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Journal of Materials Processing Technology 172 (2006) 451460

Numerical modelling and experimental validationof steel deep drawing processesPart I. Material characterization

Claudio Garca a, Diego Celentano a,, Fernando Flores b, Jean-Philippe Ponthot c

a Departamento de Ingeniera Mec anica, Universidad de Santiago de Chile, Avda. Bdo. OHiggins 3363, Santiago de Chile, ChilebDepartamento de Estructuras, Universidad Nacional de Cordoba, Casilla de Correo 916, 5000 Cordoba, Argentina

cLTAS-Milieux Continus et Thermomecanique, Universite de Liege, 1 Chemin des Chevreuils, B-4000 Liege-1, Belgium

Received 9 August 2004; accepted 8 November 2005

Abstract

This work presents an experimental characterization of the mechanical behaviour of the EK4 deep drawing steel. The experimental procedure

encompasses spectrometry, metalography, tension testing and hardness measurements. Special attention is devoted to the derivation of the elastic

and plastic parameters involved in the assumed constitutive model based on the anisotropic Hill-48 yield criterion. The simulation of the deformation

process during the whole tensile test is subsequently performed with the aim of assessing the adequateness of the proposed methodology. It should

be mentioned that the material parameters obtained with this procedure are the basic data for the modelling and experimental validation of different

deep drawing applications presented in Part II of this work.

2005 Elsevier B.V. All rights reserved.

Keywords: Forming process; Deep drawing; Material characterization

1. Introduction

A relevant aspect that has to be considered in the analysis

of any forming process is the knowledge of the mechanical

behaviour of the materials involved in it. This is particularly

difficult in metal sheet deep drawing operations due to the dif-

ferent complex mechanisms (finite strain plasticity, hardening,

damage, viscous effects, etc.) that develop in the blank material

during its deformation. Therefore, an adequate material charac-

terization is a crucial task that needs to be carried out in order

to design, control and/or eventually optimize the production of

a prescribed component.

Themanufacturing of steel sheets by means of press-working

represents a large industry that produces numerous parts for a

wide variety of applications. Specifically, the steelcommercially

known as EK4is nowadays extensively used in many deep draw-

ing operations (see, for instance [1]). Some particular features

Corresponding author.

E-mail addresses: [email protected] (C. Garca),

[email protected] (D. Celentano), [email protected] (F. Flores),

[email protected] (J.-P. Ponthot).

of it are specified in [2]: maximum admissible values for car-bon and nitrogen contents, ultimate tensile strength (UTS), yield

strength and hardness and, in addition, a minimum bound for

the fracture elongation. However, it is a well-known fact that

the final properties of the material under the EK4 denomination

are not a world-wide standard since rolled sheets provided by

various steel companies may behave completely different when

being deformed. This drawback definitely restricts the flexibility

of the process since usually expensive trial and error calibration

tests are required to tune the appropriate operation parameters.

The aim of this work is to present an experimental charac-

terization of the mechanical behaviour of a specific EK4 deep

drawing steel broadly employed in the industry [1]. Spectrom-

etry, qualitative metalographic observations, tensile testing and

hardness measurements are undertaken to this end. In particular,

the tensile tests have been conducted according to the standard

specifications [36] using specimens oriented at 0, 45 and 90

with respect to the sheet rolling direction. The experimentally

derived elastic and plastic parameters are the basic data for the

constitutive modeladopted to describe the material response dur-

ing the deformation process. A hyperelastic elastoplastic law

written in terms of Hencky stress and strain measures is con-

sidered [7]. Strain hardening and normal anisotropy (along the

0924-0136/$ see front matter 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.jmatprotec.2005.11.015

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452 C. Garca et al. / Journal of Materials Processing Technology 172 (2006) 451460

sheet thickness) effects are both considered via the Hill-48 asso-

ciate plasticity model [8]. Afterwards, this model is used in the

simulation of the material behaviour during the whole test via

a large strain shell formulation discretized within the finite ele-

ment context. Finally, the numerical results are validated with

the experimental measurements.

2. Material constitutive model

The constitutive model adopted in this work to describe the

sheet material behaviour is briefly presented below. More details

about this model can be found in [7], and references therein.

2.1. Strain and stress measures

The so-called Hencky deformation tensor is chosen here

as the main kinematic state variable since it is an objective

(Lagrangian) strain measure and constitutes, in addition, a natu-

ral extension of the unidimensional logarithmic strain. Consid-

ering the sheet as a thin shell with a curvilinear local coordinatesystem over its middle surface, this deformation is defined as:

e = LT ln()L (1)

where and L, respectively, include the eigenvalues and the

associated eigenvectors of the right stretch tensor U such that

U2 =FTF, F being the deformation gradient tensor (T is the

transpose symbol). Note that the strain related to the sheet thick-

ness variation is simply e3 =ln(t/t0), where t/t0 is the thickness

ratio between the deformed and initial configurations. The other

two eigenvalues define the principal (in-plane) stretches in the

shell surface. Consistently, the Hencky stress tensor T, whose

expression is given in Section 2.2, is used in this context.

2.2. Stressstrain law

A linear hyperelastic stressstrain law expressed in terms

of Hencky measures is considered to describe the material

behaviour:

T = C : (e ep) (2)

where C is the isotropic elastic constitutive tensor correspond-

ing to the plane stress condition and ep is the plastic contribution

ofe. As can be seen, an additive elasticplastic decomposition

of the Hencky strain is assumed based in the fact that the elas-tic deformations are usually small in many steel deep drawing

processes. The evolution law ofep is given below.

2.3. Plastic behaviour

It has been long recognized that the use of the associate rate-

independent plasticity theory including the classical anisotropic

Hill-48 yield criterion is a useful framework to simulate sheet

responses subjected to deep drawing operations [8]. Although

the further assumption of planar isotropy (or normal anisotropy)

appreciably simplifies the formulation, this situation was found

to be approximately fulfilled in many applications (see e.g.

[1012]). In this context, the yield function written in terms of

the Cauchy stress tensor (that can be derived from T through

a proper kinematic transformation; see Section 4.1) under the

plane stress condition (i.e. 33 = 13 = 23 = 0) reads:

F= 211 + 222

2R

1+ R1122 +

2(1+ 2R)

1+ R212 (C

p)2 = 0

(3)

where R is the average Lankfords coefficient accounting for the

non-isotropic plastic behaviour along the shell thickness and Cp

is the isotropic hardening function given by:

Cp = Ap(ep0 + e

p)np

(4)

where Ap and np are the hardening parameters, ep is the effec-tive plastic deformation and e

p0 is an assumed initial value such

that y = Ap(e

p0)

np

, with y being the yield strength defining the

material initial elastic domain. The effective plastic deformation

rate is computed as ep=

2/3 ep : ep with the following evo-

lution equation for ep:

ep=

F

(5)

where is the rate (or increment in this framework) of the plas-

tic consistency parameter calculated according to the standard

procedures of the plasticity theory [13].

The yield strength together with the hardening parametersand average Lankfords coefficient are all derived, as described

in Section 3, from the experimental data measured during tensile

tests applied to the EK4 steel.

3. Experimental procedure

Theexperimentalprocedure adoptedin thiswork to characterize themechan-

ical behaviour of the EK4 steel consisted in the following steps.

3.1. Chemical composition

This routine taskaimed at checking the adequate composition of the selected

materialis carried outby means of an optical spectrometer. Theaveragechemical

compositionof theEK4 steelis shown in Table1. Itis seen that thecarboncontent

present in this materialis less than 0.08%whichis themaximum bound typically

required for this kind of deep drawing steels [2].

3.2. Metalography

Metalographic observations in the sheet plane and thickness for the unde-

formed material are, respectively, shown in Figs. 1 and 2. A ferritic matrix with

Table 1

Average chemical composition of EK4 steel (% in weight)

C (%) 0.0457 Mn (%) 0.1936 P (%) 0.0093 S (%) 0.0045 Cr (%) 0.0375

Ni (%) 0.0114 Al (%) 0.0049 Cu (%) 0.0302 Co (%) 0.0080 Fe (%) 99.58

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C. Garca et al. / Journal of Materials Processing Technology 172 (2006) 451460 453

Fig. 1. Metalography of EK4 steel in the sheet plane.

Fig. 2. Metalography of EK4 steel in the sheet thickness.

uniformly distributed equiaxial grains can be appreciated in both planes pre-

cluding in this way the presence of any grain alignment.

3.3. Tensile test

The geometric configuration of the EK4 steel sheet sample to be tested

according to the standard specifications [3,4] is sketched in Fig. 3. The distance

between thetwo markers denotes theinitial extensometer lengthtaken as 50mm

in this case. Therespectiveinitial values forthe width andthickness in thework-

ing zone are w0 = 12.5 mm and t0 = 0.6 mm. The specimens have been cut along

three differentorientations (0,45 and90) with respect to therolling directionof

the sheet during its manufacturing process. A nearly gradual reduction in width

is considered in order to trigger the necking development that has to take place

approximately at the middle of the extensometer length. This tapered profile

fits the usual standards since the difference between the adopted maximum and

minimum width values existing in the extensometer length is lower than 1%

[3,4].

The average engineering stressstrain curves obtained with 18 samples (six

for each orientation) considering a load cell of 2.5 mm/min are plotted in Fig. 4.

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Fig. 3. Tensile specimen (dimensions (mm)).

Fig. 4. Average engineering stressstrain curves of EK4 steel for specimens

oriented at 0, 45 and 90 with respect to the rolling direction.

The material uniformity was reflected in the remarkably low dispersion found

in the measurements for each orientation. As usual, the engineering stress is

defined as P/A0, where P is the axial load and A0 = w0t0 is the initial transversal

area, whilethe engineering strainor elongation is computed as (LL0)/L0, with

L and L0 being the current and initial extensometer lengths, respectively. At the

beginning of the deformation process, the material behaves elastically. Once

the yield strength is reached, a plastic deformation without hardening develops

within theelongationrangeof 0.481%.Thisresponse, usuallyknownas Luders

band formation [14], is a typical phenomenon observed for low carbon steels

whose negative effect on the drawing properties has been limited in this case by

means of a cold rolling applied to the material during its manufacturing process.

Then, the plastic hardening starts and the load increases up to a maximum value

Table 2

Parameters obtained from the tensile test applied to EK4 steel samples

Youngs

modulus

(GPa)

Yield

strength

(MPa)

Maximum

load (kN)

UTS

(MPa)

Fracture

elongation

(%)

Average 212 183 2.168 285 46.7

Range () 6.80 2.80 0.030 3.00 4.1

Table 3

Average original and final dimensions of EK4 steel tensile samples

Original Final Ratio (final/original)

Width (mm) 12.5 7.67 0.61

Thickness (mm) 0.600 0.397 0.66

Extensometer length (mm) 50.0 72.3 1.44

Transversal area (mm2) 7.50 3.05 0.41

that takes place in a wide elongation plateau (from 25 to 35%, approximately).

For higher levels of elongation, as it is well-known, the load decreases since

the effect of the reduction of the transversal area at the necking zone is stronger

than that of the hardening mechanism. As can be observed in Fig. 5, a highly

localised(ductile) necking develops until the rupture of the specimen that occurs

at a elongation of about 47%. The average experimentally measured values

(considering the three orientations) for the Youngs modulus, yield strength,

maximum load, ultimate tensile strength (UTS) and elongation at the fracture

stage are summarized in Table 2. It is seen that these values fit the admissible

bounds usually considered for the EK4 steel [2], i.e. maximum yield strength of

210 MPa, UTS ranging from 270 to 350 MPa and minimum fracture elongation

of 38%. Moreover, some average original and final characteristic dimensions of

the sample are presented in Table 3.

The experimental characterization of the plastic behaviour that is subse-

quently accomplished is mainly aimed at deriving the hardening parameters Ap

and np and the average Lankfords coefficient R involved in the constitutive

model described above. Details of the methodology followed to obtain such

constants are given below.

Since the engineering stressstrain relationship cannot provide a proper

description of the hardening material response during the whole tension pro-

cess, an alternative stressstrain curve defined in terms of an equivalent stress

eq andan equivalentstrain eq isusedtothisend [5]. Accordingto theprocedure

Fig. 5. Geometric configurations of the sample: (a) necking zone and (b) rupture.

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Fig. 6. Mean equivalent stress vs. equivalent deformation.

originally proposedby Bridgman[15], these variables can be, respectively, com-

puted as eq =fBP/A and eq = eq/E+ p, wherefB(p) 1 is an assumed known

correction factor (i.e. the conditions fB =1 and fB < 1, respectively, account

for the uniaxial and triaxial stress distributions that occur before and after thenecking formation) applied to the mean true axial stress P/A, A is the current

transversal area at the necking zone (A = wt, where w and t are the current

width and thickness, respectively), Eis the Youngs modulus and p =ln(A0/A)

is the true (logarithmic) deformation. As can be seen, w and tare the additional

variables to be measured. It should be mentioned that the correction factor fBhas been exclusively obtained for isotropic materials tested in specimens that

only develop diffuse necking (e.g. either cylindrical or strip samples with a

relativelyhigh t0/w0 ratio [16]). Therefore,this methodologycannot be straight-

forwardly applied to the present case. However, a simplified procedure can be

adopted. It basically consists in plotting a eq eq curve with the available

experimental data within the deformation range that exhibits uniaxial stress dis-

tribution (035% for this material), i.e. prior to the onset of necking where the

condition fB = 1 holds. Although this simpler approach is not strictly valid for

the whole deformation range up to rupture, it ensures a reasonably accurate

material characterization. Fig. 6 shows the experimental data (correspondingto 10 specimens) for the eq eq relationship together with the potential

correlation derived from them via the application of a classical least-squares

technique to Eq. (4). The resulting hardening parameters are: Ap = 566 MPa and

np = 0.345.

The measurement of the average Lankfords coefficient accounting for the

normal anisotropy of the sheet has been performed according to the standard

specifications [6]. The recommended expression to compute this parameter,

defined as the ratio between the width and thickness deformations for sheets

specimens oriented at 0, 45 and 90 with respect to the rolling direction, is

R0,45,90 =ln(w/w0)/ln(l0w0/lw) which, as can be seen, assumes the incompress-

ible nature of the plastic flow. In order to minimize the experimental errors

associated with the measurement of the sample dimensions, the adequate use of

this equation is restricted to the elongation range corresponding to the develop-

ment of uniaxial stress with hardening, i.e. around 1025% in this case. The

average Lankfords coefficient is given by R = 1/4(R0 + 2R45 +R90). Theassumption of planar isotropy can be assessed by the so-called earing index

defined as R = 1/4(R0 2R45 +R90). The Lankfords coefficient together with

the earing index for the EK4 steel are summarized in Table 4. As can be

appreciated, this steel presents a relatively high value of R that makes it appro-

priate for deep drawing applications. Moreover, the small value of R found

for this steel indicates that it exhibits a low planar anisotropy and, therefore,

Table 4

Lankfords coefficients and earing index of EK4 steel

R0 R45 R90 R R

Average 1.62 1.50 2.00 1.65 0.16

Range () 0.13 0.06 0.12

Fig.7. Engineering stressstraincurves for differentload cellvelocities V(spec-

imens oriented at 0 with respect to the rolling direction).

this fact justifies the use of a simplified yield crietrion as that presented in

Section 2.3.

The influence of strain rate effects on the mechanical behaviour of the EK4steel has been evaluated by measuring the variation of both the yield strength

and UTS when performing tensile test at different load cell velocities. The cor-

responding results are shown in Fig. 7 and Table 5. Although a noticeable strain

rate dependency can be observed on the yield strength, the maximum difference

in the plastic responses (given for instance by the UTS values) is around 10%.

This relatively small difference allows the application of the hardening parame-

ters obtainedabove under lowload cell velocities to other deformation stiuations

undergoing higher strain rates (e.g. a reasonable bound within which many real

applications fit is ep

< 0.1 s1 [1012,14]).

A photographic tracking of the necking formation that develops at high

levels of tensile elongation is sketched in Fig. 8 with the sake of qualitatively

characterizing the fracture of this material. The necking evolution starts with

two symmetric diffuse shear bands that progressively become more localised

as the elongation increases. The severe thickness reduction that mainly takes

place at the center of the sample causes a sudden initiation of a crack thatimmediately propagates in an unstable form along only one shear band that can

be visually detected as a narrow groove inclined around 68 with respect to the

axial direction of the specimen. It is seen that the grid drawn on the sample

distorts until the very instant of crack propagation since from that time onwards

thefurther deformation is completely confined to theshearbandwhere, as noted

in [16], the plane stress condition is no longer valid.

3.4. Hardness test

The objective of the hardness measurements carried out in this work is two-

fold: first, to check thesuitableness of thehardnessvalue foundfor theEK4 steel

considered here and, second, to verify the well-known relationship stating the

direct proportion between hardnessvaluesand strain hardening effects expressed

by the variable Cp (see Eq. (4)). The hardness Rockwell unit in the scale 30T

(HR30T) is used in the measurements done to original and deformed samples.

Table6 summarizes theseresults. It is appreciated thatthe averagehardness value

found for the original sample fits the standard requirements (according to [2],

the corresponding maximum value is 50). Moreover, higher hardness values are

Table 5

Average parameters of EK4 steel obtained at different load cell velocities

Load cell velocity (mm/min) Yield strength (MPa) UTS (MPa)

2.5 183 285

100 220 309

250 232 312

300 236 314

400 243 316

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Fig. 8. Necking formation evolution for different levels of elongation (magnification: 30).

Table 6

Hardness values of EK4 steel measured in the HR30T scale

Original sample Deformed sample

Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Zone 6 Zone 7

Average 38.5 39.3 53.8 52.5 48.4 51.1 51.7 40.4

Range () 1.3 0.7 1.8 2.9 3.6 3.2 2.8 3.6

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observedin those zones thatexperienced larger strain hardening effects ratifying

the validity of the above mentioned relationship.

4. Numerical simulation of the tensile test

The performance of the adopted constitutive model including

the material parameters obtained via the experimental character-

ization describedabove areboth assessed in thesimulation of the

mechanical response during the tensile test of EK4 steel sheet

samples. Some relevant features of the discretized formulation

used in the analysis are concisely given below. A full description

of it can be found in [7,9].

4.1. Principle of virtual work

The Hencky strainstress pair eTdescribed in Section 2 can

be easily transformed to the usual Green-Lagrange strain E and

second PiolaKirchhoff stress S. These last conjugated tensors

present, as it is well-known, some practical advantages when

solving the equilibrium equation through the principle of virtualwork whose expression is [17]:

0

ETSd0 =

0

uTB0 d0 +

0

uTt0 d0 (6)

where 0 is the material configuration of a body subjected to

a body force B0 and a traction force t0 on its boundary 0 and

u is the displacement vector field. The mentioned strainstress

transformations are, respectively, given by [7]:

E =1

2L

T(2 1)L (7)

and

S= LTSLL (8)

with 1 being the the unity tensor and

[SL] =1

2

L

TTL

[SL] =2ln(/)

(2 2)

L

TTL

(9)

Furthermore, the Cauchy stress tensor can be computed as

[17]:

=1

det(F)FSFT

4.2. Discretized formulation

A simple rotation-free thin (KirchhoffLove) shell triangle

is used in the simulation [7]. The basic idea of this discretized

formulation is to combine the standard finite element interpo-

lation with finite volume concepts which allows to express the

curvatures over a control domain in terms of the displacement

gradients along the domain edges. These gradients are in turn

written as a function of the deflections of the nodes belonging to

an element patch surrounding the control domain and this leads

Fig. 9. Basic shell triangle (BST): element patch definition.

Table 7

Material properties of EK4 steel used in the simulations

Youngs modulus (E) 212 GPa

Poissons ratio (v) 0.32

Yield strength (Cth) 183 MPa

Hardening parameter (Ap) 566 MPa

Hardening exponent (np) 0.345

Average Lankfords coefficient (R) 1.65

Fig. 10. Finite element mesh used in the simulation (mesh composed of 511

shell elements and 292 nodes).

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to a relationship between curvatures and nodal displacements.

This three-node shell triangle termed BST (for basic shell trian-

gle; see Fig. 9) has showed an excellent behaviour for both linear

and non-linear analysis of shell structures. Moreover, a very

good performance was observed when modelling large strain

plasticity responses as those typically found in sheet stamping

problems by using, in particular, the hyperelastic material con-

stitutive model described in Section 2. Different aspects related

to the accuracy and efficiency of this formulation are reported

in [7,9].

4.3. Results and discussion

Based on the experimental characterization of the EK4 steel

described above, the main objective of the present analysis is to

validate the predictions of the material constitutive model with

the experimental measurements obtained in the tensile test of

sheet specimens.

The material properties considered in the numerical analysis

are shown in Table 7. The spatially non-uniform finite element

mesh shown in Fig. 10 has been chosen in order to describe

Fig. 11. Comparison between experimental and numerical results. (a) Engineering stressstrain curve. Results at the section undergoing extreme necking: (b) mean

true axial stress vs. true deformation; (c) load vs. true deformation; (d) ratio of current to initial width (mean values along thickness); and (e) ratio of current to initial

thickness (mean values along width).

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correctly the large stress and deformation gradients expected in

the necking zone. Due to symmetry, only one-fourth of the spec-

imen is studied. Fitting the specifications of Fig. 3, the neck in

the middle of the specimen is triggered by assuming a very small

linear width variation along its length. The axial displacement

imposed at the top boundary is denoted as Uand it is increased

up to a value that corresponds to the fracture elongation (see

Table 2). Moreover, four layers have been considered to per-

form the numerical integration of the constitutive law along the

sheet thickness [9].

Fig. 11 shows a comparison between experimental and

numerical results for the engineering stressstrain relationship

and some variables at the section undergoing extreme necking:

the mean true axial stress and load both against the true (loga-

rithmic) deformation and, in addition, the width and thickness

ratios versus the elongation. These curves include experimen-

tal measurements corresponding to specimens oriented at 0, 45

and 90 with respect to the rolling direction. An overall good

agreement between the numerical predictions and the average

experimental values can be observed in these curves. The rel-atively small differences can be considered as approximately

bounded within the experimental uncertainty range.

The experimentalnumerical discrepancies appearing in the

engineering stressstrain curve can be mainly attributed to the

inaccuracy of the potential correlation at the beginning of the

plastic region. The experimentally measured load decreases

from an elongation of 35% or, equivalently, to a logarithmic

deformation of 0.38 onwards. The corresponding deformations

provided by the simulation are very similar to these values. As

already commented on in Section 3.3, a geometrical instability

(instead of a constitutive one) caused by the necking formation

occurs since the mean true axial stress continues increasing until

the fracture stage. Moreover, note that the well-known simplified

condition [14] stating that the related logarithmic deformation

at the point of maximum load has to be equal to the hardening

exponent is approximately verified in this case. At high levels

of deformation, the regions of the specimen outside the necking

zone are being elastically unloaded.

The numerical predictions for the mean w/w0 and t/t0 ratios

match the normal anisotropy condition quantified by the Lank-

fords coefficient (see Section 3.3). This is apparent at the begin-

ning of the test, where a uniaxial stress is achieved, since the

reduction in t/t0 is, as expected, lower than that of w/w0. In this

case, the following relationship (ww0)/w0 R(t t0)/t0 is

approximately fulfilled. Once the neck is formed, a reverse trendis found, i.e. a stronger variation in t/t0 is observed such that

(ww0)/w0 < R(t t0)/t0.

Although the hardening parametershave beenexperimentally

derived within an elongation range upperly bounded by the point

for which the load starts decreasing, the reasonably good fitting

Fig. 12. Effective plastic deformation contours for an elongation of 43.2%: (a) whole sample and (b) detail at the necking zone.

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of thenumerical results to theexperimental ones shownin Fig.11

confirms the validity of the material characterization carried out

in Section 2 even beyond such point.

The effective plastic deformation contours at a deformation

stage close to the rupture of the specimen are sketched in Fig. 12.

A non-uniform distribution is clearly obtained due to the com-

plex deformation pattern of the neck. A diffuse shear band

development canalso be appreciated where its slope nearly coin-

cides with that observed in the experiments (see Section 3.3). It

should be mentioned, however, that the constitutive model used

in the simulation cannot cope with the modelling of damage and

crack propagation effects.

5. Conclusions

A detailed experimental characterization of the EK4 deep

drawing steel has been presented. In particular, the low disper-

sion found in the experimental data measured during the tensile

test allowed a consistent derivation of some material parame-

ters which, in turn, are involved in a constitutive model that wassubsequently used to simulate the material response during such

deformation process. The experimental validation of the numer-

ical results has been satisfactory. Hence, the procedure followed

to characterize the mechanical behaviour of this material has

proved to be a basic and useful tool to undertake a realistic

experimentally-based modelling of complex sheet forming pro-

cesses.

Acknowledgements

The supports provided by the Chilean Council of Research

and Technology CONICYT (FONDECYT Projects No.

1020026 and 7020026) and the Department of Technological

and Scientific Research at the University of Santiago de Chile

(DICYT-USACH) are gratefully acknowledged. The authors

wish to express their appreciation to the Aeronautical Techni-

cal Academy at Santiago de Chile and the Compa na Tecno-

Industrial (CTI) for the provision of experimental facilities.

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