DP 980was expected to accentuate the unloading modulus effect, assuming a dislocation pile-up and release mechanism as theprincipal source.

The DP 980 and DP 780 alloys used in this study had previously been characterized to obtain accurate 1-D plastic con-stitutive equations (Sung, 2010; Sung et al., 2010) with standard mechanical properties shown in Table 1. The standard ten-sile tests represented in Table 1 were carried out at General Motors North America (GMNA, 2007) according to ASTM E8-08at a crosshead speed of 5 mm/min. The normal plastic anisotropy parameters r1 and r2 refer to results from alternate testprocedures applied to sheets of original thickness and reduced thickness (Sung, 2010). In either case, the r values are closeto 1 and do not vary greatly with testing direction, justifying an assumption of plastic isotropy as a first approximationadopted in the current work.

2.2. Tensile testing

Standard parallel tensile specimens (ASTM-E646) with gage length 75 mm and width 12.5 mm were cut in the rollingdirection and used for uniaxial tensile testing. Unless otherwise stated, a nominal strain rate of 10�3/s was imposed. AnMTS 810 testing machine and an Electronic Instrument Research LE-05 laser extensometer were used. The laser extensom-eter works by performing a linear scan 100 times per second and detecting the positions of flags attached to the specimenoriginally separated by a fixed distance, usually 50 mm for full-sized ASTM-E646 tensile specimens. For a strain resolution of0.01%, the scan rate limits the resolvable strain rate to approximately 10�2/s. Experience shows that even at this strain rate,transient effects are not captured perfectly because of lag in the detection system.

2.3. Compression / tension testing

Compression/tension testing was performed using methods appearing in the literature (Boger et al., 2005). Two flat back-ing plates and a pneumatic cylinder system were used to provide side force to constrain the exaggerated dog-bone specimenagainst buckling in compression. Side forces of 3.35 kN were applied.

The stabilizing side force necessitates correction for two effects in order to obtain uniaxial stress–strain curves compara-ble to standard tensile testing: (1) friction between the sample surface and supporting plates, which reduces the effectiveaxial loading force, and (2) biaxial stress state. Analytical schemes for making corrections for each of these were employed,as presented elsewhere (Balakrishnan, 1999; Boger et al., 2005). A friction coefficient of 0.165 was determined using a leastsquares value of the slope dT/dN, where T and N are the measured tensile force and applied normal force in a series of other-wise identified experiments.

2.4. Draw-bend springback (DBF) testing

The draw-bend springback test (Wagoner et al., 1997; Carden et al., 2002; Wang et al., 2005), shown schematically inFig. 1, reproduces the mechanics of deformation of sheet metal as it is drawn, stretched, bent, and straightened over a dieradius entering a typical die cavity. It thus represents a wide range of sheet forming operations, but has the advantage ofsimplicity and the capability of careful control and measurement, particularly important for the sheet tension force. Itwas developed from draw-bend tests designed for friction measurement (Vallance and Matlock, 1992; Wenzloff et al.,1992; Haruff et al., 1993). However, while it mimics many practical operations, it is complex to analyze (Li and Wagoner,1998; Geng and Wagoner, 2002; Li et al., 2002a; Wagoner and Li, 2007) because of reversing strain paths (need to account

Table 1Mechanical properties of DP780 and DP980 steels.

t (mm) 0.2% YS (MPa) UTS (MPa) eu (%) et (%) n Dr r1 r2

DP780 1.4 499 815 12.7 17.9 0.19 �0.11 0.97 0.84DP980 1.43 551 1022 9.9 13.3 0.15 �0.23 0.76 0.93

Key: t = sheet thickness; UTS = ultimate tensile strength; eu, et = uniform, total elongation (engineering strain); n = strain hardening power, Hollomonequation, obtained from 0.04 < e < 0.06; Dr = in-plane anisotropy parameter; r1, r2 = duplicate measures of normal plastic anisotropy parameters up to theuniform elongation.

1128 L. Sun, R.H. Wagoner / International Journal of Plasticity 27 (2011) 1126–1144


for Baushinger effect, e.g.), a wide range of simultaneous strains and strain rates, the 3-D nature of the deformation in view ofanticlastic curvature (Wang et al., 2005) and the dependence of testing results on anisotropy (Geng and Wagoner, 2002).

The draw-bend test system has two hydraulic actuators set on perpendicular axes and controlled by standard mechanicaltesting controllers. A 25 mm-wide strip cut in the rolling direction was lubricated with a typical stamping lubricant, ParcoPrelube MP-404, and wrapped around the fixed tool of radius 6.4 mm (R/t of 4.3). The front actuator applied a constant pull-ing velocity of 25.4 mm/s to a displacement of 127 mm while the back actuator enforced a pre-set constant back force, Fb, setto 0.3–0.9 of the 0.2% offset yield stress. After forming, the sheet metal was released from the grips and the springback angleDh as shown in Fig. 1 was recorded to reveal the magnitude of the springback. The details of the experiment and itsinterpretation have been presented elsewhere (Carden et al., 2002).

2.5. Draw-bend springback (DBF) simulation

Simulations of draw-bend springback tests used a three-dimensional finite element model with 5 layers of solid elements(ABAQUS element C3D8R) through the sheet thickness, 215 elements in length and 5 elements in width. The friction coef-ficient between the specimen and roller was taken as 0.04 in the simulations, which was determined by comparing simu-lated and measured front and back forces.

Some simulations with 7 layers and 10 layers of elements through the thickness (both having 5 layers in the width direc-tion and 397 elements in the length direction) were compared with the corresponding 5-layer results in order to test meshsensitivity. The standard deviations of the computed springback angles with the experiments were 3.2�, 3.3�, and 2.2�,respectively, for the 5, 7, and 10-layer models. These are all approximately equal to the estimated experimental scatter of2�. The 5-layer QPE simulations required an average of 22 h using a single-processor PC while the 7-layer and 10-layerQPE simulations required 11 h and 27 h, respectively, using a 32-processor workstation. (The details of these results willbe presented in Table 3, introduced later.) Because of the long CPU times using the refined meshes, and the minor changesin results from mesh refinement, the remainder of the simulations made use of the original mesh.

3. Experimental results: tensile tests

Results for tensile tests of DP 780 and DP 980 with intermediate unloading cycles are shown in Fig. 2. Fig. 2a and b showsthat unloading and reloading are nonlinear, forming hysteresis loops that are more pronounced for higher flow stresses priorto unloading. The loop expansion occurs for both strain hardening to higher stress or a higher initial yield stress because ofmicrostructural differences (i.e. by comparing DP780 with DP980). It can also be seen that, to a close approximation, the(stress, strain) point at the start and end of unloading are in common for unloading and loading legs. That is, to a first approx-imation, all of the strain in the loop is recoverable and the plastic flow stress is not affected by unloading/loading cycle (Infact, as will be shown later in Fig. 11, the unloading–loading cycle does increase the subsequent flow stress slightly, an effectthat will be ignored in the first model developed. It can be included optionally by minor changes in the model.).

Fig. 2c is an expanded view of the fourth cycle for DP 980 shown in Fig. 2b. More detail is apparent. The shape of theunloading leg can now be seen as close to linear initially, with a slope approximately equal to Young’s modulus(208 GPa). At a critical stress rc1, the slope is reduced and progressively becomes smaller until the external stress is removed.

Fb

θ V=25.4 mm/sStroke=127 mm

Start

Final shape

grip

Start

Finish

Finish

grip

Bending, unbending and friction

Fig. 1. Schematic of draw-bend springback test.



The reloading curve has similar properties, in reverse: an initial reloading linear portion with slope consistent with Young’smodulus and a reduction after a critical stress rc2 is reached. This appearance is similar to that reported for other alloys(Cleveland and Ghosh, 2002; Luo and Ghosh, 2003; Yang et al., 2004). The area of the loop formed represents work dissipatedby the strain shown between the two Young’s modulus construction lines (labeled eQPE on Fig. 2c). A chord modulus of145 GPa (which is a composite of the linear and nonlinear portions of unloading or loading curves) is shown for comparison.Note that chord modulus is 30% less than the atomic-bond-stretching value, which could potentially produce 30% morespringback than expected using the standard Young’s modulus. (This is apparently the largest deviation reported for a steel,confirming the choice of DP980 as a good material to illustrate the effects.)

A conceptual breakdown of the axial strains (and axial strain increments) from the tensile test at the point of unloading(i.e. at what will be called the pre-strain throughout this paper) will be used to motivate the current development as sug-gested by Fig. 2c:

e ¼ ee þ eQPE þ ep ð2Þ

where ee is the elastic strain, ep is the plastic strain, and eQPE is a new category of strain deemed here the ‘‘Quasi-Plastic-Elas-tic’’ (‘‘QPE’’) strain. The corresponding 1-D infinitesimal increments are dee, dep deQPE and tensor generalizations are dee, dep

deQPE. QPE deformation has the following apparent characteristics:

� It is recoverable (along with, and similar to, elastic strain).� It is energy-dissipating (along with, and similar to, plastic strain).

Thus, QPE strain straddles recovering and energy-dissipating categories of strain as follows for the 1-D case:

ð3Þ

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

Tru

e S

tres

s (M

Pa)

True Strain

DP780-1.4

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

Tru

e S

tres

s (M

Pa)

True Strain

DP980 -1.43

See Fig. 2c

0

200

400

600

800

1000

1200

0.069 0.072 0.075 0.078

Tru

e st

ress

(M

Pa)

True strain

DP980 -1.43

Measuredunload-load loop

208 GPa

208 GPa

145 GPa

εQPE

εe

εp

εrecov

σc1

σc2

a b

c

Fig. 2. Results of uniaxial tensile tests with unloading–loading cycles. (a) Four-cycle test of DP 780. (b) Four-cycle test of DP 980. (c) Expanded view of thefourth unload–load cycle for DP 980 showing alternate moduli and conceptual components of strain.



The elastic and plastic strain components have their usual, idealized definitions:

� Elastic strain ee = r/E0(E0 = 208 GPa) is recoverable and energy conserving� Plastic strain, ep, is non-recoverable and energy dissipating

Fig. 3 shows how the QPE effect evolves with plastic straining and strain hardening. As shown in Fig. 3a and b, the workdissipated by QPE deformation increases with plastic deformation and has a single proportional relationship to the stress atunloading for the two alloys tested. Fig. 3c and d shows that the QPE strain differs in the two alloys at a given pre-strain, butmaintains a nearly constant fraction of �1/3 of elastic strain, independent of pre-strain (and thus flow stress) and choice ofmaterial.

Fig. 4 compare various kinds of cyclic unloading-loading tensile tests. Fig. 4a and b shows that repeated loading andunloading cycles increase the flow stress slightly compared with a single-cycle test or monotonic tensile testing, the differ-ence apparently approximating the stress increment expected if the accumulated eQPE were included in the total plasticstrain. (The first implementation of a model presented and utilized in this paper will ignore this effect for simplicity,treating eQPE as not affecting the state of the material.) Fig. 4b shows that the details of a loading–unloading cycle areunchanged by the existence of previous such cycles, except for the slight increase of flow stress and proportional increaseof QPE strain and dissipated energy (Fig. 3b–d). Fig. 4c compares a first and second unloading cycle without interveningplastic deformation. The second cycle exhibits slightly less QPE strain and dissipated energy, thus justifying ignoring theeffect for a first model implementation, particularly for a small number of cycles.

In order to test the hypothesis that the modulus effect is related to rate-dependent anelasticity (Zener, 1948), loading–unloading tests like those shown in Fig. 2 were conducted at strain rates 0.1, 1 and 10 times the one employed in Fig. 2(i.e. at strain rates of 10�2/s, 10�3/s, and 10�4/s). Fig. 5 shows that, to a first approximation, the hysteresis loops do not varywith strain rate, shown particularly clearly for 10�4/s and 10�3/s. This behavior distinguishes the nonlinear strain recoveryfrom time-dependent anelastic deformation. The data shows some scatter and drift, particularly at 10�2/s, which is typical

0

0.5

1

1.5

0.02 0.04 0.06 0.08

Dis

sip

ated

En

erg

y (1

06 J/m

3)

Pre-strain

DP980-1.43

DP780-1.4

0

0.5

1

1.5

600 800 1000 1200

Dis

sip

ated

En

erg

y (1

06 J/m

3)

Stress at unloading (MPa)

DP980-1.43

DP780-1.4

DP980-1.43 (Unfinished cycle)

0

0.002

0.004

0.006

0.02 0.04 0.06 0.08

Str

ain

Pre-strain

DP980-1.43

DP780-1.4

εQPE

εe

a b

c d

0.2

0.3

0.4

0.5

0.02 0.04 0.06 0.08

Pre-strain

ε QP

E/ε

eAverageof ε

QPE/ε

e=0.35

DP980-1.43

DP780-1.4

Fig. 3. Variation of QPE response with deformation history for DP 780 and DP 980. (a) Relationship between dissipated energy and pre-strain of DP 780 andDP 980. (b) Relationship between dissipated energy and stress at unloading of DP 780 and DP 980. (c) Relationship of elastic strain, QPE strain and pre-strainof DP 780 and DP 980. (d) The ratio of QPE strain to elastic strain as a function of pre-strain, DP 780 and DP 980.



for strains measured using the laser extensometer at higher strain rates. Note that the strain continues to advance duringinitial unloading at 10�2/s, a result of the known limited response time of the laser extensometer. (Also note that continuoustensile tests such as these cannot be used to reveal strain rate sensitivity of the flow stress for such high-strength/low ratesensitivity materials. As noted in the literature (Sung et al., 2010), this is a consequence of unavoidable small random vari-ations of flow stress from specimen to specimen, larger than the effect of strain rate sensitivity.)

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08

Tru

e S

tres

s (M

Pa)

True Strain

DP980-1.43

Multiple cycles

Single cycle

see Fig.4b

Monotonic tension

0

200

400

600

800

1000

1200

0.05 0.0525 0.055 0.0575 0.06

Tru

e S

tres

s (M

Pa)

True Strain

DP980-1.43

Multiple cycles

Single cycle

Monotonic tension

0

200

400

600

800

1000

1200

0.05 0.0525 0.055 0.0575 0.06

Tru

e S

tres

s (M

Pa)

True Strain

DP980-1.43

First cycleSecond cycle

Monotonic tension

a b

c

Fig. 4. The effect of repeated cycles on the loading–unloading test, DP 980. (a) Four-cycle test vs. single-cycle test. (b) Expanded view as shown indicated on(a), fourth cycle vs. first cycle. (c) Comparison of identical first and second loading cycle in a loading–unloading test without intervening plasticdeformation.

0

200

400

600

800

1000

1200

0 0.025 0.05 0.075 0.1

Tru

e S

tres

s (M

Pa)

True Strain

DP980-1.43

Strain rate

10-2/s

Strain rate

10-3/s

Strain rate

10-4/s

See Fig. 5b

0

200

400

600

800

1000

1200

0.05 0.0525 0.055 0.0575 0.06

Tru

e S

tres

s (M

Pa)

True Strain

DP980-1.43

Strain rate

10-2/s

Strain rate

10-3/s

Strain rate

10-4/s

a b

Fig. 5. Results of uniaxial tensile tests with loading–unloading cycles conducted at three nominal strain rates (10�2/s, 10�3/s, 10�4/s). (a) Overall view. (b)Close-up of the fourth cycle.



In summary, a set of simple tensile experiments for DP 980 suggests the presence of a special kind of continuum strain,here deemed QPE strain, eQPE, that is recoverable but energy dissipating. The QPE strain is, to a first approximation, strain rateindependent, does not change the material plastic state appreciably, and is proportional to the flow stress or elastic strain.There is a critical stress change required to induce QPE straining (i.e. nonlinear response), both on unloading and reloading.These characteristics serve as a basis to devise a practical constitutive model incorporating all three types of deformation, aspresented next.

4. Quasi-Plastic-Elastic (QPE) model

In order to develop a general QPE constitutive model consistent with the characteristics discussed above, tensor geomet-ric concepts from two-yield-surface (TYS) plasticity theories (Krieg, 1975; Dafalias and Popov, 1976; Tseng and Lee, 1983;Ohno and Kachi, 1986; Ohno and Satra, 1987; Geng and Wagoner, 2000, 2002; Yoshida and Uemori, 2002, 2003b; Leeet al., 2007) are employed. (See, for example, (Lee et al., 2007) for a brief introduction to TYS.) The differences will be madeapparent below. Most generally, the yield surface is the inner surface for the structure of TYS models; the outer surface forQPE. In TYS models, a continuously varied hardening function is defined in terms of the distance between the yield surfaceand a bounding surface in order to establish a smooth stress–strain curve in the plastic state. In the proposed QPE model, theelastic-QPE surface (the inner surface) translates to reproduce a stress–strain modulus that is a continuously varying func-tion of strain during unloading and reloading.

To begin the development, consider an inner surface in stress space f1 defining an elastic-QPE transition and a standardyield surface f2 defining a transition from elastic or QPE deformation to plastic deformation, as shown in Fig. 6 for the plane-stress condition, r3 = 0:

f1 ¼ /1ðr� aÞ � R1ðpÞ ¼ 0f2 ¼ /2ðr� � a�Þ � R2ðpÞ ¼ 0

ð4Þ

where R1 and R2 represent the sizes of the QPE surface f1 and yield surface f2, respectively, which are centered at a and a⁄

respectively. R1 and R2 define the respective surface sizes based on a standard uniaxial tensile reference state. The variable pis the equivalent plastic strain2 defined for von Mises yield functions by dp ¼ ð23 dep : depÞ1=2. The applied stress is r (it may beanywhere within f1 for a purely elastic state, or on f1 otherwise, as shown in Fig. 6) and r⁄ is a point on f2 corresponding to r onf1 sharing normals such that n = n⁄which are defined as n ¼ @f1

@r =@f1@r

�� and n� ¼ @f2@r =

@f2@r

�� . (The notation = ||� � �|| indicates the normof the vector or tensor.)

σ

σ1

σ2

σ*

αα∗

nn*

f1

f2

Fig. 6. Definitions of variables in the two-surface QPE model.

2 For simplicity, f1 and f2 are of the von Mises form in the current development, which closely represents the plastic isotropy of the dual-phase steels testedhere. However, there is no limitation to applying the QPE theory using anisotropic yield functions as long as a consistent definition of effective strain isincorporated, and as long as the two functions f1 and f2 are of the same form (to assure that the evolution laws can enforce a single tangent contact pointbetween them). f1 is initially smaller than f2 and the evolution rules assure that this relationship continues, so at each equivalent point throughout thedeformation, the curvature of f1 is larger than that of f2. The functions f1 and f2 could also be of differing forms, but in that case a restriction would be requiredthat f1 has higher curvature than f2 at each equivalent point. The evolution laws would be considerably more complex in the case of f1 and f2 having differentforms.



Three fundamental deformation models and corresponding evolution rules are envisioned in the QPE model depending onthe applied stress, stress increment, and locations of f1 and f2:

4.1. Elastic mode

r is inside the surface f1, or else r is on the surface f1 and projects inward, i.e. dr::dn < 0. The sizes and centers of f1 and f2

are unchanged by straining in the elastic state. Only elastic strain ee occurs and the material behaves according to a classicallinear elastic principle, with the elastic strain increment as follows:

dr ¼ C0 : dee ð5Þ

where C0 is the constant elastic modulus tensor representing atomic bond stretching as measured, for example, by soundvelocities. The sizes and locations of f1 and f2 are constant. Eq. (5) defines dee within any state of the material (i.e. whetherplastic loading, QPE loading or purely elastic loading is taking place currently).

4.2. Plastic mode

The inner surface f1 is in contact with the yield surface f2 at a point congruent with the applied stress, i.e. r = r⁄, and drprojects outward (dr::dn > 0). Plastic strain ep, elastic strain ee and QPE strain eQPE occur. The size of f1 evolves and maintainscongruency of r and r⁄while f2 evolves according to any plastic hardening law3. The governing equations are thus as follows:

dr ¼ C0 : dee ¼ C : ðde� depÞ ð6aÞde ¼ dee þ deQPE þ dep ð6bÞdee=kdeek ¼ deQPE=kdeQPEk ð6cÞ

where C is an apparent elastic + QPE stiffness tensor evaluated, as shown below, at the last transition to plastic loading.(More complex formulations are possible, but the differences would be small because the form of C, Eqs. (11) and (12), be-low, is close to its limiting value whenever plastic deformation is occurring.)

4.3. QPE mode

Three conditions must be satisfied simultaneously – r is on the surface f1, dr is outward (dr::dn > 0), and f1 and f2 are notin contact. During QPE deformation, the size and location of f2 are unchanged (in the first implementation). The size of f1 isconstant, but its location evolves. Energy is dissipated and both elastic strain ee and QPE strain eQPE occur. The translation off1 during QPE deformation follows two-yield surface evolution rules to ensure that when a plastic state occurs (i.e. when f1

and f2 first make contact) the points r and r⁄ coincide and therefore that n and n⁄ are congruent (Lee et al., 2007). To assurethis, the following evolution rule is adopted (again adopting plane-stress conditions or deviatoric space for uniqueness underincompressive plasticity):

da ¼ dlðr� � rÞ ð7Þr� a

R1¼ r� � a�

R2ð8Þ

The consistency condition df1 = 0 leads to an explicit form of Eq. (7) as follows:

dahn : dri

n : ðr� � rÞ ðr� � rÞ ð9Þ

where the brackets in the term hn:dri denote the rule that hn:dri = 0 if n:dr 6 0, otherwise, hn:dri = n:dr.The relationship between stress increment and total strain increment (de = dee + deQPE) while in the QPE state is ex-

pressed as follows:

dr ¼ C0 : dee ¼ C : de ð10aÞde ¼ dee þ deQPE ð10bÞdee=kdeek ¼ deQPE=kdeQPEk ð10cÞ

where C is an apparent elastic + QPE stiffness tensor function, an explicit form for which relies on a varying ‘‘apparentYoung’s modulus’’, E, that represents the slope of the stress–strain curve in uniaxial tension (i.e. thus taking into accountelastic and QPE strain):

3 In the current work, a Chaboche-type model of plastic yield surface evolution is adopted, such that f2 translates and expands accordingly.



E ¼ E0 � E1 1� exp �bZkde� depk

� �� ð11Þ

where the integral is evaluated from the initiation of a new QPE loading process (i.e. at the moment when r arrives at surfacef1 from its interior, dr is outward (dr::dn > 0), and f1 and f2 are not in contact) to the current QPE state4. E0 is the traditionalYoung’s modulus for atomic bond-stretching and E1 and b are material parameters to be determined from measured unloadingand loading behavior. The form of Eq. (10) insures that at the transition between elastic and QPE straining the Young’s modulustakes the value E0 as the stress approaches from either side. Poisson’s ratio is assumed to remain constant, such that C dependsonly on E via Eq. (11). The explicit expression for C for isotropic elasticity and isotropic QPE model is therefore as follows:

Cijkl ¼mE

ð1þ mÞð1� 2mÞ dijdkl þE

ð1þ mÞ ðdikdjl þ dildjkÞ ¼EE0

� �C0ijkl ð12Þ

where dij is Kronecker delta and C0ijkl represents the Cartesian components of constant elastic tensor C0. Note that C is parallelto C0, which insures that dee and deQPE are parallel (Eq. (10c)).

In view of Eq. (10), an explicit form for the increment of QPE strain that occurs during plastic deformation, deQPE, is givenby

deQPE ¼ ðS � S0Þ : dr ð13Þ

where S and S0 are compliance tensors with components that are inverses of matrices representing tensors C and C0.In order to specify an exact form of the QPE model for testing, it is necessary to choose explicit forms that will in general

depend on the material. For the first version implemented here, the size of inner surface R1 is determined from a Voce func-tion (Voce, 1948; Follansbee and Kocks, 1988) of equivalent plastic strain:

R1 ¼ A1ð1� B1 expð�D1pÞÞ ð14Þ

Note that for initial tensile loading of a virgin material, a and a⁄ will be zero and R1 and R2 will represent the proportionallimit (elastic-QPE transition) and the yield stress (QPE-plastic transition). This means that the size of f1 will be considerablysmaller than that of f2, thus avoiding geometrical problems of the intersections of the two surfaces initially or later.

Fig. 7 shows linear–nonlinear transition stresses for loading and unloading vs. pre-strain for DP 980. The parameters in Eq.(14) have been determined based on these data. Alternatively, other forms such as Hollomon (Hollomon, 1945) and mixedHollomon/Voce models (Sung, 2010), or even a linear model could be utilized to describe the isotropic hardening behavior off1. However, it should also be noted that the exact transition stresses are largely a matter of judgment and the isotropic hard-ening of f1 in the plastic state is small (as shown in Fig. 7) and could be ignored with little error.

The evolution of f2 in the current implementation is according to a modification of the popular Chaboche model (Chab-oche, 1986). The evolution of yield surface back stress r⁄ is decomposed into two parts, a nonlinear term r�1 (Chaboche, 1986)and a linear term r�2 (Lee et al., 2007), as follows:

150

175

200

225

250

0 0.025 0.05 0.075 0.1

Size

of e

last

ic d

omai

n (M

Pa)

Plastic strain

UnloadLoad

QPE model

DP980-1.43

Fig. 7. Relationship between size of elastic domain (R1) and plastic strain, DP 980.

4 Note: The integral in Eq. (11) will in general differ for differing unloading paths because the integral begins when the stress first approaches f1 from inside.This effect is similar to that of possible plastic effects during unloading (Li and Wagoner, 1998; Li et al., 2002a; Khan et al., 2009).



a� ¼ a�1 þ a�2 ð15aÞ

da�1 ¼23

C1dep � ca�1dp ð15bÞ

da�2 ¼23

C2dep ð15cÞ

The incorporation of the linear term in the standard Chaboche model allows for a permanent offset of subsequent flow stressfollowing path reversals, such as are observed for some materials (Geng et al., 2002).

The isotropic hardening of f2 mirrors that of f1 with its own constants:

R2 ¼ A2ð1� B2 expð�D2pÞÞ ð16Þ

A numerical algorithm for implementing the QPE model to update r;R1;R2;a;a� for a specified strain increment (such as isneeded in an Abaqus/Standard UMAT subroutine) is outlined in the Appendix Fig. A1.

5. Determination of material parametric values

Parameters for the proposed QPE model, three standard elastic-plastic constitutive models and one special elastic-plasticmodel with 3 Chaboche backstress terms were determined by least-squares fitting (except as otherwise indicated) to thetensile data presented for DP 980. The results are shown in Table 2. All of the models incorporate isotropic elasticity andplasticity (von Mises yield function). The differences occur in the handling of loading to and unloading from a plastic state,and the plastic hardening law. Shorthand labels refer to constitutive approaches as follows:

0

200

400

600

800

1000

1200

0 0.025 0.05 0.075 0.1

Tru

e S

tres

s (M

Pa)

True Strain

DP980-1.43

Measured

QPE model

See Fig. 8b

See Fig. 8c

0

200

400

600

800

1000

1200

0.032 0.034 0.036 0.038 0.04 0.042

Tru

e S

tres

s (M

Pa)

True Strain

DP980-1.43

Chord model

Measured

QPE model

C0/3-Param

0

200

400

600

800

1000

1200

0.069 0.072 0.075 0.078

Tru

e S

tres

s (M

Pa)

True Strain

DP980-1.43

Chord model

Measured

QPE model

C0/3-Param

a b

c

Fig. 8. Comparison of experimental data and simulations for loading–unloading tests, DP 980. (a) Four-cycle test vs. QPE model prediction, overview. (b)Close-up of the second cycle in (a), along with simulations using other constitutive models. (c) Close-up of the fourth cycle in (a), along with simulationsusing other constitutive models.



QPE/Chaboche: QPE loading/unloading, modified Chaboche plastic hardening (2 backstresses)Chord/Chaboche: Chord modulus loading/unloading (chord modulus varying with plastic strain), modified

Chaboche plastic hardening model (2 backstresses)C0/Chaboche: Standard elastic constants Co, modified Chaboche plastic hardening (2 backstresses)Chord/Iso: Chord modulus loading/unloading (chord modulus varying with plastic strain),

isotropic plastic hardeningC0/3-Param: Standard elastic constants Co, a special 3-parameter Chaboche plastic hardening

model used to fit the nonlinear unloading behavior

The first four constitutive models are for comparative simulations of the draw-bend springback test for DP 980 steel. Thefifth constitutive model is aimed at simulating the unloading–loading behavior of DP 980 steel in tension using an elastic-plastic Chaboche model with 3 backstress evolution terms.

The parameters for the three states (elastic, QPE, and plastic) in the QPE model were determined using separate proce-dures and data, with results for DP 980 as shown in the QPE column of Table 2. The elastic properties (Eo, m) are standardhandbook values (ASM, 1989). A monotonic tensile test was used to establish the proportional strain hardening behaviorfor all plastic models using a true strain range of 0.02–0.11 (the uniform limit for DP 980). This is the only test that isrequired for fitting of the isotropic hardening plastic model.

The QPE properties (El, b, A1, B1, D1) were determined by the method of least-squares using the data shown in Fig. 2 withfinal standard deviation for all 4 cycles reported in Table 2. The constants El and b were determined using only the fourthunload–load cycle. The constants A1, B1, and D1 were fit using visually-identified transition points from linear to nonlinearbehavior upon loading and unloading, as presented in Fig. 7. Fig. 8 compare the overall fit of the QPE part of the model to theexperimental data. Note that although only the fourth cycle was used to fit the QPE parameters, the fit of the predictions for

Table 2Best-fit parameters of alternate constitutive models.

Parameters (unit) Type QPE/Chaboche Chord/Chaboche C0/Chaboche Chord/Iso C0/3-Param

E0 (MPa) Elastic 208000 208000 208000 208000 208000m Elastic 0.291 0.291 0.291 0.291 0.291E1 (MPa) QPE, Eq. (11) 117500b QPE, Eq. (11) 645A1 (MPa) QPE, Eq. (14) 385B1 QPE, Eq. (14) 0.587D1 QPE, Eq. (14) 3.96Std. dev. 4 cycles 15.6 88.8 162.1 88.8 57.7C1 (MPa) Chaboche Eq. (15) 37900 37900 38000 0c Chaboche Eq. (15) 105 105 100 0C2 (MPa) Chaboche Eq. (15) 1270 1270 1193 0A2 (MPa) Chaboche Eq. (16) 662 662 646 868B2 Chaboche Eq. (16) 0.183 0.183 0.209 0.412D2 Chaboche Eq. (16) 12.7 12.7 11.9 96.6ry (MPa) 541 541 511 510 234.5Std. dev. C–T test 12.4 12.7 14.6 13.5K (MPa) Chord Eq. (17) 56200 56200D3 Chord Eq. (17) 175 175Cp

1 ðMPaÞ 3-param Eq. (18) 76800

cp1

3-param Eq. (18) 127

Cp2 ðMPaÞ 3-param Eq. (18) 304000

cp2

3-param Eq. (18) 2360

Cp3 ðMPaÞ 3-param Eq. (18) 1190

Ap (MPa) 3-param Eq. (18) 456Bp 3-param Eq. (18) 0.485Dp 3-param Eq. (18) 0.4

Table 3Comparison of measured and simulated springback angles (in degrees) for the draw-bend springback test and various constitutive models.

Fb 0.3 0.6 0.8 0.9 hr0i CPU (h)

Experiment 63.5 53.9 45.9 37.5 0QPE (5 layer) 69.2 54.1 44.2 39.6 3.2 18–30 (1 CPU)QPE (7 layer) 61.0 49.3 41.8 37.8 3.3 9–14 (32 CPUs)QPE (10 layer) 63.4 51.9 42.2 37.3 2.1 21–35 (32 CPUs)Chaboche, E = 208 GPa 53.7 43.9 36.5 32.9 8.8 3–4 (1 CPU)Chaboche, Chord 74.1 56.1 48.4 43.6 6.3 4–6 (1 CPU)Isotropic Hard., Chord 90 71.8 58.4 52.4 18.7 4–5 (1 CPU)



other loops (the second loop is shown) is equally satisfactory. Fig. 7 compares the visually-identified transition stresses be-tween linear and nonlinear behavior for individual loading and unloading legs with the ones from the model fit.

The Chaboche plastic evolution parameters (C1, c, C2, A1, B1, D1) were determined by the method of least squares usingdata from the single standard tensile test (for the initial elastic-plastic transition and large-strain hardening) and two com-pression–tension tests with pre-strains of approximately 0.04 and 0.08, as shown in Fig. 9. (A third test with a reversal at0.06 absolute pre-strain as shown in Fig. 9 was not used for the fitting.) The yield stress ry was determined by curve fit,so there is some deviation from the value defined by a standard 0.2% yield offset definition. If any QPE straining occurs(i.e. if unloading follows any path except that linear elastic unloading according to elastic constants C0), the plastic hardeningcoefficients for the Chaboche model with or without QPE strains are slightly different, the differences being related to plasticstrain differences approximately 1/3 the magnitude of the elastic strains.

The parameters shown in Table 2 for the other constitutive models were found by fit, wherever possible, using the sameprocedures and data as for the corresponding states in the QPE/Chaboche model. Other parameters require introduction andpresentation of fitting procedures, as follows.

The chord modulus has a single value at a given plastic strain before unloading from a tensile state. It is found from Dr/Defrom two points for tensile unloading, one just before unloading and the other at zero applied stress. The discrete chord mod-uli found at pre-strains such as shown in Fig. 2 were fit to a continuous function of pre-strain as follows:

E ¼ E0 � Kð1� expð�D3pÞÞ ð17Þ

In order to test the idea that nonlinear unloading and reloading can be modeled by a generalized Chaboche model, param-eters for the evolution of two nonlinear backstress terms ap

1;ap2 and one linear backstress term ap

3, were fit using the proce-dures adopted for the two-parameter Chaboche model. That is, the 3-parameter Chaboche plastic evolution properties(Cp

1; cp1;C

p2; c

p2;C

p3;A

p;Bp;Dp) were determined by the method of least squares using data from the single standard tensile test,two compression–tension tests with pre-strains of approximately 0.04 and 0.08 and one unloading–reloading test at a pre-strain 0.02.

The new terms and equations take the following forms:

ap ¼ ap1 þ ap

2 þ ap3 ð18aÞ

dap1 ¼

23

Cp1dep � cp

1ap1dp ð18bÞ

dap2 ¼

23

Cp2dep � cp

2ap2dp ð18cÞ

dap3 ¼

23

Cp3dep ð18dÞ

Rp ¼ Apð1� Bp expð�DppÞÞ ð18eÞ

6. Comparison of QPE and other simulations with experiments

Fig. 8b and c, introduced earlier to illustrate the agreement of the QPE model with unloading–loading cycles, also showthat the chord model and 3-parameter Chaboche plastic model do not provide good predictions for loading–unloadingcycles. The 3-parameter Chaboche model in particular gives a significant variation from the reloading measurement. This

0

200

400

600

800

1000

1200

0 0.05 0.1 0.15 0.2 0.25

Abso

lute

Tru

e St

ress

(MPa

)

Accumulated Absolute True Strain

Monotonic tension

DP980-1.43

C-T tests

Fig. 9. Comparison of QPE model predictions with monotonic tension and compression–tension (C–T) tests.



is a result of reloading behavior in the nonlinear kinematic hardening model which must be similar to the original elastic-plastic transition in the uniaxial tensile test, regardless of the number of backstress terms used to reproduce these transi-tions. Therefore, the abrupt transition from elastic + QPE reloading to monotonic plastic loading cannot be described ade-quately using the same parameters as initial tensile loading or reverse loading (as shown in Fig. 9), in spite of fittingmany parameters to match the nonlinear unloading behavior.

Fig. 10 compares partial unloading cycles predicted by the QPE model fit independently of this experiment, and the mea-surement. The agreement captures the observed behavior qualitatively and quantitatively, with much less hysteresis than forfull unloading cycles.

As discussed previously, a plastic-like hardening effect of QPE strain can be added readily to the first implementation ofthe model by incorporating QPE strain into the isotropic hardening of yield surface in Eq. (16)

Rnew2 ¼ A2ð1� B2 expð�D2ðpþ tpQPEÞÞÞ ð19Þ

where pQPE is the equivalent QPE strain, defined here as dpQPE ¼ ð23 deQPE : deQPEÞ1=2 and t is an optional weighting parameter ofQPE hardening to plastic hardening rates. Fig. 11 shows that the ‘‘Modified QPE Model’’ gives a better agreement with themeasured four-cycle loading unloading test than original one for an assumed value of t = 1. A larger value of t would allowa slightly better match with the data at large strains, but the overall reproducibility of stress magnitudes from test-to-test ofDP 980 do not justify fitting this parameter.

Table 3 compares the springback angles for the draw-bend tests and simulations for DP 980. The standard deviations werecalculated as follows:

hri ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1ðhmodel � hexptÞ2

N

sð20Þ

where hmodel and hexpt are the simulated and experimental springback angles, respectively. N = 4, the number of results com-pared. The results in Table 3 are for the original 5-layer mesh, except for the two rows using refined meshes for mesh sen-sitivity tests as described in the Experimental Procedures section. The QPE model shows the best overall agreement withmeasured data among the four constitutive models. The standard deviation of the QPE/Chaboche model is one half that ofChord/Chaboche model (typically the most sophisticated / complex model usually applied to springback prediction), indicat-ing that unloading in some elements after draw-bending must be taking place to non-zero residual stresses (as expected andas verified by Fig. 12). It is this partial internal unloading, or even reverse internal loading upon release of external loads thatdemands use of a model such as QPE to account for nonlinear unloading effects consistently. Comparison of Fig. 12a and billustrates the fact that the loaded state of the drawbend-springback specimen is not affected significantly by the QPE model,but rather only the unloading behavior.

As simpler (and more standard) constitutive models are adopted, the simulation predictions become progressively lesssatisfactory. Moving from the Chord/Chaboche model to the C0/Chaboche model (in effect using the atomic bond-stretchingmodulus) increases the prediction error by approximately 30% (as would be expected by ignoring the QPE strains observed intension). The last comparison, Chord/Chaboche with Chord/Iso, shows the effect of ignoring the Bauschinger effect andsubsequent transient hardening upon strain reversals. In this case, the error of Chord/Iso is 3 times that of Chord/Chaboche(Table 3).

0

200

400

600

800

1000

1200

0.03 0.04 0.05 0.06

True

Str

ess

(MPa

)

True Strain

DP980-1.43

QPE model

Measured (loading)

Measured(unloading)

Fig. 10. Comparison of QPE model predictions with partial unloading–reloading cycles.



7. Conclusions

A series of experiments for two dual-phase steels was conducted to reveal the nature of nonlinear unloading from a plas-tic state. Those one-dimensional results were used to derive a consistent three-dimensional constitutive model that wasthen implemented in Abaqus Standard via UMAT and compared with existing models for springback. The following conclu-sions were reached:

� A new kind of strain component, here called QPE strain, was identified as having the following characteristics: it is recov-erable (like elastic strain) but is energy dissipating (like plastic strain). Its existence accounts for the hysteresis loopsobserved during unloading–loading cycles following tensile deformation.� QPE strain is, to a first approximation, strain rate (over the measured range of _e ¼ 10�4 � 10�2=s, at least) independent,

and its magnitude is proportional to the flow stress or elastic strain. Remarkably, the same stress–strain behavior isobserved for two microstructurally and strength distinct alloys (DP 780 and DP 980) when compared when unloadingfrom the same flow stress.� The observed deviation from linearity for DP 980 steel is larger than reported for other steels, a reduction of up to 30% of

the apparent Young’s modulus (average chord modulus from plastic state to zero stress) was measured.� A two-surface constitutive model (QPE Model) was derived with an inner surface to describe elastic-QPE transitions,

including explicit evolution laws for DP 980. The outer surface and its evolution can come from any plastic constitutivemodel; a Chaboche-type model was used here.� The QPE model reproduces tensile experiments within the experimental scatter (including experiments not used for fit-

ting parameters). In the first implementation, QPE strain does not cause plastic hardening. An alternate form was alsointroduced to treat QPE strain as inducing plastic hardening.

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

True

Stre

ss (M

Pa)

True Strain

DP980 -1.43

Measured

QPE

Modified QPE

Fig. 11. Comparison of a modified QPE model (incorporating plastic hardening by QPE straining) with the QPE model and experimental measurements.

-1000

-500

0

500

1000

1500

-0.715 -0.3575 0 0.3575 0.715

Thickness (mm)

Tang

entia

l str

ess

(MPa

)

QPE/Chaboche

Chord/Chaboche

C0 /Chaboche

DP 980-1.43F

b=0.6 σ

y

-400

-200

0

200

400

600

800

-0.715 -0.3575 0 0.3575 0.715Thickness (mm)

Tang

entia

l str

ess

(MPa

)

QPE/Chaboche

Chord/Chaboche

C0/Chaboche

DP 980-1.43F

b=0.6 σ

y

a b

Fig. 12. Tangential stress distributions through the thickness. (a) After forming, under load. (b) After springback, unloaded.



� The QPE model significantly improves springback prediction by capturing the nonlinear nature of unloading stress–strain.It reduces springback prediction error by a factor of 2 as compared with a chord model, i.e. one suggested widely in theliterature based on substituting an apparent unloading modulus for the handbook / atomic bond stretching elasticmodulus.� A standard Chaboche-type plastic model, even with multiple backstress evolutions, cannot be fit to reproduce the QPE

effect adequately.

Acknowledgements

This work was supported cooperatively by the National Science Foundation (Grant CMMI 0727641), the Department ofEnergy (Contract DE-FC26-02OR22910), the Auto/Steel Partnership, the Ohio Supercomputer Center (PAS-080), and theTransportation Research Endowment Program at the Ohio State University. The authors would also like to thank Dr. KwansooChung, Seoul National University, for many helpful discussions.

Appendix A. Numerical algorithm for QPE model

The numerical implementation of the QPE model via UMAT in Abaqus Standard is shown schematically in figure AppendixA. The following steps were followed:

(a) Given the total strain increment De, the trial stress is calculated

rtrialnþ1 ¼ rn þ Cn : De ðA1Þ

where Cn is the modulus at step n.Check the stress state: pure elasticity, QPE or plastic state.if following equation is satisfied

f2 ¼ /2ðrtrialnþ1 � a�nÞ � R2 > 0 ðA2Þ

Then the stress is in the plastic state. Chaboche model is applied (where a�n is the current back stress of yield surface).

input nσ , Δε , , ,e p QPEn n nε ε ε

trial Stress

1 :trialn n n+ = + Δσ σ C ε

check if 1trialn+σ is inside

the yield surface

Yes

check if 1trialn+σ is outside

the inner surface

plastic state update 1n+σ with

Chaboche model

NO

Yes

QPE state update 1n+C and 1n+σwith QPE model

0n =C C

1 :trialn n n+ = + Δσ σ C ε

NO

check if 0n =C C

elastic state

1 1trial

n n+ +=σ σ

Yes

NO

output 1n+σ , 1 1 1, ,e p QPEn n n+ + +ε ε ε

Fig. A1. Flow Chart of the numerical implementation scheme for the QPE model.



else if

f2 ¼ /2ðrtrialnþ1 � a�nÞ � R2 6 0 ðA3Þ

The stress state is either in pure elasticity or QPE state. (go to (b))(b) When Eq. (A3) is satisfied and

if f 1 ¼ /1ðrtrialnþ1 � anÞ � R1 < 0 ðA4Þ

The stress state is pure elastic (where an is the current back stress of inner surface). Update the stress and back stress insidethe inner surface.

rnþ1 ¼ rtrialnþ1

anþ1 ¼ anðA5Þ

if f 1 ¼ /1ðrtrialnþ1 � anÞ � R1 > 0 ðA6Þ

The stress is in the QPE state. (go to (c))(c) First update the QPE stiffness Cn+1 with Eqs. (11)

Update the stress

rnþ1 ¼ rn þ Cnþ1ðe; eQPEÞ : De ðA7Þ

where Cn+1 is a function of e, eQPE, the two point Gaussian quadrature is used to obtain the average Cn+1(e, eQPE).Update back stress an+1 in QPE state.From Eqs. (7)–(9)

anþ1 ¼ an þ DlðR2 � R1

R1rn �

R2

R1an þ �anÞ ðA8Þ

Assume von Mises function for the inner surfaceffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

X1�X

2

2þ

X1�X

3

2þ

X2�X

3

2þ 6

X2

4þX2

5þX2

6

� �s¼ R1 ðA9Þ

From (A8)

Xi

�X

j

!2

¼ ðyij � DlzijÞ2 ði; j ¼ 1;2;3Þ ðA10Þ

Xk

!2

¼ ðyk � DlzkÞ2 ðk ¼ 4;5;6Þ ðA11Þ

where

yij ¼ ½rnþ1ðiÞ � rnþ1ðjÞ� � ½anðiÞ � anðjÞ� ði; j ¼ 1;2;3Þ ðA12Þyk ¼ rnþ1ðkÞ � anðkÞ ðk ¼ 4;5;6Þ ðA13Þzij ¼ bðiÞ � bðjÞ ði; j ¼ 1;2;3Þ ðA14Þzk ¼ bðkÞ ðk ¼ 4;5;6Þ ðA15Þ

b ¼ R2 � R1

R1rn �

R2

R1an þ �an ðA16Þ

Substitute (A10) and (A11) into (A9)

aDl2 þ bDlþ c ¼ 0 ðA17Þ

a ¼ z212 þ z2

13 þ z223

� �þ 6 z2

4 þ z25 þ z2

6

� �b ¼ �2ðz12y12 þ z13y13 þ z23y23Þ � 12ðz4y4 þ z5y5 þ z6y6Þc ¼ y2

12 þ y213 þ y2

23

� �þ 6 y2

4 þ y25 þ y2

6

� �� 2R2

1

ðA18Þ

Substitute the solution of (A17) into (A8), an+1 is obtained.We can decrease the step from an to an+1 to increase the accuracy, like

rg ¼ rn þðrnþ1 � rnÞ

Ng ðA19Þ

where g is a integer from 1 to N. N = 100 is used in this paper.



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