international journal of plasticity 25 (2009) 1777–1817

8/14/2019 International Journal of Plasticity 25 (2009) 17771817

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Anisotropic hardening and non-associated flow

in proportional loading of sheet metals

Thomas B. Stoughton a,*, Jeong Whan Yoon b,c,*

a Manufacturing Systems Research Lab, MC 480-106-359, General Motors R&D Center, Warren, MI 48090-9055, USAbAlcoa Technical Center, 100 Technical Dr., Alcoa Center, PA 15069-0001, USAc Department of Mechanical Engineering, Uiversity of Aveiro, 3810-193 Aveiro, Portugal

a r t i c l e i n f o

Article history:

Received 17 November 2008

Received in final revised form 27 January

2009

Available online 13 February 2009

Keywords:

Anisotropic hardening

Constitutive law

Non-associated flow rule

Yield function

a b s t r a c t

Conventional isotropic hardening models constrain the shape of

the yield function to remain fixed throughout plastic deforma-

tion. However, experiments show that hardening is only approx-

imately isotropic under conditions of proportional loading, giving

rise to systematic errors in calculation of stresses based on mod-

els that impose the constraint. Five different material data for

aluminum and stainless steel alloys are used to calibrate and

evaluate five material models, ranging in complexity from a

von Mises model based on isotropic hardening to a non- associ-

ated flow rule (AFR) model based on anisotropic hardening. A

new model is described in which four stressstrain functions

are explicitly integrated into the yield criterion in closed form

definition of the yield condition. The model is based on a non-

AFR so that this integration does not affect the accuracy of the

plastic strain components defined by the gradient of a separate

plastic potential function. The model not only enables the elim-

ination of systematic errors for loading along the four loading

conditions, but also leads to a significant reduction of systematicerrors in other loading conditions to no higher than 1.5% of the

magnitude of the predicted stresses, far less that errors obtained

under isotropic hardening, and at a level comparable to experi-

mental uncertainty in the stress measurement. The model is

expected to lead to a significant improvement in stress predic-

tion under conditions dominated by proportional loading, and

this is expected to directly improve the accuracy of springback,

tearing, and earing predictions for these processes. In addition,

it is shown that there is no consequence on MK necking

localization due to the saturation of the yield surface in pure

0749-6419/$ - see front matter 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2009.02.003

* Corresponding authors.

E-mail addresses: [email protected] (T.B. Stoughton), [email protected] (J.W. Yoon).

International Journal of Plasticity 25 (2009) 17771817

Contents lists available at ScienceDirect

International Journal of Plasticity

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j p l a s
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shear that occurs with the aluminum alloys using the present

model.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

A constitutive model for sheet metal forming is developed that substantially improves accuracy in

the stressstrain prediction for linear monotonic loading processes under biaxial loading conditions at

all levels of strain. This is achieved by generalizing the concept of isotropicscalar hardening to a con-

cept based on anisotropicscalar hardening. Although such a model is theoretically limited to true lin-

ear loading, due to its dependence on a scalar hardening variable, it is expected to provide significantly

more accurate results in applications where isotropic hardening is used and considered to be approx-

imately valid. The model is considered to be especially useful in the analysis of the first stamping pro-

cess of an automotive sheet metal product, where conventional thinking is that the forming process at

each point on the metal is, with few exceptions, nearly linear and monotonic.Conventional isotropic scalar hardening is based on the equality of a linear homogeneous yield

function r~r;a1; . . . ;an of the stress tensor components to a scalar hardening function rep of theeffective plastic strain, where a1; . . . ;an denote material constants in the yield function that definethe shape of the six-dimensional yield surface. The material constants yield function can be deter-

mined by a set of tests, usually a set of uniaxial tension tests and/or equal biaxial tension, while

the hardening function is defined by the stressstrain relation at a specific loading condition deter-

mined by the normalization of the yield function. The hardening function is also usually approximated

bya parametric function, most often of the formproposed by Swift or Voce, but may be represented by

a spline fit to experimental data, for example. Although isotropic hardening is a rough approximation

to metal behavior, there are significant differences in the rate of hardening between uniaxial and biax-

ial loading conditions, and systematic differences in the rate of hardening along different directions inuniaxial tension, as shown for illustration in the AA 5182-O in Fig. 1. The predictions are based on an

advanced model that matches the initial anisotropy in the yield behavior, but deviate from the exper-

imental data as plastic strain is increased to finite levels.

Hill and Hutchinson (1992) described a framework and motivation for anisotropic scalar hardening

of the yield function under associated flow rules, including the discussion of possible loss of convexity

and the necessity of a relationship between the plastic work contours and the shape of the yield func-

tion. Kuroda and Tvergaard (2000) accounted for certain types of distortion of the yield function

attributed to rotations of the crystalline structure and captured by prescribed evolution equations

for the antisymmetric part of the velocity gradient or plastic spin. Abedrabbo et al. (2006a,b, 2007)

used a more empirical approach, allowing general distortion of the yield surface by first characterizing

the anisotropic properties of aluminum alloy sheets for various temperatures and different strain

rates. Then, the anisotropy coefficients of Yld96 and Yld2000-2d models (Barlat et al., 1997, 2003)

and the hardening parameters were described as a function of discrete temperatures and interpolated

the parameters smoothly using a curve fitting method. These papers showed that the consideration of

the hardening evolution produces very good correlations with experimental data for thermo-mechan-

ical coupled forming simulation. Plunkett et al. (2007) also proposed anisotropic modeling of textured

metals. In the work, the discrete sets of anisotropy coefficients as well as the size of elastic domain

were considered to be the functions of the accumulated plastic strain in order to consider the texture

evolution and showed that the consideration of the texture evolution leads to the right prediction of

anisotropic behavior of Taylor cylindrical impact for a hcp material. Recently, Aretz (2008) also pro-

posed a simple isotropic-distortional hardening model. In the model, the anisotropic coefficients were

obtained in different discrete levels of plastic work density in order to describe the hardening direc-

tionality and discussed the important impact on localized necking prediction.

As described above, anisotropic scalar hardening is not a new concept and has been used by

Abedrabbo et al. (2006a,b, 2007), Plunkett et al. (2007), Aretz (2008), and others to introduce

change in the shape of the yield function during plastic deformation, as is implied by the discrep-

1778 T.B. Stoughton, J.W. Yoon / International Journal of Plasticity 25 (2009) 17771817


3/41

ancies such as those shown in Fig. 1. Yield surface distortion has been accounted for in these

works by measuring the yield stress at fixed and discrete levels of plastic work under the same

loading conditions that are conventionally used to define the coefficients of the yield function.

Then, using these measurements, the parameters of the yield function are defined at each level

of plastic work, independently from the yield function defined at lower levels of work hardening.

Then, using either a linear C1 or nonlinear C2 or higher continuous interpolation of the yield

function parameters, the yield function constants are replaced by functions of the effective plastic

strain, a1ep; . . . ;anep. While this solution is effective and the accuracy of reproducing thestressstrain responses improves as the plastic strain increment between subsequent work

contours is decreased, the stressstrain response in directions other than the one selected for

the normalization of the scalar yield function is dependent on the interpolation function, and

0

50

100

150

200

250

300

350

400

450

0.0 0.1 0.2 0.3 0.4 0.5

TrueStress

(MPa)

True Plastic Strain

Stress-strain for AA 5182-O

0 Stoughton [2002]

45 Stoughton [2002]

90 Stoughton [2002]

EB Stoughton [2002]

0 -exp45 - exp

90 - exp

EB -exp

Fig. 1. Experimental stressstrain data for AA 5182-O for uniaxial tension at 45 and 90 degrees to the rolling direction of the

sheet coil and for equibiaxial tension based on data listed in Table 1. The solid lines are predictions of a conventional hardeningmodel based on Stoughton (2002) non-AFR calibrated to the uniaxial tension data along the rolling direction.

T.B. Stoughton, J.W. Yoon/ International Journal of Plasticity 25 (2009) 17771817 1779


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mathematically, the result is in agreement with experiment only at the selected contours of plas-

tic work.

An alternate solution to improve conformance of the model to anisotropic hardening is to use a

normalized quadratic yield function, based on Hills (1948) criterion. The quadratic form enables

the model to explicitly incorporate four experimental stressstrain responses in different loading con-

ditions. While this ensures an exact replication of the stressstrain response in these calibration con-

ditions, the solution has a serious problem under the associated flow rule (AFR) due to limitations of

the quadratic yield function to describe both the anisotropic yield and the anisotropic strain behavior

of most sheet metals. So, calibrating the yield function from yield stresses is not a practical solution for

engineering applications, where accurate plastic strains are considered to be more important than

accurate prediction of forming stresses. The recognition of this problem was documented in Bramley

and Mellor (1966), Pearce (1968), and many others since, which showed conclusively that quadratic

functions of the stress, calibrated to agree with anisotropic plastic strain measurements, do not ade-

quately describe the anisotropy of the yield behavior for metals with low r values, or even the full

anisotropy of metals with higher r values, under an AFR. This limitation led to developments during

the past 40 years of more sophisticated models under AFR using higher order functions of the stress

tensor.Lademo et al. (1999), Stoughton (2002), Stoughton and Yoon (2004, 2006, 2008), Cvitanic et al.

(2008), and others proposed to use non-AFR models for metal deformation as a less complex alterna-

tive to using higher exponent functions of the stress, in order to achieve a more accurate description of

the anisotropic response than can be obtained with the quadratic form under AFR. It was shown that

the yield function and yield criterion for isotropic hardening can still be defined in quadratic form and

capture the initial yield condition quite well, even in the biaxial condition for metals with low rvalues.

Non-associated flow rule has been also introduced to improve the accuracy of the prediction. Kuroda

and Tvergaard (2001) proposed a phenomenological plasticity model, in which a smooth yield surface

for an anisotropic solid is combined with a vertex-type plastic flow rule and allows the non-normality

of plastic flow based on the observation from experiments and polycrystal calculation for an abrupt

strain path change to determine the shape of the subsequent yield surface. Yoshida et al. (2007) suc-cessfully applied Kuroda and Tvergaard (2001) model for changing strain paths on the forming limit

stresses of sheet metals combined with the MarciniakKuczynski model.

In the present work, an anisotropic hardening model based on non-AFR is proposed for application

to proportional loading conditions, where the plastic strain directions are defined by a separate qua-

dratic plastic potential with constant coefficients defined by the plastic strain ratios in different direc-

tions of uniaxial tension, identical in form and calibration to the Hills (1948) model, but without

imposing the AFR. So with this change it is possible to precisely reproduce the anisotropy in both yield

and plastic strain behaviors, with the later assumed to be unaffected by plastic deformation. The pro-

posed model uses a normalized quadratic yield function with coefficients replaced by the experimen-

tal stressstrain response along different uniaxial directions, as well as the stressstrain response in

equal biaxial tension. While this ensures an exact replication of the stressstrain response in these cal-ibration conditions, this method will be shown to provide nearly an order of magnitude reduction in

root-mean-square (RMS) error between model and experiment over all loading conditions, compared

to conventional models based on isotropic hardening.

The model is described for plane stress conditions in Section 2. Section 3 explains how the model

components are calibrated from experimental data, and Section 4 describes the metrics used to char-

acterize the predictive power and overall accuracy of the model for all loading conditions and levels of

strain, which is expected to reflect the attainable accuracy of an analysis involving a full range of linear

loading conditions and levels of plastic strain. Section 5 discusses the accuracy of the model observed

for three aluminum alloys and two alloys of stainless steel. The model is also compared to accuracy

obtained with four more conventional models based on isotropic hardening including the von Mises

and Hills (1948) yield functions under AFR, and two versions of non-AFR based on quadratic functions.An interesting anomaly of stress saturation and negative work hardening is observed in the shear

loading condition for the aluminum alloys using the anisotropic hardening model. This is investigated

in Section 6 using MK analysis to determine if the saturation leads to any instability issues beyond the

expected conventional prediction of localization by the MK Method. The paper is concluded with a

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brief summary and conclusions. Details of finite element implementation are given in Appendix A,

including extension of the model to full stress loading conditions.

2. Constitutive equations for anisotropic hardening under non-AFR

One of the most common models of metal deformation still widely used in industry is based on the

quadratic plastic potential of the form proposed by Hill (1948), which for plane stress conditions is

often normalized for convenience to uniaxial tension along the 1-axis as follows:

rp~r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir211 kpr222 2mpr11r22 2qpr12r21

q;

kp 1 1=r901 1=r0 ; mp

1

1 1=r0 ; qp 1=r0 1=r90

1 1=r01 2r45

2

1

The parameters, r0; r45, and r90 are the ratios of the plastic strain rate across the width of a uniaxial

tension test to the plastic strain rate through the thickness for tension at, respectively, 0, 45, and

90 degrees to the rolling direction of the sheet coil aligned with the 1-axis of the coordinate system.

The above three and additional experimental rvalues in 15-degree increments of the uniaxial tension

to the rolling direction of the sheet coil are listed for three aluminum alloys in Table 1 and two steel

alloys in Table 2. Coefficients of the plastic potential are given in Table 3 for the five models considered

in this paper.

According to classical plasticity, the plastic potential is used in the flow rule to define the direction

of the rate of change of all components of the plastic strain tensor

dEpij

dt @rp~r

@rij_k; 2

Table 1

Properties of three aluminum alloys used in automotive applications fit to Voce Hardening Model parameters. Numbers in the Test

Column refer to the orientation of the uniaxial tension test and EB refers to an Equal-Biaxial hydraulic bulge test. Initial yield stress

is AB.

Test A B C r

AA5182-O 0 366.84 251.07 11.166 0.957

15 366.87 252.76 10.462 0.903

30 361.29 248.42 10.062 0.916

45 358.74 247.11 9.719 0.934

60 355.56 244.73 9.638 0.947

75 360.91 248.76 9.569 0.981

90 362.39 248.11 9.981 1.058EB 437.28 312.26 6.179 0.948

AA 6022-T4E32 0 328.16 194.30 10.978 0.823

15 324.09 191.30 10.598 0.732

30 325.37 191.88 9.803 0.529

45 325.12 192.02 9.223 0.411

60 320.13 189.71 9.165 0.483

75 316.23 188.26 9.765 0.550

90 315.99 188.71 10.147 0.678

EB 360.44 219.76 6.729 1.244

AA 6022-T43 0 339.05 202.50 10.357 1.029

15 336.05 198.74 10.053 1.010

30 336.95 199.26 9.557 0.703

45 335.40 199.29 8.975 0.53260 328.64 194.78 8.864 0.553

75 325.09 194.23 9.042 0.689

90 322.13 193.64 9.196 0.728

EB 363.44 234.67 7.278 1.149

T.B. Stoughton, J.W. Yoon / International Journal of Plasticity 25 (2009) 17771817 1781
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where _k is the plastic compliance factor that controls the magnitude of the rate of change of the plastic

strain tensor. Since the plastic potential is a linear homogeneous function of the stress, it follows from

Eq. (2), and the following sequence, that the rate of plastic work is equal to the product of the mag-

nitude of the plastic potential and the plastic compliance factor

dwpdt

rij dEp

ijdt

rij @rp~r@rij

_k rp~r _k: 3

Although it is not necessary to explicitly define the compliance factor for implementation in finite ele-

ment analysis, it is useful to note that the quadratic form of the plastic potential, with the flow rule,

Table 3

Parameters of the plastic potentials of the five models described in this paper for the five metal alloys described in Tables 1 and 2.

Note that the plastic potential of all but von Mises model are identical.

Metal Plastic potential

Parameter von Mises Hill (1948) Stoughton (2002) Cvitanic (2008) New model

AA 5182-O kP 1 0.9512 0.9512 0.9512 0.9512vP 1 0.4890 0.4890 0.4890 0.4890

qP 1 1.3956 1.3956 1.3956 1.3956

AA 6022-T4E32 kP 1 1.1173 1.1173 1.1173 1.1173

vP 1 0.4515 0.4515 0.4515 0.4515

qP 1 1.1063 1.1063 1.1063 1.1063

AA 6022-T43 kP 1 1.2038 1.2038 1.2038 1.2038

vP 1 0.5071 0.5071 0.5071 0.5071

qP 1 1.2275 1.2275 1.2275 1.2275

718-AT kP 1 0.9036 0.9036 0.9036 0.9036

vP 1 0.6466 0.6466 0.6466 0.6466

qp 1 1.7051 1.7051 1.7051 1.7051

719-B kP 1 0.9175 0.9175 0.9175 0.9175vP 1 0.6840 0.6840 0.6840 0.6840

qP 1 1.1488 1.1488 1.1488 1.1488

Table 2

Properties of two stainless steel alloys provided by POSCO fit to both Voce and Swift Hardening Model parameters. Initial yield

stress is AB.

Test A B C r K n

718AT 0 532.38 321.50 9.592 1.830 746.66 0.264

15 520.39 307.40 10.428 1.763 719.85 0.256

30 519.28 306.09 9.897 1.834 724.87 0.255

45 520.70 301.30 10.149 2.294 724.25 0.248

60 528.90 304.16 9.839 2.708 727.23 0.246

75 519.45 297.34 10.516 2.562 723.93 0.244

90 516.28 299.79 10.412 2.517 725.06 0.250

EB 709.54 471.79 9.980 0.803 965.00 0.267

719B 0 540.59 329.71 10.828 2.165 785.45 0.267

15 546.69 336.19 11.214 1.900 807.20 0.271

30 548.90 334.50 11.883 1.570 818.91 0.268

45 555.51 332.95 12.029 1.591 822.13 0.261

60 548.21 326.74 12.271 1.979 813.59 0.259

75 537.45 323.54 11.363 2.745 783.73 0.262

90 534.93 323.33 11.425 2.930 784.31 0.264EB 724.88 465.30 9.095 0.860 938.60 0.250



7/41

also leads to a specific definition of the plastic compliance factor in terms of the parameters of the

plastic potential and the components of the plastic strain rate tensor as follows:

_k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikp _Ep11 2 _Ep22 2 2mp _Ep11 _Ep22

kp

m2p

2q0

_Ep12 2s

: 4

It is therefore a consequence of the flow rule and the linear homogeneous plastic potential, that the

compliance factor is a specific linear homogeneous function of the rate of change of the plastic strain

tensor, with parameters of this function determined solely by the parameters of the plastic potential.

It is therefore reasonable to identify the plastic compliance as the rate of change of an effective plastic

strain, and more to the point, the integral of which

k Z

_kdt 5

can be used as the work hardening variable in the stressstrainrelation. This definition was adopted in

Stoughton (2002) and in subsequent work on non-AFR in Stoughton and Yoon (2004, 2006, 2008).

However, recently Cvitanic et al. (2008) proposed a modification to the non-AFR by making a distinc-tion between the plastic compliance factor and the rate of change of an additional variable, propor-

tional to the plastic compliance, to be used instead to define the work hardening variable. The

purpose of this second parameter, which they identified to be the effective plastic strain, was to intro-

duce a relationship between the magnitude of the yield function and the rate of plastic work, which

works in unison with Eq. (3), but is not maintained in the model proposed in Stoughton (2002),

Stoughton and Yoon (2004, 2006, 2008). This alternate formulation has an impact on the prediction

of the stressstrain responses for isotropic hardening, and since we want to compare the accuracy

of the new anisotropic hardening model proposed in this paper with respect to the accuracy that is

obtainable under isotropic hardening, it is important to look at both alternatives for the definition

of the work hardening variable.

So, for clarity in the following discussion, we will adopt and retain the compliance factor from the

original non-AFR formulation proposed in Stoughton (2002) to control the stressstrain relation in the

anisotropic hardening model without necessarily introduction of or connection with the so-called

effective plastic strain. Then in comparison to isotropic hardening, we will continue to use the com-

pliance factor to define the work hardening variable in the Stoughton and Yoon version of non-AFR,

and finally, introduce the variable called the effective plastic strain only in the Cvitanic et al. (2008)

formulation to control the isotropic hardening.

Before we introduce the anisotropic non-AFR model, as a segue, it is useful to start with the isotro-

pic hardening non-AFR model proposed in Stoughton (2002), which using the integral of the plastic

compliance, for plane stress conditions can be expresses as

r0k

P ry

~r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir211

kyr222

2myr11r22

2qyr12r21q ;

ky r0r90

2; my 1

21 r0

r90

2 r0

rb

2 !; qy

1

2

2r0r45

2 r0

rb

2 ! 6

The parameters in the yield function,r0; r45, and r90 are the initial yield stresses in uniaxial tension atrespectively, 0, 45, and 90 degrees to the rolling direction of the sheet coil, and rb is the initial yieldstress in equal biaxial tension. With the normalization to uniaxial tension along the 1-axis, the func-

tion r0k on the left-hand side of the yield criterion is the stressstrain response in uniaxial tensionalong the rolling along the 1-axis, where by definition, r00 r0. Coefficients of the yield function aregiven in Table 4 for four isotropic hardening models used in this paper for comparison to the new

anisotropic hardening model described in the remainder of this section.

While the non-AFR model described in Eqs. (1)(6) is suitable for describing the anisotropy of the

strain ratios and the initial yield stress, it does not allow for the effects of anisotropic hardening, due to

the fact that hardening is controlled by a single function of the compliance factor, calibrated in this

normalization to the experimental data along the 1-axis, or rolling direction of the sheet coil. The ef-

fect of this constraint is seen in Fig. 1 by comparison of the experimental stressstrain data for an AA



8/41

5182-O at 45 and 90 degrees to the rolling direction and in equal biaxial tension, with predictions un-

der isotropic hardening. We can see in this figure that the initial yield behavior is matched perfectly,

calibration of the model to the hardening along the rolling direction in uniaxial tension leads to

increasing deviations in equal biaxial condition, and to a lesser, but still finite degree, in uniaxial ten-sion along the other loading axes. Generally, these systematic differences are problematic for applica-

tions such as springback prediction and compensation as well as earing prediction. Especially, the

need for precision is driven by the high cost of changing tool shapes after the tool die is cast to account

for springback effects.

A solution to the problem of non-isotropic hardening under conditions of proportional loading is

obtained by first rescaling the yield function defined in Eq. (6) by the stressstrain relation r0k,and then noting that the four initial yield stresses r0; r45; r90, and rb that are used to define theparameter constants of the model can be explicitly replaced by independent yield functions,

r0k; r45k; r90k, and rbk, respectively. After squaring the function and rearrangement of theterms, this leads to thefollowing dimensionless quadratic homogeneous yield function for plane stress

conditions

fy~r; k r11r20k

r22r290k

r11 r22 r11r22 r12r21

r2bk

4r12r21r245k

: 7

Similar to the yield criterion given in Eq. (6) for isotropic scalar hardening, the constitutive equation in

this anisotropic scalar hardening model is constrained so that

fy~r; k 6 1 8for all possible elastic and elasticplastic deformations. Furthermore, the deformation is unambigu-

ously defined to be elasticplastic if and only if

fy~r; k

1

8a

and

@fy~r; k@rij

drij

dt> 0: 8b

Table 4

Parameters of the yield function of the four isotropic hardening models described in this paper for the five metal alloys described in

Tables 1 and 2. Note that the parameters of von Mises and Hill (1948) are identical to those of the plastic potential in Table 3, as

required under the AFR. The parameters of the Stoughton (2002) and Cvitanic et al. (2008) models are also identical with the

difference in the definition of the work hardening variable. The new model replaces the parameters constants in the yield

function by explicit functions of the hardening variable.

Metal Yield function

Parameter von Mises Hill (1948) Stoughton (2002) Cvitanic (2008) New model

AA 5182-O kY 1 0.9512 1.0262 1.0262 N/A

vY 1 0.4890 0.5844 0.5844 N/A

qY 1 1.3956 1.7223 1.7223 N/A

AA 6022-T4E32 kY 1 1.1173 1.1061 1.1061 N/A

vY 1 0.4515 0.6003 0.6003 N/A

qY 1 1.1063 1.5702 1.5702 N/A

AA 6022-T43 kY 1 1.2038 1.1294 1.1294 N/A

vY 1 0.5071 0.5025 0.5025 N/A

qY 1 1.2275 1.4507 1.4507 N/A

718-AT kY 1 0.9036 0.9488 0.9488 N/A

vY 1 0.6466 0.5811 0.5811 N/A

qY 1 1.7051 1.4543 1.4543 N/A

719-B kY 1 0.9175 0.9932 0.9932 N/A

vY 1 0.6840 0.6666 0.6666 N/A

qY 1 1.1488 1.4656 1.4656 N/A



9/41

For any other condition satisfying Eq. (8), the deformation is defined to be purely elastic.

If the deformation satisfies Eqs. (8a) and (8b), and therefore involves changes in the plastic strain,

the compliance factor in the flow rule (Eq. (2)) controlling the magnitude of the increase in plastic

strain is defined by imposing Eq. (8a) as a constraint of deformation, as follows:

_k 1^h@fy~r;

k@rij

drij

dt> 0; 9

where

^h dfy~r;k

dk 2 r11

r20kh0 r22

r290kh90

r11 r22 r11r22 r12r21

r2bk hEB

4r12r21r345k

h45

!;

hh 1rhk

drhkdk

; hEB 1rbk

drbkdk

:

10

The complete constitutive equation that governs elasticplastic deformation is then given in tensorform by

drij

dt Cijkl dEkl

dt C

ijkl @rp~r@rkl

^h @fy~r;k@rab

Cabcd@rp~r@rcd

@fy~r; k@rmn

CmnopdEopdt

; 11

where Cijkl

is the elastic stiffness tensor. Note the similarity of form of this constitutive equation with

that for conventional isotropic hardening, even though both the yield function, fy~r; k, and plasticmodulus, h, are now dimensionless functions.

The compliance factor used in the flow rule can be defined either in terms of the stress rate tensor

from Eq. (9), or in terms of the total strain rate tensor by combining Eqs. (9) and (11) to obtain the

following equivalent form:

_k ^h @fy~r;k

@rabCabcd

@rp~r@rcd

1@fy~r; k@rij

CijkldEkldt

: 12

Predictions of this model are shown in Figs. 2, 4, 7, 9, 11, and 13 for the metals listed in Tables 1 and 2.

While the model described in Eqs. (1)(12) is limited to plane stress condition, it can be extended to

full stress condition with additional stress component terms in the generalization of Eqs. (7) and (10).

Vector forms of the tensor relations and other implementation details applicable to both plane-stress

and full stress conditions are described in Appendix A.

3. Model calibration

Tables 1 and 2 provides Voce Law fit parameters to experimental true stress as a function of

plastic strain. While the Voce Law is not generally suitable for most steels, it is reported to more

closely match the response of the present stainless steel alloys listed in Table 2, so parameters of

both the Voce and Swift hardening functions are given. The data in Tables 1 and 2 for uniaxial ten-

sion are obtained in a fit of the true stress, rU, as a function of the plastic strain along the tensionaxis at a direction h to the 1-axis of the sheet, or rUh; EU. The fitted functions in the two tables at0, 45, and 90 degrees are not identical to functions required in the material model, in Eq. (7),

which must be defined in terms of the compliance factor, not the magnitude of the true plastic

strain. The compliance factor is related to the plastic strain tensor through Eq. (4). Therefore, mod-

el calibration requires specification of the transformation from the raw data to the form required

in the model.

For uniaxial tension to a true stress rU along angle h to the rolling direction of the sheet coiling, thestress tensor components are given explicitly by

http://-/?-http://-/?-


10/41

r11r22r12

264

375 rU

cos2hsin

2hcosh sinh

264

375: 13

By using the relationship of _Ep

U _k

@rprIJ

, the plastic strain rate _EU along the direction h is projected as

_EUh _Ep11 cos2h _Ep22 sin2h 2 _Ep12 cosh sinh

_k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos4h kp sin4h 2qp mp cos2h sin2hq : 14Consequently, for proportional loading in uniaxial tension, the relation between the plastic strain

along the 1-axis and the plastic compliance factor is

0

50

100

150

200

250

300

350

400

450

0.0 0.1 0.2 0.3 0.4 0.5

TrueStress

(MPa)

True Plastic Strain

Stress-strain for AA 5182-O

0 - New Model

45 - New Model

90 - New model

EB- New Model0 -exp

45 - exp

90 - exp

EB -exp

Fig. 2. Solid lines show the AA 5182-O stressstrain response using the anisotropic hardening model described in the current

paper for uniaxial tension at 0, 45, and 90 degrees to the rolling direction of the sheet coil and for equibiaxial tension. The datapoints are experimental data.



11/41


12/41

will look at metrics designed to measure the improvement over all loading conditions momentarily, it

is interesting to look at the improvement in the four uniaxial tension test conditions listed in Tables 1

and 2 that are not involved in the model calibration, specifically, those at 15, 30, 60, and 75 degrees to

the rolling direction where the models results can be classified as actual predictions.

Fromthe yield function defined in Eq. (7) and the general stress state for uniaxial tension defined in

Eq. (13), the magnitude of the uniaxial stress in a given direction as a function of the plastic compli-

ance is given by the following function:

rAHU h; k 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos2hr2

0kh

sin2hr2

90kh

cos2h sin22h

r245

kh

r ; 24

where the three functions rhkh are defined by Eq. (16) for Power Law hardening and Eq. (18) forVoce Law hardening, and

kh

EUh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos4h kp sin4h 2qp mp cos2h sin2h

q: 25

The error between model and the experimental stressstrain data is defined by the following

difference:

dhEU rAHU h;EUjhEU 1: 26

Since stress and strain conditions at the end of forming a cold stamped metal product vary between

zero strain and up to strains within a safety margin of necking and fracture conditions, a more useful

metric of error is the average RMS error over a finite strain range,

Dhe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

e

Ze

0

jdhEUj2dEUs

: 27

Since the errors in uniaxial tension along 0, 45, and 90 degrees with the new model are explicitly zero,

a useful metric for comparing predictions of this model with others, is the average RMS error over uni-

axial tension directions at 15, 30, 60, and 75 degrees,

Dpredicte ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

4D15e2 D30e2 D60e2 D75e2

r: 28

For comparison, the same error functions are obtained for the predictions of several constitutive mod-

els published in the literature, including isotropic hardening based on von Mises, Hills (1948), and

Barlats (2003) Yld2000-2d models, all based on AFR, and isotropic hardening based on the modelsproposed in Stoughton (2002), Cvitanic et al. (2008), both based on the non-AFR approach. The pre-

dicted stress strain responses for the von Mises model, which replaces the function rAHU in the errorfunction given by Eq. (26), and then integrated in Eq. (27), and summed in Eq. (28), is

rMisesU EU j0EUh; 29awhere j0EU is the true stress vs. true strain response in uniaxial tension along the rolling direction ofthe sheet, described in this paper by either the Voce Law or Power Law functions. The corresponding

function for the Hills (1948) model is

rHillU EU j0

EUhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos4hkp sin

4

h2qpmp cos2h sin2

hp !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos4h kp sin4h 2qp mp cos2h sin2h

q ; 29bfor Yld2000-2d model,



13/41

rBaU EU j0

EUhcU

cU

; 29c

where cU is a complex function equal to the constant defined in Table 5 for uniaxial loading in 15 de-gree increments to the rolling direction; for Stoughton (2002) model,

rStUEU j0

EUhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos4hkp sin4h2qpmp cos2h sin2h

p !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos4h ky sin4h 2qy my cos2h sin2h

q ; 29d

and for Cvitanic et al. (2008) model,

rCvU EU j0

EUhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos4hky sin4h2qymy cos2h sin2h

p !

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos4h ky sin4h 2qy my cos2h sin2h

q : 29eNote that although the Hill (1948), Stoughton (2002), Cvitanic et al. (2008) models predict stress

strain responses that are functionally similar, the first two models explicitly use the plastic potentialparameters kp;qp; mp for scaling the plastic strain in the experimental stressstrain functionj0EU, while the latter two models explicitly use the yield function parameters ky;qy; my for scalingthe predicted stress response in uniaxial tension (see the denominators).

An unbiased metric to compare accuracy of models for arbitrary uniaxial tension is to replace the

metric in Eq. (28) with the following average:

DUeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

12D0e2 2D15e2 2D30e2 2D45e2 2D60e2 2D75e2 D90e2

r;

30where the weighting factors of 2 on the 15, 30, . . ., 75 degree functions reflect the inclusion of contri-

butions at angles of 105, . . ., 150, and 175 degrees. Since the data for uniaxial tension along the rolling

direction are used to calibrate the hardening in all six models, D0e is explicitly zero for all models.Furthermore, D45e D90e 0 for only the new model, while these functions are generally nonzerofor the other models.

Table 5

Parameters of the Barlat 2000-2d model for the AA5182-O described in Barlat et al. (2003) and scale factors required for error

calculations using Eqs. (29c) and (33c).

Coefficient AA5182-O AA 6022-T4E32 AA 6022-T43 718 AT 719 B

A1 0.9872 0.8611 0.8687 0.6071 0.5764

A2 1.1009 0.9380 0.9637 0.5936 0.6399

A3 0.8710 0.7511 0.9261 0.5023 0.4306

A4 1.0246 0.9350 0.9582 0.5132 0.4938

A5 1.0317 0.9002 0.9212 0.5355 0.5143

A6 0.8802 0.7935 1.0131 0.4871 0.3798

A7 1.0564 0.8226 0.8632 0.5894 0.5600

A8 1.2778 1.0784 0.9017 0.5112 0.6053

k0 1.0000 1.0000 1.0000 1.0000 1.0000

k15 0.9912 1.0032 1.0039 1.0082 1.0124

k30 0.9739 1.0040 1.0070 1.0264 1.0388

k45 0.9640 0.9940 0.9970 1.0400 1.0550

k60 0.9683 0.9768 0.9743 1.0400 1.0442

k75 0.9805 0.9591 0.9509 1.0317 1.0169

k90 0.9870 0.9510 0.9410 1.0270 1.0030kB 1.0799 1.0509 0.9430 1.1274 1.2309



14/41

In order to further extend the error analysis beyond uniaxial tension, to introduce biaxial loading

directions, we need the explicit expressions for the error in equal biaxial tension, Dbe. For the newmodel, the true stress in equal biaxial tension is given by

rAHb EB jbkh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 kp 2mpq jb EBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 kp 2mpp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 kp 2mpq ! jbEB: 31So the error, defined by the difference between model and experiment,

dbEB rAHb EBjbEB 1; 32

is explicitly zero. However, this is not the case for the isotropichardening models, which are calibrated

to the uniaxial tension data along the rolling direction. For these models, based on the stressstrain

function along the rolling direction, j0EB, the stressstrain predictions are given by

rMisesb EB j0EB; 33a

rHillb EB j0

EBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1kp2mp

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 kp 2mp

p ; 33brBab EB

j0EBcB

cB

; 33c

where cB is a complex function equal to the constant defined in Table 5 for equal biaxial loading;

rStb EB

j0EB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1kp2mpp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ky 2my

p 33dand

rCvb EB j0

EBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ky2my

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ky 2my

p : 33eNote that as in the case of the uniaxial stressstrain predictions given in Eqs. (29c) and (29d), the

Hill (1948), Stoughton (2002), Cvitanic et al. (2008) models predict equal biaxial stressstrain re-

sponses that are functionally similar. As in the uniaxial case, the first two models explicitly use

the plastic potential kp;qp; mp parameters for scaling the plastic strain in the experimental stress

strain function j0EB, while the latter two models explicitly use the yield function parame-tersky;qy; my for scaling the predicted stress response in equal biaxial tension (see thedenominators).

The error in the prediction of equal biaxial tension is defined by replacing rAHb in Eq. (32) with oneof the four stressstrain predictions in Eq. (33) and similar to Eq. (27) defining the average RMS error

over a finite strain range by

Dbe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

e

Ze0

jdbEBj2dEBs

: 34

Although additional experimental data in other biaxial stress conditions would provide a more real-

istic measure of overall model error, the following metric is used to estimate the model accuracyfor all uniaxial and biaxial loading conditions:

De ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

2DUe2 Dbe2

r: 35



15/41

5. Application of the models to seven sets of material data

For the purposes of this paper, the material parameters given in Tables 1 and 2 are taken to

represent the stressstrain response for all strains, and for the purposes of argument, the Voce

and Swift Law hardening characterizations for the two stainless steels are treated as though they

characterize different materials. Under this assumption, the accuracy of any of the six material

models can be evaluated based on how well the models reproduce the stressstrain responses

in the loading conditions defined in these tables, using the metrics defined in the previous sec-

tion. These error estimates are limited only to the calculation of the stress and do not account for

errors in predicted strain ratios. The error in predicted strain ratios may be quite large in the

case of the von Mises model, since the r values listed in Tables 1 and 2 differ substantially from

the implied isotropic response with r equal to unity. However, for the other four models, the r

values in the table are used to define the plastic potential, and therefore, these models are ex-

pected to result in negligible errors in predicted strain ratios, at least in uniaxial tension, and

as long as experimental strain ratios in uniaxial tension remain constant. While errors in pre-

dicted strain ratios should in general be considered in the evaluation of the von Mises model,

for the purposes of this paper we will focus on the errors in the prediction of stresses for all five

models.

Figs. 1 and 2, respectively, show the predicted stressstrain behavior using the non-AFR model

proposed in Stoughton (2002) based on isotropic hardening, and the new non-AFR model based on

anisotropic hardening for AA 5182-O for equal biaxial tension and uniaxial tension along 0, 45, and

90 degrees to the rolling direction. Note that the new model, shown in Fig. 2, which uses the

experimental data as input, exactly reproduces the experimental response. It clearly demonstrates

that the model has the ability to overcome a serious limitation of isotropic hardening models, even

those that are able to exactly capture the anisotropy of the initial yield stress, as is the case with

the Stoughton (2002) model. This problem of anisotropic hardening is common to all the metals

considered in this paper, including the stainless steel alloys. Fig. 3 shows the accumulated RMS er-

rors, calculated from Eq. (27) for the three sets of uniaxial data and Eq. (34) for equal biaxial datafor the two models.

Fig. 4 shows the predicted stressstrain behavior for AA 5182-O for uniaxial tension at 15, 30,

60, and 75 degrees to the rolling direction using the new model. Note that the new model does not

use this experimental data, so these results are true predictions. While careful inspection of these

curves with experiment shows a remarkable agreement, it is better to quantify this agreement

using the metrics in the previous section, and compare the accuracy of the new model with that

obtained using the five isotropic hardening models. Fig. 5 shows the accumulated RMS error for

the four uniaxial tension directions using the metric defined in Eq. (28) for all six models. In addi-

tion to the simple fact that the RMS error is significantly smaller in the case of the new model, it is

also interesting to note that the RMS error changes with plastic strain for all isotropic hardening

models, but the error for the anisotropic hardening does not appear to change significantly overthe plastic strain range from 0% to 50%. This would be expected if the differences in the stress

strain response in the data for the anisotropic hardening model in Fig. 4 reflect stress measure-

ment uncertainty, since this uncertainty is not expected to be a function of plastic strain. The fact

that the nominally 1% magnitude of the RMS error obtained with the new model is near to the

expected measurement uncertainty, supports the interpretation that this anisotropic hardening

model may be achieving the maximum possible level of accuracy that current measurement meth-

ods can detect.

Fig. 6 shows the accumulated RMS error for all loading conditions including biaxial loading for the

AA 5182-O alloy using the error metric defined in Eq. (35). This figure shows that after only a few per-

cent strain, all the isotropic hardening models result in stress errors that average between 5% and 10%

of the stress magnitude. This loss of accuracy applies even those models that capture the exact initialyield stresses with precision in biaxial tension and the three primary uniaxial test conditions, such as

the Barlats (2003) Yld2000-2d, and the Stoughton (2002), Cvitanic et al. (2008) models. Perhaps most

surprising is that it appears that the best of the five isotropic hardening models for AA-5182-O,



16/41

between plastic strains of about 2% to about 35% strain, is the von Mises model. This might be dis-

missed from the well-known limitation of the quadratic yield function to describe the behavior of alu-

minum alloys. However, the comparable errors obtained with Yld2000-2d model suggests that the

problem is unrelated to the exponent, but is due to distortion of the shape of the yield function. Fur-

thermore, the very small error of nominally 1% of the stress magnitude obtained with the new model,

which is also based on a quadratic yield function, supports this conclusion.

Figs. 7, 9, 11, and 13 show the predicted stressstrain curves from the new model along with exper-

imental data for the two other aluminum alloys listed in Table 1 and the two stainless steel alloys

listed in Table 2, all based on Voce Law hardening. Following each figure, respectively, in Figs. 8, 10,

12, and 14 are the accumulated RMS errors obtained using the six models. The observations are essen-

tially identical to those in the discussion of the RMS error functions shown in Fig. 6 for the AA 5182-O,

where the new model is observed to be in excellent agreement with experimental data. As a summary,

all of the materials are compared in Fig. 15, which shows the peak value of the accumulated RMS error

45 deg

Stoughton[2002]

90 deg

Stoughton[2002]

Equibiaxial

Stoughton[2002]

0 deg

New Model

45 deg

New Model

90 deg

New Model

0 deg

Stoughton [2002]

Equibiaxial

New Model0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.0 0.1 0.2 0.3 0.4 0.5

RMS

Error

True Plastic Strain

AA 5182-O

Accumulated RMS Error

Fig. 3. RMS error in predicted stresses for AA 5182-O defined by Eq. (27) for uniaxial tension of AA 5182-O at 45 and 90 degrees

to the rolling direction and Eq. (34) for equal biaxial tension. Solid lines (all at zero error) are for the anisotropic hardening

model. Dashed lines are for the Stoughton (2002) model based on isotropic hardening.



17/41

obtained over the range from 0% to 50% plastic strain. Remarkably, the RMS error obtained with the

new anisotropic hardening model appears to remain static at approximately the 1% error level, consis-

tent with the postulated cause due primarily to measurement uncertainty.

We can also see in Fig. 15 that the von Mises model is a poor characterization of the stainless

steel alloys, with an RMS error in stress prediction on the order of up to 20%. It is also seen in this

figure that Hills model under the AFR has an RMS error for both stainless steel and aluminum of

about 10%. This error in the Hill model is actually higher than the error obtained using von Mises

model on the aluminum alloys, which averages about 6.5%. All of these results are consistent with

previous observations about the inapplicability of von Mises model to steel and the inapplicability

of Hills model to aluminum alloys. However, what is perhaps surprising is that, in comparison ofHills model to von Mises for the steel alloys, despite the addition of 3 parameters to capture the

material anisotropy, there is only about a factor of 2 reduction in the peak RMS error over the

strain range from 0 to 50% strain. One could argue from this that the cost/benefit of using Hills

0

50

100

150

200

250

300

350

400

450

0.0 0.1 0.2 0.3 0.4 0.5

TrueStress(MPa)

True Plastic Strain

Stress-Strain Relations for AA 5182-O

15 - New Model

30 - New Model

60 - New Model

75 - New Model

15 - exp

30 - exp

60 - exp

75 - exp

Fig. 4. Solid lines show the predicted AA 5182-O stressstrain responses using the anisotropic hardening model described in

the current paper for uniaxial tension at 15, 30, 60, and 75 degrees to the rolling direction of the sheet coil. The data points are

experimental data.



18/41

fully anisotropic model over von Mises is not low enough to warrant its use in engineering

applications.

What is also especially interesting are the large errors using the Yld2000-2d, and the two non-AFR

models proposed in Stoughton (2002) and in Cvitanic et al. (2008). These later models introduced sep-

arate quadratic functions of the form proposed by Hill in 1948 for the purpose of improving the pre-

dictions of anisotropic yielding and anisotropic plastic strain ratios, and the Yld2000-2d was

developed for the same purpose under the AFR. Indeed all models result in zero error at the initial yield

point for uniaxial tension along the rolling, transverseand diagonal directions, as well as in equal biax-ial tension as seen in Figs. 3,714, where the RMS error defined by Eq. (35) at zero plastic strain is due

only to small contributions of error from the uniaxial tension data at angles of 15, 30, 60, and 75 de-

grees. However, as the plastic strain increases, the errors in these models based on isotropic hardening

von Mises

Hill [1948]

Stoughton [2002]

Cvitanic et al. [2008]

New Model

Barlat et al. [2003]

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.1 0.2 0.3 0.4 0.5

RMS

Er

ror

Uniaxial Plastic Strain

AA 5182-O

Average Error at 15o, 30

o, 60

o, and 75

oUniaxial Tension

Fig. 5. Accumulated errors in predictionof stresses for AA5182-O in uniaxial tension averaged over directions at 15, 30, 60, and

75 degrees to the rolling direction, using the metric defined in Eq. (28) for five material models.



19/41


20/41

model for aluminum alloys over the strain range considered. Furthermore, it may be well argued

as a practical matter, that the advanced Barlat or non-AFR models with RMS errors on the order of

4 to 8% are not sufficiently better than the RMS errors using Hills (1948) models under the AFR,

where for the stainless steel, the RMS errors are between 8% and 13%. These errors with the ad-

vanced models are still sufficiently large that a significant fraction of problems with formability

and springback will be left undiscovered until prototype tools are constructed and the physical try-

out process begins.

Finally, we look at the RMS error of the new anisotropic hardening model based on a non-AFR

with constant quadratic plastic potential and variable quadratic yield functions. The errors in this

model come from errors in predicting the stressstrain response in uniaxial tension at 15, 30, 45,

and 90 degrees. We can see from Fig. 5 in the case of the AA 5182-O alloy that the new model

substantially improves the prediction of the stress response in these four uniaxial tension direc-

0

50

100

150

200

250

300

350

400

0.0 0.1 0.2 0.3 0.4 0.5

TrueStress

(MPa)

True Plastic Strain

Stress-strain Response for AA 6022-T4E32

0 - New Model

15 - New Model

30 - New Model

45 - New Model

60 - New Model

75 - New Model

90 - New Model

EB- New Model

0 -exp

15 - exp

30 - exp

45 - exp

60 - exp

75 - exp

90 - exp

EB -exp

Fig. 7. Experimental and predicted stress strain responses for AA 6022-T4E32 with Voce Law hardening. Predictions based on

new anisotropic hardening model.



21/41

tions. This figure provides an unbiased picture to argue that a substantial improvement has been

made since it compares true predictions from all five models. However, a less biased measure of

accuracy is to include data in a representative set of loading conditions, and for this measure we

calculate the RMS errors over all uniaxial and biaxial data that are available using Eq. (35), and

shown in the later figures, with the results for all materials summarized in Fig. 15. It is found

that the peak RMS error with the new model varies from 0.4% for the AA 6022-T4E32 to 1.2%

for the 718-AT stainless steel. In comparison, these errors are 920 times higher for the four iso-

tropic hardening models over the strain range from 0 to 50% plastic strain. Furthermore, the best

of the four isotropic hardening models at any strain level is still 5 to 15 times higher than the

error obtained with the anisotropic hardening model at that strain. Consequently, the new model

is shown to result in approximately an order of magnitude reduction in errors in stress

prediction.

von Mises

Hill [1948]


Stoughton [2002]

New Model


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.1 0.2 0.3 0.4 0.5

RMS

Error

Plastic Strain

AA 6022-T4E32Average Error Overall

Fig. 8. Accumulated errors in prediction of stresses for AA 6022-T4E32 in uniaxial and equal biaxial tension using the metric

defined in Eq. (35).



22/41

It is common practice to plot contours of the yield function in the 2D space of the in-plane

principal stresses in order to compare models to experiment and to each other. The yield functions

for 718 AT steel alloy using all five isotropic hardening models, and the new model based on non-

AFR and anisotropic hardening are compared in Fig. 16. The smaller set of ellipses that intersect

the 1-axis at 212.4 MPa correspond to the initial yield surfaces of these models. Note that the yield

function of the new model is identical to the yield function of the two non-AFR models based on

isotropic hardening. At higher strains, corresponding to an equivalent strain-hardening parameter

equal to 0.40 strain along the rolling direction, the yield function of the anisotorpic hardening

model develops a different shape from the two isotropic hardening non-AFR models, both of whichretain the same shape of the initial yield surface. Similarly, the von Mises, Hills (1948), and

Barlats (2003) Yld2000-2d models are shown to expand isotropically. The same analysis is applied

to the AA 5182 alloy in Fig. 17.

0T 2


23/41

While the representation of the yield functions in plots shown in Figs. 16 and 17 are often used

to draw conclusions about the accuracy of the models, it is not so obvious what significant state-

ments can be made. For example, there is probably no importance to the observation that the Hills

(1948) yield function exactly matches the initial uniaxial yield stress along the 2-axis and exactly

matches the equal biaxial yield stress at an equivalent strain of 0.40, since it does not match the

initial equal biaxial yield stress or the uniaxial yield stress along the 2-axis at an equivalent strain

of 0.40.

What is perhaps a more useful purpose of the plots in Figs. 16 and 17 is to use them to point

out distinctions between the models. The first distinction is that for the non-AFR models, there is

another critical component of the model, the plastic potential, which is identical in all three non-

AFR models and is equivalent to the shape of the Hills (1948) model in these figures, whose

parameters are calculated from three r values measured in uniaxial tension at 0, 45, and 90

von Mises

Hill [1948]

Cvitanic et al. [2008]Stoughton [2002]

New Model


0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.1 0.2 0.3 0.4 0.5

RMS

Error(M

Pa)

Plastic Strain

AA 6022-T43Average Error Overall

Fig. 10. Accumulated errors in prediction of stresses for AA 6022-T43 in uniaxial and equal biaxial tension using the metric

defined in Eq. (35).



24/41

degrees. By assumption in all three models, even for the anisotropic hardening model, the shape of

the plastic potential does not evolve, so the plastic potential remains equivalent to the Hills (1948)

model for all plastic strain.

The second distinction is that, in the case of the von Mises, Hills (1948), Barlats (2003)

Yld2000-2d, and Cvitanic et al. (2008) models, the yield function contours are also the con-

tours of equivalent plastic work. However, in the new model, as well as in Stoughton

(2002), it is the plastic potential, i.e., a function whose shape is identical to the illustrations

of Hills (1948) yield function, that describe the contours of equivalent plastic work. One con-

sequence of this is that different amounts of plastic work are required to reach different

points on the yield function in these models beyond the initial yield point, and this can lead

to distinct behaviors under nonproportional loading, giving rise to, for example, the energetic

0

100

200

300

400

500

600

700

800

0.0 0.1 0.2 0.3 0.4 0.5

Truestress

(MPa)

True Plastic Strain

Stress-strain Response for Stainless Steel 718-AT

0 - New Model

15 - New Model

30 - New Model

45 - New Model

60 - New Model

75 - New Model

90 - New Model

EB - New Model

0 -exp

15 -exp

30 -exp

45 -exp

60 -exp

75 -exp

90 -exp

EB -exp

Fig. 11. Experimental and predicted stress strain responses for 718-AT stainless steel with Voce Law hardening. Predictions

based on new anisotropic hardening model.



25/41

nature of non-AFR materials. However, as discussed in Stoughton and Yoon (2006, 2008), the

non-AFR does not in general imply instability of the type that can lead to bifurcation or non-

uniqueness of solution, and there are a set of restrictions on the yield function and plastic

potential to ensure stability, which are satisfied in the models described in this paper. While

there is no thermodynamic or other theoretical reason why the yield function should serve as

the contour of equivalent plastic work, beyond imposing the condition of the AFR or equiva-

lently, the principle of maximum plastic work, this decoupling of the yield surface from the

contours of plastic work was the reason for modification of the Stoughtons (2002) model tothat proposed by Cvitanic et al. (2008), which restores the yield functions service as the con-

tour of equivalent plastic work.

von Mises

Hill [1948]


Stoughton [2002]

New Model


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.1 0.2 0.3 0.4 0.5

RMS

Error(M

Pa)

Plastic Strain

Stainless Steel 718-AT (Voce)Average Error Overall

Fig. 12. Accumulated errors in prediction of stresses for Posco Stainless Steel alloy 718-AT in uniaxial and equal biaxial tension

using the metric defined in Eq. (35) using Voce Law hardening. Note the change in scale for the steel data compared to

aluminum shown in Figs. 46.



26/41

6. Instability and strain localization

Fig. 17 shows the predicted evolution of the shape of the yield function at selected levels of plastic

work for the 718 AT steel alloy using Voce hardening for the new model. Although the shape changes

are difficult to detect in this figure byeye, the differences are significant and are on the order of 10%, as

discussed in the previous section. A similar picture of roughly-isotropic evolution of the yield surface

contours is seen for the other steel alloy.

The picture of yield surface evolution is significantly different for the aluminum alloys, showinga strong saturation in the stress levels in the second and fourth quadrants in a diagram of normal

stress at the strain levels, as shown in Fig. 18 for the AA 5182 alloy. There is no evidence of sat-

uration in the first and third quadrants at these strain levels. For comparison, recall from Fig. 17,


27/41

that there is also no evidence of saturation in any quadrant for the stainless steel, even though a

Voce law was also used in that case. Similar saturation of the yield surface contours is obtained

with the new model for the other two aluminum alloys in Table 1. This directional saturation arises

from the definition of the weighted plastic modulus in Eq. (10), which in pure shear at an angle h

to the rolling direction, is given by

^hshearh 2dr0epr3

0epdep

dr90epr3

90epdep

cos2h drEBep

2r3EB

epdep 2sin2h dr45ep

r345

epdep1

r2

0ep

1

r2

90ep

cos2

h

1

2r2

EBep

2sin2hr2

45ep

: 36

Fig. 19 plots the plastic shear modulus for shearing along the rolling direction, ^hshearh 0, which is

calculated to pass through a zero at plastic strains of 0.449, 0.509, and 0.595 for AA 5182-O, AA 6022-

T4E32, and AA 6022-T43, respectively.


28/41

The interest in the condition of saturation at finite strain is that a zero (or negative) slope of the

plastic modulus may lead to numerical instability and indeterminacy of the solution, as discussed

in Stoughton and Yoon (2006). Since the point of saturation is restricted to high strain, the effect

could play a role in a mechanism for metal failure. While the absence of saturation in the harden-

ing in steel prevents this effect from playing a general role in the limit of stable ductile deforma-

tion for all metals, it is possible that the instability for a particular loading condition (such as that

seen in conditions near to pure shear in Fig. 17, for example) could trigger metal failure in other

states of stress for the same metal, even though saturation does not explicitly occur in these other

stress states.

One example where this work softening in shear could lead to metal failure is through a neck-ing localization obtained in an MK analysis, in which the stress state within the defect may tend

toward the shear condition. To see if this is possible under any circumstances for the aluminum

alloys, an MK analysis of the AA 5182-O alloy was performed and the results are shown in strain

space in Fig. 20 and in stress-space in Fig. 21 for an initial defect parameter of 0.99. For compar-

von Mises

Hill[1948]

Stoughton

[2002]

Cvitanic et al.

[2008]

New Model

Barlat et al.

[2003]

0.00

0.05

0.10

0.15

0.20

0.25

AA 5182 -

O

AA 6022 -

T4E32

AA 6022 -

T43

718-AT

(Voce)

719 -B

(Voce)

718-AT

(Swift)

719 -B

(Swift)

Peak Value of Accumulated RMS Error

Fig. 15. Peak value of Eq. (35) obtained over the strain range from 0% to 50% strain for seven sets of material data and six

material models.



29/41

ison, a second series of MK analyses was performed using the Stoughton (2002) non-associated

flow model based on isotropic hardening. In the later case, the model does not allow the plastic

modulus to become or pass through zero under any loading condition. Since the effect of the initial

defect on the two models is different due to the differences in the hardening rate, the size of the

initial defect parameter for the isotropic hardening model was increased to 0.994 in an attempt to

improve the correlation of the two predicted forming limits near plane strain in the strain based

diagram shown in Fig. 20. Other than differences in the initial defect size, the same necking crite-

rion was used for both models in the MK Analysis.

Although anisotropic hardening clearly affects the shape of the FLC in both strain- and stress-

space, there is no evidence that the passage of the plastic modulus through zero and into the do-

main of work softening in and near the condition of pure shear loading, leads to any change in the

strain condition defined to be the onset of necking in the MK Analysis. Ultimately, the localization

near pure shear is identified to occur at plastic strains much higher than the point at which the

plastic modulus becomes negative, as explained in the following observations: Fig. 18 shows thatthe stress saturates in shear at h 0 where r11 r22 201:11 MPa. This saturation occurs at aplastic strain of 0.449, as shown in Fig. 18. MK Analysis is found to predict the necking localization

in shear at a strain of 0.892, which is nearly twice the strain beyond the point of work softening.


30/41

Necking in shear is predicted at the stress r11 r22 200:66 MPa as seen in Fig. 20, which islower than the maximum stress in shear, due to the work softening above the plastic strain of

0.449. It is also noted that the plastic modulus at the instant of MK Analysis instability in pureshear has a value of 0.0060, but the modulus was even more negative at lower plastic strains

prior, as evident in Fig. 19, where the minimum modulus of0.0100 is seen for the AA 5182-O

at a plastic strain of approximately 0.57. Consequently, the instability is evidently also not trig-

gered by a critical value of the modulus. Therefore, it is concluded that the MK instability analysis

is unaffected by the work softening in shear that is predicted by the model for the aluminum al-

loys (above plastic strains of 0.449 for the AA 5182-O), and that the MK Analysis instability con-

dition arises without respect to the appearance and degree of work softening in pure shear (see

Fig. 22).

7. Discussion

We have shown that significant improvement in the accuracy of stress predictions under propor-

tional loading conditions can be obtained using a simple modification of the quadratic yield function

under non-AFR. Estimated RMS errors in stress values for five metals are reduced from nominally 10%

e wand 40% equivalent true plastic strain yield surfaces for AA 5182-O with Voce Law hardening using the new


31/41

error to nominally 1% over the strain range from 0% to 50%. The resulting error is comparable to the

expected measurement uncertainty of stressstrain data.The quadratic yield function was used in this model because it is possible to explicitly incorpo-


32/41

number work contours are included in the definition of the functional parameters, it is expected

that the accuracy of prediction for all linear loading histories will be comparable to the order of

magnitude improvement demonstrated with the quadratic model for the limited data in Tables1 and 2.

Furthermore, while this model, or a similar approach for another variable yield function, can be ex-

pected to yield remarkably more accurate results for simulations involving linear loading, it is ex-

pected that kinematic hardening effects will undermine this improvement for nonlinear loading.

Nonlinear loading is expected to introduce errors comparable to those obtained using less sophisti-

cated models under linear loading, similar to the deterioration of the error after yielding observed

in Figs. 414 using the non-AFR models proposed by Stoughton (2002) and Cvitanic et al. (2008).

Despite the fact that kinematic effects are the cause of this expected deterioration of accuracy,

existingconventional kinematic hardening model that are based on a constant shape of the yield func-

tion do not have the flexibility to accommodate the anisotropic hardening that occurs under condi-

tions of proportional loading. Consequently, to achieve the improvement in accuracy that is

possible with models such as the one described in this paper for application to linear loading, it ap-

pears necessary to incorporate additional flexibility in the kinematic hardening model to allow explicit

evolution of the shape of the yield function. This may be the biggest challenge yet facing the metal

forming community.


33/41

8. Conclusions

In this paper, a model is developed for proportional loading for any biaxial [or triaxial, in Appendix

A] stress condition. The model is demonstrated to lead to an order in magnitude reduction in errors of

prediction of the anisotropic stressstrain relationships in uniaxial and equal biaxial tension. Despite

the flexibility and power of the model its complexity is low due to the use of quadratic functions forthe plastic potential and yield functions. Consequently, with the improvement in accuracy, even

though it is validated for linear loading conditions, it may be of interest to the metal forming industry,

whichcurrently uses primarily Hills (1948) model, also limited to linear loading conditions, wherethe

RMS error in stress predictions are nominally 10 times higher.

Fig. 20. Plastic modulus defined in shear stress r22 r11 for the three aluminum alloys in Table 1 using the new model.

http://-/?-http://-/?-http://-/?-http://-/?-


34/41

While the model may not correlate as well in other states of biaxial stress, especially for alumi-

num alloys near the plane-strain condition, this paper serves to document what improvements are

possible with carefully calibrated models with higher flexibility to accommodate the complexity of

the metal response, and to support similar methods of accounting for distortional hardening using

methods proposed by Abedrabbo et al. (2006a,b), Plunkett et al. (2007), and Aretz (2008). In addi-

tion, the reduction of error demonstrated emphasizes the need for a similar approach to include dis-

tortion of the shape of the yield function in advanced models of kinematic effects for application to

non-proportional loading.

Finally, it was also observed that for the aluminum alloys, the yield function defined by the new

model reaches a maximum and then work softens in the second quadrant of stress-space, with the

peak stress occurring earliest in the pure shear stress condition. The model was investigated using

MK analysis to determine if the effect resulted in any unusual localizations or instability. It was found

that the maximum shear stress condition did not contribute to the localization condition, which was

Fig. 21. Strain-basedneckinglimits for linear loading predicted by MKAnalysis forAA5182-O using new model with saturation

in shear stress (compared to Stoughton (2002) model with no saturation).



35/41

found to occur at nearly twice the effective plastic strain at which the maximum load in shear

occurs. Therefore, it was concluded that there is no consequence on MK necking localization due to

the saturation of the yield surface in pure shear that occurs with the aluminum alloys using this

model.

Acknowledgements

The authors thank Mr. John C. Brem of Alcoa Technical Center for providing the data used in

this paper and the reviewers for helpful suggestions to improve the scope of the paper. The

authors are also grateful for the financial support from Ministrio da Cincia e Ensino Superior(FCT-Portugal) under PTDC/CTM/74286/2006 program and NSF-CMMI-0800197 (NSF-USA) grant

program.

Fig. 22. Stress-based necking limits predicted by MK Analysis for AA 5182-O using new model with saturation in shear stress

(compared to Stoughton (2002) model with no saturation).



36/41

Appendix A

The full stress model represents the six components of the stress and strain tensors as 6-compo-

nent vectors,

~r

r1r2r3r4r5r6

0BBBBBBBB@

1CCCCCCCCA

r11

r22

r33

r12

r23

r31

0BBBBBBBB@

1CCCCCCCCA; ~E

E1E2

E3

E4

E5

E6

0BBBBBBBB@

1CCCCCCCCA

E11E22

E33

2E12

2E23

2E31

0BBBBBBBB@

1CCCCCCCCA: A1

Although a fully anisotropic model is used for the plastic response of the material, the elastic response

is often characterized by an isotropic response using a single elastic modulus Eand one Poissons ratio

m. This is partly because elastic properties are difficult to measure, so book values are often used, and itis rare to find anisotropic constants for polycrystals, which in any case depend on texture. So in the

body of the paper, we will use isotropic elasticity, and defer the generalization to anisotropic ortho-tropic elasticity to the end of this Appendix. Let C1 represent the elastic matrix relating the stress

to elastic strain tensors, ~Ee C1~r or

Eei C1ij rj; A2

where for isotropic elasticity,

C1

C111 C112 C

113 0 0 0

C121 C

122 C

123 0 0 0

C131 C132 C

133 0 0 0

0 0 0 C1

44 0 00 0 0 0 C155 0

0 0 0 0 0 C166

0BBBBBBBBBB@

1CCCCCCCCCCA

1

E

1 m m 0 0 0m 1 m 0 0 0m m 1 0 0 0

0 0 0 21 m 0 00 0 0 0 21 m 00 0 0 0 0 21 m

0BBBBBBBB@

1CCCCCCCCA:

A3This matrix has an inverse

C

C11 C12 C13 0 0 0

C21 C22 C23 0 0 0

C31 C32 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

0 0 0 0 0 C66

0BBBBBBBBB@

1CCCCCCCCCA

E1

m

1 m12m

m12m

m12m 0 0 0

m12m 1 m12m m12m 0 0 0m

12mm

12m 1 m12m 0 0 00 0 0 1

20 0

0 0 0 0 12 0

0 0 0 0 0 12

0BBBBBBBB@

1CCCCCCCCA

A4so that ~r C~Ee, or in component form

ri CijEej : A5The conventional problem in finite element simulation of metal deformation processes involves a gi-

ven state of stress at time t

~r

r11;r22;r33;r12;r23;r31

r1;r2;r3;r4;r5;r6

A6

at a material point that has been subjected to a past deformation history resulting in accumulatedplastic damage characterized by an effective plastic strain ep. The boundary value problem involvesspecification of an increment or rate of change applied to the strain state represented bythe strain rate

vector and the task is to determine the consequent change to all other variables. The equations will be



37/41

written in rate form, but can be converted to increments, as is required for time integration, by mul-

tiplying all rate variables by the time increment, Dt, then using Euler or higher order time integration

schemes. In rate form, the change in the strain tensor is characterized by

_~E

_E11; _E22; _E33;2 _E12;2 _E23;2 _E31

_E1; _E2; _E3;2 _E4;2 _E5;2 _E6

:

A7

From this strain rate, the problem is to determine how much, if any, of the strain increment is plasticand how much is elastic. Then, the elastic component of the strain rate can be directly used to deter-

mine the rate of change of, or increment to, the stress tensor. At the same time, the rate of change or

increment of the effective plastic strain must be determined so that the effective plastic strain can be

updated for use in determining the status of the yield criterion for the next time increment. Alterna-

tively, the status of the yield criterion can be determined from the updated stress tensor

international journal of plasticity 25 (2009) 1777–1817

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