Anisotropichardeningequationsderivedfrom...

Anisotropic hardening equations derived fromreverse-bend testing

Lumin Geng, Yao Shen, R.H. Wagoner*

Department of Materials Science and Engineering, The Ohio State University, 2041 College Rd., Columbus,

OH 43210, USA

Received in final revised form 6 March 2001

Abstract

The plastic response of materials during reverse loading has practical consequences for

common sheet forming operations in terms of loads, localization behavior, and springback.However, it is difficult to measure the reverse loading (Bauschinger effect) in sheet materialsbecause of their propensity to buckle. A simple reverse-bend test was constructed and used toinvestigate the cyclic loading of three automotive body alloys. The results showed that con-

sideration of the Bauschinger effect is essential to obtaining agreement with such results. Aninverse procedure was used to determine anisotropic hardening law parameters. Lawsobtained in this way were compared with ones generated by more sensitive tension-compres-

sion tests appearing in the literature for the same alloys. The two laws were significantly dif-ferent, but both produced accurate simulations of reverse-bend test load–displacement curves.Several artificial material models were then constructed to simulate the reverse-bend test and

thus to probe its sensitivity to material constitutive equation details. For materials whosereverse-loading response varies with the level of prestrain, as is the case for each of the threealloys tested, a wide range of constitutive response is capable of producing identical reverse-

bending behavior. The results show that inverse procedures applied to the reverse-bend test donot provide unique results, and thus the usefulness of the reverse-bend test for such investi-gations is limited. # 2002 Published by Elsevier Science Ltd.

Keywords: Cyclic plasticity; Reverse bending; Tension–compression; Anisotropic hardening; Aluminum; Steel

1. Introduction

In many sheet metal forming processes, sheet specimens can experience cyclicloading conditions. The hardening behavior of materials is more complicated under

International Journal of Plasticity 18 (2002) 743–767

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* Corresponding author. Tel.: +1-614-292-2079; fax: +1-614-292-6530.

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

such conditions than under monotonic loading. For materials that exhibit a Bauschingereffect, the yield stress in reverse loading is usually lower than in the case of con-tinuous loading, and the subsequent hardening rate is higher. In such cases, theconventional isotropic hardening model is no longer an adequate approximation.Therefore, for some materials under reverse loading, consideration of the Bauschingereffect is required for a realistic stress distribution, which is in turn essential for pre-diction of springback under bending/unbending conditions (Li et al., 1999b; Geng,2000; Geng and Wagoner, 2000).The Bauschinger effect refers to a decrease of flow stress upon a stress reversal

(Sowerby et al., 1979; Dafalias, 1984; Abel, 1987; Mughrabi, 1987), although in someferrous alloys the flow stress may increase (Ghosh and Backofen, 1973; Laukonis andGhosh, 1978; Wagoner and Laukonis, 1983). Measurement of the effect is usuallydone under cyclic torsion conditions for bulk materials (Svensson, 1966; Hecker,1971, 1973; Lipkin and Swearengen, 1975; Anand and Gurland, 1976; Chang andAsaro, 1978; Drucker and Palgen, 1981), although reverse shear tests (Miyauchi,1984a,b; Vreede, 1992) and in-plane reverse uniaxial tests (Hoge et al., 1973; Tan etal., 1994; Kuwabara et al., 1995) have also been presented for sheet or plate materials.A special version of the in-plane tension–compression test was devised (Balakrishnan,1999) for sheet materials to avoid buckling for compressive strains up to 0.08. Correc-tions were required for biaxiality and friction introduced by the fixture (Balakrishnan,1999). A reverse-bend test (Jiang, 1997; Shen, 1999; Zhao, 1999; Zhao and Lee,1999) provides a simple alternative for testing sheet metals under reverse loading,and has been proposed for measuring the Bauschinger effect in sheet materials, but itdoes not provide direct stress–strain data.To model the material hardening behavior revealed by these tests, anisotropic

phenomenological models are needed, various forms of which have been proposed.Classical kinematic hardening models (Prager, 1956; Ziegler, 1959) have simple for-mulations with sharp reverse yield at fixed stresses. Hodge’s mixed hardening model(Hodge, 1957) reproduces the permanent softening upon reverse loading, againwithout transient yield or hardening. Multilayer (Besseling, 1958) and multiyieldsurface (Mroz, 1967, 1969) models reproduce transient aspects at the expense ofmany fit parameters, and even more complex formulations have appeared (Baltovand Sawczuk, 1965; Williams and Svensson, 1971; Rees, 1981).In two-surface models, the yield surface translates and expands within an enclosing

bounding surface. A set of differential equations is defined to describe the translationand expansion of these surfaces (Krieg, 1975), or the plastic modulus can be definedas a function of the distance between the two (Dafalias and Popov, 1976). Anupdate procedure is essential whenever a drastic change in loading direction occurs.The nonlinear-kinematic hardeningmodel (Armstrong and Frederick, 1966; Chaboche,1986) is a generalization of Prager and Ziegler’s linear kinematic hardening model: arecall term is introduced into the classical kinematic hardening formulation toreproduce the smooth elastic-plastic transition upon the change of loading path. Reviewsof such models have been presented (Chaboche, 1986; Khan and Huang, 1995).The nonlinear kinematic hardening model was fit to published in-plane tension–

compression test results because it approximates faithfully the nonlinear stress–strain

744 L. Geng et al. / International Journal of Plasticity 18 (2002) 743–767

curves in the complex loading paths, and it is readily implemented. The two materialparameters in the model can be defined as functions of prestrain, in accord withdetailed measurements (Balakrishnan, 1999).The current work focuses on the consistency of anisotropic hardening laws

derived from tension–compression and reverse-bending tests, and on the uniquenessof laws determined from fitting reverse-bend tests via an inverse procedure. TheChaboche law (Chaboche, 1986) constants are first fit to tension–compression testcurves. The best-fit parameters are found to vary markedly with prestrain, i.e. thestrain at the stress reversal. Next, a similar fit is performed to reverse-bend data, withleast-squares Chaboche constants found by a trial-and-error inverse optimization.Finally, the constitutive equation obtained from tension-compression tests is used tosimulate the reverse-bend test. As will be shown, the laws obtained in these alternateways fit the reverse-bend test equally well, but diverge when compared with tension–compression data. Therefore, the reverse-bend test cannot be used to measure theBauschinger effect uniquely because the measured load–displacement points eachinvolve a range of stresses and strains simultaneously.

2. Experimental procedures

2.1. Materials

Three automotive body alloys were tested and analyzed: 6022-T4 aluminum alloy,high-strength low-alloy steel (HSLA), and drawing-quality silicon-killed steel(DQSK). Chemical composition of the three alloys is listed in Table 1. 6022-T4 is analuminum–magnesium–silicon heat-treatable alloy. The strength is derived from ahigh temperature solution heat treatment followed by a rapid quench and a pre-cipitation aging (Mg2Si) treatment at room temperature. HSLA is a microalloyedand precipitation strengthened pearlite-ferritic as-rolled steel with Mn for sulfidecontrol and strength, and Nb for carbide formation and grain refinement to increasestrength. DQSK is a very low-carbon steel, similar to HSLA but with approximatelyhalf of the carbon and Mn, processed to obtain a high R-value (for improved deepdrawability). The microstructures of the three alloys used have been presented(Balakrishnan, 1999). The average grain diameters of 6022-T4, HSLA and DQSKare 150, 35 and 10 microns, respectively. The uniaxial tensile flow curves for thethree materials are shown in Fig. 1 (Balakrishnan, 1999).

Table 1

Chemical compositions of three alloys, in weight percent

C Mn P S Si Cu Ni Cr Mo Sn Al N Cb Fe Mg Ti

DQSK 0.035 0.22 0.009 0.010 0.007 0.03 0.01 0.02 0.01 0.002 0.031 0.004 0.001 Balance – –

HSLA 0.070 0.477 0.010 0.004 – 0.02 0.02 0.03 – 0.001 0.022 0.005 0.039 Balance – –

6022-T4 – 0.07 – – 1.24 0.09 – – – – Balance – – 0.13 0.58 0.02

L. Geng et al. / International Journal of Plasticity 18 (2002) 743–767 745

2.2. The reverse-bend test

Reverse-bend tests were carried out using a specially-constructed three-pointbending device (Fig. 2). The upper fixture or punch consists of two vertically-aligned, non-rotating rods in close contact with the center of the strip specimen. Thelower fixture consists of two freely-rotating bearings into each of which are mountedfour rotatable rollers, an upper pair and a lower pair. The lower pair of rollers is at afixed position in the bearing while the upper pair may be adjusted vertically toaccommodate materials of various thicknesses. The rollers are located such that amaterial 1mm thick passes through the bearing centerline. A load cell is mountedabove the punch to measure the axial force, and a computer records the punch dis-placement and force.Shear-cut specimens and electrical discharge machined (EDM) specimens of

nominally 25.4mmwidth were tested. The EDM samples exhibited smaller experimentalscatter than those of the shear cut specimens, but otherwise showed no differences. The

Fig. 1. Tensile strain hardening for three sheet alloys.

Fig. 2. Schematic of the reverse-bend device (Shen, 1999).


bending moment or load was normalized to nominal specimen width (b0) andthickness (t0) (Lange, 1985) to account for specimen-to-specimen scatter of thesequantities, as follows:

LN ¼ Lb0b

t0t

� �2ð1Þ

where L, b, t are the measured load, width and thickness, respectively, and LN is thenormalized load.Two configurations of rollers were compared. In one case, all four rollers were

utilized in contact, in the other case only the ones opposing the punch force wereused. No significant difference was found in the results from these two configurations,thus validating the use of an analytical model introduced in the next section.Fig. 3 shows typical load–displacement curves generated for the three materials. The

elastic range can be modeled as a cantilever beam to obtain the Young’s modulus fromthe initial linear slope of the load (p)–displacement (w) curve (Gere and Timoshenko,1984; Jiang, 1997):

E ¼4bt3

l 3�

dp

dw

� �ð2Þ

where b, t and l are the width, thickness and distance between the centers of the twobearings, respectively. The slopes of initial and reverse load–displacement curves areshown in Fig. 3, each fit to lines with correlation coefficients greater than 0.9999.The Young’s moduli calculated according to Eq. (2), along with those from literature,are reported in Table 2. As is frequently observed, the moduli determined from

Fig. 3. Typical bend/reverse-bend results for three materials. Initial slopes and proportional limits are shown.


mechanical testing are lower than those measured sonically. The difference pre-sumably lies in microplastic or anelastic effects.The difference between the displacements at full load and after unloading is a

measure of springback. The presence of the Bauschinger effect is demonstrated bythe reverse proportional limit being much lower than that of the forward flow stress.Deviation from linearity during unloading is observed for DQSK and HSLA, indi-cating the presence of plastic deformation in the unloading process.Although the present work focused on a single reverse cycle, a few multi-cycle

bending tests were performed for the three materials. The bending and reverse-bending curves tend to saturate after two cycles for HSLA, four cycles for DQSK,and six cycles for 6022-T4.

3. Modeling of the reverse-bend test

Various models of the reverse-bend test were constructed using techniques devel-oped for analysis of draw-bend springback (Wagoner et al., 1997; Li and Wagoner,1998). In particular, attention was paid to element size, number of integration pointsthrough the thickness (shell elements), number of elements through the thickness(solid elements), and convergence and contact tolerances. SHEET-S (Wagoner et al.,1989; Saran et al., 1991; Wang and Wagoner, 1991a; Keum et al., 1992; Wang andWagoner, 1991b) was used for two-dimensional simulations, both plane stress andplane strain; and ABAQUS (Anon., 1998) for three-dimensional simulations. The 2-D simulations with SHEET-S utilized either Bernoulli or Timoshenko beam elementswith either plane stress or plane strain constraints.The material models for initial bend simulations made use of von Mises yield

functions and isotropic strain hardening models. For the DQSK and 6022-T4materials, the hardening curves shown in Fig. 1 were employed. For HSLA, thenegative slope associated with the yield point phenomenon introduced convergenceproblems. This softening is usually associated with the appearance of Ludersbands (Sinha, 1989). Because of this non-uniformity of strain, there is no directway to extract the true local stress-strain curve from the macroscopic tensile testin this range (Kochs, 1981; Duncan et al., 1999a, b; 1993). For HSLA, theeffective stress–strain curve was adjusted following a modification of the flowcurve suggested by Kocks (1981) and Duncan et al. (1993, 1999a). This curve,shown in Fig. 4, also had convergence difficulties at the elastic-plastic transition,

Table 2

Young’s moduli as determined by bending, tension, and handbook values

Material 6022-T4 HSLA DQSK

Young’s modulus (GPa) Bending [Eq. (2)] 65�2 189�11 187�5

Tension (Carden, 1997) 69 212 202

Handbook (Anon., 1949) 69–73 207 207


so an intermediate extrapolation that matched experiments well was generated bytrial and error. This curve, shown in Fig. 4, was used for subsequent analysis.A cantilever beam bending model (i.e. one-half of the symmetric physical problem)

was constructed, with the driving action at the end of the beam provided by contactwith two rods matching those carrying the reaction forces in the experiment. Thetwo rods were free to rotate about a common axis located between their centerlines.For reverse-bending, the rods are located on the opposite side of the specimen. Thismodel was found to reproduce the boundary conditions adequately.For two-dimensional simulations, half of the specimen was meshed (Fig. 5a);

while for three-dimensional simulations, a quarter of the specimen was meshed totake advantage of both axes of symmetry (Fig. 5b). In view of previous experiencewith simulation of springback (Li and Wagoner, 1998; Li et al., 1999a, 2000), 51integration points were used through the thickness with shell elements (ABAQUS

Fig. 4. Alternative hardening curves in the yield point phenomenon region for HSLA.

Fig. 5. Finite element meshes for (a) 2-D beam and (b) 3-D shell elements. The meshes for 3-D solid ele-

ments are obtained by extending the 3-D shell mesh to nine layers in the thickness direction.


S4R) and nine layers were used through the thickness with the solid elements(ABAQUS C3D8R). As shown in Fig. 6 for 6022-T4 aluminum alloy, both 3-Dsolid and shell elements reproduce the experimental bending results accurately, whilethe plane-strain beam elements over-predict the load and plane-stress elementsunder-predict it.The origin of the difference between 2-D and 3-D simulations lies with the pre-

sence of anticlastic curvature (Yu and Zhang, 1996; Carden et al., 2000; Li et al.,2000), the secondary curvature that occurs because of differential lateral strain fromthe outer fiber to inner fiber of the sheet caused by the principal bending. Fig. 7shows the simulated secondary curvature for the specimen under load and afterbeing unloaded for a section 8mm away from the symmetry plane of the model(bend center of the physical specimen). Anticlastic curvature is observed in theexperiments as well, but measurement accuracy is poor because of the small specimenwidth and small magnitude of the secondary curvature.

4. Anisotropic hardening models

Throughout the current work, a modified nonlinear kinematic/isotropic hardeningmodel is used to represent plastic hardening. The model is based on several devel-opments appearing in the literature. Armstrong and Frederick (1966) added to theclassical kinematic hardening model of Prager (1956) a recall term to reproduceobserved transient hardening following an abrupt change of load path. Chaboche(1986) generalized this approach to include multiple evolution equations of the yieldsurface location (�, also known as the back stress) and incorporated a classical iso-tropic hardening component as well. The update of the backstress in the Prager

Fig. 6. Differences among 2-D and 3-D simulations of the reverse-bend test.


equations takes place along the direction of plastic strain, a procedure which has sincebeen shown (Ziegler, 1959) to give inconsistent results unless specially formulated(Khan and Huang, 1995). Therefore, in the current work, the Armstrong–Frederick–Chaboche approach for nonlinear kinematic/isotropic hardening is adopted, withthe exception that the steady-state evolution of � is given by Ziegler (1959):

d� ¼C

�0�� ð Þd"p � ��d"p ð3Þ

where � is the back stress; d"p is the equivalent plastic strain rate; and C and � arematerial parameters. �0 is the yield surface size, which is constant in a purely kinematichardening model, but which is allowed to evolve with plastic strain in a combinedkinematic-isotropic model (as is adopted here). C, �0, and � determine the plastichardening rate following an abrupt change of loading path.For a given yield function

F ¼ fð� � �Þ � �0 ¼ 0; ð4Þ

and the consistency condition is expressed as follows:

@f

@�: �:�

@f

@�: �:�d�0d"p

d"p ¼ 0 ð5Þ

For simplicity, only the von Mises yield function is considered in the currentwork. Substituting (3) into (5), and using

Fig. 7. Transverse curvatures from FEM simulation. The curvature is calculated with shell elements

(S4R) for a cross-section 8 mm from the middle point.


@f

@�: � � �ð Þ ¼ �0

obtains

@f

@�: �:� C� �

@f

@�: � þ

d�0d"p

� �d"p ¼ 0 ð6Þ

For uniaxial loading, the plastic modulus is expressed as follows:

Hp ¼d�

d"p¼ C� ��sgnð� � �Þ þ

d�0d"p

ð7Þ

where sgn (x)=1 if x>0, or sgn(x)=-1 if x<0.For the case of a stress reversal, the second term in Eq. (6) changes sign. The

parameters C and � at the time of the reversal may be determined with the use of thefollowing relationships:

C ¼Hf "

pð Þ þHr "pð Þ

2�d�0d"p

; ð8Þ

� ¼Hr "

pð Þ �Hf "pð Þ

2�ð9Þ

where Hf and Hr are the plastic moduli for the forward and reverse loading at themoment of stress reversal, respectively. The yield surface size, �o, and the back stress,� in Eq. (9), may be found from the measured forward and reverse flow stresses:

�f ¼ �o þ � ð10Þ

� ¼ �f � �o ¼ �f ��f � �r

2¼

�f þ �r2

ð11Þ

where �f and �r are the forward and reverse flow/yield stresses at the reversal. Theother quantity in Eq. (8), d�o/dp, may be approximated by �so/�Ep, from stressreversals carried out at other prestrains.A hardening law with similar formulation is available in the commercially available

FEA code ABAQUS (Anon., 1998), where � is required to be constant, and C maychange with the temperature or field variables. To relax this constraint, a nonlinearkinematic hardening law with C and � as functions of prestrain was implemented inABAQUS via the user subroutine UMAT. A backward Euler algorithm is used forintegrating the plastic strain increments, in which a trial stress is obtained byassuming that the strain increment is purely elastic deformation. Residuals of stress,back stress and yield function are defined, and are linearized using the first term of aTaylor series expansion about the trial states. From these expansions, the stresses, backstresses and other state variables are updated. Iteration continues until the residual ofthe yield function is less than the specified tolerance (typically the residual is required to


be less than 10�12 times the yield stress). Upon convergence, the consistent tangentmodulus matrix is computed and used with a Newton–Raphson equilibrium iterationprocedure.

5. Anisotropic hardening law determined from tension–compression tests

Special tension–compression tests were developed to probe the Bauschinger effectof sheet metal under in-plane deformation at large strains (Balakrishnan, 1999).These tests make use of a standard tensile specimen and special fixtures to apply astabilizing load parallel to the thickness direction. Corrections are made to accountfor friction introduced by the stabilizing fixtures and for the small biaxial loading that isrequired. Tests were conducted starting with either a tensile prestrain followed bycompressive loading after reversal (labeled ‘‘T-C’’), or a compressive prestrain followedby tensile loading (labeled ‘‘C-T’’). Prestrains of up to approximately 0.08 wereachievable in compression, and the full range of strain in tension may be achieved.Fig. 8 shows Balakrishnan’s (1999) data for the C-T tests at several prestrains, alongwith nonlinear kinematic hardening curves obtained using Eqs. (8) and (9). (The T-Cresults, which are not shown in Fig. 8 for clarity, are very similar and the materiallaws derived from those tests are compared later).The series of C-T or T-C reverse curves is used to obtain the necessary coefficients

of the nonlinear kinematic/isotropic hardening model as follows. The location of theyield surface, or backstress, �, is determined from the measured forward and reverseflow stresses at the reversal according to Eq. (11), and the size of the yield surface,so, is determined according to Eq. (10). The last term in Eq. (8), d�o/d, is determinedby reference to the reverse loading curve at an adjacent prestrain by approximatingd�o/dp, by ��o/�p. With these terms in hand, Eqs. (8) and (9) can be used withthe measured plastic moduli, Hf and Hr, to determine the remaining parameters Cand g. While this is a sufficient method in principle, exact values of Hf and Hr arenot readily obtained without ambiguity. Therefore, while initial values of Hf and Hr

are obtained from apparent slopes of the curves, these quantities are adjusted in orderto obtain a satisfactory fit throughout the transient period following the reversal. Thechanges of Hf and Hr introduced by this procedure are within the accuracy of thedirect fit of the slopes. The result of this analysis is presented at each prestrain inTable 3, and Fig. 9 shows the assumed piece-wise linear interpolation of parametersneeded to complete construction of a general constitutive equation for any prestrain.Fig. 8 shows that the derived material laws reproduce the major features of the

reverse loading curves. At lower prestrains, typically less than 0.02, the reverse flowcurves tend to rejoin the monotonic loading curve after a transient strain range. Forsuch cases, the nonlinear kinematic hardening law fits the reverse flow curves well.As the prestrain increases, permanent softening is observed after the reversal. Thenonlinear kinematic hardening law cannot reproduce this behavior well because theflow stress approaches the monotonic loading curve after the transient period.The nonlinear kinematic hardening law can be considered to consist of a yield

surface and a limiting surface (Chaboche, 1986), with the limiting surface expanding


Fig. 8. Nonlinear kinematic hardening model as fit to uniaxial compression–tension test data (Balak-

rishnan, 1999).


isotropically. To reproduce the permanent softening of the reverse flow, the reverseflow modulus Hr must be set to be very small so that the approach rate of thereverse flow to the monotonic flow curve will be gradual. In this way, long-termsoftening can be accommodated for the reverse curve at the expense of reproducingthe transient accurately. The experimental uniaxial tension–compression curveshows that the transient range decreases as the prestrain increases. Therefore, for thereverse flow at high prestrain, a reasonably close fit can be achieved by ignoring theearly transient portion while closely approximating the long-term offset. While byno means perfect, particularly at the elastic–plastic transition, the standard non-linear kinematic hardening model fit in this way adequately reproduces the mainfeatures of reverse hardening, as shown in Fig. 8.To complete the model needed for FEM simulation, the variation of C and � must

be known explicitly with prestrain. At zero prestrain, Hr is taken to be the same asHf . For intermediate prestrains, linear interpolation is used. For prestrains beyondthe measured range, three assumptions were made: (a) a saturation of permanentsoftening is adopted; (b) the rate of reverse yield stress change with prestrain is takento be the same as the plastic modulus of the monotonic loading curve, and (c) thevalue of Hr levels off after the highest prestrain measurement available. The varia-tions of parameters thus obtained are shown in Fig. 9. The differences betweencompression–tension tests (C-T) and tension–compression tests (T-C) are minorcompared with the variation with prestrain. (Note that all such fitting is done con-sidering only a single strain reversal, rather than multiple cycles.)

6. Anisotropic hardening law determined from reverse-bend tests

The richness of data available from a reverse-bend test is limited compared withthat obtained from a uniaxial tension–compression test. At each instant, a range ofstrains is sampled, not only through the thickness (where strain varies approximatelylinearly with position) but also with location along the length of the specimen.

Table 3

Material parameters obtained from the compression–tension test data (Balakrishnan, 1999)

p � (Mpa) C (Mpa) �

Al 6022-T4 (C-T) 0.0111 51.3 12896.0 260.3

0.0275 41.2 6224.4 175.3

0.0466 19.3 �575.4 �2.3

DQSK(C-T) 0.0132 18.8 1579.7 53.4

0.0405 13.8 2119.9 116.2

0.0514 18.4 �453.9 �8.5

HSLA (C-T) 0.0211 84.5 9618.0 135.3

0.0344 56.1 6369.4 134.0

0.0485 35.5 �124.4 �0.7


Fig. 9. Variation of nonlinear kinematic hardening parameters with prestrain. Points obtained by analysis

of tension–compression test data (Balakrishnan, 1999). H0 is the plastic modulus at the initial yield of the

monotonic loading curve.


Fig. 10 shows the distribution of the equivalent plastic strain obtained at the end ofthe bending stage for the elements located along the longitudinal centerline at thesurface of the sheet specimen. The maximum equivalent plastic strain is approxi-mately 0.03 for HSLA and DQSK, and about 0.046 for 6022-T4; and the equivalentplastic strain decreases rapidly with the distance from the punch. A few elementsclose to the punch and near the sheet surface experience relatively large straining,while most of the elements achieve strains less than 0.01–0.02. The reverse-bend testis thus likely to reflect material hardening behavior within a limited strain range,typically less than 0.02. The resulting load–displacement curve depends on a rangeof strains at each instant, so it is not possible to discriminate accurately the effect ofprestrain at the strain reversal.1

With this limitation in mind, the reverse-bend test may be simulated using finiteelement procedures and models discussed in a previous section. For given values ofC and �, the accuracy of the fit can be assessed by comparing the simulated andmeasured load–displacement curves. C and � can be adjusted in a trial-and-errorfashion (or using more systematic inverse procedures (Zhao, 1999; Zhao and Lee,1999)) to obtain best-fit values. This process is facilitated because C and � have

Fig. 10. Distribution of equivalent plastic strain in the outside fiber along the sheet specimen centerline

after bending.

1 It is possible in principle to discern large changes that occur in strain reversals at higher prestrains by

changing the stroke amplitude of the bend/reverse-bend test and finding best-fit coefficients for each

amplitude. However, at each amplitude, the fundamental problem remains that the data represents a

range of prestrains up to a fixed maximum determined by the stroke amplitude.


qualitatively different effects on the uniaxial flow curve (Shen, 1999). Larger � (forfixed C) produces a higher reverse yield stress and shorter transient period. Larger C(for fixed �) decreases the reverse yield stress and results in a longer transient strain.Thus, by matching reverse yield stress and transient strain, or yield stress and initialslope, acceptable values of C and � may be found.A trial-and-error procedure was adopted in the current work, and replication of

the measured load–displacement results was obtained with excellent agreement(Fig. 11). In the process of the fitting, however, it was noted that multiple choices ofC and g were capable of reproducing nearly identical fits to the experiments.Therefore, it appears that even constant values of the two parameters C and � cannotbe identified unambiguously, thus obviating the usefulness of inverse methods(Zhao, 1999; Zhao and Lee, 1999) or more complex material laws (for example,where C and � depend on other variables).

7. Results

Fig. 11 presents the simulation results for the reverse-bend test making use ofthree constitutive equations. The curve labeled ‘‘C-T model’’ was generated usingthe constitutive equation from uniaxial compression–tension tests, as described pre-viously. (The curves using T-C models are omitted from Fig. 11 for clarity. In allcases, the curves lie within two line-widths of the C-T curves presented.) The curveslabeled by values of C and � were generated using the constitutive equationsobtained from the bend test results themselves; that is, they represent simulationswith best-fit (but not unique, as discussed above) coefficients obtained from the sameexperiments. All equations fit the original bend region equally because there is nostrain reversal. For the reverse leg, both anisotropic hardening constitutive equationsare in close agreement with the experimental results, while simulations based on astandard isotropic hardening law diverge.Results shown in Fig. 11 demonstrate that taking into account the Bauschinger

effect is essential for accurate calculation of the stress distribution through a sheetspecimen undergoing reverse-bend deformation. In sheet metal forming, bend andreverse-bend deformation occur whenever the sheet is drawn over a die radius orthrough a draw bead. Because springback is proportional to moment and inverse tomodulus, an accurate stress distribution is essential in the prediction of the amountof springback associated with these types of forming process.Fig. 12 examines the accuracy of the constitutive equations derived from bend

tests (i.e. best-fit constant C and �) in reproducing the uniaxial tension–compressiontests. The agreement is poor, even qualitatively, except at the smallest prestrains.For prestrains greater than 0.02, none of the key characteristics is reproducedproperly: i.e. the yield stress, initial hardening slope, transient strain, and long-termoffset from the monotonic hardening law. Thus, while the two anisotropic hardeningconstitutive equations are indistinguishable in terms of reverse-bend simulations,which average over a range of strains simultaneously, they diverge when stress andstrain can be distinguished uniquely, as in the tension–compression test.


Fig. 11. Comparison of measurements to simulation results of the reverse-bend test using isotropic and

anisotropic hardening models for (a) 6022-T4, (b) DQSK and (c) HSLA.


Fig. 12. Comparison of constitutive equations obtained from reverse-bend tests and tension–compression

(T-C) and compression–tension (C-T) measurements for (a) 6022-T4, (b) DQSK and (c) HSLA.


8. Discussion

The reverse-bend test is an attractive alternative to tension–compression andreverse-shear tests for investigating the Bauschinger effect in sheet materials. It iseasy to perform with standard equipment, the set-up time and specimen preparationare minimal, and the fixtures are readily constructed. However, as noted earlier,stress and strain are obtained only indirectly, and must be averaged over a range ofvalues at each instant.In attempts to overcome this difficulty (Jiang, 1997; Yoshida et al., 1998; Shen,

1999; Zhao, 1999; Zhao and Lee, 1999), a hardening model (such as a nonlinearkinematic hardening model) is typically assumed, and an optimization procedure isthen used to obtain best-fit values of the material parameters. Because of the relativelymeager data available for the inverse procedure, the material parameters are typicallytaken as constants. Although more parameters may be added and better fits possiblyobtained, a large multiplicity of nearly equal inverse results may be obtained for a

Fig. 13. Comparison of Modification I with the original C-T model. Notice that there are no changes to

the reverse flow curves for prestrains greater than 0.015.


single test result. As shown in Figs. 8 and 9, the material parameters are not constants,thus the essential assumption needed for the inverse procedure is violated. Worse,even using constant parameters in the material hardening law allows several sets togive equally good fits to the test results.Three artificial modifications of the C-T model for DQSK were constructed to

assess the sensitivity of the reverse-bend test results to details of the constitutivematerial response. Each modification makes use of the nonlinear kinematic hard-ening model, but each employs an altered set of constants to represent alternatereverse loading responses. For Modification I, the yield surface size and strainhardening rate differ considerably only for prestrains less than 0.015 (Fig. 13a andb), while for Modification II the changes are apparent only for prestrains greaterthan 0.015 (Fig. 14a and b). For Modification III (Fig. 15a and b), the yield surfacesize is modified throughout the range of prestrains.Fig. 16 shows the simulated force–displacement curves for reverse-bend tests carried

out using the modified material models. (Note that the strain distributions at the

Fig. 14. Comparison of Modification II with the original C-T model. Notice that there are no changes to

the reverse flow curves for prestrains less than 0.015.


maximum displacement are presented in Fig. 10). For comparison, the experimentalpoints (which are almost undistinguishable from the simulations using the C-T model;Fig. 11b) are also shown.Modification I gives reverse-bending loads which differ by 10%from the other models and experiments, while the others are virtually indistinguishable.Comparison of results from Modifications I and II confirms the supposition statedearlier that the reverse-bend test is mainly sensitive to the small-prestrain constitutiveresponse, for strains less than 0.015. Modification II, while having markedly differentstress–strain response for larger prestrains, shows no significant difference in simu-lated reverse bend tests compared with simulations based on the baseline materialmodel (‘‘C-T’’). Thus, it should not be expected that the large-prestrain materialresponse can be discerned by the reverse-bend test.Modification III shows that even alternative constitutive laws which differ con-

siderably at small prestrains may reproduce the measured reverse-bend test force–displacement curves with excellent agreement. This is particularly important in viewof the qualitatively different physical behavior, which is apparent at various rangesof prestrain, Figs. 8 and 9. At small prestrains, the transient range is limited. After an

Fig. 15. Comparison of Modification III with the original C-T model. The reverse flow curves are differ-

ent at various prestrains.


incremental strain of 0.01–0.02, the monotonic hardening curve is recovered. Atlarger prestrains, the transient region is extended and further hardening shows apermanent offset from the monotonic curve. As demonstrated by Modification III,the details of many such models could be chosen to reproduce the reverse-bend testload–displacement curves accurately. Therefore, while the reverse-bend test can detect,in an average sense, the existence of a Bauschinger effect, it cannot be used reliably inan inverse manner to obtain details of the correct constitutive material laws.

9. Conclusions

A nonlinear kinematic hardening model has been used to model the Bauschingereffect in three sheet alloys. The required hardening parameters were obtained byfitting load–displacement curves obtained from a reverse-bend test (Shen, 1999), andby re-analyzing tension–compression data appearing in the literature (Balakrishnan,1999). Both of the models were used with finite element modeling to simulate thereverse-bending test.The following conclusions were reached:

1. The constitutive models obtained from the tension–compression test and aninverse procedure applied to reverse-bend results reproduce reverse-bend test

Fig. 16. Comparison of the simulation results of the reverse-bend test with Modification I, II, III and the

original C-T model.


measurements equally well. Thus, there is no inconsistency of behavior in thetwo deformation modes.

2. The constitutive models obtained by fitting to the reverse–bend test and ten-sion–compression test show significant differences when evaluated in terms oftheir stress–strain response following a stress reversal.

3. Constitutive models obtained by an inverse procedure applied to the reverse-bendtest are not unique, even under the assumption of independence of prestrain.Widely varying constitutive equations can produce identical load–displacementcurves from reverse-bend tests.

4. The reverse-bend test is primarily sensitive to the material response for pre-strains less than 0.015, and is nearly insensitive to the material behavior athigher prestrains.

5. Consideration of the Bauschinger effect is essential for accurate calculation ofmoments and springback.

6. The nonlinear kinematic hardening law cannot represent accurately the Bauschin-ger effect at larger prestrains, where a long-term offset of stress or strain isobserved.

Acknowledgements

The financial support of a PNGV subcontract via NIST/ATP and the Center forAdvancedMaterials andManufacturing of Automotive Components (CAMMAC) isgratefully acknowledged. The in-plane compression/tension test data were providedby V. Balakrishnan. Computer time was provided by the Ohio SupercomputerCenter (PAS 080). The authors would like to thank Christine D. Putnam forproofreading and editing assistance.

References

Abel, A., 1987. Historical perspectives and some of the main features of the Bauschinger effect. Materials

Forum 10 (1), 11–26.

Anand, L., Gurland, J., 1976. Strain-hardening of spheroidized high carbon steels. Acta Metall 24, 901–909.

Anon, 1998. ABAQUS/Standard User’s Manual Version 5.8, Vol. 1, Hibbitt. Karlsson & Sorensen Inc,

Pawtucket, RI.

Anon, 1949. Properties of SomeMetals and Alloys: Chemical Composition,Mechanical Properties, Hardness

and Physical Constants. International Nickel Company, Development & Research Div., New York.

Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger

effect. G.E.G.B. Report RD/B/N 731.

Balakrishnan, V., 1999. Measurement of In-plane Bauschinger Effect in Metal Sheets. Masters Thesis,

The Ohio State University.

Baltov, A., Sawczuk, A., 1965. A rule of anisotropic hardening. Acta Mechanica 1, 81–92.

Besseling, J.F., 1958. A theory of elastic, plastic, and creep deformation of an initially isotropic material

showing strain hardening, creep recovery, and secondary creep. Journal of Applied Mechanics 25, 529.

Carden, W.D., 1997. Springback after Drawing and Bending of Metal Sheets, Masters thesis, The Ohio

State University.

Carden, W.D., Geng L.M., Matlock D.K., Wagoner R.H., 2000. Measurements of springback. Interna-

tional Journal of Mechanical Sciences, in press.


Chaboche, J.L., 1986. Time-independent constitutive theories for cyclic plasticity. International Journal of

Plasticity 2 (2), 149–188.

Chang, Y.W., Asaro, R.J., 1978. Bauschinger effects and work-hardening in spheroidized steels. Metal

Science 12, 277–284.

Dafalias, Y.F., Popov, E.P., 1976. Plastic internal variables formalism of cyclic plasticity. Journal of

Applied Mechanics 98, 645–651.

Dafalias, Y.F., 1984. Modeling cyclic plasticity: simplicity versus sophistication. In: Desai, C.S., Galla-

gher, R. (Eds.), Mech. Engineering Materials. Wiley, New York, p. 155.

Drucker, D.C., Palgen, L., 1981. On stress-strain relations suitable for cyclic and other loadings. J. Appl.

Mech 48, 479.

Duncan, J.L., Sue-Chu, M., Wang, X.J., 1993. Mechanical Behavior of Materials IV. Pergamon Press,

New York, pp. 631–639.

Duncan, J.L., Ding, S.-C., Jiang, W.-L., 1999aa. Moment-curvature measurement in thin sheet—part I:

equipment. Int. Journal of Mechanical Science 41, 249–260.

Duncan, J.L., Ding, S.-C., Jiang, W.-L., 1999b. Moment-curvature measurement in thin sheet—part II:

yielding and kinking in aged steel sheet. Int. Journal of Mechanical Science 41, 261–267.

Geng, L.M., Wagoner, R.H. 2000. Springback analysis with a modified hardening model, SAE technical

paper 2000-01-0768. In: Sheet Metal Forming: Sing Tang 65th Anniversary Volume, SP-1536. SAE, pp.

1–11.

Geng, L.M., 2000. Application of Plastic Anisotropy and Non-isotropic Hardening to Springback Pre-

diction, Ph.D. Thesis, The Ohio State University.

Gere, J.M., Timoshenko, S.P., 1984. Mechanics of Materials. PWS Engineering, Boston, MA, pp. 244, 369.

Ghosh, A.K., Backofen, W.A., 1973. Strain hardening and instability in biaxially stretched sheets.

Metallurgical Transactions 4, 1113–1123.

Hecker, S.S., 1971. Yield surface in prestrained aluminum and copper. Metallurgical Transactions 2,

2077–2086.

Hecker, S.S., 1973. Influence of deformation history on the yield locus and stress-strain behavior of alu-

minum and cooper. Metallurgical Transactions 4, 985–989.

Hodge, P.G., 1957. A new method of analyzing stresses and strains in work hardening plastic solids. J.

Appl. Mech 24, 482–483.

Hoge, K.G.R., Brady, L., Cortez, R., 1973. A tension-compression test fixture to determine Bauschinger

effect. Journal of Testing and Evaluation 1 (4), 288–290.

Jiang, S., 1997. Springback Investigations. Masters Thesis, The Ohio State University.

Keum, Y.T.M., Saran, J., Wagoner, R.H., 1992. Practical die design via section analysis. J. Mater. Proc.

Technol 35, 1–36.

Khan, A., Huang, S.H., 1995. Continuum Theory of Plasticity. John Wiley & Sons, New York.

Kocks, U. F., 1981. Chalmers Anniversary Volume, Progress in Materials Science, pp. 185–241.

Krieg, R.D., 1975. A practical two surface plasticity theory. Journal of Applied Mechanics, Transactions

of the ASME 47, 641–646.

Kuwabara, T., Morita, Y., Miyashita, Y., and Takahashi, S., 1995. Elastic-plastic behavior of sheet metal

subjected to in-plane reverse loading. In: Proceedings of Plasticity ’95, Dynamic Plasticity and Struc-

tural Behaviors, The 5th International Symposium on Plasticity and Its Current Application, Tani-

mura, S., Khan, A.S. (Eds.) pp. 841–844.

Lange, K. (Ed), 1985. Handbook of Metal Forming. McGraw-Hill, New York (Section 19.1.3).

Laukonis, J.V., Ghosh, A.K., 1978. Effects of strain path changes on the formability of sheet metals.

Metallurgical Transactions A 9A, 1849–1856.

Li, K., Wagoner, R.H., 1998. Simulation of springback (Simulation of Materials Processing). In: Huetink,

J., Baaijens, F.P.T.(Eds.). Balkema, Rotterdam, pp. 21–32.

Li, K., Geng, L., Wagoner, R.H., 1999a. Simulation of springback: choice of element (Advanced Tech-

nology of Plasticity 1999, Vol. III.). In: Geiger, M.. Springer-Verlag, Berlin, pp. 2091–2098.

Li, K.P., Geng, L., Wagoner, R.H., 1999b. Simulation of springback with the draw/bend test. In: Meech,

J. (Ed.), Proceedings of IPMM’99, Vol. 1. The Second International Conference on Intelligent Proces-

sing and Manufacturing of Materials. IEEE, Piscataway, NJ, pp. 91–104.


Li, K.P., Carden, W.P., Wagoner, R.H. 2000. Simulation of springback. Int. J. Mech. Sci., in press.

Lipkin, J., Swearengen, J.C., 1975. On the subsequent yielding of an aluminum alloy following cyclic

prestraining. Metallurgical Transactions A 6A, 167–177.

Miyauchi, K., 1984a. A proposal of a planar simple shear test in sheet metals. In: Scientific Papers of the

Institute of Physical and Chemical Research, Vol. 78 (3), pp. 27–40.

Miyauchi, K., 1984b. Bauschinger effect in planar shear deformation of sheet metals. Advanced Tech-

nology of Plasticity 1984, 1. 523-628.

Mroz, Z., 1967. On the description of anisotropic workhardening. J. Mech. Phys. Solids 15, 163–175.

Mroz, Z., 1969. An attempt to describe the behavior of metals under cyclic loads using a more general

workhardening model. Acta Mechanica 7, 199–212.

Mughrabi, H., 1987. Johann Bauschinger, pioneer of modern materials testing. Materials Forum 10, 1. 5-10.

Prager, W., 1956. A new method of analyzing stresses and strains in work hardening plastic solids. J.

Appl., Mech 23, 493–496.

Rees, D.W.A., 1981. Anisotropic hardening theory and the Bauschinger effect. Journal of Strain Analysis

16, 2. 85-95.

Saran, M.J., Keum, Y.T., Wagoner, R.H., 1991. Section analysis with irregular tools and arbitrary draw-

in conditions for numerical simulation of sheet forming. Int. J. Mech. Sci. 33, 893–910.

Shen, Y., 1999. The reverse-bend test: indirect measurement of the Bauschinger effect in metal sheets.

Masters thesis. The Ohio State University.

Sinha, A.K., 1989. Ferrous physical metallurgy. Butterworths, Boston. pp. 54–55.

Sowerby, R., Uko, D.K., Tomita, Y., 1979. A review of certain aspects of the Bauschinger effect in metals.

Materials Science and Engineering 41, 43–58.

Svensson, N.L., 1966. Anisotropy and the Bauschinger effect in cold rolled aluminum. J. Mech. Engrg Sci.

8, 162–172.

Tan, Z., Magnusson, C., Persson, B., 1994. Bauschinger effect in compression-tension of sheet metals.

Materials Science & Engineering A: Structural Materials: Properties, Microstructure and Processing, A

183 (1–2), 31–38.

Vreede, P., 1992. A Finite Element Method for Simulations of 3-Dimensional Sheet Metal Forming.

Enschede, The Netherlands.

Wagoner, R.H., Laukonis, J.V., 1983. Plastic behavior of aluminum-killed steel following plane-strain

deformation. Metall. Trans. A 14A, 1487–1495.

Wagoner, R.H., Wang, C. T. and Nakamachi, E., 1989. Quick analysis of sheet forming using sectional

FEM. In: Proc. 1st Japan Int’l. SAMPE Symp. Exhibition, SAMPE, pp. 695–700.

Wagoner, R.H., Carden, W.D., Carden, W.P., Matlock, D.K. 1997. Springback after drawing and bend-

ing of metal sheets. In: Proc. IPMM ’97—Intelligent Processing and Manufacturing of Materials, Vol.

1. Chandra, T., Leclair, S.R., Meech, J.A., Verma, B., Smith, M. and Balachandran, B. University of

Wollongong, Intelligent Systems Applications, pp. 1–10.

Wang, C.T., Wagoner, R.H. 1991a. Plane-strain deep drawing: finite element modeling and measure-

ments. SAE paper no. 910774. In: Autobody Stamping Technology Progress, Soc. Auto. Engrs., SP-

865, Warrendale, PA, pp. 133–142.

Wang, C.T., and Wagoner, R.H. 1991b. Square-punch forming: section analysis and bending study. SAE

paper no. 910772. In: Autobody Stamping Technology Progress, Soc. Auto. Engrs., SP-865, Warren-

dale, PA, pp. 119–127.

Williams, J.F., Svensson, N.L., 1971. A rationally based yield criterion for work hardening materials.

Meccanica 6, 104.

Yoshida, F., Urabe, M., Totopov, V.V., 1998. Identification of material parameters in constitutive model

for sheet metals from cyclic bending tests. International Journal of Mechanical Science 40 (2–3), 237–249.

Yu, T.X., Zhang, L.C., 1996. Plastic Bending, Theory and Applications. World Scientific, Singapore.

Zhao, K.M., 1999. Cyclic Stress-strain Curve and Springback Simulation, PhD Thesis, The Ohio State

University.

Zhao, K.M., Lee, J.K. 1999. On simulation of bend/reverse-bend of sheet metals. In: MED-Vol. 10,

Manufacturing Science and Engineering. ASME, pp. 929–933.

Ziegler, H., 1959. A modification of Prager’s hardening rule. Quart. Appl. Math 17, 55–65.


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