International Journal of Plasticity 22 (2006) 342–373 Www.elsevier.com/Locate/Ijplas

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International Journal of Plasticity 22 (2006) 342–373

Forming of aluminum alloys at elevatedtemperatures – Part 2: Numerical modeling

and experimental verification

Nader Abedrabbo a, Farhang Pourboghrat a,*, John Carsley b

a Department of Mechanical Engineering, Michigan State University,

East Lansing, MI 48824-1226, United Statesb General Motors Research & Development Center, Warren, MI 48090, United States

Received 30 January 2005

Available online 31 May 2005

Abstract

The temperature-dependent Barlat YLD96 anisotropic yield function developed previously

[Forming of aluminum alloys at elevated temperatures – Part 1: Material characterization. Int.

J. Plasticity, 2005a] was applied to the forming simulation of AA3003-H111 aluminum alloy

sheets. The cutting-plane algorithm for the integration of a general class of elasto-plastic con-

stitutive models was used to implement this yield function into the commercial FEM code LS-

Dyna as a user material subroutine (UMAT). The temperature-dependent material model was

used to simulate the coupled thermo-mechanical finite element analysis of the stamping of an

aluminum sheet using a hemispherical punch under the pure stretch boundary condition. In

order to evaluate the accuracy of the UMAT�s ability to predict both forming behavior and

failure locations, simulation results were compared with experimental data performed at sev-

eral elevated temperatures. Forming limit diagrams (FLDs) were developed for the AA3003-

H111 at several elevated temperatures using the M-K model in order to predict the location of

the failure in the numerical simulations. The favorable comparison found between the numer-

ical and experimental data shows that a promising future exists for the development of more

accurate temperature-dependent yield functions to apply to thermo-hydroforming process.

� 2005 Elsevier Ltd. All rights reserved.

0749-6419/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2005.03.006

* Corresponding author. Tel.: +1 517 432 0819; fax: +1 517 353 1750.

E-mail address: [email protected] (F. Pourboghrat).

mailto:[email protected]

N. Abedrabbo et al. / International Journal of Plasticity 22 (2006) 342–373 343

Keywords: Thermo-mechanical; Temperature; Material anisotropy; UMAT, Cutting-plane algorithm;

Yield function, FLD

1. Introduction

Numerical analysis is critically important to understanding the complex deforma-

tion mechanics that occur during sheet forming processes. Finite element analysis(FEA) and simulations are used in automotive design and formability processes to

predict deformation behavior accurately during stamping operations (Chung et al.,

1992, 1996). Although available commercial FEA codes offer a library of material

models applicable to a variety of applications, they often do not offer highly special-

ized material models developed for a specific material and process. Also, very few

available material models are capable of handling complex forming simulations that

incorporate the temperature-dependence of materials. The process becomes increas-

ingly complicated when materials exhibit anisotropic behavior. Currently availablecommercial codes do not offer material models that are appropriate for simulating

the thermo-mechanical forming processes of anisotropic materials such as aluminum

sheet alloys. The importance of using an appropriate material model for aluminum

during hydroforming processes using Barlat�s YLD96 model have been emphasized

previously (Zampaloni et al., 2003; Abedrabbo et al., 2005b).

Use of anisotropic material models in FEA requires thorough material character-

ization under multiple loading conditions. Since the material anisotropy and harden-

ing behavior, i.e., material response to loading conditions, change with elevatedtemperatures, the anisotropy coefficients and the hardening behavior must be deter-

mined as a function of temperature to perform accurate thermo-mechanical numer-

ical analysis for these materials.

Prior research available for simulation of warm forming processes focuses only on

the effect of elevated temperature on the evolution of the flow (hardening) stress.

These include Li and Ghosh (2003), Ayres (1979), Ayres and Wenner (1979), Painter

and Pearce (1980), Takata et al. (2000), Naka et al. (2001) and Boogaard et al.

(2001). The evolution of the yield surface of aluminum alloys as a function of tem-perature and the effect on the anisotropy coefficients were not fully explored. In most

cases, either Hill�s 1948 model (Hill, 1948) or the von Mises isotropic yield functions

was used. Boogaard et al. (2001) characterized the behavior of AA5754-O for which

two types of functions representing the flow stress were used: the modified power law

model and the Bergstrom model. The yield surface used in this case was assumed to

remain constant with respect to changing temperatures. Only the coefficients of the

power law model were curve-fit exponentially as a function of temperature. The pre-

dictions of the material model, however, underestimated the values of the punch loadin both models (power law and Bergstrom models). Canad-ija and Brnic (2004)

presented an associative coupled thermoplasticity model for J2 plasticity model to

represent internal heat generated due to plastic deformation. In it, temperature-

dependent material parameters developed were used.

344 N. Abedrabbo et al. / International Journal of Plasticity 22 (2006) 342–373

Generally, heat generation due to dissipated mechanical work during plastic

deformation leads to temperature rise in the specimen (Wriggers et al., 1992; Armero

and Simo, 1993). This temperature rise is a local phenomenon however and depen-

dent on the forming speed. During the characterization process of the material (Abe-

drabbo et al., 2004a) the anisotropy coefficients were evaluated at the sametemperature and the same forming speed, and therefore any thermal strain effects

present were included in the material model. In the current thermo-forming analysis,

since the forming dies were maintained at a constant elevated temperature and the

speed at which the test was performed was slow, the magnitude of thermal strains

turned out to be of the same order as elastic strains and negligible compared to plas-

tic strains (see Fig. 19). Therefore, to save on computational time during thermo-

forming analysis, the effect of thermal strain in the integration of the elasto-plastic

constitutive model was neglected. This is particularly important when performingthermo-forming analysis of large industrial components.

In this paper, the temperature-dependent anisotropic material model developed in

Part 1 of this paper (Abedrabbo et al., 2005a) was implemented as a user material

subroutine (UMAT) into the explicit finite element code LS-Dyna. Within the frame-

work of rate independent plasticity and plane stress conditions, an efficient algorithm

based on the operator split methodology proposed by Simo and Ortiz (1985) was

implemented. The UMAT was then used in a coupled thermo-mechanical finite ele-

ment analysis of stamping of the aluminum sheet with a hemispherical punch underpure stretch boundary conditions, and the deformation behavior and failure loca-

tions were compared against experimental results.

A description of Barlat�s YLD96 yield function and its implementation as a

UMAT into the finite element code will be discussed. The experimental data of

the pure stretch forming of an aluminum sheet with a hemispherical punch at ele-

vated temperatures and its comparison with the numerical analysis of the forming

process at elevated temperatures using the new material model also will be presented.

2. Anisotropic yield function

The YLD96 anisotropic yield function presented by Barlat et al. (1997a) is one of

the most accurate yield functions for aluminum alloy sheets (Barlat et al., 2003), be-

cause it simultaneously accounts for yield stress and r-value directionalities. This

yield function is based on a phenomenological description in which the behavior

of the material was generalized to include a stress tensor with six components asshown in the following equation:

U ¼ a1jS2 � S3ja þ a2jS3 � S1ja þ a3jS1 � S2ja ¼ 2�ra ð1Þwith a = 6 and a = 8 for BCC and FCC materials, respectively. S1, S2 and S3 are the

principal values of the stress tensor Sij, which is defined later. �r is the flow stress.

The isotropic plasticity equivalent (IPE) stress is:

S�¼ L�r�; ð2Þ


where, for orthotropic symmetry, L is defined below with anisotropy coefficients ck,

Lij ¼

ðc2þc3Þ3

�c33

�c23

0 0 0

�c33

ðc3þc1Þ3

�c13

0 0 0

�c23

�c13

ðc1þc2Þ3

0 0 0

0 0 0 c4 0 0

0 0 0 0 c5 0

0 0 0 0 0 c6

26666666664

37777777775. ð3Þ

For the plane stress case (rz = ryz = rzx = 0), Eq. (2) reduces to:

Sij ¼

Sx

Sy

Sz

Sxy

26664

37775 ¼

ðc2þc3Þ3

�c33

�c23

0

�c33

ðc3þc1Þ3

�c13

0

�c23

�c13

ðc1þc2Þ3

0

0 0 0 c6

266664

377775

rx

ry

0

rxy

26664

37775. ð4Þ

The principal values of Sij, as defined in Eq. (1), are found as follows:

S1;2 ¼Sx þ Sy

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSx � Sy

2

� �2

þ S2xy

sð5Þ

and S3 = �(S1 + S2) because of the deviatoric nature of Sij. The principal values are

ordered such that S1 P S2 P S3. The anisotropy coefficients ai of the yield function

(1) are defined as:

a1 ¼ axcos2bþ aysin

2b;

a2 ¼ axsin2bþ aycos

2b;

a3 ¼ az0cos22bþ az1sin

22b.

ð6Þ

In the above equations, c1, c2, c3, c6, ax, ay, az0 and az1 are coefficients that describe

the anisotropy of the material. The parameter 2b represents the angle between theline OA and the axis of the principal IPE stress, S1, as shown in Fig. 1, and is equal

to:

2b ¼ tan�12Sxy

Sx � Sy

� �. ð7Þ

With this yield function, it is possible to increase the yield stress at pure shear with-

out increasing the other plane strain yield stresses. It should be noted that by setting

the anisotropy coefficients c1 = c2 = c3 = c6 = a1 = a2 = a3 = 1.0 and a = 2 (qua-

dratic), the von Mises isotropic yield function is recovered.

To calculate the anisotropy coefficients of the yield function, several material tests

need to be performed (Barlat et al., 1997b). In Part 1 of this paper, AA3003-H111was fully characterized using several uniaxial tensile tests in multiple directions at

Table 1

Barlat YLD96 material model anisotropy coefficients for AA3003-H111 as a function of temperature using

two different order polynomial fit functions

YLD96

coefficient

Third order fit Fifth order fit

c1 =1.0446 + 0.0045834T � 3.9615E

� 5T2 + 9.7409E � 8*T3

=0.90902 + 0.0097045T � 6.2517E � 5T2 � 3.3687E

� 7T3 + 3.7316E � 9T4 � 7.6631E � 12T5

c2 =0.9867 + 0.000373T + 1.1717E

� 7T2 + 3.0052E � 9T3

=0.90308 + 0.0023977T + 3.5398E � 5T2 � 8.9808E

� 7T3 + 5.4066E � 9T4 � 9.8763E � 12T5

c3 =0.9860 + 0.0003334T � 1.7857E

� 6T2 + 1.593E � 9T3

=0.98895 + 0.0010519T � 3.7432E � 5T2 + 4.7418E

� 7T3 � 2.3519E � 9T4 + 3.9325E � 12T5

c6 =1.0592 + 0.000855T � 1.3728E

� 5T2 + 3.696E � 8T3

=0.76161 + 0.017737T � 0.0003099T2 + 2.2357E � 6T3

� 7.2652E � 9T4 + 8.8146E � 12T5

ax =1.0146 � 0.010636T + 8.3876E

� 5T2 � 2.0664E � 7T3

=1.8838 � 0.05496T + 0.00073166T2 � 3.8433E � 6T3

+ 7.5607E � 9T4 � 3.0894E � 12T5

ay =1.4253 � 0.012009T + 6.4874E

� 5T2 � 1.3432E � 7T3

=2.683 � 0.079188T + 0.0011348T2 � 7.0955E � 6T3

+ 1.9271E � 8T4 � 1.8287E � 11T5

az1 =0.86396 + 0.007906T + 3.0245E

� 6T2 � 6.2854E � 8T3

=2.7451 � 0.096914T + 0.0017925T2 � 1.2899E � 5T3

+ 4.0712E � 8T4 � 4.7071E � 11T5

Temperatures in �C.

Fig. 1. Determination of the anisotropy parameter 2b from Mohr�s circle.


several elevated temperatures and two polynomial functions were determined to rep-

resent the anisotropy coefficients as a function of temperature, as shown in Table 1.

Fig. 2 is a 2D-plot of Barlat�s YLD96 yield function for AA3003-H111 at several

elevated temperatures. Fig. 2(a) shows the change in the shape (stress values in Fig.

2(a) are normalized with respect to �r) and Fig. 2(b) shows the change in the size of

the yield surface. It is evident from Fig. 2 that temperature has a distinct effect on the

yield surface�s shape and size, which must be accounted for during thermo-forming

analysis.

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

25˚C93˚C149˚C204˚C260˚C

σy/σ−

σx

/ σ−

-60

-40

-20

0

20

40

60

-60 -40 -20 0 20 40 60

25˚C93˚C149˚C204˚C260˚C

σ y (

MP

a)

σx

(MPa)

(a)

(b)

Fig. 2. 2D-Plot of Barlat�s yield function for AA3003-H111 at several elevated temperatures. (a) Stresses

normalized with the flow stress to show the change in the yield surface shape. (b) Plot of actual stresses to

show the change in the size of the yield surface.



3. Flow stress

Flow stress ð�rÞ represents the size of the yield function during deformation. An

appropriate equation describing changes in the flow stress of the material depends

on deformation conditions such as temperature and strain rate. The AA3003-H111 aluminum sheet was characterized in Part 1 of this paper (Abedrabbo et al.,

2005a) in which the following modified power law flow rule was proposed that in-

cludes temperature effects and assumed isotropic hardening behavior in the material:

�rð�ep; _e; T Þ ¼ KðT Þð�ep þ e0ÞnðT Þ_eesr0

� �mðT Þ

; ð8Þ

where K (strength hardening coefficient), n (strain-hardening exponent) and m

(strain-rate sensitivity index) are material constants. �ep is the effective plastic strain,

e0 is a constant representing the elastic strain to yield and esr0 is a base strain rate.

This model was primarily selected over other types of hardening laws (e.g., Voce)

because it represented the hardening behavior of the current material accurately, and

it incorporates strain rate effects. Gronostajski (2000) describes different types of

hardening laws that could be used (e.g., Backofen, Grosman) to represent other

materials, including those with hardening saturation behavior.

The flow rule coefficients in Eq. (8) are presented in Table 2 as a function oftemperature.

To summarize, the yield function represented by Eq. (1) can be written as:

Uð�r;�ep; _e; T Þ ¼ �rðr�; T Þ � Hð�ep; _e; T Þ ¼ 0; ð9Þ

whereH is the hardening rule defined by Eq. (8). T is the temperature calculated dur-

ing the thermal analysis step and supplied as input to UMAT during the structural

analysis part.

4. Stress integration for elasto-plasticity using the YLD96 anisotropic yield function

As mentioned earlier, in this work, the effect of thermal strain in the integration of

the elasto-plastic constitutive model will be neglected for reasons that its magnitude

is very small and neglecting it will substantially improve the computational speed of

the UMAT. Fig. 19 shows the computed magnitude of thermal, elastic and plastic

strain increments in thermo-forming process. It can be seen that thermal strain is

Table 2

Summary of equations used to fit hardening parameters for the power law flow rule

Hardening parameter Rolling direction, 0� Unit

K(T) =�0.5058*T + 210.40 MPa

n(T) =�0.0004*T + 0.2185

m(T) = 0.0018*exp(0.0147*T)

Temperatures in �C.


negligible compared to plastic strain and neglecting it in the stress integration pro-

cess is justified.

Stress integration of the elasto-plastic yield functions is explored in numerous

publications (Armero and Simo, 1993; Auricchio and Taylor, 1999; Tugcu and

Neale, 1999; Hashiguchi, 2005). The temperature-dependent YLD96 model devel-oped in the previous section was implemented within the framework of rate-indepen-

dent plasticity and plane stress conditions using an efficient integration algorithm

proposed by Simo and Ortiz (1985) and further analyzed by Ortiz and Simo

(1986). The current analysis is based on incremental theory of plasticity (Chung

and Richmond, 1993; Yoon et al., 1999; Han et al., 2003).

These algorithms, which fall within the class of cutting-plane methods of con-

strained optimization, was proposed to bypass the need for computing the gradients

of the yield function and the flow rule as required by the closest point projection iter-ative methods (Simo and Hughes, 1998). The general closest point projection proce-

dure usually leads to systems of nonlinear equations, the solution of which by the

Newton–Raphson method requires evaluation of the gradients of system equations.

While the previous approach might be applicable to simple plasticity models (e.g.,

von Mises), its application to complex yield functions such as Barlat�s YLD96 is

not only exceedingly laborious, but also computationally extensive and makes the

FEM code run slower for industrial applications. In the following, the stress integra-

tion algorithm for the YLD96 yield function will be presented and the implementa-tion of it as a UMAT into the explicit finite element code LS-Dyna will be described.

In a displacement finite element formulation, the nature of the FEM code is strain

driven. The cutting-plane algorithm falls within the operator splitting methodology

(Ortiz, 1981) in which the strain is decomposed into two parts: elastic and plastic.

The method proposed by Simo and Ortiz (1985) and Ortiz and Simo (1986), how-

ever, is based on the total deformation theory. In this method, the history of total

strain and total plastic strain are saved as history variables for the next step. This

adds an unnecessary step, and in some cases where loading and unloading occurs,it might affect the accuracy of the code. Using the incremental theory of plasticity

eliminates this step.

The incremental theory of plasticity (Chung and Richmond, 1993; Yoon et al.,

1999) was applied to the elasto-plastic formulation based on the materially embedded

coordinate system. Under this scheme, the strain increments in the flow formulation

are the discrete true (or logarithmic) strain increments, and the material rotates by the

incremental angle obtained from the polar decomposition at each discrete step (Yoon

et al., 2004). In addition, a multiplicative decomposition theory can be also utilized,especially when material deformation follows minimum plastic work path (or loga-

rithmic strain path). Multiplicative theory formulation coincides with the current

additive decomposition theory based on the incremental theory (Han et al., 2003).

In the general commercial codes, e.g., LS-Dyna and Abaqus, the strain increment

ð _e�nþ1Þ, the previous stress state value ðr

�nÞ and any history variables saved at the pre-

vious stress update step are provided at the beginning of each time step. The new

strain increment is then assumed to be elastic and an elastic predictor stress state


‘‘trial stress’’ is calculated through the customary elasticity relations. Using the cut-

ting-plane algorithm, the actual stress state is then restored (plastic corrector) and

other plastic variables are calculated.

The basic steps in the numerical procedure for iteratively integrating the elasto-

plastic constitutive equations for rate independent plasticity with associated flow ruleare:

_e�e

nþ1¼ _e�nþ1

; ð10Þ

_r�¼ C�: _e�e

nþ1; ð11Þ

Associative flow rule : _e�p ¼ _k

oUo r�

; ð12Þ

Yield function : U 6 0; ð13ÞNormality parameter : _k P 0; ð14Þ

Kuhn–Tucker condition : _kU ¼ 0; ð15ÞConsistency condition : _k _U ¼ 0; ð16Þ

where r�; _e�e and _e

�p are the stress, elastic strain and plastic strain, respectively. C

�is

the fourth order elastic tensor which is assumed to be a constant. The associated flow

rule is expressed by Eq. (12) in which _k is the plastic multiplier and U is the yield

function as defined by Eq. (9). The yielding criterion check and the loading–unload-

ing conditions are expressed in the standard Kuhn–Tucker form (Simo and Hughes,

1998) in which the constraints in Eqs. (13)–(16) are satisfied.The exact implementation of the iterative procedure for integrating the elasto-

plastic constitutive equations for YLD96 is explained next. In the finite element anal-

ysis using isoparametric elements, stress is updated at the Gauss (or Lobatto) points

and the incremental deformation is given (Ortiz and Simo, 1986). The function of the

user material subroutine (UMAT) is therefore to update the state variables (stress,

plastic strain) from a converged configuration Bn to their corresponding values on

the updated configuration Bn + 1.

To obtain the associated plastic strain increment the normality rule is utilized.From the associative flow rule:

_e�p ¼ _k

o�rðr�Þ

o r�

. ð17Þ

The numerical procedure in updating the stress state involves finding the unknown _k(normality parameter). Using _k, all kinematics and stresses are updated at the end of

the iteration procedure. In incremental theory, it should be noted that D�ep ¼ _k as

follows:

D�ep ¼r�: _e�p

�rðr�Þ ¼

r�: _k

o�rðr�Þ

o r�

�rðr�Þ ¼

k�rðr�Þ

�rr�

¼ _k; ð18Þ


where �rðr�Þ is a first order homogeneous function, i.e., �rðr

�Þ ¼ r

�

o�rðr�Þ

o r�, and D�ep is the

equivalent plastic strain increment. In order to calculate D�ep, the calculation of �rðr�Þ

ando�rðr�Þ

o r�

is required. The explicit forms for these terms are presented in Appendix A.

At the beginning of each time step t, the strain increment ð _e�nþ1Þ, the previous total

stress state value ðr�Þ and any history variables saved at the previous stress update

step are provided by the FEM code as input. In the elastic predictor step, the strain

increment is assumed to be elastic and combined with the previous converged valuesof the state variables a trial elastic stress state is calculated. If the new stress state lies

outside the yield surface, this trial state is taken as the initial condition for the plastic

corrector part.

First, a trial stress state is calculated using the elasticity tensor

r�ðtrialÞnþ1¼ r�nþ C�: _e�nþ1

. ð19Þ

Using this trial stress, the yield function and its first derivative are calculated (see

Appendix A)

�rðtrialÞnþ1 ¼ �rðr�ðtrialÞnþ1Þ;

o�rðtrialÞnþ1o r�

¼ o�rðtrialÞnþ1o r�

��ðr�ðtrialÞnþ1

Þ

.ð20Þ

The size of the yield locus Hð�epnþ1; _e; T Þ is calculated using the hardening flow rule as

presented in Eq. (8).

Next, a check is performed to test whether the calculated trial stress state lies out-side the yield surface, i.e., plastic state, by rewriting Eq. (9) as

Uð�rðtrialÞnþ1 ;�epnþ1; _e; T Þ ¼ �rðr�ðtrialÞnþ1Þ � Hð�epnþ1; _e; T Þ 6 0. ð21Þ

If the condition is met, then the trial stress state is elastic and the trial stress is the

actual stress state that is returned to the FEM code. If the condition is not met, then

the material has yielded and the stress state is plastic. The Newton–Raphson method

is then used to iteratively return the trial stress state to the yield surface by calculat-ing the normality parameter _k using sub-steps i. After calculating the normality

parameter _k, the sequential update procedure for the next stress state is done as

follows:

r�ðiþ1Þnþ1¼ C�: e

�nþ1� e�p

nþ1

� �ð22Þ

and since,

e�p

nþ1¼ e�p

nþ De

�p ð23Þ

then


r�ðiþ1Þnþ1¼ C�: e

�nþ1� e�p

n

� �� C�: De

�p ð24Þ

Combined with Eq. (17) and setting r�ðiÞnþ1¼ r�ðtrialÞnþ1

, this becomes

r�ðiþ1Þnþ1¼ r�ðiÞnþ1� _kC

�:o�rðiÞnþ1ðr�Þ

o r�

. ð25Þ

With this new stress state, the yield function and the hardening rule are calculated atthis new step and the yielding check is performed again

Uðiþ1Þ �rðiþ1Þnþ1 ;�epðiþ1Þnþ1 ; _e; T� �

¼ �r r�ðiþ1Þnþ1

� �� H �epðiþ1Þnþ1 ; _e; T

� �6 0. ð26Þ

The iterative procedure is repeated for a number of sub-steps until the plastic con-

sistency is restored within a prescribed tolerance, i.e.,

Uðiþ1Þ �rðiþ1Þnþ1 ;�epðiþ1Þnþ1 ; _e; T� �

6 d; ð27Þ

where d is a small number, e.g., d = 10�7.

The tangent modulus involved in the Newton–Raphson iteration to solve nonlin-

ear equilibrium equations was obtained by consistently linearlizing the responsefunction obtained from the integration algorithm (Simo and Ortiz, 1985; Ortiz

and Simo, 1986). To solve for the normality parameter _k, the system equation as

shown in Eq. (21) is linearized using Taylor�s expansions:

0 ¼ UðiÞ �rðiÞnþ1;�epðiÞnþ1

� �þ oUðiÞ

o r�

r�ðiþ1Þnþ1� r�ðiÞnþ1

� �þ oUðiÞ

o�ep�epðiþ1Þnþ1 � �epðiÞnþ1

� �. ð28Þ

From Eq. (25)

r�ðiþ1Þnþ1� r�ðiÞnþ1¼ � _k ¼ C

�:o�rðiÞnþ1ðr�Þ

o r�

ð29Þ

and

�epðiþ1Þnþ1 � �epðiÞn ¼ _k. ð30Þ

Therefore, Eq. (28) becomes

0 ¼ UðiÞ �rðiÞnþ1;�epðiÞnþ1

� �þ oUðiÞ

o r�

: � _kC�:o�rðiÞnþ1ðr�Þ

o r�

0@

1Aþ _k

oUðiÞ

o�ep. ð31Þ

The normality parameter _k is then found from

_k ¼ UðiÞð�rðiÞnþ1;�epðiÞnþ1Þ

o�rðiÞnþ1ðr�Þo r�

: C�:

o�rðiÞnþ1ðr�Þo r�� oH ðiÞ

o�ep

. ð32Þ


In finite element commercial codes for the plane stress case, the thickness strain

needs to be updated and reported to the FEM code. This is done with the following

equation:

_ezz ¼_epxx þ _epyy

� �þ m _exx þ _eyy

� 2m _epxx þ _epyy

� �m� 1

; ð33Þ

where the plastic strain is calculated using Eq. (17) as follows:

_e�p

nþ1¼ _k

o�rðr�Þ

o r�

. ð34Þ

The cutting-plane algorithm used to update the stress state according to the previous

equations is summarized in Table 3. The cutting-plane algorithm described above

can be interpreted geometrically as shown in Fig. 3. The linearized yield function

as shown in Eq. (28) defines a tangent ‘‘cut’’ at each new iteration of the yield func-

tion, and then the new variables are projected to define the next iteration. It was

found that this implementation of the code is relatively fast and a converged solutionis reached within 1–3 iterations, which is considered efficient for large-scale finite ele-

ment problems. Since this implementation of the constitutive equation is used in the

explicit part of LS-Dyna, there is no need to calculate the consistent tangent modulus

since it is only required in the implicit formulation.

Accuracy of the developed user material subroutine (UMAT) was tested by com-

paring the calculated values of the plastic anisotropy parameters (R0, R45 and R90)

using the developed code with the ones extracted from the experimental uniaxial

tests. Results of this comparison were presented in Part 1 of this paper, and wereshown to be satisfactory.

The algorithm used to iteratively return the trial stress state to the yield surface is

found to be sensitive to the number of (strain) increments used to calculate the stress

state; this is shown in Fig. 4. As step size increases, the strain increment value de-

creases and vice versa. As can be seen, with a small step size of 30 (strain incre-

ment = 0.01), the predicted stress values do not match the experimental results

very well. As step size is increased, the calculated stress values start to converge to-

ward the actual stress values. The stress values calculated using 3000 steps (strainincrement = 0.0001), match the experimental stress values perfectly. Therefore, an

important factor affecting the accuracy of the result is the number of steps or the size

of the strain increment used in the stress integration algorithm. In an implicit code,

where larger strain increments can be used, this integration algorithm will not per-

form satisfactorily. However, in an explicit finite element analysis, where small step

sizes are used, to ensure convergence and numerical stability, good accuracy could be

expected.

It should be noted that planar anisotropy was incorporated into the formulationfor sheet forming simulations using the plane stress version of Barlat�s YLD96

model. When the deformation of the workpiece is not limited to the plane of the

sheet, it is important to impose the requirement that for the plane stress application,

Table 3

Stress update algorithm based on incremental theory of plasticity

(i) Geometric update: (given by FEM code and user history variables)

_e�nþ1

; r�n;�epn

(ii) Elastic predictor:

r�ð0Þnþ1¼ r�nþ C�: _e�nþ1

�epð0Þnþ1 ¼ �epn

�rð0Þnþ1 ¼ �rðr�ð0Þnþ1

; T Þ

H ð0Þ ¼ �rð�epð0Þnþ1 ; _e; T ÞUð0Þnþ1 ¼ �rð0Þnþ1 � H ð0Þ

(iii) Check for yielding:

Uð0Þnþ1 P 0

NO: Material is elastic. Set trial state to be final state:

e�p

nþ1¼ r�pð0Þnþ1

r�nþ1

¼ r�ð0Þnþ1

�epnþ1 ¼ �epð0Þnþ1

YES: Material is plastic. Set i = 0

(iv) Plastic corrector:

_k ¼ UðiÞ

o�rðiÞðr�Þ

o r�

: C�:

o�rðiÞðr�Þ

o r�� oH ðiÞ

o�ep

r�ðiþ1Þnþ1¼ r�ðiÞnþ1� _k C

�:o�rðiÞðr

�Þ

o r�

24

35

_e�pðiþ1Þ

nþ1¼ _k

o�rðiþ1Þðr�Þ

o e�

�epðiþ1Þnþ1 ¼ �epðiÞnþ1 þ _k

�rðiþ1Þnþ1 ¼ �rðr�ðiþ1Þnþ1Þ

H ðiþ1Þ ¼ Hð�epðiþ1Þnþ1 ; _e; T Þ(v) Convergence check:

�rðiþ1Þnþ1 � H ðiþ1Þ� �

6 d

Where dis a small number, e.g., 10�7.


Fig. 3. Geometrical interpretation of the cutting-plane algorithm. The trial stress state �rðtrialÞnþ1 is returned

iteratively to the yield surface.

Table 3 (continued)

NO: set i i + 1 and GO TO (iv)

YES: set states to converged values and exit

r�nþ1

¼ r�ðiþ1Þnþ1

_e�p

nþ1¼ _e�pðiþ1Þ

nþ1

�epnþ1 ¼ �epðiþ1Þnþ1


the in-plane material axes have to remain in the plane of the sheet during the defor-

mation (Tugcu and Neale, 1999). In the current application using the LS-Dyna FEA

code, the initial anisotropy of the material is introduced by defining two local vectorsin the plane of the material. Then, in the LS-Dyna code, all transformations into the

element local system are performed prior to entering into this user material subrou-

tine. Transformations back to the global system are performed after exiting the user

material subroutine (Hallquist, 1999).

5. Experimental results

Experimental tests were conducted using a modified Interlaken 75-ton double ac-

tion servo press. Detailed information about the press and the forming procedure

can be found in previous papers by the authors (Zampaloni et al., 2003; Abedrabbo

0

50

100

150

200

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Tensile TestUMAT- 30 StepsUMAT- 300 StepsUMAT- 3000 Steps

Tru

e S

tres

s(M

Pa)

True Strain

Fig. 4. Accuracy of the developed UMAT with respect to step size compared to true stress–strain results

of a uniaxial test performed at 25 �C.


et al., 2005b). The experimental setup was used to form 101.6 mm (4 in.) diameter

hemispherical cups from 1 mm (0.04 in.) thick, 177.8 mm (7 in.) diameter roundblanks of AA3003-H111under pure stretch condition. The blank was placed over

a draw bead and clamped with a blank holding force (BHF) of approximately

267 kN (60,000 lbf) to insure no material will draw in during the pure stretch

experiments.

Heating elements with an active control device were added to the LDH machine

in order to raise the temperature of the dies and the sheet to the desired elevated tem-

perature. The active control was achieved by using two thermocouples linked to the

die and blank system. Other thermocouples were installed to measure directly thetemperature of both the blank and the punch during the forming process. These

are critical measurements needed in order to be able to perform accurate numerical

analysis of the experiment.

The procedure for performing the experiment at a specific elevated temperature is

as follows. The blank was clamped between the dies with three round heating ele-

ments positioned around the perimeter of the dies. The system was then insolated

to minimize heat loss to the environment. Using the temperature control device,

the desired temperature was set and maintained for a period of about 20 min or untila constant and isothermal condition was achieved. Temperature was monitored

using several thermocouples within the system. Initially, the temperature of the

punch was not controlled independently. A mechanism to control the temperature

of the punch is currently under development. For the current research, the punch

temperature was found to be cooler than the blank, see Table 5. With an isothermal


blank condition, the punch was then actuated to stretch the blank. During the pro-

cess, the punch force–displacement curve was recorded. This process was repeated

several times for each temperature to establish repeatability.

Pure stretch experiments were performed at several elevated temperatures in the

range of 25–204 �C (77–400 �F). Fig. 5 shows the pure stretch results at these temper-atures along with the failure punch depths and failure locations. Fig. 6 shows the

punch load vs. punch depth at several elevated temperatures.

From Figs. 5 and 6, as the test temperature increased:

i. Forming depth increased from 27 mm (1.06 in.) at room temperature to 37 mm

(1.46 in.) at 204 �C (400 �F). This represents a 37% increase in forming depth.

ii. The location of failure (where material ruptures) changed dramatically during

the forming process (Fig. 5). At room temperature, failure occurred at the con-tact edge with the punch, but as temperature increased the failure point

migrated from the punch contact region toward the blank holder contact edge.

At temperatures of 177 �C (350 �F) and 204 �C (400 �F), failure occurred at the

die contact edge.

iii. The punch load required for forming the part reduced as temperature

increased. This was expected since the tensile strength of the material reduced

with temperature as explained in Part I of this paper.

Fig. 5. Pure stretch experimental results using the 101.6 mm (4 in.) hemispherical punch at several elevated

temperatures, along with failure punch depth. The failure locations are also shown.

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25 30 35 40Punch Depth ( mm)

Pu

nch

Lo

ad (

kN) 204 ˚C

25 ˚C

38 ˚C 93 ˚C 149 ˚C

177 ˚C

Fig. 6. Punch force vs. punch depth for the hemispherical punch experiments at several elevated

temperatures.


The greater forming depths achieved at elevated temperatures are attributed to

the temperature gradient between the blank and the punch. In the experimental set-

up, the punch itself was not heated directly and was found to be at a lower temper-ature than the blank (see Table 5). This condition was advantageous because areas of

the blank in contact with the punch were lower in temperature and thus stronger

than the unsupported areas of the blank. As the punch traveled, the unsupported re-

gions of the blank (regions not in contact with the punch) stretched more due to their

lower tensile strength. At the highest possible temperature for this setup, 204 �C(400 �F), the unsupported regions continued to stretch until critical thinning oc-

curred at the edge of the sheet and blank holder.

6. Coupled thermal structural finite element model

Finite element analysis (FEA) was performed using the commercial explicit finite

element code LS-Dyna (Hallquist, 1999) to understand the deformation behavior of

the aluminum sheet during the thermo-forming process. The UMAT option was

used to build the user material subroutine in FORTRAN (COMPAQ VISUAL

FORTRAN PROFESSIONAL EDITION 6.6B�), which was then linked withthe library files supplied by LSTC. The finite element model used in the simulations

was first created using Unigraphics and imported as IGES (Initial Graphics Ex-

change Specification) files. Hypermesh� was used to create the finite element mesh,

Table 4

Thermal properties of material used in numerical analysis

Material Density (kg/m3) Specific heat capacity (J/kg K) Thermal conductivity (J/m K)

Rigid dies (FE) 7.85E3 450.0 70.0

Blank (AL) 2.71E3 904.0 220.0

Table 5

Measured temperatures of dies, punch and blank during experiments

Test temperature (�C) Die temperature (�C) Punch temperature (�C) Blank temperature (�C)

25 25 25 25

38 38 38 38

93 93 90 92

149 149 120 142

177 176 144 170

204 204 170 200


assign the boundary conditions and to build the LS-Dyna input deck for the analy-

sis. The full size finite element model used approximately 48,000 four- and three-

node shell elements. The punch, die, and the blank holder were created using rigid

materials (Material 20 in LS-Dyna). First trials with the adaptivity option in LS-

Dyna to reduce calculation time revealed errors and problems in convergence of

the thermal analysis. Therefore, the blank was modeled with a fine mesh of about

30,500 four-node shell elements without using adaptive meshing schemes.

The thermal analysis was performed first, during which the temperature of eachelement was calculated and supplied as input to the UMAT. Using the temperature

value for each element, the temperature-dependent anisotropic material model coef-

ficients were calculated. Before every structural iteration step, two thermal analysis

steps were performed with a controlled time step to insure that the temperature up-

date was adequate.

In this research, a linear fully implicit transient thermal analysis was performed

with the diagonal scaled conjugate gradient iterative solver type in LS-Dyna. The

die and blank materials were assumed to behave with isotropic thermal properties.Table 4 shows the thermal properties defined in the analysis for the die and blank.

The lower die, blank holder and punch were assigned a constant temperature

boundary condition, while the blank was given an initial temperature boundary con-

dition equal to the upper and lower dies. The temperature of the punch was set to the

lower temperature based on experimental data, see Table 5. Thermal properties were

assigned to contact surfaces to enable heat transfer at appropriate areas of contact

between the blank and tooling during the analysis. Subsequently, areas of the blank

that made contact with the punch lost heat to the punch while the unsupported re-gions of the blank remained at higher temperatures.


7. Failure criteria

Failure criteria used in the analysis are based on forming limit diagrams (FLDs).

FLDs for AA3003-H111 at multiple temperatures were calculated with the M-K

model (Marciniak and Kuczynski, 1967) using Barlat�s YLD96 anisotropic yieldfunction and appropriate coefficients for each temperature. Yoa and Cao (2002) de-

scribe methods for extracting FLD for prediction of forming limit curves using an

anisotropic yield function.

In the current process, it was assumed that the loading path is sufficiently close to

being linear that the use of a strain-based FLD to assess accurately the failure of the

sheet is justified. For a general forming process in which the loading path may not

be linear, it would be necessary to either integrate the M-K model into the FEM

analysis to assess each element separately according to its loading path, or to usea stress-based FLD, which is less sensitive to strain path (Stoughton, 2001). A review

of different types of FLD and their use in FEA can be found in Stoughton and Zhu

(2004).

The FLD curves for the current material were calculated using two types of hard-

ening laws: Voce hardening law and Hollomon�s power law. It was found that there

is a noticeable difference between the predictions of the two models. As seen in Fig.

7, Voce hardening law predicts a lower failure limit curve as compared to the power

law. A recent paper by Aghaie and Mahmudi (2004) also reports a similar differencebetween the two models and also shows that a FLD based on the Voce hardening

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40

Minor Strain

Maj

or

Str

ain

Power LawVoce

Fig. 7. Forming limit diagrams for AA3003-H111 at 25 �C (77 �F). Calculations based on the M-K model

and Barlat�s YLD96 anisotropic material model using two types of hardening law: Voce and power law.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

-0.60 -0.40 -0.20 0.00 0.20 0.40 0.60

Minor Strain

Maj

or

Str

ain

25˚C38˚C93˚C149˚C177˚C204˚C232˚C260˚C

Fig. 8. Forming limit diagrams (FLDs) for AA3003-H111 based on the M-K model, Barlat�s YLD96

anisotropic yield function, and Voce hardening law at several elevated temperatures.


law better predicts the experimental data. In general, the forming limits predicted by

the Voce hardening law offer a more conservative prediction of the failure as com-

pared to the power law.

In this paper, FLD curves based on the Voce hardening law were used. Fig. 8shows the FLD curves for AA3003-H111 at different temperatures. As seen from this

figure, the forming limit curves increase with temperature, suggesting that AA3003-

H111 can be stretched to higher levels before failure occurs.

8. Numerical analysis

8.1. Structural analysis

In the early stages of the research, a structural only finite element analysis (FEA)

was performed of the warm forming process, where all parts were assumed to be in

an isothermal condition during the forming process, i.e., every part in the process

had the same temperature and there was no temperature gradient.

Preliminary results from this model did not compare well with the experiments.

That is, both failure location and punch load vs. punch depth curve were different

from the experiments. Fig. 9 shows the predicted punch load vs. punch depth curve

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40

Punch Depth(mm)

Pu

nch

Lo

ad (

kN)

Experimental Result @ 204˚C

Numerical Result @ 204˚C

Fig. 9. Initial results of punch force vs. punch depth for the hemispherical punch compared to numerical

predictions at 204 �C (400 �F) using only structural FEM analysis.


as compared with the experimental curve at the temperature of 204 �C (400 �F). Inthe numerical analysis the blank was assumed to have an isothermal condition,

i.e., the temperature of the blank did not change as it contacted the dies and thepunch. In the structural FEM analysis, the anisotropy coefficients for the material

model were set to be constant and equal to those at 204 �C (400 �F).Upon careful observation of the experimental process and using multiple thermo-

couples to record the temperature of the blank, punch, and dies, it was noted that the

punch is at a lower temperature than the blank and the dies. This was due to the fact

that the punch itself was not directly heated; rather its temperature was indirectly

raised through heat transfer. Therefore, as the punch moved into the die cavity

and contacted the blank, parts of the blank contacting the punch lost some of its heatto the punch. Fig. 10 shows a schematic of the effect of having the band heaters

placed on the outside and temperature drop toward the center of the die cavity.

Next, to better understand the effect of temperature gradient on the deformation,

the blank in the numerical analysis was divided into four sections, with each section

having constant material properties at different temperatures. The center of the

blank was assumed to be at the lowest temperature, while the edge of the blank

was assumed to be at the maximum temperature. This model crudely approximated

the measured temperature gradient in the sheet in our experimental setup when bandheaters were placed on the outside of the die. Fig. 11 shows the setup used for the

blank and the temperatures assigned to each section.

Fig. 10. A schematic showing the effect of temperature drop away from the band heaters. The dies which

are contacting the band heaters are at a higher temperature than the punch (which is not heated directly),

and the center of the sheet.

Fig. 11. A blank divided into multiple sections with each section having different temperatures.

Temperatures used in the analysis are shown in the graph.


A structural only analysis was then performed using constant material properties

appropriate for that section�s temperature. Fig. 12 shows a comparison of the pre-

dicted punch load vs. punch depth curve with the experimental curve at 204 �C(400 �F). As could be seen from the graph, there is a significant improvement, as

compared to Fig. 9, in the match between the experimental and numerical curves.

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40

Punch Depth(mm)

Pu

nch

Lo

ad (

kN)

Experimental Result @ 204˚C

Numerical Result @ 204˚C Using Sections

Fig. 12. Punch force vs. punch depth for the hemispherical punch compared to numerical predictions for

the sectioned blank in Fig. 11.


Although the punch force–punch depth curve nicely matched the experimental re-sults, the model�s prediction of the punch depth at the time of failure did not com-

pare well with the experiment.

To obtain accurate predictions of both force–displacement as well as failure of the

sheet a fully coupled thermo-mechanical finite element analysis of warm forming

process is required.

8.2. Fully coupled thermo-mechanical analysis

Finite element analysis of pure stretch experiments was performed with the ther-

mal-structural finite element model described previously using the temperature-

dependent user material subroutine (UMAT) of Barlat�s YLD96 anisotropic yield

function. The purposes of the numerical analysis were first to check the validity of

the assumption that thermal strains are negligible in thermo-forming process, and

then to verify the accuracy of both the FEM model and the developed user material

subroutine to predict failure in the aluminum sheet at elevated temperatures.

Test samples from the various temperatures are shown in Fig. 5. Temperatures ofthe dies, punch, and blank were measured with thermocouples as listed in Table 5.

Since the material model is temperature dependent, these temperature values were

input to the numerical model to insure accurate analysis corresponding to the exper-

imental tests.

Fig. 13. Finite element results from a fully coupled thermo-mechanical simulation of thermo-forming at

25 �C (77 �F). Failure location is shown on the left side. The right side shows the distribution of the major

and minor strains on the FLD. The sheet failed where the strains crossed the FLD curve at the punch

depth of 28 mm (1.1 in.).


149 �C (300 �F). Failure location is shown on the left side. The right side shows the distribution of the

major and minor strains on the FLD. The sheet failed where the strains crossed the FLD curve at the

punch depth of 35 mm (1.37 in.).


204 �C (400 �F). Failure location is shown on the left side. The right side shows the distribution of the

major and minor strains on the FLD. The sheet failed where the strains crossed the FLD curve at the

punch depth of 38 mm (1.5 in.).


Fig. 16. Numerical simulation results of pure stretch using the 101.6mm (4 in.) hemispherical punch at

several elevated temperatures. Punch depth at failure and failure locations based on FLD prediction are

also shown.

Fig. 17. Experimental results of pure stretch using the 101.6 mm (4 in.) hemispherical punch at several

elevated temperatures. Punch depth at failure and failure locations are shown.



Fig. 13 shows the results of the fully coupled thermo-mechanical pure stretch sim-

ulation at 25 �C (77 �F). The major and minor strains of each element were projected

onto the FLD shown on the right, while the failure location predicted by the analysis

is shown schematically on the left. The punch depth at which the sheet failed in this

analysis was 28 mm (1.1 in.). This compares well with the experimental data shownin Fig. 5 of about 27 mm (1.06 in.) for the test at 25 �C (77 �F).

Figs. 14 and 15 show results of the fully coupled thermo-mechanical simulation at

149 �C (300 �F) and 204 �C (400 �F), using the temperatures for the dies and sheet as

specified in Table 5. The failure locations and strain distribution are also shown. The

punch depth at which the sheet failed in the 149 �C (300 �F) case was 35 mm

(1.37 in.) and for the 204 �C (400 �F) case was 38 mm (1.5 in.). This compares well

with the experimental results of Fig. 5 (34 and 37 mm, respectively).

In Fig. 16, numerical simulation results with failure location and punch depth atfailure are shown at several elevated temperatures. Fig. 17 shows the corresponding

experimental results at the same temperatures.

As seen from these figures, the fully coupled thermo-mechanical finite element

analysis model was able to predict accurately both the failure depth and location

in the sheet at various temperatures. Table 6 compares experimental and numerical

punch depths at failure for all measured temperatures.

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40

Punch Depth(mm)

Pu

nch

Lo

ad (

kN)

25˚C -Experimental25˚C -Numerical149˚C-Experimental149˚C-Numerical204˚C-Experimental204˚C-Numerical

Fig. 18. Punch force vs. punch depth for the hemispherical punch stretch experiments at several elevated

temperatures. Numerical results closely match the experimental results.

-9

-8

-7

-6

-5

-4

-3

-2

-1

00.00 5.00 10.00 15.00 20.00 25.00

Time (ms)

Str

ain

Val

ues

(L

og

Sca

le) Elastic Strain Increment

Plastic Strain IncrementThermal Strain Increment

Fig. 19. Computed values of elastic, plastic and thermal strain increments for an element at different

times. The results shown correspond to thermo-forming analysis at the elevated temperature of 204 �C(400 �F).

Table 6

Numerical prediction of failure depth compared to experimental results

Temperature (�C) Experimental punch depth at failure (mm) Numerical punch depth at failure (mm)

25 27 28

25 29 30

93 31 32

149 34 35

177 36 36

204 37 38


Fig. 18 shows a comparison between experimental and numerical results of the

punch load vs. punch depth curve at several elevated temperatures. As could be seen

from this plot, the developed fully coupled thermo-mechanical model is capable of

accurately predicting the punch load curves.Finally, Fig. 19 shows the plot of thermal, elastic and plastic strain increments cal-

culated for the case of thermo-forming at the elevated temperature of 204 �C(400 �F). It could be seen that thermal strains are between 2 and 3 orders of magni-

tude smaller than plastic strains and the assumption to neglect them in the stress inte-

gration algorithm was well justified. As shown from the previous results, the

numerical model and the UMAT developed in this research were able to predict


accurately both the behavior and failure locations of an aluminum sheet during

warm forming process at several elevated temperatures.

9. Conclusions

The temperature-dependent anisotropic material model developed in Part 1 of this

paper for the aluminum alloy 3003-H111 was successfully implemented as a user

material subroutine (UMAT) in the explicit finite element code LS-Dyna. A fully

coupled thermo-mechanical finite element model was developed for the analysis of

the warm forming of the aluminum sheet with a hemispherical punch under pure

stretch condition. The thermal part of the numerical analysis was used to solve

the heat transfer behavior of the aluminum sheet based on the assigned boundaryconditions. Temperatures from the thermal analysis were then applied to the

UMAT, and the anisotropic behavior of AA3003-H111 was then determined in

the structural analysis portion of the solution.

The adaptive meshing capability in LS-Dyna used in the coupled thermo-mechan-

ical analysis of thermo-forming produced errors at contact regions. Temperatures of

new elements created using adaptive meshing became negative. The authors believe

that this could be a coding problem in LS-Dyna.

The modified power law, Eq. (8), was used as flow stress in the thermo-forminganalysis, while the FLD used for AA3003-H111 was calculated based on the Voce

hardening law. This was done for following two reasons: (a) at elevated temperatures

the aluminum sheet exhibited strain rate characteristics (Abedrabbo et al., 2005a),

while the conventional Voce model does not account for strain rate; (b) failure loca-

tions predicted by FLD curves based on power law were incorrect, as also reported

recently by Aghaie and Mahmudi (2004). It is the opinion of the authors that if a

modified Voce model with strain rate effect is used for thermo-forming finite element

analysis, the results will be similar to those shown in this paper based on the powerlaw hardening, Eq. (8).

Finite element analysis with the developed thermo-mechanical constitutive model

accurately predicted both the deformation behavior and the failure location in the

blank and compared favorably to the experimental results. The current research

shows the importance of using both thermal analysis and an accurate anisotropic

temperature-dependent material model in a fully coupled mode in order to model

accurately the warm forming process.

Although the current thermo-forming analysis was only verified for biaxialstretching, its application to more complex parts is expected also to yield accurate

results. This is because the accuracy of the YLD96 yield function at room tempera-

ture already has been thoroughly verified by many researchers, therefore, to expect a

similar performance at elevated temperature is not unreasonable. However, to verify

the accuracy of this model�s prediction further, the authors plan to conduct thermo-

forming of a deep-drawn automotive part in the near future and compare the results

with numerical predictions. The results of the numerical and experimental compar-

isons will be reported separately.


Acknowledgments

The authors thank General Motors for the support of this research project. The

authors especially thank Drs. Paul Krajewski and Anil Sachdev from the GM Re-

search and Development Center and Dr. Frederic Barlat from ALCOA for theirassistance and helpful discussions in support of this research.

Appendix A. Explicit derivation of YLD96 and its derivative

First, an expression for �rðr�Þ must be obtained. The plane stress can be expressed

as

r�k¼

rxx

ryy

0

rxy

26664

37775. ðA:1Þ

The deviatoric stress is defined as

½nk� ¼

Sxx

Syy

Szz

Sxy

26664

37775 ¼

ðc2þc3Þ3

rxx � c33ryy

ðc3þc1Þ3

ryy � c33rxx

� c23rxx þ c1

3ryy

c6rxy

266664

377775. ðA:2Þ

Although that r3 = 0, Szz 6¼0 and therefore must be included in the calculations of the

principal stresses. It should be noted also that Szz = �(Sxx + Syy).

The principal values of the deviatoric stress are

½gk� ¼S1

S2

S3

264

375 ¼

SxxþSyy2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSxx�Syy

2

� �2

þ S2xy

r

SxxþSyy2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSxx�Syy

2

� �2

þ S2xy

r�ðS1 þ S2Þ

2666664

3777775; ðA:3Þ

where S3 = �(S1 + S2) because of the deviatoric nature of Sij.

The principal values are ordered such that S1 P S2 P S3. The anisotropic coeffi-

cients ai of the yield function (1) are defined as:

½ak� ¼a1a2a3

264

375 ¼

axcos2bþ aysin2b

axsin2bþ aycos2b

az0cos22bþ az1sin22b

264

375. ðA:4Þ


In the above equations, c1, c2, c3, c6, ax, ay, az0 and az1 are coefficients that describe

the anisotropy of the material. The angle b is the angle between the principal stress

Sx and the principal shear stress Sy, as shown in Fig. 1, and is equal to:

2b ¼ tan�12Sxy

Sxx � Syy

� �. ðA:5Þ

Therefore, �rðr�Þ defined in Eq. (1) can be written as

�rðr�Þ ¼ 1

2W

� �1a

¼ 1

2a1jS2 � S3ja þ a2jS3 � S1ja þ a3jS1 � S2jað Þ

� �1a

. ðA:6Þ

The derivative of the yield functiono�rðr�Þ

o r�

is obtained by applying the chain rule

o�rðr�Þ

o r�

¼ 2�1a

a�r

1a�1ð Þ

( )X3

i

X4

j

o�rðr�Þ

ogi r�

ogionj

onjokþo�rðr

�Þ

oai r�

oaiob

obonj

onjok

0@

1A; ðA:7Þ

where k = 1–3.The components of the previous equation are as follows:

o�rðr�Þ

ogi¼

a a3ðS1 � S2ÞjS1 � S2ja�2 � a2ðS3 � S1ÞjS3 � S1ja�2n o



266664

377775. ðA:8Þ

By defining

u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4S2

xy þ ðSxx � SyyÞ2q

ðA:9Þ

then,

ogionj¼

12

1þ Sxx�Syyu

� �12

1� Sxx�Syyu

� �0

2Sxyu

12

1� Sxx�Syyu

� �12

1þ Sxx�Syyu

� �0

�2Sxyu

�1 �1 0 0

26664

37775; ðA:10Þ

onjok¼

ðc2þc3Þ3

� c33

0

� c33

ðc1þc3Þ3

0

� c23

� c13

0

0 0 c6

266664

377775. ðA:11Þ

The derivative with respect to ai is

o�rðr�Þ

oai r�¼jS2 � S3ja

jS3 � S1ja

jS1 � S2ja

264

375; ðA:12Þ


oaiob¼

ðay � axÞ sin 2bðax � ayÞ sin 2b

2ðaz1 � az0Þ sin 4b

264

375; ðA:13Þ

obonj¼

�Sxyu2

Sxyu2

0Sxx�Syy

u2

266664

377775. ðA:14Þ

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Documents

International Journal of Plasticity 22 (2006) 342–373 Www.elsevier.com/Locate/Ijplas