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Page 1: Numerical Determination

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Journal of Materials Processing Technology 216 (2015) 472–483

Contents lists available at ScienceDirect

Journal of Materials Processing Technology

jo ur nal home p ag e: www.elsev ier .com/ locate / jmatprotec

umerical determination of the forming limit curves of anisotropicheet metals using GTN damage model

bdolvahed Kamia, Bijan Mollaei Dariania,∗, Ali Sadough Vaninia, Dan Sorin Comsab,orel Banabicb

Mechanical Engineering Department, Amirkabir University of Technology, 424 Hafez Ave, 15875-4413 Tehran, IranCERTETA Research Centre, Technical University of Cluj-Napoca, Str. C. Daicoviciu nr. 15, 400020 Cluj-Napoca, Romania

r t i c l e i n f o

rticle history:eceived 1 July 2014eceived in revised form 15 October 2014ccepted 20 October 2014vailable online 29 October 2014

eywords:TN damage model

a b s t r a c t

In this paper, the Gurson–Tvergaard–Needleman (GTN) damage model is used to determine the forminglimit curve of anisotropic sheet metals. The mechanical behavior of the matrix material is described usingHill’48 quadratic yield criterion and an isotropic hardening rule. For this purpose, a VUMAT subroutinehas been developed and used inside the ABAQUS/Explicit finite element code. The implementation of theconstitutive model in the finite element code is presented in detail. Finally, the forming limit curve of anAA6016-T4 sheet metal is constructed using the developed VUMAT subroutine and running numericalsimulation of Nakjima tests. The quality of the numerical results is evaluated by comparison with an

orming limit curveormabilitynisotropy

experimental forming limit curve. Furthermore, theoretical forming limit curves of the AA6016-T4 sheetare obtained using Marciniak–Kuczynski (M–K) and modified maximum force criterion (MMFC) models.The results show that the forming limit curve predicted by the anisotropic GTN model is in better agree-ment with the experimental results especially in the biaxial tension region. This fact indicates that theGTN model is a useful tool in analyzing the formability of anisotropic sheet metals.

. Introduction

The forming limit curve (FLC) is a very useful and common tooln industries involved in sheet metal forming. This curve is actually

plot of the major principal strain vs. minor principal strain char-cterizing the onset of sheet necking. Consequently, FLC divides theossible combinations of the major and the minor strains into safend unsafe regions. More precisely, the strain combinations whichtand below the FLC are considered as safe (acceptable), while thetrain combinations located above the FLC are considered as unsafe.

Since the introduction of the FLC by Keeler and Backhofen1963), many attempts have been made to construct it usingxperimental, theoretical and numerical methods. Because of thexpenses involved by the experimental procedures of FLC construc-ion, the theoretical (Hill, 1952; Marciniak and Kuczynski, 1967)

nd numerical (Li et al., 2010) methods have been more attractiveo researchers. One of the suitable theoretical approaches for deter-

ination of the FLC is the Gurson–Tvergaard–Needleman (GTN)

∗ Corresponding author. Tel.: +98 21 64543413/66419736.E-mail addresses: [email protected], [email protected] (A. Kami),

[email protected] (B.M. Dariani), [email protected] (A. Sadough Vanini),[email protected] (D.S. Comsa), [email protected] (D. Banabic).

ttp://dx.doi.org/10.1016/j.jmatprotec.2014.10.017924-0136/© 2014 Elsevier B.V. All rights reserved.

© 2014 Elsevier B.V. All rights reserved.

damage model (Gurson, 1977; Tvergaard, 1981, 1982; Tvergaardand Needleman, 1984). The original formulation of this model hasbeen proposed by Gurson (1977) by assuming that the degra-dation of the load carrying capacity and finally the fracture ofductile metals are caused by the evolution of voids. Gurson’smodel takes into account only the growth of pre-existing voids,without assuming any generative mechanisms. In order to over-come this limitation, Tvergaard (1981, 1982) and Tvergaard andNeedleman (1984) have proposed mathematical descriptions of thevoid nucleation and coalescence. The final modified model is knownas Gurson–Tvergaard–Needleman (GTN) damage model.

As the metallic sheets are commonly produced by rolling, theyexhibit high levels of anisotropy. Due to this characteristic, it is veryimportant to include the anisotropy of the matrix material in theGTN model. There are few works that have dealt with this aspect.Liao et al. (1997) utilized a similar approach to that proposed byGurson (1977) to derive an approximate potential formulation forthe prediction of damage in the metallic sheets. The original fea-ture of their model consists in the fact that the equivalent stress isdescribed by Hill’s quadratic (Hill, 1948) and non-quadratic (Hill,

1979) anisotropic expressions. Liao et al. (1997) used anisotropyparameters defined as the ratio of the transverse plastic strain rateto the through-thickness plastic strain rate under in-plane uni-axial loading along different directions. Wang et al. (2004) replaced
Page 2: Numerical Determination

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t((t(ao(

ttcosa1psiBttpbaoe((

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2A

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p

A. Kami et al. / Journal of Materials P

he directional parameters in the model proposed by Liao et al.1997) with the average anisotropy parameter. Chen and Dong2008) extended the GTN model to characterize the matrix materialhrough Hill quadratic (Hill, 1948) and Barlat–Lian 3-componentBarlat and Lian, 1989) expressions of the equivalent stress. Chennd Dong (2009) proposed extensions of the GTN potential basedn Hill’s quadratic anisotropic expression of the equivalent stressHill, 1948).

Of course, it is possible to solely use the GTN damage modelo predict the fracture of the ductile materials during deforma-ion. On the other hand, the GTN model could be also used toonstruct the forming limit curve. Brunet et al. (1996) studied theccurrence of necking in square cup deep drawing of a mild-steelheet and also extracted the limit strains of the sheet using annisotropic Gurson–Tvergaard criterion (Gurson, 1977; Tvergaard,981). Brunet et al. (1998) and Brunet and Morestin (2001) pro-osed a necking criterion based on the load-instability and planetrain localization assumptions in which the failure of the materials defined by Gurson–Tvergaard damage model with Hill (1948) andarlat and Lian (1989) anisotropy models. He et al. (2011) predictedhe forming limit stress diagram of 5052 aluminum alloy based onhe GTN model. Abbasi et al. (2012a, 2012b) used GTN model toredict the forming limit curve of an IF-steel and a tailor weldedlank made from IF-steel, respectively. Furthermore, the GTN dam-ge model has been employed to predict the forming limits ofther sorts of sheets like the AA5052/polyethylene/AA5052 (Liut al., 2013; Liu and Xue, 2013) and AA3105/Polypropylene/AA3105Parsa et al., 2013) sandwich sheets, dual-phase and multi-phaseUthaisangsuk et al., 2009; Ramazani et al., 2012) steels.

In this paper, a GTN model based on Hill’s quadratic expres-ion of the equivalent stress is used to construct the formingimit curve. The model is implemented as a VUMAT routine inhe ABAQUS/Explicit finite-element code (ABAQUS analysis user’s

anual, 2011). Furthermore, the plastic strain and void volumeraction distributions near the fracture section are analyzed. The

aterial parameters involved in the constitutive relationships areetermined by means of an identification procedure that combineshe response surface methodology (RSM) and the simulation of aniaxial tensile test.

. Formulation of the constitutive model and itsbaqus/Explicit implementation

Abaqus/Explicit allows the implementation of solid materialodels by means of the VUMAT routine. Because Abaqus/Explicit

ses corotational components of the Cauchy stress and logarithmictrain as input/output when communicating with VUMAT, plainime derivatives of such tensor quantities can be involved in theormulation of the rate-type constitutive relationships, withoutny concern about their objectivity. The model presented belowssumes that the Abaqus/Explicit corotational frame also reflectshe plastic orthotropy of the sheet metal, being initially coincidentith the frame defined by the rolling direction – RD (axis 1), trans-

erse direction – TD (axis 2) and normal direction – ND (axis 3).he following symbols will denote macroscopic strain and stressuantities:

εij components of the corotational logarithmic strain tensor sep-

rable into elastic ε(e)ij

and plastic ε(p)ij

terms, i.e.

ij = ε(e)ij

+ ε(p)ij

(1)

ij components of the corotational Cauchy stress tensorp hydrostatic pressure:

= −���

3(2)

ing Technology 216 (2015) 472–483 473

� Hill’48 equivalent stress:

� =√

�ijPijk��k�, (3)

where Pijkl are components of a fourth-order tensor by means ofwhich the constitutive model approximates the plastic orthotropyof the sheet metal. In general, Pijkl (i, j, k, � =1, 2, 3) are subjected tothe constraints

Pijk� = Pjik� = Pij�k = Pk�ij, Piik� = 0. (4)

Two other strain/stress parameters will be associated to the fullydense matrix material:

ε(p) equivalent plastic strain (ε(p)≥0, ˙ε(p)≥0)

Y yield stress defined as function of ε(p) by means of a hardeninglaw Y = Y[ε(p)] > 0.

The elasticity of the sheet metal is described by the isotropicHooke’s law

�ij = E

1 + �

[ε(e)

ij+ �

1 − 2�ε(e)

��ıij

], (5)

where E and v are Young’s modulus and Poisson’s ratio, respectively,while ıij denotes Kronecker’s symbol.

The plastic part of the constitutive model is based on the GTNpotential (Chen and Butcher, 2013)

={

Y[ε(p)]

}2

+ q1f ∗(f )

{2 cosh

{−q2

3p

2Y[ε(p)]

}− q1f ∗(f )

}− 1, (6)

where

f ∗(f ) =

⎛⎜⎜⎝f, if f ≤ fC,

fC + f ∗F − fCfF − fC

(f − fC ), if fC < f < fF ,

f ∗F , if f ≥fF ,

with f ∗F = 1/q1,

(7)

is a porosity parameter depending on the void volume fraction f.The quantities denoted as q1, q2, fC, and fF in Eqs. (6) and (7) arematerial constants. The inequality ≤ 0 defines all the admissiblestress states of the sheet metal. More precisely, < 0 is associatedto the elastic states and = 0 corresponds to the elastoplastic ones.

The flow rule associated to the potential can be expressed inthe form

ε(p)ij

= �∂˚

∂�ij, with

(� = 0, if < 0,

�≥0, if = 0,(8)

or, if Eqs. (6), (3) and (4) are taken into account,

ε(p)ij

= 1�

ε(p,dev)Pijk��k� + 13

ε(p,vol)ıij, (9)

where

ε(p,dev) = �∂˚

∂ �, ε(p,vol) = −�

∂˚

∂p, (10)

and

∂˚

∂ �= 2 �

{Y[ε(p)]}2,

∂˚

∂p= −3q1q2

f ∗(f )Y[ε(p)]

sinh

{−q2

3p

2Y[ε(p)]

}·(11)

Eq. (10) allows deducing the following consistency condition

that accompanies the constraint = 0 in the elastoplastic states ofthe sheet metal:

ε(p,dev) ∂˚

∂p+ ε(p,vol) ∂˚

∂ �= 0. (12)

Page 3: Numerical Determination

4 rocess

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oa

f

f(

f

wd

f

e

A

w

aTpfsmptteptpi

tt

cna

t

w

t

a

tt

re

where

t+t�(e)ij

=t+t�(e)

ij

Y[t+tε(p)], (30)

t+t�ij =t+t�ij

Y[t+tε(p)], (31)

74 A. Kami et al. / Journal of Materials P

The evolution of the parameter ε(p) is controlled by the equiva-ent plastic work rule (Chen and Butcher, 2013)

ijε(p)ij

= (1 − f )Y[ε(p)] ˙ε(p)

. (13)

With the help of Eqs. (9) and (2)–(4), Eq. (13) becomes

¯ ε(p,dev) − pε(p,vol) = (1 − f )Y[ε(p)] ˙ε(p)

. (14)

As for the parameter f, its time derivative cumulates the growthf the existing voids f (g) and the nucleation of new voids f (n) (Chennd Butcher, 2013):

˙ = f (g) + f (n). (15)

˙ (g) is related to the volumetric part of the plastic strain rate tensorsee also Eqs. (9) and (4)),

˙ (g) = (1 − f )trε(p) = (1 − f )ε(p)��

= (1 − f )ε(p,vol), (16)

hile f (n) is a function of the equivalent plastic strain and its timeerivative:

˙ (n) = A[ε(p)] ˙ε(p)

. (17)

The multiplier A[ε(p)] on the right-hand side of Eq. (17) has thexpression

[ε(p)] =

⎛⎜⎝ fN

sN

√2�

exp

{−1

2

[ε(p) − εN

sN

]2}

, if p < 0,

0, if p≥0,

(18)

here fN, sN and εN are material constants.The constitutive model described above has been implemented

s a VUMAT routine of the Abaqus/Explicit finite element code.he implementation follows the principles detailed in the papersublished by Aravas (1987) and Zhang (1995). Abaqus/Explicit per-orms the simulation of nonlinear processes by dividing them intomall time increments [t, t + t]. The configuration of the finite ele-ent mesh corresponding to the moment t is a reference state, the

arameters of its numerical integration points being known quan-ities passed to the VUMAT routine as input. In the particular case ofhe constitutive model described above, the reference state param-ters are the components of the stress tensor t�ij, the equivalentlastic strain t ε(p) and the void volume fraction tf. Together withhese quantities, Abaqus/Explicit also provides as input the com-onents of the logarithmic strain increment associated to the time

nterval [t, t + t] :

+tεij =∫ t+t �

t

εijd�. (19)

The main task of the VUMAT routine consists in evaluating theurrent state parameters t+t�ij,

t+tε(p) and t + tf. The compo-ents t+t�ij can be expressed using incremental forms of Eqs. (5)nd (1) in combination with Eq. (19):

+t�ij = t+t�(e)ij

− E

1 + �

[t+tt ε(p)

ij+ �

1 − 2�t+tt ε(p)

��ıij

], (20)

here

+t�(e)ij

= t�ij + E

1 + �

[t+tt εij + �

1 − 2�t+tt ε��ıij

], (21)

nd

+tε(p)ij

=∫ t+t

� ε(p)ij

d�. (22)

t

Eq. (21) defines a stress tensor that would occur in the cur-ent configuration if the incremental logarithmic strain were purelylastic. In practice, the components t+t�(e)

ijare the first quantities

ing Technology 216 (2015) 472–483

that must be evaluated. As soon as t+t�(e)ij

are available, Eqs. (2)and (3) allow the calculation of the corresponding pressure andequivalent stress, respectively:

t+tp(e) = −t+t�(e)��

/3, t+t�(e) =√

t+t�(e)ij

Pijk�t+t�(e)

k�. (23)

At this stage it is possible to decide whether the increment of thelogarithmic strain is elastic or elastoplastic. The decision is madeafter evaluating the trial GTN function (see Eqs. (6) and (7))

t+t˚(e) ={

t+t�(e)

Y[t ε(p)]

}2

+ q1f ∗ (t f )

{2 cosh

{−q2

3 t+tp(e)

2Y[t ε(p)]

}− q1f ∗ (t f )

}− 1.

(24)

If t + t˚(e) ≤ 0, the incremental logarithmic strain is purely elas-tic and the parameters of the current configuration can be set asfollows: t+t�ij = t+t�(e)

ij, t+tε(p) = t ε(p), and t + tf = tf. On the

other hand, if t + t˚(e) > 0, the material evolves through elasto-plastic states during the time interval [t, t + t]. In such a case,the plastic terms on the right-hand side of Eq. (20) are differentfrom zero. These terms will be evaluated using a backward Eulerapproximation of Eq. (22):

t+tt ε(p)

ij≈ t+tε(p)

ijt. (25)

When combined with Eq. (9), Eq. (25) becomes

t+tt ε(p)

ij≈

t+tt ε(p,dev)

t+t�Pijk�

t+t�k� + 13

t+tt ε(p,vol)ıij, (26)

where

t+tt ε(p,dev) ≈ t+tε(p,dev)t, t+t

t ε(p,vol) ≈ t+tε(p,vol)t (27)

are increments generated as backward Euler approximations and(see Eq. (3))

t+t� =√

t+t�ijPijk�t+t�k� (28)

is the current equivalent stress. The plastic strain incrementsdefined by Eqs. (26)–(28) and (4) allow rewriting Eq. (20) in thenormalized form

t+t�ij + E

Y[t+tε(p)]

[t+tt ε(p,dev)

(1 + �)t+t � Pijk�t+t�k� +

t+tt ε(p,vol)

3(1 − 2�)ıij

]≈ t+t�(e)

ij, (29)

and

t+t ˆ� =t+t�

Y[t+tε(p)]=

√t+t�ijPijk�

t+t�k�. (32)

Page 4: Numerical Determination

rocess

i

w

t

t

a

t

ct

tt

tt

t

tt

ss(

tt

A. Kami et al. / Journal of Materials P

In the case of an elastoplastic evolution of the material, Eq. (29)s accompanied by the following constraints:

First consistency condition t + t = 0, i.e. (see also Eqs. (6), (7),

(2), (31) and (32))

t+t �2 + q1∗(t+tf )

[2 cosh

(−3

2q2

t+tp)

− q1 f ∗ (t+tf )]

= 1,

(33)

here

+tp = −t+t���

3(34)

Incremental form of the second consistency condition defined byEq. (12) together with Eqs. (11), (27), (32) and (34):

t+t � t+tt ε(p,vol)

− 32

q1q2f ∗(t+tf ) sinh(

−32

q2t+tp

)t+tt ε(p,dev) ≈ 0 (35)

Incremental form of the equivalent plastic work rule defined byEq. (14), the backward Euler approximation

+tε(p) − t ε(p) =∫ t+t�

t

˙ε(p)

d� ≈ t+t ˙ε(p)

t, (36)

s well as Eqs. (27), (32) and (34):

t+t �t+tt ε(p,dev) − t+tpt+t

t ε(p,vol) ≈ (1 − t+tf )[t+tε(p) − t ε(p)]

(37)

Incremental form of Eq. (15),

+tf = t f + t+tt f (g) + t+t

t f (n), (38)

ombined with the following backward Euler approximations ofhe terms t+t

t f (g) and t+tt f (n) (see Eqs. (16)–(18), (27) and (36)):

+tf (g) =∫ t+t

t

� f (g) d� ≈ t+t f (g) t ≈ (1 − t+tf )t+tt ε(p,vol),

(39)

+tf (n) =∫ t+t

t

� f (n)d�≈t+t f (n)t ≈ A[t+tε(p)][t+tε(p)−t ε(p)].

(40)

Eqs. (38)–(40) can be used to express t+tt ε(p,vol) as a function of

he state parameters t+tε(p) and t + tf :

+tε(p,vol) ≈ 11 − t+tf

{t+tf − t f − A[t+tε(p)] [t+tε(p) − t ε(p)]}.(41)

Eq. (41) is always valid because 0 ≤ t + tf < 1. In any elastoplastictate, the current value of the normalized equivalent stress alsoatisfies the condition t+t � > 0. Under these circumstances, Eq.37) allows expressing t+t

t ε(p,dev) as follows:

+tε(p,dev) ≈ 1t+t � {t+tpt+t

t ε(p,vol) + (1 − t+tf )[t+tε(p)−t ε(p)]}.(42)

ing Technology 216 (2015) 472–483 475

Eqs. (30), (32), (34), (41) and (42) transform Eqs. (29), (33)and (35) into a nonlinear set having t+t�ij,

t+tε(p) and t + tfas unknowns. The VUMAT routine solves this set in a numericalmanner, using a forward finite difference Newton scheme com-bined with a line search strategy. The reference state parameterst �ij = t�ij/

tY, t ε(p) and tf define the start point of the solutionprocedure. This initial guess ensures a rapid convergence of theNewton iterations for sufficiently small logarithmic strain incre-ments passed as input to the VUMAT routine. The performancesof the solution procedure tend to degrade in the final stages of thenumerical simulation, when t + tf is greater than fC and the carryingcapability of the metallic sheet suffers a sudden drop. The continu-ous diminishment of the material strength leads to an acceleratedaccumulation of the logarithmic strain. The VUMAT routine is ableto overcome almost all the convergence difficulties that may occurin such cases by activating an adaptive subincrementation proce-dure. The principle of the subincrementation consists in splittingthe components of the full logarithmic strain increment t+t

t εij intosmaller fractions. The VUMAT routine successively halves t+t

t εij

until the convergence of the Newton scheme is achieved or thesubincrement of the logarithmic strain becomes too small (forexample, less than 0.5% of the full increment passed as input). Inthe latter case, the numerical simulation is stopped after issuing anerror message. If the Newton scheme has converged, the VUMATroutine doubles the subincrement of the logarithmic strain anduses the previously obtained state parameters to define the ini-tial guess for a new iterative solution of Eqs. (29), (33) and (35).This procedure ends when the full increment components t+t

t εij

have been restored by adding small fractions. As soon as t+t�ij

and t+tε(p) are known, the components of the actual stress tensorin the current configuration can be also evaluated (see Eq. (31)):t+t�ij = t+tYt+t�ij.

3. Experimental determination of a forming limit curve

The performances of the constitutive model implemented asa VUMAT routine have been assessed by calculating the forminglimit curve of an AA6016-T4 metallic sheet (1 mm thickness) andcomparing the numerical predictions with reference data obtainedfrom a series of Nakajima tests (Banabic, 2010). As one may noticein Figs. 1 and 2, such experiments are punch stretching processesthat induce different strain evolutions in the central area of thespecimen. Each strain path is characterized by the ratio minor vs.major principal surface strain. Modifying the width of the specimenshaft (denoted as w in Fig. 2) allows the reproduction of differentload states ranging from uniaxial traction (small values of the shaftwidth) to balanced-biaxial traction (case of a fully circular speci-men). These loads cover both branches of a forming limit curve. Inaccordance with the methodology described in the InternationalStandard ISO 12004-2 (2008), six discrete load states have beenanalyzed, the limit strains associated to each of these states beingobtained by averaging the results of five Nakajima tests. The valuesof the width parameter w adopted by the authors are 30, 55, 70, 90,120, 145, and 185 mm, the last of them corresponding to a fully cir-cular specimen. In Fig. 2 the horizontal symmetry line correspondsto the sheet rolling direction.

Fig. 3 shows the position-dependent method used by theARAMIS software for determining the limit values of the majorprincipal strain ε1 and minor principal strain ε2. This procedure isdetailed in the International Standard ISO 12004-2 (2008). In prin-ciple, the position-dependent method assumes that necking affects

a banded region of the specimen, while the strain distributionremains virtually unchanged outside this band. From a practicalpoint of view, the ARAMIS software requests the user to defineat least 3–5 cross sections perpendicular to the crack path. These
Page 5: Numerical Determination

476 A. Kami et al. / Journal of Materials Processing Technology 216 (2015) 472–483

Fig. 1. Principle of the Nakajima test with a hemispherical punch, Ø100 mm.

w

Fig. 2. Dimensional characteristics of the circular and notched specimens used inthe Nakajima tests with hemispherical punch, Ø100 mm.

Fig. 3. Definition of the standard method used by ARAMIS system for determinationof limit strains (International Standard ISO 12004-2, 2008).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-0.2 -0.1 0 0. 1 0. 2 0.3

Maj

or S

trai

n

Mino r St rain

Fig. 4. Experimental forming limit curve of the AA6016-T4 metallic sheet.

sections are then transferred to the frame showing the distributionof the major surface strain ε1 just before the fracture occurrence.The limits of the necking region along each cross section are thendetermined by localizing the maxima of the second derivative ofε1 with respect to the arc-length. In the next stage, the distributioncurves outside these limits are used to generate an inverse parabolicregression for ε1 as a dependence on the arc-length. The limit valueof the major principal strain ε1 is represented by the maximum ofthe inverse parabolic function. The corresponding limit value of theminor principal strain ε2 is determined in the same position, usingan inverse parabolic regression constructed for ε2 as a dependenceon the arc-length. In the last stage, the ARAMIS software averagesthe individual results obtained for the whole set of cross sectionsto obtain the most representative limit strains ε1 and ε2.

The experimental forming limit curve obtained in this manner isshown in Fig. 4. According to the specification of the InternationalStandard ISO 12004-2 (2008), the forming limit curve is generatedas a linear interpolation of the individual experimental points (rep-resented by diamond marks in Fig. 4). It is worth mentioning thatthe same position-dependent methodology has been used to pro-cess the surface strain distributions predicted by the constitutivemodel implemented in Abaqus/Explicit (see Section 5). In this way,a full comparability of the numerical and experimental results hasbeen ensured.

4. Numerical simulation of the Nakajima tests

The commercial Abaqus/Explicit finite element code and theVUMAT implementation of the anisotropic GTN damage modelhave been used to calculate the forming limit curve of an AA6016-T4 sheet (1 mm thickness). The mechanical properties of thismaterial have been determined from uniaxial tensile tests per-formed on rectangular specimens (200 × 20 × 1 mm) cut at 0, 45and 90◦ with respect to the rolling direction. The extensometergauge length (80 mm) has been used to measure the elongation.The tensile testing methodology is detailed by Paraianu and Banabic(2013). The Lankford coefficients obtained from these tests are thefollowing ones: r0 = 0.5529, r45 = 0.4091 and r90 = 0.5497. In thisstudy, the hardening behavior of the metallic sheet is describedby Swift’s law (Swift, 1952):

Y(εp) = K(ε0 + εp)n, (43)

where K, ε0 and n are material constants. Their values have beendetermined using an identification procedure based on the leastsquares method and the experimental hardening curves associatedto the rolling direction. The following material parameters have

Page 6: Numerical Determination

A. Kami et al. / Journal of Materials Processing Technology 216 (2015) 472–483 477

Table 1Values of Hill’48 coefficients.

bnrt

o(

wb

F

acbtobaTto

cf

R

nbell2a

up

TI

0

1000

2000

3000

4000

5000

6000

0 5 10 15 20 25

Forc

e (N

)

Dis placmen t (mm)

ExperimentNumerical Simulat ion

Hill’48 coefficients F G H L M N

Values 0.647705 0.643956 0.356043 1.5 1.5 1.174250

een obtained in this manner: K = 525.77 MPa, ε0 = 0.011252, and = 0.2704. Furthermore, Young’s modulus E = 70 GPa and Poison’satio � = 0.33 have been also evaluated by averaging the results ofhe uniaxial tensile tests associated to the 0◦, 45◦ and 90◦ directions.

As mentioned in Section 2, the GTN potential defined by Eq. (6)perates with Hill’48 expression of the equivalent stress � – see Eq.3). For practical purposes, � can be rewritten in the form

¯ = [F(�22 − �33)2 + G(�33 − �11)2 + H(�11 − �22)2 + 2L�223 + 2M�2

31 + 2N�212]

1/2.

(44)

here F, G, H, L, M, and N are material constants uniquely definedy the Lankford coefficients r0, r45 and r90:

= H

r90, G = H

r0, H = r0

r0 + 1, L = M = 3

2,

N = (r0 + r90)(2r45 + 1)2r90(r0 + 1)

· (45)

The values of these coefficients are listed in Table 1.The GTN model has nine parameters, q1, q2, q3, f0, fC, fF, fN, SN,

nd εN that require to be calibrated. To increase the efficiency of thealibration procedure for these parameters, only f0, fC, fF and fN haveeen allowed to change in the ranges specified in Table 2, whereashe other parameters have been kept constant. The optimum valuesf the parameters included in the identification procedure haveeen obtained using central composite (face centered type) designnd RSM. The central composite design resulted in 27 experiments.hese experiments are numerical simulations of the uniaxial tensileest in the rolling direction performed with different combinationsf f0, fC, fF and fN values.

The optimum values of f0, fC, fF and fN parameters have beenalculated by minimizing the cost function R which is defined asollows (Kami et al., 2014):

=n∑

i=1

|Fexp − Fnum|i. (46)

In the above formula, Fexp and Fnum are experimentally andumerically obtained tensile test forces, respectively. A total num-er of 1000 equally spaced points (n = 1000) has been selected onach tensile curve. The results of the optimization procedure areisted in Table 2. According to the recommendations found in theiterature (Needleman and Tvergaard, 1984; Benseddiq and Imad,008), the values of the remaining parameters have been adopted

s follows: q1 = 1.5, q2 = 1, q3 = q2

1 = 2.25, SN = 0.1, and εN = 0.3.The optimization procedure has been validated by simulating a

niaxial tensile test performed along the rolling direction. A com-arison between the numerical simulation and the experimental

able 2dentification ranges and optimum values of the GTN material parameters.

Factor Values

Minimum Maximum Optimum

f0 0.0001 0.001 0.00035fN 0.001 0.05 0.05fC 0.005 0.05 0.05fF 0.06 0.2 0.15

Fig. 5. Comparison between experimental and numerical force vs. elongation curvesof the uniaxial tensile test in the rolling direction.

results is shown in Fig. 5. The excellent agreement between theresults indicates that the optimization procedure succeeded toprovide reliable values of f0, fc, ff and fN parameters for the AA6016-T4 sheet metal.

The value fN = 0.05 obtained from identification is larger thanthe actual inclusion fraction of the aluminum alloy AA6016-T4.Similar results have been obtained by other researchers (fN = 0.036in Brunet et al., 1998; fN = 0.075 in Brunet and Morestin, 2001and fN = 0.095 in Thuillier et al., 2011) for AA6016-T4 metallicsheets when using a calibration procedure based on macroscopicexperimental data. An alternate identification method of the GTNmodel relies on the analysis of the cross section of fractured spec-imens using optical microscopy and automatic image processing.However, some studies (Zhang et al., 2000; Chhibber et al., 2008)showed that the volume fraction of inclusions obtained from metal-lographic examination cannot be directly applied to the GTN modelbecause not all the inclusions will nucleate voids. Consequently, itis recommended to identify the nucleation parameters accordingto the macroscopic fracture behavior. The same situation is spe-cific to the initial void volume fraction (f0). On the basis of theseobservations, the macroscopic fracture behavior has been used tocalibrate the GTN parameters in this study.

Fig. 6 shows that the predicted distribution of the Lankford coef-ficient is in perfect agreement with the experimental values. Thissituation is normal because r0, r45 and r90 are used by the identifi-cation procedure. On the other hand, Fig. 7 depicts that in the caseof the normalized yield stress y� (see Table 3), the agreement is notso good. Except for y0 = 1 (which is used in the identification pro-cedure), the other experimental values are not fitted. The largesterror (corresponding to y45) is about 8.5%. In the case of the nor-malized yield stress y90 the error is only about 1.9%. The differencesbetween the experimental and predicted values of the normalizedyield stresses y45 and y90 are inevitable when the Hill’48 yield cri-terion is identified in the manner specific to sheet metals (i.e., byenforcing the exact reproduction of the experimental values y ,

0r0, r45, and r90). However, these errors do not affect dramaticallythe predictions of the GTN model, because the Nakajima speci-mens are oriented along the rolling direction (as requested by the

Table 3Normalized yield stress in the plane of the AA6016-T4 metallic sheet.

Reference angle, � (◦) Yield stress, Y�

(MPa)Normalized yieldstress, y� = Y�/Y0

0 158.07 145 152.25 0.96318190 154.75 0.978997

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478 A. Kami et al. / Journal of Materials Processing Technology 216 (2015) 472–483

FT

Ii

jAIom

fobe

FA

ig. 6. Predicted distribution of the Lankford coefficient in the plane of the AA6016-4 metallic sheet.

nternational Standard ISO 12004-2, 2008) and the punch stretch-ng process induces insignificant shearing effects.

To construct the numerical forming limit curve, several Naka-ima tests have been simulated using the finite element codebaqus/Explicit and the VUMAT implementation of the GTN model.

n each case, the major and minor principal surface strains at thenset of necking have been determined by processing the strainaps with the ARAMIS software (see the discussion in Section 3).Fig. 8 shows the finite element model of the Nakajima test per-

ormed on a specimen having the shaft width w = 55 mm. Because

f the anisotropic behavior of the material, the specimens haveeen modeled with full geometry, as shown in Fig. 8. In the finitelement model, the die has been locked, while the punch and the

ig. 7. Predicted distribution of the normalized yield stress in the plane of theA6016-T4 metallic sheet.

Fig. 8. Finite element model of a Nakajima test.

blank holder have been allowed to move only in the vertical direc-tion.

Two steps have been defined in the simulation model. Duringthe first step, a holding force of 100 kN has been applied to thereference point of the blank holder. In the second step, the punchhas been translated in the vertical direction while keeping constantthe holding force.

The die, punch and the blank holder have been modeled as ana-lytical rigid surfaces, while the specimen has been meshed usinghexahedral elements with 8 nodes (C3D8R). Two layers of elementshave been used in the thickness direction. As one may notice inFig. 8, the mesh has been refined in the central region of the spec-imen. The frictional interactions between the metallic sheet andtools have been described using Coulomb’s model. The followingvalues of the friction coefficient have been adopted: 0.03 on thepunch surface (where several Teflon foils separated by grease havebeen placed under the specimen), and 0.1 on the die and clampingring surfaces (where no special care should be taken to reduce thefrictional interactions).

5. Comparison between experimental and numericalresults

The forming limit curve has been constructed according to thespecifications of the International Standard ISO 12004-2 (2008).Different strain paths have been tested by using different geome-tries of the specimens (see Fig. 2). The forming limit curve obtainedby numerical simulations has been validated by comparison withthe results of the Nakajima test.

Fig. 9 shows the distribution and evolution of the major loga-rithmic strain determined by numerical simulation of the punchstretching experiment performed on a notched specimen withw = 130 mm. One may notice that the strain distribution in the cen-tral area is almost uniform in the early stages of the process. Whilethe deformation proceeds, the strain tends to localize in smallerregions near the dome apex. Finally, this strain accumulation leadsto necking and fracture (see Fig. 9(g)). As shown in Fig. 9, the neck-ing and fracture do not occur at the dome apex. This phenomenonis the effect of the frictional contact between the punch and thespecimen which limits the straining of the sheet regions in contactwith the punch. Fig. 9(h) shows a specimen fractured during thelaboratory tests. The shape of the fracture region in the numerical

simulations (Fig. 9(g)) compares well with the experimental results.The fractured dome height predicted by the numerical simulation(38.47 mm) is also in very good agreement with the experimentaldata (38 mm).
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A. Kami et al. / Journal of Materials Processing Technology 216 (2015) 472–483 479

F n of a( perim

tvvtuichf

ig. 9. Distribution of the major logarithmic strain obtained by numerical simulatiomm) (a) 6.26, (b) 12.56, (c) 18.85, (d) 25.15, (e) 31.44, (f) 37.75, (g) 38.47 and (h) ex

Since the GTN damage model has been used to predict the frac-ure in the numerical simulations, it is also worth looking at theoid volume evolution during the simulation. For this purpose, theoid volume fraction at different stages has been extracted fromhe Abaqus output database. Fig. 10 depicts the values of void vol-me fraction at different deformation stages at the nodes shown

n Fig. 9(f) and (g). In the early stages of the punch stretching pro-ess (frames #182, 192 and 202) the void volume fraction is almostomogeneously distributed over the fracture region. Starting with

rame #213 (onset of necking), a noticeable increase in the void

Nakajima test performed on the notched specimen with w = 130 mm: punch depthental results.

volume fraction can be observed at the level of a small section. Itis clear from Fig. 10 that after necking the increase in the void vol-ume fraction is limited to the nodes near the necking region andthe value of the void volume fraction for nodes outside this arearemains constant between frames #213 and 214.

Frame #192 in Fig. 10 corresponds to the start of the void coales-

cence process. The distribution of the voids in the specimen atthis stage is shown in Fig. 11(a). One may easily notice that thevoid distribution is almost uniform in the polar region of the speci-men. Fig. 11(b) shows the voids distribution at the onset of necking
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480 A. Kami et al. / Journal of Materials Processing Technology 216 (2015) 472–483

0.00

0.03

0.06

0.09

0.12

0.15

0 10 20 30 40 50

Void

Vol

ume

Frac

tion

(SD

V2)

Sequ ence Nu mber

Frame #21 4Frame #21 3Frame #20 2Frame #19 2Frame #18 2

Fs

(osvsvtoe

TtmrnatnFfittTwc1c0dApc

ig. 10. Evolution of voids in the path normal to the fracture section for notchedpecimen with w = 130 mm.

frame #213). From this figure one may notice that the high valuesf void volume fraction are concentrated in a small region of thepecimen (necking region) while at other regions there are smalleralue of voids and they are uniformly distributed. Furthermore, ashown in Fig. 11(a) and (b), there are some regions where the voidolume fraction is slightly smaller than the initial value. In general,he decrease is insignificant (3·10−5) and seems to be a consequencef the high compressive loads acting on the surface of the specimen,specially in the blank holding area.

The numerical simulations have been continued up to fracture.hree groups of nodes distributed along paths normal to the frac-ure section have been selected and the values of the major and the

inor strains at these nodes have been measured on the frame cor-esponding to the stage just before the onset of fracture. One of theode paths used to measure the principal strains is shown in Fig. 9(f)nd (g). Having the fracture section as shown in Fig. 9(f) facili-ates the selection of node paths for strain measurement. The firstode path passes through the middle of the fracture section (seeig. 9(g)), while the other two paths are located on each side of therst path with an approximate distance of 2 mm, being also parallelo each other. Fig. 12 shows the values of the strains associated tohe nodes of the middle (first) path at the onset of necking (Fig. 9(f)).he measured strains have been analyzed with the ARAMIS soft-are to determine the corresponding point on the forming limit

urve. The ARAMIS software uses Bragard’s method (Bragard et al.,972; D’Haeyer and Bragard, 1975) to find the limit strain. Thease shown in Fig. 12 corresponds to major and minor strains of.339 and 0.215, respectively. The same procedure has been used to

etermine the major and the minor strains for the other two paths.veraged values of the calculated strains for each set of three nodeaths have been finally used in the construction of the forming limiturve.

Fig. 11. Void volume distribution (a) before coalescence and (b) at th

Fig. 12. Determination of the major and minor strains at the onset of necking onthe notched specimen with w = 130 mm.

The forming limit curve obtained using the ARAMIS software ispresented in Fig. 13. This figure indicates that the results obtainedby numerical simulation using the GTN damage model are in goodagreement with the experimental data. The comparison becomeseven more favorable when confronted with the predictions of theMarciniak–Kuczynski (M–K) model and the modified maximumforce criterion (MMFC) – see Fig. 13. One may notice from thediagram that the quality of the GTN predictions is far better, espe-cially along the right branch of the forming limit curve, where bothM–K and MMFC models overestimate the formability of the metallicsheet.

The strain distribution corresponding to the fracture momenthas been also used for the determination of the forming limit curveby means of Bragard’s method. The results obtained in the case ofthe notched specimen with w=130 mm (see Fig. 9(g)) are illustratedin Fig. 14. By comparing the values presented in this figure withthe limit strains in Fig. 12, one may notice that the strain valuesare not far from those obtained by processing the frame associatedto the necking stage of the specimen. This is a consequence of thefact that after necking the strains evolve only in a small region ofthe specimen. Although there are differences in the strain values

corresponding to necking and fracture zones (see Figs. 12 and 14),the limit strains calculated by the ARAMIS software are almost thesame for both cases, because the middle points are excluded fromthe calculations. In this way, the forming limit curve constructed

e onset of necking on the notched specimen with w = 130 mm.

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A. Kami et al. / Journal of Materials Processing Technology 216 (2015) 472–483 481

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.2 -0. 1 0 0.1 0. 2 0.3 0. 4 0.5 0. 6

Maj

or S

trai

n

Minor Strain

Experimen tGTN wit h Hill'48M-KMMFC

uicodo

fFutfvo

Ft

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-0.1 -0.0 5 0 0.05 0. 1 0.1 5 0. 2 0.25 0.3

Maj

or S

trai

n

Minor Strain

FLC obta ined from necking fra meFLC obtained from fracture frame

Fig. 13. Comparison between the FLC obtained by different methods.

sing the fracture frame (see Fig. 15) is very similar to the form-ng limit curve obtained by processing the necking frame. In someases, because of the strain relaxation after fracture, the limit strainsbtained from the fracture frame are somewhat lower than thoseetermined on the basis of the necking frame (see the right branchf the forming limit curve in Fig. 15).

Although it is possible to detect the moment of necking andracture from the visual tracking of the deformation process (seeig. 9), studying the void volume fraction and strain values providesseful information about the deformation of the specimens. For

his purpose, as shown in Fig. 16, five elements on one side theracture section have been selected and the changes of the voidolume fraction and the logarithmic strain at the integration pointf these elements have been studied. These elements (C3D8R type)

ig. 14. Determination of the major and minor strains at the fracture moment inhe notched specimen with w = 130 mm.

Fig. 15. Comparison between the forming limit curves obtained at the onset ofnecking and after fracture.

have one integration point at the center of the element. To avoidunnecessary repetitions, the discussion will make reference only tothe specimens having the dimensions w = 30 mm and w = 90 mm.

Fig. 17 illustrates the strain evolution at the integration pointsshown in Fig. 16 during the Nakajima test in the notched specimenwith w = 30 mm. Fig. 17 indicates that the deformation remains uni-formly distributed until the time of 0.029 s but after this momentthe curves start to separate from each other. On the other hand,after reaching the time of 0.03 s, the strain evolution in the integra-tion point 5 cease while in the other integration points the straincontinues to increase. The increase in the integration point 1 has ahigher acceleration as compared with other integration points. Thisphenomenon is a sign of localization in the specimen. By reachingthe time of 0.0307 the fracture occurs in the specimen and, becauseof this, the strain values in all integration points remain constantup to end of the simulation. A small drop in the major strain val-ues after fracture time is due to the elastic recovery. As one maynotice from Fig. 17, the minor strain have negative values which isa characteristic of the uniaxial tensile test.

Fig. 18 shows the evolution of the void volume fraction in thenotched specimen with w = 30 mm. The trend of the curves in thisfigure is very similar to the trend of the major strain curve. So,a similar discussion can be made about the void evolution. By

Fig. 16. Five locations near the fracture section used to study the strain and voidevolution (w = 30 mm).

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482 A. Kami et al. / Journal of Materials Processing Technology 216 (2015) 472–483

Fig. 17. Strain evolution during the Nakajima test in a notched specimen withw = 30 mm.

0.00

0.03

0.06

0.09

0.12

0.15

0.02 0.02 2 0.02 4 0.02 6 0.02 8 0.0 3 0.032

Void

Vol

ume

Frac

tion

(SD

V2)

Time (s)

Integration poin t #1Integration poin t #2Integration poin t #3Integration poin t #4Integration poin t #5

Fw

astv

stFt

Fw

0.00

0.03

0.06

0.09

0.12

0.15

0.02 0.023 0.026 0.02 9 0.03 2 0.03 5 0.038

Void

Vol

ume

Frac

tion

(SD

V2)

Time (s)

Integration point #1Integration point #2Integration point #3Integration poin t #4Integration poin t #5

ig. 18. Evolution of voids during the Nakajima test in a notched specimen with = 30 mm.

pproaching the fracture time (0.0307), the void volume fractionuffers a sudden increase in the integration point 1. This accelera-ion is the result of the void coalescence. After the fracture, the voidolume fraction remains constant in all integration points.

The strain evolution in the notched specimen with w = 90 mm ishown in Fig. 19. The behavior of the curves in this figure is similar

o that presented in Fig. 17. However, there are some differences.irst, a larger deformation occurs in the specimen with w = 90 mmhan in the case of the specimen with w = 30 mm. Second, the values

ig. 19. Strain evolution during the Nakajima test in a notched specimen with = 90 mm.

Fig. 20. Evolution of voids during the Nakajima test in a notched specimen withw = 90 mm.

of the minor strain in Fig. 19 are positive i.e. there is biaxial stretch-ing in this case. Finally, the difference between the curves corre-sponding to the minor strain is almost unnoticeable in Fig. 19. Thisindicates a uniform stretching along the second principal direction.

The evolution of voids corresponding to the specimen withw = 90 mm is shown Fig. 20. In this figure, the sudden increase inthe void volume fraction at integration point 1 indicates the onsetof necking. Furthermore, the continuation of the simulation with-out any change in the void volume fraction in all curves indicatesthe fracture occurrence in the specimen.

6. Conclusions

In this research, the anisotropic GTN damage model with Hill’48quadratic yield criterion was used to construct the forming limitcurve of metallic sheets. The following results were obtained:

(1) The response surface methodology with the numerical sim-ulation of the uniaxial tensile test provided the GTN modelparameters in a very quick and efficient manner. Using thisprocedure of identification, the values of f0, fN, fC and fF werecalculated as 0.00015, 0.05, 0.05 and 0.15.

(2) The forming limit curve of the AA6016-T4 aluminum alloy wascomputed by simulating a sequence of Nakajima tests with thefinite element code Abaqus/Explicit and a VUMAT implemen-tation of the GTN damage model. The numerical forming limitcurve showed a good agreement with the experimental results.In general, the limit strains predicted by the GTN model weremore accurate than those calculated with the M–K and MMFCmodels.

(3) The numerical forming limit curve of the AA6016-T4 aluminumalloy was also determined by processing the distribution of theprincipal strains associated to the fracture stage of the Nakajimatest. The results showed that this curve is almost coincidentwith the forming limit curve obtained using the onset of neck-ing.

(4) The analysis of the plastic strain and void volume fraction dis-tributions showed that they are useful tools for recognizing ofthe onset of necking and the final fracture. So, in the numericaldetermination of the forming limit diagrams, these curves canbe used to make the decision about the moment and locationof the limit strain measurement.

References

ABAQUS/Standard user’s manual, 2011. Version 6.11. Hibbitt, Karlsson & SorensenInc.

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rocess

A

A

A

B

B

B

B

B

B

B

C

C

C

C

D

G

H

H

H

A. Kami et al. / Journal of Materials P

bbasi, M., Shafaat, M.A., Ketabchi, M., Haghshenas, D.F., Abbasi, M., 2012a. Applica-tion of the GTN model to predict the forming limit diagram of IF-Steel. J. Mater.Process. Technol. 26, 345–352.

bbasi, M., Bagheri, B., Ketabchi, M., Haghshenas, D.F., 2012b. Application ofresponse surface methodology to drive GTN model parameters and determinethe FLD of tailor welded blank. Comput. Mater. Sci. 53, 368–376.

ravas, N., 1987. On the numerical integration of a class of pressure-dependentplasticity models. Int. J. Numer. Methods Eng. 24, 1395–1416.

anabic, D., 2010. Formability of sheet metals. In: Banabic, D. (Ed.), Sheet MetalForming Processes. Springer, Heidelberg, pp. 141–211.

arlat, F., Lian, K., 1989. Plastic behavior and stretchability of sheet metals. Part I: ayield function for orthotropic sheets under plane stress conditions. Int. J. Plast.5, 51–66.

enseddiq, N., Imad, A., 2008. A ductile fracture analysis using a local damage model.Int. J. Press. Vessel. Pip. 85, 219–227.

ragard, A., Baret, J.C., Bonnarens, H., 1972. A simplified method to determine theFLD onset of localized necking. Rapp. Cent. Rech. Metall. 33, 53–63.

runet, M., Morestin, F., Mguil, S., 1996. The prediction of necking and failure in 3D. Sheet forming analysis using damage variable, analytical and experimentalstudies of necking in sheet metal forming processes. J. Phys. IV Fr. 06, C6-473-C6-482.

runet, M., Mguil, S., Morestin, F., 1998. Analytical and experimental studies ofnecking in sheet metal forming processes. J. Mater. Process. Technol. 80–81,40–46.

runet, M., Morestin, F., 2001. Experimental and analytical necking studies ofanisotropic sheet metals. J. Mater. Process. Technol. 112, 214–226.

hen, Z., Dong, X., 2008. Comparison of GTN damage models for sheet metal forming.J. Shanghai Jiaotong Univ. (Sci.) 13, 739–743.

hen, Z., Dong, X., 2009. The GTN damage model based on Hill’48 anisotropic yieldcriterion and its application in sheet metal forming. Comput. Mater. Sci. 44,1013–1021.

hen, Z., Butcher, C., 2013. Micromechanics Modelling of Ductile Fracture. Springer,Dordrecht.

hhibber, R., Arora, N., Gupta, S.R., Dutta, B.K., 2008. Estimation of Gursonmaterial parameters in bimetallic weldments for the nuclear reactor heattransport piping system. Proc. Inst. Mech. Eng. C: J. Mech. Eng. Sci. 222,2331–2349.

’Haeyer, R., Bragard, A., 1975. Determination of the limiting strains at the onset ofnecking. Rap. Cent. Rech. Metall. 42, 33–35.

urson, A.L., 1977. Continuum theory of ductile rupture by void nucleation andgrowth Part I – yield criteria and flow rules for porous ductile media. J. Eng.Mater. Technol. 99, 2–15.

e, M., Li, F., Wang, Z., 2011. Forming limit stress diagram prediction of aluminumalloy 5052 based on GTN model parameters determined by in situ tensile test.

Chin. J. Aeronaut. 24, 378–386.

ill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. R. Soc.Lond. Proc. A 193, 281–297.

ill, R., 1952. On discontinuous plastic states, with special reference to localizednecking in thin sheets. J. Mech. Phys. Solids 1, 19–30.

ing Technology 216 (2015) 472–483 483

Hill, R., 1979. Theoretical plasticity of textured aggregates. Math. Proc. Camb. Philos.85, 179–191.

International Standard ISO 12004-2, 2008. Metallic Materials-sheet and StripDetermination of Forming Limit Curves. Part 2: Determination of FormingLimit Curves in the Laboratory. International Organization for Standardization,Geneva, Switzerland.

Kami, A., Mollaei Dariani, B., Sadough Vanini, A., Comsa, D.S., Banabic, D., 2014. Appli-cation of a GTN damage model to predict the fracture of metallic sheets subjectedto deep-drawing. Proc. Rom. Acad. Ser. A 15, 300–309.

Keeler, S.P., Backhofen, W.A., 1963. Plastic instability and fracture in sheet stretchedover rigid punches. Trans. Am. Soc. Met. 56, 25–48.

Li, B., Nye, T.J., Wu, P.D., 2010. Predicting the forming limit diagram of AA 5182-O. J.Strain Anal. Eng. 45, 255–273.

Liao, K.C., Pan, J., Tang, S.C., 1997. Approximate yield criteria for anisotropic porousductile sheet metals. Mech. Mater. 26, 213–226.

Liu, J., Liu, W., Xue, W., 2013. Forming limit diagram prediction ofAA5052/polyethylene/AA5052 sandwich sheets. Mater. Des. 46, 112–120.

Liu, J.G., Xue, W., 2013. Formability of AA5052/polyethylene/AA5052 sandwichsheets. Trans. Nonferr. Met. Soc. 23, 964–969.

Marciniak, Z., Kuczynski, K., 1967. Limit strains in processes of stretch-forming sheetmetal. Int. J. Mech. Sci. 9, 609–620.

Needleman, A., Tvergaard, V., 1984. An analysis of ductile rupture in notched bars.J. Mech. Phys. Solids 32, 461–490.

Paraianu, L., Banabic, D., 2013. Characterization of the plastic behaviour of AA6016-T4 aluminium alloy. Adv. Eng. Forum 8–9, 293–300.

Parsa, M.H., Ettehad, M., Matin, P.H., 2013. Forming limit diagram determinationof Al 3105 sheets and Al 3105/polypropylene/Al 3105 sandwich sheets usingnumerical calculations and experimental investigations. J. Eng. Mater. Technol.135, 031003.

Ramazani, A., Abbasi, M., Prahl, U., Bleck, W., 2012. Failure analysis of DP600 steelduring the cross-die test. Comput. Mater. Sci. 64, 101–105.

Swift, H.W., 1952. Plastic instability under plane stress. J. Mech. Phys. Solids 1, 1–18.Thuillier, S., Le Maoût, N., Manach, P.Y., 2011. Influence of ductile damage on the

bending behaviour of aluminium alloy thin sheets. Mater. Des. 32, 2049–2057.Tvergaard, V., 1981. Influence of voids on shear band instabilities under plane strain

conditions. Int. J. Fract. 17, 389–407.Tvergaard, V., 1982. On localization in ductile materials containing spherical voids.

Int. J. Fract. 18, 237–252.Tvergaard, V., Needleman, A., 1984. Analysis of the cup-cone fracture in a round

tensile bar. Acta Metall. 32, 157–169.Uthaisangsuk, V., Prahl, U., Bleck, W., 2009. Characterisation of formability behaviour

of multiphase steels by micromechanical modeling. Int. J. Fract. 157, 55–69.Wang, D.A., Pan, J., Liu, S.D., 2004. An anisotropic Gurson yield criterion for porous

ductile sheet metals with planar anisotropy. Int. J. Damage Mech. 13, 7–33.

Zhang, Z.L., 1995. On the accuracies of numerical integration algorithms for Gurson-

based pressure-dependent elastoplastic constitutive models. Comput. MethodsAppl. Mech. Eng. 121, 15–28.

Zhang, Z.L., Thaulow, C., Odegard, J., 2000. A complete Gurson model approach forductile fracture. Eng. Fract. Mech. 67, 155–168.