Experimental evaluation and constitutive modeling of non-proportional deformation for asymmetric steels

International Journal of Plasticity 27 (2011) 82–101

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International Journal of Plasticity

journal homepage: www.elsevier .com/locate / i jp las

Experimental evaluation and constitutive modelingof non-proportional deformation for asymmetric steels

Rahul K. Verma a,b, Toshihiko Kuwabara b,*, Kwansoo Chung c, A. Haldar a

a Research and Development, Tata Steel Limited, Jamshedpur 831001, Indiab Division of Advanced Mechanical Systems Engineering, Graduate School of Tokyo University of Agriculture and Technology, 2-24-16, Naka-cho,Koganei-shi, Tokyo 184-8588, Japanc Department of Material Science and Engineering, Research Institute of Advanced Materials, Seoul National University, 599, Kwanak-rho, Kwanak-ku,Seoul 151-742, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 August 2009Received in final revised form 31 March 2010Available online 9 April 2010

Keywords:Two-stage formingNon-proportional deformationTension–compression asymmetryCombined isotropic–kinematic hardening

0749-6419/$ - see front matter � 2010 Elsevier Ltddoi:10.1016/j.ijplas.2010.04.002

* Corresponding author.E-mail address: [email protected] (T. Kuwa

Recently non-proportional deformation has received increased attention from researchersworking in the area of experimental and computational modeling of metal deformation.However, most of them are numerical in nature with limited experimental data available,making it further difficult to model non-proportional deformation. In the present work,two-stage uniaxial tests, along with uniaxial cyclic and biaxial tests for different stressratios, have been performed to evaluate deformation behavior of ultra-low carbon highstrength automotive steel. Behaviors like cross-effect and hardening stagnation, whichare attributed to the evolution of complex dislocation structures, were observed in thissteel. It was also noticed that this steel exhibits tension–compression asymmetry. As forconstitutive modeling, a modified asymmetric yield function is proposed to be used witha combined isotropic–kinematic hardening model. Also methods to account for the harden-ing stagnation during reverse loading and the cross-effect during two-stage deformationare proposed. The resulting constitutive model showed reasonably good agreement withexperimental results.

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1. Introduction

High strength and advanced high strength steels are finding wide applications in the automotive industry. The main moti-vation for such increased usage is to reduce vehicle weight with improved safety. However, such steels have their own draw-backs like lower formability and large springback. If such problems are properly predicted in advance, it may be possible for adie designer to take remedial action during the die design stage. However, for predicting such behaviors effectively, it is re-quired to evaluate the deformation behavior under non-proportional deformation conditions and develop constitutive equa-tions suitable for predicting the observed behaviors (Hill, 2000; Kuwabara, 2007).

In-plane uniaxial tension–compression cyclic tests were performed by Kuwabara et al. (1995, 2009), Yoshida et al. (2002),Lee et al. (2005), Boger et al. (2005) and Cao et al. (2009). Yoshida et al. (2002) performed the tests by sandwiching a numberof specimens together whereas Lee et al. (2005) and Boger et al. (2005) used solid flat plates as buckling constraints.Kuwabara et al. (1995, 2009) performed tension–compression tests using comb-shaped dies whereas Cao et al. (2009)designed a special wedge type fixture for carrying out tension–compression tests on thin sheets. It has been commonlyobserved, during uniaxial cyclic tests, there is early reverse yielding (the Bauschinger effect) accompanied by sharp initial

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R.K. Verma et al. / International Journal of Plasticity 27 (2011) 82–101 83

hardening (the transient behavior). Also, the reverse stress–strain curves quite often do not converge to the original mono-tonic loading curve, leaving a permanent gap, often referred to as hardening stagnation (or permanent softening).

Though many researchers have performed in-plane tension–compression tests, very few have actually carried out exper-iments for evaluating the effect of more general strain path changes on the deformation behaviors of sheet metals. Doucetand Wagoner (1989) carried out tension tests along the transverse direction on samples pre-strained along the rolling direc-tion under plane strain condition. Hiwatashi et al. (1997, 1998) carried out non-proportional biaxial loading experiments forthe evaluation of crystal plasticity based numerical models. Kuwabara et al. (2002) performed experiments involving abruptstrain path change to determine the yield surface in the vicinity of the path change point without any influence of interme-diate elastic unloading and were successful to detect a yield vertex at a loading point. Hu et al. (1992), Thuillier and Rauch(1994), Peeters et al. (2000) and Bouvier et al. (2005) employed either shear followed by shear or tension followed by sheardeformation, in two stages, for evaluating the deformation behaviors. Reil and Boogaard (2007) and Noman et al. (2010) per-formed tension-shear tests with continuous strain path change with increasing amounts of sharpness. The material chosenby them were IF steel and LH800, respectively, and it was reported that the cross-effect is present for sharp change in thestrain path. For gradual change the cross-effect was absent. Khan et al. (2009a,b,c) carried out a detailed study for measuringthe evolution of yield surfaces for both proportional and non-proportional loading conditions using a series of torsion–ten-sion and tension–bulge tests on cylindrical hollow specimen prepared from low work hardening and high work hardeningaluminum alloys. They observed that the initial yield surface was closer to von Mises surface; however, subsequent yieldsurfaces showed distortion, expansion and positive cross-effect and a ‘‘nose” in the loading direction.

As to constitutive modeling of deformation behavior of sheet metals, most sheet metal forming simulations in industryuse phenomenological plasticity, in which the elastic region is bounded by a convex surface. This surface, referred to as theyield surface, represents the onset of plastic deformation. With further deformation, after the onset of plastic deformation, amaterial strain-hardens as the dislocation density increases, while the initial crystals get distorted and re-oriented. Becauseof the strain hardening and crystals distortions, re-orientations and other micro-mechanisms, further evolution of the yieldsurface takes place in its size and shape as well as in its positioning in the stress space. However, for simplification, isotropichardening is most commonly assumed, in which only the yield surface size change is modeled. This assumption is reasonablyeffective for near monotonous and proportional loading. If, along with the size change, the position change of the yield sur-face (kinematic hardening) is also considered, uni-directional cyclic deformation, such as early reverse yielding accompaniedby sharp initial hardening, can also be modeled. However, even such models (based on the combined isotropic–kinematichardening models) may not be effective enough to account for more general non-proportional and non-monotonous defor-mations as discussed above.

To phenomenologically model the Bauschinger effect and the transient behavior during tension–compression (or com-pression–tension), several basic models have been proposed considering kinematic hardening, as comprehensively reviewedby Khan and Huang (1995). The models proposed can be broadly classified as single-surface models (Prager, 1956; Ziegler,1959; Chaboche and Rousselier, 1983; Chaboche, 1986, 1991) and two-surface models (Dafalias and Popov, 1976; Krieg,1975) or multi-surface models (Mroz, 1967). The Chaboche model uses a single yield surface; therefore, its computationis cost effective and subsequently many modified non-linear kinematic hardening models were proposed based on the Chab-oche model (Ohno and Wang, 1993a; Geng and Wagoner, 2000; Chun et al., 2002; Chung et al., 2005).

It has been found that the combined isotropic–kinematic hardening models based on the Chaboche type model, withoutany modification, generally saturate to the original isotropic hardening model and are unable to simulate the hardening stag-nation (Kim et al., 2006). To simulate the work hardening stagnation during reverse loading, Hu et al. (1992) proposed a uni-axial constitutive model; however, this model is not suitable for modeling the transient behavior (Yoshida and Uemori, 2002)and it takes non-standard tests to identify material parameters, making it less popular in industry. Yoshida and Uemori(2002) and Lee et al. (2007) proposed constitutive models based on two-surface plasticity, making them computationallymore expensive compared to single surface models. Peeters et al. (2000) and Rauch et al. (2007) proposed models basedon crystal plasticity to simulate the work hardening stagnation.

Another important behavior observed for materials deforming in twinning modes, like magnesium alloys (Lou et al.,2007; Lee et al., 2008) and pure titanium (Ishiki et al., 2008), is that the flow stress in tension is not identical to that in com-pression, resulting in an asymmetric yield surface. In the literature, this is commonly referred to as strength differential ef-fect (SDE). In the studies by Spitzig and co-workers (Spitzig et al., 1975, 1976; Spitzig, 1979), the SDE was seen as a directconsequence of the dependence of the flow stress on the hydrostatic stress in the specimen. Kuroda and Kuwabara (2002)and Kuroda (2003) numerically simulated an increase in flow stress under superimposed hydrostatic pressure, as well asSDE, using rate-dependent crystal plasticity theory with non-Schmid effects. Similar effects have also been reported recentlyfor other cubic materials like for copper alloys (Kuwabara et al., 2009) and for tungsten (Ripoll et al., 2009).

Several attempts were made to model the asymmetric yield behavior (Teodosiu and Hu, 1998; Iyer and Lissenden, 2003;Cazacu and Barlat, 2004; Lee et al., 2008; Bai and Weirzbcki, 2008; Plunkett et al., 2008; Cazacu et al., 2009). Hosford (1966)proposed a modified Hill’s (1948) yield function by incorporating linear terms whereas Iyer and Lissenden (2003) proposed aJ2J3 class material model. Teodosiu and Hu (1998) proposed an anisotropic hardening model involving three internal vari-ables: the first describing the directional strength of planer dislocation structures, the second for its polarity and the thirdwas a back stress. The selection of the internal variables was motivated by the contribution of dislocation structures and ini-tial texture to anisotropic hardening. Cazacu and Barlat (2004) proposed an anisotropic yield criterion in terms of the gen-eralized second and third invariants of stress deviator, which involves a unique parameter that captures

84 R.K. Verma et al. / International Journal of Plasticity 27 (2011) 82–101

tension–compression asymmetry. This parameter depends on anisotropy coefficients and yielding in tension and compres-sion. Bai and Weirzbcki (2008) proposed a model incorporating both pressure sensitivity and Lode parameter dependence.Plunkett et al. (2008) extended the criterion proposed by Cazacu et al. (2006) using multiple linear transformations on thestress deviator to describe the tension–compression asymmetry in yielding and r-values; however, it should be noted that itmakes any problem computationally more expensive. Cazacu et al. (2009) derived an analytic anisotropic strain rate poten-tial, which can describe both the anisotropy and tension–compression asymmetry of the plastic response of textured metalsand was the exact work-conjugate of the anisotropic stress potential of Cazacu et al. (2006) when the degree of homogeneitywas a = 2.

The prediction accuracy of hardening behaviors, mainly those observed during non-proportional loading, has been lim-ited partly because of lack of constitutive model but mainly because of limited experimental data. It is clear that not muchwork has been done on experimental evaluation of deformation behavior during non-proportional deformation which makesdevelopment of mathematical models for such cases restricted. Regarding material models, there have been attempts withreasonable success; however, most of them were verified for (near) proportional loading conditions, in which strain pathswere quite often similar to those used to characterize the material. More importantly, most of the models represented justone or two observed deformation behaviors, viz., strength differential effect or Bauschinger effect with permanent softening.For multi-path deformations, which involve major deformation path changes, the above models may or may not performsatisfactorily.

In the present work, in an effort to investigate the deformation behavior and also to validate material models for casesinvolving major deformation path changes, two-stage deformation tests involving four different combinations of deforma-tion modes were performed on an ultra-low carbon steel (an interstitial free high strength steel commonly used for exteriorpanels of automobiles). Besides, tension–compression–tension (TCT), compression–tension–compression (CTC) and biaxialtests for various stress ratios were also performed, hence covering a wide range of deformation path changes. It was observedthat the present material exhibits the tension–compression asymmetry as well as the hardening stagnation and the cross-effect. Since most of available yield criteria are symmetric, a non-symmetric yield criterion was proposed here by modifyingthe criteria proposed by Hill (1948) and Hoffman (1967). The combined isotropic–kinematic hardening model proposed byChung et al. (2005) along with the present yield criterion properly predicted asymmetry in yielding. As for the hardeningstagnation, Ahn et al. (2009) recently proposed a method to model the hardening stagnation by modifying the kinematichardening part of the isotropic–kinematic hardening model; however, the method based on kinematic hardening only wouldnot be general enough; therefore, a general approach involving both kinematic and isotropic hardening was discussed in thiswork. The cross-effect was also accounted for in the proposed model.

2. Experiments

The test material used in this study was an ultra-low carbon interstitial free high strength steel with 1.0 mm thickness.Except the uniaxial tension tests (in which an extensometer was used), strains were measured using strain gauges (TokyoSokki Kenkyujo Co., Japan). Two specimens were tested in each case and, if the two test results were not close enough, thentests were further repeated.

2.1. Uniaxial monotonic tension tests

Fig. 1 shows the dimension of the specimen for the monotonic tension tests. Clip-on type extensometer with 50 mmgauge length for longitudinal strain measurements and transverse averaging extensometer for transverse strain measure-ments were used. The tests were carried out under constant cross-head speed of 3.0 mm/min. The tests were performedon an universal testing machine, Instron 5582, for samples aligned along 0�, 45� (diagonal) and 90� to the rolling direction.

Fig. 1. Geometry of the specimen for the uniaxial tensile test (unit: mm).


2.2. In-plane tension–compression–tension, compression–tension–compression and monotonic compression tests

Sets of in-plane tension–compression–tension (TCT) and compression–tension–compression (CTC) tests were carried outfor evaluating the deformation behavior of the sheet under cyclic loading. Fig. 2 shows the specimen dimension used for theTCT and CTC tests. Tests were performed on specimens prepared along rolling direction, using the testing apparatus de-scribed in Kuwabara et al. (2009), for different amounts of pre-strains. Since buckling of the specimen is a major problemwhile carrying out compression tests on thin samples, this testing apparatus uses specially designed comb-shaped diesfor applying a small blank holding pressure (2–3 MPa) to a sheet specimen to prevent buckling of the specimen. In orderto prevent the specimen from galling the dies, it was lubricated on both sides with Vaseline and Teflon sheets. A run withoutthe sample is carried out to measure the friction in the die assembly, which is later compensated in the results by subtracting(or adding) it from measured load values. Besides, monotonic uniaxial compression tests were also performed in rolling andtransverse directions to evaluate the tension–compression asymmetry. The tests were carried out under a displacement con-trolled mode and two different speeds were used: in the elastic regions, the test speed was 0.5 mm/min whereas, in the plas-tic region, it was 1.5 mm/min.

2.3. Two-stage uniaxial tests

Two-stage uniaxial tests were performed to evaluate the hardening behavior of the test material subjected to abruptstrain path change. Four different types of strain path changes were tried as listed below and schematically shown in Fig. 3:

� Tension 0.02 and 0.05 followed by tension 0.1 in the transverse direction.� Tension 0.02 and 0.05 followed by compression 0.06 in the transverse direction.� Compression 0.02 followed by tension 0.1 in the transverse direction.� Compression 0.02 followed by compression 0.06 in the transverse direction.

The first tensile loading was conducted on large specimens, 290 mm long and 65 mm wide, prepared along the rollingdirection. The first compressive loading was conducted on specimens, 190 mm long and 65 mm wide. To ensure the uniax-iality of stress, the length (between the grips) to width ratios was taken to be 3.4 for tension and 2.6 for compression. It wascross-checked with finite element simulation for uniaxial compression. In the simulation, for a compressive strain of 0.025,the ratio of width stress to longitudinal stress was 0.01, which is very low suggesting the uniaxiality of the state of stress.

The uniaxiality in the bigger specimen was cross-checked by comparing the stress–strain data on the bigger specimenwith those measured using a standard specimen. Also the specimens for the second loading were prepared at a location suf-ficiently away from the grips as the deformation is close to plane strain near the grips. The second loading was always con-ducted in the transverse direction of the specimen already pre-strained in the rolling direction. Fig. 4 shows a schematicdiagram showing the dimensions and directions of the samples used in the first and the second loading.

2.4. Biaxial tensile test

Biaxial tensile tests were performed to measure the plastic deformation behavior of the test material and to compare theperformance of different yield criteria. Fig. 5 shows the geometry of the cruciform specimen used for biaxial tensile testswhich is the same as those used in Kuwabara et al. (1998, 2000, 2002). Each arm of this specimen has seven slits, each with60 mm in length and 0.2 mm in width, with 7.5 mm interval so as to avoid geometric constraint on the deformation of thesquare gauge section. The slits were cut by laser-cutting. The rolling and transverse directions of the test material are alignedalong the x- and y-axes, respectively. Kuwabara et al. (1998) provide details regarding the servo-controlled biaxial tensiletesting apparatus.

Fig. 2. Geometry of the specimen for the uniaxial cyclic tests (unit: mm).

Fig. 3. Combinations of stress paths used in the two-stage uniaxial tests.

Fig. 4. Schematic diagram showing the dimensions and directions of the samples used in the first and second loading (for tension, L = 290 mm andW = 65 mm, for compression, L = 190 mm and W = 65 mm) (unit: mm).


Biaxial tension was applied to cruciform specimens with nominal tensile stresses, rNx and rNy, in fixed proportions:rNx:rNy = 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, and 1:4. For rNx:rNy = 1:0 and 0:1, standard uniaxial tensile specimens were used. Toevaluate the work hardening behavior of the test material under biaxial tension, the notion of the contour of plastic workin the stress space was used (based on the plastic work equivalence principle). This was because, in the simulation of metalforming, modeling the flow stress as an average behavior of a material over a deformation range is more important thandetermining an initial yield locus of the material. The method for constructing contours of plastic work is as follows. A uni-axial tension test in the rolling direction of the material is first conducted, and the true uniaxial stress r0 and plastic work Wdissipated per unit volume are determined for particular values of the uniaxial logarithmic plastic strain ep

0. Next, groups oftrue stress points, rx:ry, for which the same amount of plastic work W is required, are plotted in the principal stress space toconstruct contours of plastic work corresponding to particular values of ep

0. When ep0 is taken as sufficiently small, the cor-

responding work contour can practically be viewed as a yield locus.

Fig. 5. Cruciform specimen used for the biaxial tensile test.


3. Results and discussions on experiments

3.1. Uniaxial monotonic tension and compression tests

Fig. 6 shows uniaxial tension test results in the rolling, transverse and diagonal directions whereas Table 1 shows theaverage tensile properties in 0�, 45� and 90� to the rolling direction. The normal anisotropy (r-) value was calculated between5% and 15% engineering strains whereas the strain hardening exponent (n-value) was calculated between 5% and just beforethe ultimate tensile strength (UTS) point. All the hardening curves were very close to each other even though there was asignificant directional difference in r-values.

Fig. 7 shows the uniaxial compression test results in the rolling and transverse directions. It is noted that the compressiveflow stress in the transverse direction is larger than that in the rolling direction whereas tensile flow stresses are almostidentical to each other in both directions. Note that the tensile and compressive flow stresses are also almost identical toeach other in the rolling direction. The asymmetric behavior in tension and compression is often observed in HCP metals,which is well known to be caused by twining. However, it is rare for materials whose tension and compression mainly in-volve dislocation slip, such as steels, though, recently for copper alloys (Kuwabara et al., 2009) and for tungsten wires (Ripollet al., 2009), such behavior has been reported.

3.2. In-plane TCT and CTC tests

Figs. 8 and 9 show the results of the TCT and the CTC tests, respectively, carried out for different amounts of pre-strain:five pre-strains, 0.02, 0.04, 0.06, 0.08 and 0.1, for TCT tests and two pre-strains, 0.04 and 0.08, for CTC tests. The stress–straincurves during reverse loading do not converge to the original monotonic hardening curve, leaving a permanent gap often

Fig. 6. Uniaxial tension test results in rolling, transverse and diagonal directions.

Table 1Uniaxial tension test data.

YS (MPa)(0.2% offset)

UTS (MPa) Strain hardening exponent (n) Normal anisotropy(r-value, 5–15%)

%El (at 50 mm gauge length)

Rolling (0�) 196 362 0.235 1.59 43.0Diagonal (45�) 193 359 0.238 1.52 43.5Transverse (90�) 192 358 0.241 1.23 42.4

Fig. 7. Uniaxial compression test results in rolling and transverse directions.


referred to as work hardening stagnation (or permanent softening). It happens because of the complex dislocation micro-structure evolution. As explained by Peeters et al. (2000), during reverse loading, first the dislocation annihilation (cancel-lation of opposite sign dislocation) takes place rapidly creating early re-yielding and sharp hardening rate (transientbehavior). Following this, the dislocation cell structure, which is developed during previous (moderate to high) deformation,is broken and the distribution of dislocation becomes more homogeneous causing the hardening stagnation. Once all the dis-location cells are broken, creation of new dislocation cells causes a resumption of the positive rate of hardening. Thoughthere was hardening stagnation for all the pre-strain values, significant work hardening stagnation was observed here whenpre-strains were equal to or greater than 0.06, since the creation of dislocation cell might have taken place only when therewas moderate to high amount of deformation. It should be noted that Rauch et al. (2007) attributed the reason for suchbehavior to simply a complex evolution of dislocation density rather than breakage and creation of dislocation cells.

Fig. 10 shows the TCT and CTC test results for 0.04 and 0.08 pre-strains, in which the CTC curves have been rotated by180� for comparison purpose. No significant strength difference between tension and compression was observed in the roll-ing direction.

Fig. 8. True stress–true strain curves measured in the TCT tests for five pre-strains.

Fig. 9. True stress–true strain curves measured in the CTC tests for two pre-strains.

Fig. 10. Comparison of measured stress–strain curves for the TCT and CTC tests.


Many researchers have reported the decrease in the unloading modulus with increased plastic deformation and non-lin-earity in the unloading modulus and was observed in the present work as well; however, the average unloading moduluswas calculated between the two points, one at 95% of the stress value from where unloading starts and the other at 10%and is presented in Table 2.


Fig. 11 shows the tensile stress–strain curves measured during the second loading; the curves are offset by the amounts ofpre-strains already applied in the first loading. In addition, the monotonic uniaxial tension test result in the transverse direc-tion is also depicted in the figure. Fig. 12 shows the compressive stress–strain curves measured in the second loading withthe monotonic uniaxial compression test result in the transverse direction. In both figures, initially during cross-loading, theflow stress overshoots the original monotonic stress–strain curve (commonly referred to as cross-effect), except the onlycase (compression followed by compression in the transverse direction). With increasing the strain further, the rate of hard-ening becomes stagnant and then finally converges to the monotonic stress–strain curve (the amount of deformation afterwhich it converges to the monotonic stress–strain curve depends on the amount of pre-strain). The reason for overshootingmay be attributed to the dislocation cell structure created during the first loading (Peeters et al., 2000, 2001). During thesecond loading, which is orthogonal to the first loading direction, newly generated dislocations have to cut through the

Table 2Average unloading modulus for various tensile pre-strains.

Pre-strains 0.02 0.04 0.06 0.08 0.10Unloading modulus (GPa) 163 163 162 159 157

Fig. 11. Test results measured in the second loading in the two-stage uniaxial tests with various pre-strains.

Fig. 12. Uniaxial compression test results measured in the second loading in the two-stage uniaxial tests with various pre-strains.


dislocation cell structure, causing higher initial flow stress. When a channel is cut through, further dislocation movementbecomes (through this channel) easier and the hardening becomes stagnant for a while. On further deformation, the materialforgets the deformation history during the first loading and the work hardening curve converges to the original monotonicstress–strain curve. The final convergence of the stress–strain curve to the original monotonic curve was true for all the sixcases considered in the present study; however, it is not known whether the same will be true for more general deformationpath changes (and also for other materials) or not. Also the reason for the only exception case (compression followed bycompression in the transverse direction) is not clear.

Finally, the results of the T–C–T (and C–T–C) tests and the two-stage uniaxial tests of the current steel alloy are summa-rized as follows. When the loading direction changes by 180� there is strain hardening stagnation (permanent softening)whereas in case of the direction change by 90� there is no softening but there is cross-effect followed by stagnation for awhile before the hardening curve converges to the original hardening curve.

3.4. Biaxial tensile tests

Fig. 13 compares the measured contours of plastic work with the calculated yield loci based on Hill’s quadratic (Hill,1948), Yld2000-2d (Barlat et al., 2003) and the present yield functions under the plane stress condition. Details will be dis-cussed in Section 4.2.

4. Constitutive modeling

In order to account for the Bauschinger and transient behaviors as well as permanent softening and the asymmetry intension and compression along with the cross-effect, an improved constitutive law was proposed by modifying the

Fig. 13. Yield surfaces for plastic strains 0.002, 0.02 and 0.04.


combined isotropic–kinematic hardening model proposed by Chung et al. (2005). The combined hardening model proposedby Chung et al. (2005) is briefly reviewed here to make discussions complete.

4.1. Combined isotropic–kinematic hardening law

The yield function evolution law based on the combined isotropic–kinematic hardening law can be represented by

f ðr� aÞ � �rmiso

�eð Þ ¼ 0 ð1Þ

where f is an mth order homogeneous yield function, a is the back stress and �riso is the effective stress which represents thesize of the yield surface and also the yield surface shape in the stress field. Note that �riso is defined in Eq. (1) for the stressalready translated by the back stress a. Therefore, �riso is obtained from the initial effective stress by replacing r with r � a,while its conjugate effective plastic strain increment, d�e, is obtained from the following plastic work equivalence principle:

ðr� aÞ � dep ¼ �risod�e ð2Þ

Here, r is the Cauchy stress and dep is the plastic strain increment.The back stress describes the translation of the yield surface in the stress field and the Bauschinger and the transient

behavior during reverse loading can be described by the proper definition of the back stress evolution law. Chaboche(1986, 1991) introduced a non-linear kinematic hardening model by adding an additional term to the linear kinematic hard-ening model proposed by Ziegler (1959) to describe the transient behavior, which was further modified by Chung et al.(2005). In the modified Chaboche model, the back stress increment is assumed to be composed of two terms, da = da1 � da2;i.e.,

da ¼ d�a1

d�ed�e

� �ðr� aÞ

�risoþ da2

d�ed�e

� �a ð3Þ

where d�a1 is the effective back stress increment obtained from the initial effective stress by replacing r with da1. If, for sim-plicity, the simple tension in the rolling direction of the sheet is considered as a reference stress state, Eqs. (1) and (3) alongwith the plastic work equivalence principle lead to

drx ¼ d�a1 � da2ax þ d�riso ð4Þ

where rx and ax are, respectively, the components of r and a in the rolling direction. In the above expression, the negativesign of the second term is for loading and the positive sign is for unloading (or reverse loading). Therefore,

d�a1dep

x� da2

depxax ¼ dðrx��risoÞ

depx

� �loading

d�a1dep

xþ da2x

depxax ¼ dðrx��risoÞ

depx

� �reverse loading

9>=>; ð5Þ

Finally, considering d�e ¼ depx as a reference quantity and performing some mathematical manipulations, Eq. (5) reduces to

Table 3Values

Tens

196


d�a1

depx¼ d�a1

d�eð� h1Þ ¼

12

drx

depx


þ drx

depx

� �loading

" #� d�riso

depx

da2

depxax ¼

da2

d�eð� h2Þax ¼

12

drx

depx


� drx

depx

� �loading

" # ð6Þ

The parameters, h1 and h2, are the slopes of back stress evolutions, which are determined from the slopes of experimentalhardening curves measured by uniaxial tension–compression tests (details are referred to Lee et al. (2005)).

4.2. Asymmetric yield function

Hill’s 1948 orthotropic yield criterion under the plane stress condition is written as;

r2x �

2r0

1þ r0rxry þ

r0ð1þ r90Þr90ð1þ r0Þ

r2y þ

r0 þ r90

r90ð1þ r0Þð2r45 þ 1Þs2

xy ¼ r20 ð7Þ

where the normal anisotropic values, r0, r45 and r90, are the ratios of the width strain to the thickness strain in the uniaxialtension tests along the rolling, 45� and the transverse directions, respectively, while r0 is the flow stress in the rolling direc-tion. Hill’s 1948 yield function is easy to implement and one of the most popular functions in industry; however, it does notaccount for asymmetry in tension and compression. Therefore, in the present work, Hill’s 1948 yield function was modifiedby incorporating additional linear terms; i.e.,

a r2x � Arxry þ Br2

y þ Cs2xy

n o1=2þ ðk1rx þ k2ryÞ ¼ r0 ð8Þ

There are six coefficients which may be evaluated using two tensile yield stresses along the rolling and the transverse direc-tions (rT

x and rTy , respectively), two compressive yield stresses along the rolling and the transverse directions (rC

x and rCy ,

respectively), tensile yield stress at 45� to the rolling direction (r45) and the balanced biaxial yield stress (rb). The expres-sions for the six coefficients are derived as

k1 ¼rC

x � rTx

2rCx

ð9Þ

k2 ¼rT

x rCy � rT

y

� �2rC

yrTy

ð10Þ

a ¼ rCx þ rT

x

2rCx

ð11Þ

B ¼rC

x rTx rC

y þ rTy

� �rC

yrTy rC

x þ rTx

� �8<:

9=;2

ð12Þ

A ¼ 1þ B� 1a

rTx

rb� ðk1 þ k2Þ

� �2

ð13Þ

C ¼ 1a

2rTx

r45� ðk1 þ k2Þ

� �2

� ð1� Aþ BÞ ð14Þ

Ryou et al. (2007) and Lee et al. (2008) have also proposed a criterion similar with Eq. (8) by incorporating linear terms toaccount for yield asymmetry, however, they assumed A2 = B while they are independent in the present modification. The sixyield stresses to calculate six anisotropic coefficients using Eqs. (9)–(14) and the values of the coefficients are listed in Tables3 and 4, respectively. It may be noticed that the values of A and B are not interdependent as it was assumed in Ryou et al.(2007) and Lee et al. (2008). When a = A = B = 1, C = 3 and k1 = k2 = 0, the new yield criterion reduces to the Mises yieldcriterion. Note also that Eq. (8) can be used along with the incompressibility condition for the thickness strain rate, whilein-plane strain rates are obtained from the normality flow rule applied to Eq. (8).

In Fig. 13, the present yield function (with a vanished shear component) is compared with Hill’s yield function deter-mined using Eq. (7) and r-values shown in Table 1 along with the yield function, Yld2000-2d, whose anisotropic coefficientsare discussed in Appendix A and listed in Table A.1. In the first quadrant, initially both Yld2000-2d (with the exponent m = 6)and the present modified yield function match with the experiment better than Hill’s 1948 yield function. For ep

0 ¼ 0:04, the

of yield stresses used for the calculation of yield function coefficients (unit: MPa).

ion (rolling) Compression (rolling) Tension (transverse) Compression (transverse) Equi-biaxial tension Tension (diagonal)

195 192 205 195 193

Table 4Coefficients of the asymmetric yield criterion.

a A B C k1 k2

1.001872 1.02697 0.97074 3.01435 �0.001872 0.032323


Yld2000-2d yield function has better agreement with the work contour everywhere except transverse compression; thepresent yield function slightly underestimates the work contour in the vicinity of equibiaxial tension region; however, it per-fectly predicts the yield asymmetry. Hill’s 1948 yield function overestimates stress values near the equibiaxial tension anddoes not predict the asymmetry. This is due to the fact that the coefficients of the Yld2000-2d yield function were calibratedutilizing equibiaxial tension data and the coefficients change as ep

0 increases, whereas for the present and Hill’s yield func-tions, the coefficients are fixed for all ep

0. Since the present yield function is asymmetric so it has the closest agreement withthe work contour in the transverse direction.

In Fig. 14, (initial) yield stress ratios (normalized by the reference stress) and r-values, measured in tension and compres-sion, are compared with calculated values for three yield functions. Note that different yield functions were calibrated usingdifferent sets of experimental data so that the capability of predicting any state of stress depends not only on the type ofyield functions but also on experimental data used to calibrate the functions. If Hill’s 1948 criterion was calibrated usingyield stresses, it could have predicted the yield stress correctly in transverse direction as well even though, in that case, cor-rect prediction of r-values might not have been possible. Similarly as the present yield function does not use the r-values so itdid not predict the r-values correctly. Also, both Yld2000-2d and Hill 1948 yield criteria are symmetric so they cannot predictthe yield stress difference in tension and compression in the transverse direction, whereas the modified yield criterion can.Finally, as none of the yield function used r-values in compression so none of them could predict it in compression. It mayalso be noticed that, during compression the r-values in rolling and transverse directions are same to the corresponding val-ues during tension.

4.3. Modified combined hardening law incorporating hardening stagnation and cross-effect

In the combined isotropic–kinematic hardening law, total hardening is decomposed into the isotropic and kinematicparts; i.e.,

Fig. 14. Prediction of yield ratio and r-values at different angle from RD in tension and in compression: (a) normalized yield stress and (b) r-values.

Fig. 15based o


�r ¼ �riso þ �a ð15Þ

where �riso is the isotropic part which signifies the size change of the yield surface, whereas �a is the kinematic part whichrepresents the translation of the yield surface. As for its decomposition for the initial hardening behavior, T–C tests are per-formed with various tensile pre-strains as schematically described in Fig. 15(a), in which the translation of the center posi-tion of the yield surface (or �a) is measured. Therefore, kinematic hardening (and also the isotropic hardening) during initialloading is obtained from T–C tests using Eq. (15). Similarly, the decomposition of the hardening behavior (into isotropic andkinematic) during reverse loading (compression) in T–C tests is experimentally obtained from T–C–T tests performed withvarious compressive pre-strains during reverse loading, as schematically illustrated in Fig. 15(b). If the reverse loading curveshows permanent softening in the T–C test, the decomposition of the total permanent softening into softening attributed tothe isotropic hardening part and to the kinematic hardening part can be performed using the T–C–T tests as just explained(Here it may be noted that there were efforts by Wilson and Bate (1986) and Prangnell et al. (1996) to identify permanentsoftening using X-ray diffraction and in situ neutron diffraction on carbon steels, respectively).

For the combined isotropic–kinematic hardening discussed in Section 4.1, h1 and h2 defined in Eq. (6) are typically positivevalues. Under such conditions, both initial loading and reverse loading curves converge to the same (isotropic hardening) va-lue of �risoð1Þ þ h1ð1Þ=h2ð1Þ, while h1ð1Þ=h2ð1Þ is a positive value, so that no permanent softening is accounted for (Kim etal., 2006). Consequently, in order to account for permanent softening during reverse loading, �risoð1Þ and h1ð1Þ=h2ð1Þshould be modified to account for permanent softening in isotropic hardening and that in kinematic hardening, respectively,which are measured in T–C–T tests with various compressive pre-strains.

It was found that, for the particular material used in this study, the isotropic part is generally well represented (up to thepoint of uniform elongation) by the Voce type hardening curve; i.e.,

�riso ¼ X þ Ye�Z�e ð16Þ

where X, Y and Z are constants. Note that, in Eq. (16), stress ultimately saturates to the value X and the rate of saturation iscontrolled by the coefficient Z. Since X determines the saturated stress value, X can be modified to account for the isotropichardening stagnation in the following manner:

Xcurrent loading ¼ Xprevious loading � d ð17Þ

Since the amount of softening during the T–C–T test depends on the amount of pre-strains,

d ¼ P þ Qe�R�epre ð18Þ

where P, Q and R are constants and �epre is the strain before loading direction change. The loading direction change can beevaluated using the criterion proposed by Lee et al. (2007). As for kinematic hardening, the following expression was usedfor Eq. (6):

h1;2 ¼ aþ be�c�e ð19Þ

where a, b and c are constants. In order to account for permanent softening attributed by kinematic hardening, the hardeningparameter h1 of the back stress evolution law can be modified by introducing a softening factor, b, as done by Ahn et al.(2009); i.e.,

hcurrent loading1 ¼ b �epre

� �� hprevious loading

1 0 < b �epre� �

6 1� �

ð20Þ

where b is a function of the effective pre-strain and b(0) = 1.0.

. Schematic view of the decomposition of isotropic and kinematic hardening (a) in initial hardening based on T–C tests and (b) in reverse loadingn T–C–T tests with various pre-strains.


In order to illustrate the performances of permanent softening only in kinematic hardening and permanent softening onlyin isotropic hardening, their results are compared for the experimental T–C–T test data (with 0.08 tensile pre-strain) asshown in Fig. 16. Besides the T–C–T test shown in Fig. 8, one more T–C–T data was measured (with 0.08 tensile pre-strainalong with 0.045 compressive pre-strain) as shown in Fig. 16. Note that both, the permanent softening only in kinematichardening and permanent softening only in isotropic hardening, properly predicted the reverse loading curve. However,the permanent softening only in kinematic hardening over-predicted the tensile reloading curve, whereas the permanentsoftening only in isotopic hardening under-estimated it for both two T–C–T curves. This is because the permanent softeningin kinematic hardening kept a larger yield surface size than the permanent softening in isotropic hardening. When perma-nent softening was divided into the ratio of 40:60 between the kinematic hardening and the isotopic hardening, respectively;mixed permanent softening provided better results compared to the other two cases (permanent softening either in kine-matic hardening only or isotropic hardening only) for all T–C–T data in this work, as also shown in Fig. 16. Therefore, mixedpermanent softening with the ratio (40:60) was applied for tensile reloading for all T–C–T data in this work. As for the valuesof b, a constant value of 0.55 was found to be fairly accurate in this work, even though it should be a function of pre-strain ingeneral. Similarly, d was found to be fairly well represented by the following equation: d ¼ 30:38� 33:8e�45:11�epre .

As for the cross-effect observed during the two-stage uniaxial tests, the cross-effect was incorporated by addingD�riso ¼ Ae�B�ecurrent to the isotropic hardening part whenever there was a loading direction change by 90�. Here, �ecurrent isthe effective strain in current loading. Even though the coefficients A and B are supposed to be dependent on pre-strain,in the present work, for 2% pre-strain the values for these coefficients were found to be 30.0 and 45.0, while for 0.05 pre-strain these were found to be 35.0 and 45.0, respectively.

5. Verifications

The combined isotropic–kinematic hardening model with the modified asymmetric yield function was implemented intoABAQUS/Standard and ABAQUS/Explicit codes using the user-subroutines UMAT and VUMAT (2007), respectively, followingthe numerical formulations developed by Chung et al. (2005). In the codes, at every time increment, the material model inthe user-subroutine is called and the plastic strain increment for a discrete incremental strain is calculated and then the Cau-chy and the back stresses are updated before returning back the control to the ABAQUS main program. To verify the perfor-mance of the new constitutive law and proper code implementation, finite element calculation was performed using the S4Rshell type element with nine integration points along the thickness direction and reduced integration. Table 5 shows the iso-tropic and the kinematic hardening parameters. Note that Young’s modulus was assumed to be the following function of theeffective plastic strain, which was obtained from the data given in Table 2:

Fig. 16.isotrop

Table 5Harden

h1 ¼

a

31,1

E ðMPaÞ ¼ 160;049þ 39;928e�122:4�e ð21Þ

Though Young’s modulus was assumed to be a function of plastic strain as a whole, its value was assumed to be a constantduring each (time) increment in the numerical procedure.

Comparison of measured T–C–T test data (with 0.08 tensile pre-strain) with predictions made by permanent softening in kinematic hardening andic hardening as well as mixed permanent softening.

ing parameters.

aþ be�c�e h2 ¼ aþ be�c�e �riso ¼ X þ Ye�Z�e d ¼ P þ Qe�R�e b

b C a b c X Y Z P Q R

92 �29,261 32.39 359.9 �249.9 76.39 394.66 �211.0 7.692 30.38 �33.8 45.11 0.55


5.1. Monotonic uniaxial tension and compression

Fig. 17(a)–(d) shows the comparison between the experimental and simulated stress–strain curves for monotonic uniax-ial tension and compression in the rolling and transverse directions. Since there was no difference between the stress–straincurves in tension and compression in the rolling direction, predictions by Hill’s 1948 and the present yield functions were thesame. Predictions of both tension and compression in the transverse direction were poor when Hill’s 1948 yield function wasused whereas the present yield function predicted stress–strain relationships accurately, properly reproducing the tension–compression asymmetry of the material used in this study. It is expected that if Hill’s yield function were determined usingmeasured yield stresses, the prediction of tension in the transverse direction would have been better.

5.2. In-plane TCT tests

Fig. 18(a)–(e) shows the experimental and simulated in-plane TCT test results. The new asymmetric yield function with-out softening model and Hill’s 1948 yield function without softening predicted similar results and their predictions werepoor. However, the new asymmetric yield function with permanent softening predicted the cyclic behaviors reasonably well.It should be noted that even Hill’s 1948 yield criterion with permanent softening will predict the same cyclic behavior in therolling direction.


Simulations for the two-stage uniaxial tests were carried out as follows. In the first method, the following procedure wasused: (i) first, the simulation was carried out for the first loading; (ii) then the unloading was simulated followed by removalof elements (to simulate the cutting of the specimen as done during physical cutting process in preparing the specimen forthe second loading) as the state of stress may get re-organized; and (iii) finally, the simulation for deformation in the trans-verse direction were performed.

Fig. 17. Experimental and simulated stress–strain curves for the monotonic uniaxial tension and compression tests. (a) Tension in rolling direction; (b)compression in rolling direction; (c) tension in transverse direction; and (d) compression in transverse direction.

Fig. 18. Experimental and simulated TCT test results for different amount of pre-strains epre:epre = (a) 0.02, (b) 0.04, (c) 0.06, (d) 0.08 and (e) 0.10.


Fig. 19(a)–(f) shows the comparison between the experimental and simulated results. Fig. 19(a), (c) and (e) shows theresults in which the second deformation was in the tension mode and Fig. 19(b), (d) and (f) shows the results in whichthe second deformation was in compression. The predictions made by the modified model based on the asymmetric yieldfunction were in reasonably good agreement with the experimental results, whereas the predictions using the model basedon Hill’s 1948 yield function and the combined isotropic–kinematic hardening models were consistently lower than theexperimental values. Note that the poor prediction in Fig. 19(b), (d) and (f), when Hill’s 1948 yield function was used,was mainly caused by the incapability of simulating the yield asymmetry; however, the prediction inaccuracy inFig. 19(a), (c) and (e) was incurred by another reason; i.e., when Hill’s 1948 yield criterion is calibrated with r-values andthe yield stress in the rolling direction, the resulting criterion often does not predict yielding in the transverse direction cor-rectly, as happened in this case.

Fig. 19. Experimental and simulated results for two-stage uniaxial tests. (a) First stage tension 0.02 followed by second stage tension; (b) first stage tension0.02 followed by second stage compression; (c) first stage tension 0.05 followed by second stage tension; (d) first stage tension 0.05 followed by secondstage compression; (e) first stage compression 0.02 followed by second stage tension; and (f) first stage compression 0.02 followed by second stagecompression.


As for the cross-effect, the incorporation of the cross-effect improved predictions for all cases, except for the case of com-pression–compression (shown in Fig. 19(f)), in which there was no cross-effect observed. This suggests that, with properprogramming in the user-subroutines to differentiate cases with cross-effect from cases without the cross-effect, thecross-effect scheme needs to be selectively triggered during numerical calculations.

6. Conclusion

In the present work, two-stage uniaxial tests, along with uniaxial cyclic tests and biaxial tests under different stress ratios,were performed to evaluate the deformation behavior of an ultra-low carbon high strength steel alloy. Experimental results

Table A.1The coefficients of Yld2000-2d determined for different levels of plastic work, represented by uniaxial plastic strain ep

0 in the rolling direction.

ep0

a1 a2 a3 a4 a5 a6 a7 a8

0.002 1.1063 0.9552 0.8177 0.9713 0.9922 1.2032 1.0490 0.94930.02 1.1056 0.9461 0.7656 0.9622 0.9747 1.1129 1.0441 0.99570.04 1.1056 0.9373 0.7196 0.9545 0.9610 1.0418 1.0399 1.0343


showed the hardening stagnation, cross-effect and asymmetry in tension and compression as well as the Bauschinger andtransient behaviors. To reproduce the tension–compression asymmetry of the test material, Hill’s 1948 yield function wasmodified and it was coupled with the combined isotropic–kinematic hardening law. As for the hardening stagnation, a com-bined method to formulate the softening in isotropic hardening and kinematic hardening was proposed. Also a simple buteffective method was proposed to incorporate the cross-effect observed. The resulting constitutive model showed reason-ably good agreement with experimental results, especially when the cross-effect was selectively triggered.

Acknowledgements

This work has been performed under the joint project between Tata Steel, Tokyo University of Agriculture and Technol-ogy, Japan and Seoul National University, South Korea. The authors gratefully acknowledge the financial support and the per-mission to publish this work. K. Chung is also thankful to the Korea Science and Engineering Foundation (KOSEF) for theKorea Government Grant (MEST) (R11-2005-065) through the Intelligent Textile System Research Center (ITRC). The authorsare also thankful to Mr. Nobuyasu Noma for measuring the r-values in compression.

Appendix A. Yld2000-2d yield function

The plane stress yield stress function Yld2000-2d proposed by Barlat et al. (2003) is

f1m ¼ U

2

� 1m

¼ �r with U ¼ eS0I � eS 0II�� m þ 2eS 00II þ eS 00I�� m þ 2eS 00I þ eS 00II�� m ðA:1Þ

where �r is the effective stress and m is an exponent associated with crystal structures. In Eq. (A.1), eS0k and eS00k (k = I, II) are theprincipal values of the deviatoric stress tensor ~s ð~s0 or ~s00Þ modified by the following linear transformations:

~s0 ¼ C0 � s ¼ C0 � T � r ¼ L0 � r;~s00 ¼ C00 � s ¼ C00 � T � r ¼ L00 � r:

ðA:2Þ

Here, C0 and C00 (therefore, L0 and L00) contain eight independent anisotropic coefficients and T transforms the Cauchy stresstensor r to its deviator s; i.e.,

C0 ¼c011 0 00 c022 00 0 c066

264375; C00 ¼

c0011 c0012 0c0021 c0022 00 0 c0066

264375 ðA:3Þ

The eight anisotropic coefficients are usually determined by eight mechanical measurements, r0, r45, r90, r0, r45, r90, rb andrb, which are simple tension yield stresses along the rolling (x-) direction, 45� off and transverse (y-) directions, r-values(width-to-thickness strain ratio in simple tension), yield stress and in-plane principal strain ratio under the balanced biaxialtension condition, respectively. The values of eight coefficients, as defined in Barlat et al. (2003), are given in Table A.1.

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Experimental evaluation and constitutive modeling of non-proportional deformation for asymmetric steels

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