Engineering, 2010, 2, 471-476
doi:10.4236/eng.2010.27062 Published Online July 2010 (
Copyright © 2010 SciRes. ENG
An Inverse Analysis of the Erichsen Test Applied for the
Automatic Identification of Sheet Materials Behavior
Adinel Gavrus1, Mihaela Banu2, Eric Ragneau1, Catalina Maier2
1Civil and Mechanical Engineering Laboratory, INSA de RENNES, UEB, Rennes, France
2Department of Mechanical Engineering, University Dunarea de Jos”, UGAL, Galati, Romania
Received March 16, 2010; revised March 16, 2010; accepted May 18, 2010
Among the technological tests, the Erichsen drawing test gives a more appropriate material behavior, near
the limit of the real manufactured process. In this paper an inverse finite element analysis of the Erichsen test
is proposed. The new idea is to use a numerical simulation of the experimental test for the rheological identi-
fication of the constitutive equations available for sheet metals alloys. The inverse analysis is based on a ro-
bust optimization algorithm and uses simultaneously the experimental test data and the corresponding nu-
merical one. A numerical inverse analysis software named OPTPAR was developed and improved for an
automatically coupling with a commercial finite element code charged to simulate the experimental test. Re-
sults obtained for a virtual steel alloy will be analyzed numerically in order to validate the finite element
model and the identification method. An application to an AA5182 aluminum alloy and a DC03 steel alloy
will be presented.
Keywords: Drawing Process, Finite Element Model, Inverse Analysis, Erichsen Test
1. Introduction
Nowadays, the occurrence of the new metallic materials
with improved properties destined to the automotive
parts generates the computation developments in order to
a better identification of their mechanical and techno-
logical characteristics. Sheet metal forming is a wide
used process in the car panel manufacturing where the
formability of the metal sheet has a great importance in
prediction of the capacity of the material to be drawn
without failure [1]. The occurrence of the new sheet
metals like high strength steels, aluminum alloys or nano
structured materials with the main target of improvement
of the mechanical properties, exhibits new technological
challenges. In this context, rapid algorithms of identifi-
cation of the mechanical and technological behavior are
needed to gain time in the competition of producing the
parts [2]. Unfortunately the whole complexity of the
process requires using numerical modeling to optimize
the forming conditions [3-6]. These numerical simula-
tions require available constitutive equations and accu-
rate rheological parameters values, which characterize
the sheet material behavior. In order to identify the
rheological behavior of the material, the most used test is
the uniaxial tensile one and the parameters values are
obtained approximately by an analytical analysis using
two important hypothesis: the homogeneity of the strain
and the small influence of the necking area [7]. In this
case the stress-strain curve is available only for small
plastic deformations. Moreover, the problem is that the
deformation history during the drawing process is dif-
ferent as compared to the uniaxial tensile one, and the
constitutive equation can not be available for more se-
vere deformation conditions: strain localization, biaxial
expansion or more pronounced localized plastic strain
values. In the past decades a preliminary investigation of
the metal sheet behavior during drawing process has
been possible via the Erichsen or Olsen test [8-9]. Actu-
ally this test is standardized and it is used to differentiate
different metallic materials used in sheet forming and
hydroforming, cold rolling, tube-sinking or incremental
sheet forming. In fact the Erichsen test can be used as an
indicator of material formability through the height of
the spherical punch that deform the blank until the frac-
ture of the material corresponding to a crack of a length
approximately equal to 5 mm. In [10] the authors has
recently developed a numerical and experimental analy-
sis of Erichsen test via a finite element simulation, but
only the Erichsen index and the maximal values of the
punch force are analyzed. Moreover, several studies
based on the behavior analysis of the metallic plate or
sheet subjected to a perpendicular impact with a hemi-
spherical projectile [11], show the necessity to explore
the material behavior in conditions close to a biaxial ex-
pansion, generally developed during the Erichsen test.
Consequently, the present paper proposes an application
of the inverse analysis principle [12] to the constitutive
parameters identification directly from the Erichsen ex-
perimental device. The experimental measurements are
represented by the recordings of the evolution of the
drawing force with the depth of the sheet. The identifica-
tion of the material coefficients is performed with a finite
element simulation of the Erichsen test via an optimiza-
tion procedure. The corresponding cost function is ex-
pressed in terms of the experimental and numerical loads
using a least squares formulation. In order to validate the
proposed method, this paper neglects the sheet anisot-
ropy (i.e. choice of materials without any anisotropy ef-
fect) and the loads are supposed to be only quasi-static (i.
e. no strain rate sensitivity). Numerical analysis and real
results will be presented using the identification high
board named OPTPAR (developed in the LGCGM labo-
ratory), automatically coupled with a commercial finite
element code (FORGE2 or MARC one [13]) charged
to simulate the Erichsen test.
2. The Erichsen Experimental Test
The design of the classical Erichsen test used in the
Europe is defined by a hemispherical punch of steel with
the diameter of 20 mm, an active die with the diameter of
27 mm, a blank holder with the diameter of 33 mm and a
sheet with the diameter of 90 mm (see Figure 1(a) and
Figure 1(b)). The ra dius of the d i e is 0.75 mm.
The hemispherical punch is pressed into the sheet un til
material fracture occurs, at which point the test is stopped
immediately and the depth of the bulge recorded. This
depth expressed in millimetres gives the Erichsen index
(IE) and obviously giv es a measure of the ductility o f the
sheet under biaxial stress conditions in the plane of the
drawing. In a pure stretching forming the sheet is de-
formed by the punch and is totally clamped. From a clas-
sical point of view this test can be used only for com-
parative purposes of sheet metals. Actually, measure-
ment of the all axial punch force variation can be con-
sidered as new potential experimental data. Until now it
is not possible to develop a rigorous analytical theory of
Erichsen test able to express the punch force in function
of the test geometry and of the rheological parameters of
the sheet material. It is the principal reason for each a
numerical finite element model it is proposed to be used.
Punch (1)
Blank Holder (2)
Figure 1. Experimental set-up of the Erichsen test: (a) 3D
view (4, 6, 7–assembly components); (b) 2D asymmetric
3. Numerical Modeling and Analysis
Starting from a finite element model, a blank, with a di-
ameter of 90 mm and 3 mm of thickness, is meshed us-
ing three nodes linear triangular elements (Figure 2).
Figure 2. The numerical finite element model.
Copyright © 2010 SciRes. ENG
Re-meshing procedure is activated during the compu-
tation process in order to eliminate numerical problems
caused by degenerate elements which can occurs during
the numerical simulation. The tools, spherical punch, die
and blank holder, are considered to be rigid and the con-
tact between the sheet and the blank holder or the hori-
zontal part of the die is chosen to be a glued one. A
Coulomb friction law with a friction coefficient of 0.1 is
used between the sheet and the punch or the die active
part. Considering an isotropic rheological law of the
sheet material we can choose two different descriptions:
(1) a Ludwick one and (2) a Voce one (see Figure 3):
00 n
00 1exp a
 
 
where 00
is the elastic yield stress, K is the material
consistency and n is the hardening parameter. For a steel
alloy we must choose na = 1.
According to a virtual super plastic and ductile steel
alloy, chosen in order to describe more pronounced plas-
tic effects according to a stretching process, the values of
the elastic and plastic material data are presented in Ta-
ble 1.
The numerical simulations have been realized using
the FORGE2 commercial code (with an initial number of
elements equal to 947) and with the MARC software
(with an initial number of elements equal to 346). After
18 mm of the punch displacement the results of the de-
formed mesh and of the cumulated plastic strain distribu-
tion are pictured in Figure 4 and Figure 5.
The similar results obtained with the both numerical
codes confirm the availability o f the nu merical modeling.
It is then possible to show that the necking phenomenon
occurs at approximately 30 degree under the punch.
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91
Equivalent Stress [MPa]
Ludwick law
Voce law
Figure 3. Rheological laws used for the numerical simulation.
Table 1. Material properties for a virtual steel sheet.
Data 210 GPa 0.3 0.1
Law 00
K n
Ludwick324 MPa 401.76 MPa 0.514
Voce 324 MPa 281.46 MPa 5.14
Figure 4. Numerical finite element results obtained with
FORGE2 (Mesh and cumulated plastic strain at t = 18 s).
Figure 5. Numerical finite element results obtained with
MARC (Mesh and cumulated plastic strain at t =18 s).
Moreover, the obtained large values of the cumulated
plastic strain (up to 150%) permits a deformation history
analysis for more severe numerical conditions, very close
to a real stretching or a drawing process of a super plastic
alloy. Considering th e variation of the axial punch force,
the results are plotted in Figure 6. These results show
that similar curve shapes are obtained from the FORGE2
code and the MARC one. The small differences are
caused by the numerical treatment of the contact phe-
nomena, because different methods are used to adjust
numerically the Coulomb law [13].
Concerning the influence of the material rheological
law, we obtain a more pronounced softening with the
Voce formulation of the stress-strain variation. The use
of the extrapolation values for the stress from a classical
Ludwick law can causes sev eral problems for the estima-
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70000.00 12.0 14.0 16.0 18.0 20.0
Axial Pun ch Force [N]
Figure 6. Results of the axial punch force variation.
tion of the process forces or for the computation of the
deformation energies.
4. Validation of the Erichsen Test Simulation
for an Aluminum Alloy
In order to validate the finite element simulation of the
Erichsen test, a deformation of an aluminum alloy AA5182
is analyzed. Experimentally, for 7.6 mm of the punch
displacement, the thickness of the sheet in the necking
area is approximately of 0.5 mm. The friction phenome-
non and Coulomb parameter are considered to be a priori
known (i.e.
= 0.1) and in order to eliminate the ani-
sotropy of aluminum alloys, a special thermal treatment
has been used. So the rheological law is described only
by a rigid plastic model (K = 0) where 00
is approxi-
mately equal to 74 MPa. Particular problems caused by
the sensitivity of the sheet deformation with respect to
other constitutive parameters (as for example the hard-
ening ones) are then avoided. Starting from a numerical
finite element simulation via the FORGE2 software, the
sheet geometry and the distribution of the plastic strain
after 7.6 mm of the punch displacement is pictured in
Figure 7.
Figure 8 shows the fracture of the material obtained
from the real experiment. It is important to note that nu-
merically we obtain approximately the same final thick-
ness of the sheet.
Figure 7. The numerical finite element results obtained
from FORGE2 (Mesh and cumulated plastic strain corre-
sponding to 7.6 mm of the punch displacement).
Figure 8. Image of the fracture obtained by experiment
(Erichsen index e qual to 7. 8 mm).
The cumulated plastic strain in the necking area is ap-
proximately 60%, which corresponds to the analytical
estimation of the logarithmic plastic strain obtained from
the division between the initial and the final thickness.
These results allow us to confirm that large plastic de-
formations occur during the process and these ones can-
not be obtained from a classical tensile test. Many ex-
periments performed in laboratory LGCGM of INSA de
RENNES, which compare the cumulated plastic strain
values during a biaxial tensile test and a uniaxial tensile
one, confirm this last assertion. Finally, complete content
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5. Parameter Identification by Inverse
Starting from the previous experimental and numerical
analysis, accurate values of the constitutive parameters
behaviour can be obtained using the inverse analysis
principle. The experimental data can be represented by
the recordings of the evolution of the punch force with
the depth of the sheet (Figure 6). The identification of
the material coefficients can be then performed with a
finite element simulation of the Erichsen test via an op-
timization procedure (OPTPAR, see Figure 9). Using the
least squares formula the cost function is expressed in
terms of the experimental forces and of the numerical
ones by:
exp exp
expexp exp
,, NN
ii i
where P is the parameter vector: , Nexp is the
number of the experimental points, Fc is the finite ele-
ment computed punch forces and the Fexp is the corre-
sponding experimental data.
For a rheological identification problem, an appropri-
ate physical domain of parameters variation must be in-
troduced and a constrained optimization problem must be
formulated as:
min max
min( ,,)
() /
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Erichsen Test
FE Model
ata Fil
Gauss Newton
(P, F
, F
Figure 9. Algorithm of the inve rse analysis method.
The high no linearity of the obj ective functio n requires
a robust numerical minimization algorithm based on the
Gauss-Newton gradient method. In this case evaluation
of the first and second objective function derivatives are
exp exp
exp 2
dP F
exp 2
dF dF
dP dP
dP F
A direct differentiation method is used to compute the
derivatives of the computed forces with respect to the
constitutive parameters. This method requires adding
numerical simulations corresp onding to very s mall varia-
tions of each parameter, which will be iden tified. To test
the numerical convergences of the inverse analysis pro-
cedure, an artificial experimental data is used starting
from supposed known material parameters presented in
Table 1. The goal of convergence analysis is to verify
the capacity of the identification algorithm to find the
true rheological parameters, starting from different initial
estimations. The optimization procedure is initiated for
both laws, using initial values sufficiently far from the
real ones (see Table 2 and Table 3).
Table 2. Numerical identified results for the first initial
guess values.
Ludwick Law Voce Law
Parameters Initial Identif. Initial Identif.
324. 324. 324. 324.
K [MPa] 500. 401.79 500. 281.47
n 0.25 0.514 1. 5.14
22.63% 0.008% 11.03% 0.003%
iterations - 15 - 13
Table 3. Numerical identified results for the second initial
guess values.
Ludwick Law Voce Law
Parameters Initial Identif. Initial Identif.
324. 324. 324. 324.
K 1000. 401.74 100. 281.38
n 0.1 0.514 10. 5.147
103% 0.004% 27% 0.07%
iterations - 11 - 9
These results show that the numerical convergence is
obtained in a small number of optimization iterations and
with a very high precision (objective function smallest
that 0.1%). The available numerical inverse analysis
model can be now used in a real experimental test.
6. Application to a Real Data
In order to test the inverse analysis of the Erichsen test in
real conditions, a rheological parameter identification is
made for a DC03 steel alloy and using a sheet with the
initial thickness equ a l to 1 mm.
The axial punch force is recorded until the material
fracture occurs. The friction law is supposed to be a
Coulomb one and the friction parameter is a priori
known (
= 0.1). For the numerical model, a mesh with
approximately 853 elements is used. The Table 4 syn-
thesis the parameter identification results obtained with
the OPTPAR software. These results show that the mate-
rial has, for the Ludwick law, a normal hardening effect
(n = 0.61) and for the Voce law a small value of the Voce
parameter (n = 2.47). The comparison between the ex-
perimental and the com puted loads is pl otted in Figure 10.
The global error is approximately 1% for the Ludwick
law and 3% for the Voce one and it is easy to see that the
computed load curves are very close to the experimental
ones. It is then possible to conclude that the constitutive
parameters defining the equivalent Von-Mises stress can
be obtained directly from the Erichsen experimental test.
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Copyright © 2010 SciRes. ENG
8. Acknowledgements
Table 4. Numerical parameter identification results for a
real DC03 steel sheet.
Ludwick Law Voce Law (na = 1)
Parameters Initial Identified Initial Identified
150. 154.17 154.17 154.17
K [MPa] 500. 516.52 500. 484.33
n 0.25 0.61 10. 2.47
19.8% 1% 34.7% 2.7%
iterations - 6 - 4
The present work was elaborated under the bilateral re-
search agreement established between INSA de REN-
NES-France and University “Dunarea de Jos” GALATI-
Romania (UGAL), and financially supported by the re-
search contracts CNCSIS 686/2007 and CEEX 24/2006
of the laboratory ITCM of UGAL.
9. References
[1] D. Banabic, “Formability of Metallic Materials (Plastic
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[2] R. Padmanabhan, M. C. Oliveir, J. L. Alves and L. F.
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Figure 10. Comparison between experimental and com-
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7. Conclusions
[7] W. G. Granzou, “Sheet Formability of Steel,” 10th Edi-
tion, ASM International, Metals Handbook, Materials
Park, Ohio, 1990.
A robust optimization algorithm for the constitutive pa-
rameters identification was developed from a numerical
simulation of the Erichsen test. This inv erse method uses
the commercial codes FORGE2 and MARC one starting
only from the variation of the axial punch force. The
inverse analyses, combined with numerical validations,
robustness studies and validation through a real experi-
ment, underlines the accuracy of the results obtained by
proposed algorithm. The main feature of this identifica-
tion methodology is linked to the possibility to take into
account the all complexity of the test. The finite element
model permits to simulate the biaxial stress conditions,
together with a non-uniform distribution of the plastic
strain and of the sheet thickness. It is then possible to
obtain deformation cond itions close to those which occur
during a real drawing or stretching process. In a future
work, complementary experimental data, as for example
the angular position of the necking radius together with
he values of the sheet thickness variation, will be added.
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the American Society of Materials, Vol. 20 , 1920, pp. 398-403.
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“Numerical and Experimental Study of the Erichsen Test
for Metal Stamping,” Advances in Production Engineer-
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Numerical Analysis of an Impacted Thin Aluminum
Plate,” Proceedings of Structures Under Shock and Im-
pact VII, Montreal, 2002, pp. 477-486.
[12] A. Gavrus, “Identification Des Parametres Rheologiques
Par Analyse Inverse,” Phd Thesis, ENSMP, Paris, 1997.
[13] MARC User’s Manual, Analysis Research Corporation,