VOLUME 48

ISSUE 1
of Achievements in Materials September
and Manufacturing Engineering 2011

Parametric Finite Element Analysis for

a square cup deep drawing process
F. Ayari a, E. Bayraktar b,*
a Laboratory of Mechanics, College of Science and Technology,
1008 Montfleury, Tunis, Tunisia
b Supmeca/LISMMA-Paris, School of Mechanical

and Manufacturing Engineering, Paris, France

* Corresponding author: E-mail address: bayraktar@supmeca.fr
Received 03.07.2011; published in revised form 01.09.2011

Analysis and modelling

Abstract
Purpose: This manuscript deals with the FEA of the sheet metal forming process that involves various
nonlinearities. Our objective is to develop a parametric study that can leads mainly to predict accurately the
final geometry of the sheet blank and the distribution of strains and stresses and also to control various forming
defects, such as thinning as well as parameters affecting strongly the final form of the sheet after forming
process.
Design/methodology/approach: The main approach of the current paper is to conduct a validation study of
the FEM model. In fact, a 3D parametric FEA model is build using Abaqus /Explicit standard code. Numerous
available test data was compared to theoretical predictions via our model. Here, several elastic plastic materials
low was used in the FEA model and then, they were validated via experimental results.
Findings: Several 2D and 3D plots, which can be used to predict incipient thinning strengths for sheets with flat
initial configuration, have been presented for the various loading conditions. Unfortunately, most professionals
in the forming process, lack this expertise, which is an obstacle to fully exploit the potential of optimization
process of metal forming structures. In this study optimization approach is used to improve the final quality of
a deep drawn product d by determining the optimal values of geometric tools parameters.
Research limitations/implications: This paper is a first part study of a numerical parametric investigation that
is dealing with the most influent parameters in a forming process to simulate the deep drawing of square cup
(such as geometric, material parameters and coefficient of frictions). In the future it will be possible to get a
large amount of information about typical sheet forming process with various material and geometric parameters
and to control them in order to get the most accurate final form under particular loading, material and geometric
cases.
Originality/value: This model is used with conjunction with optimisation tool to classify geometric parameters
that are participating to failure criterion. A mono objective function has been developed within this study to
optimise this forming process as a very practical user friend manual.
Keywords: FEM; Deep drawing; Plasticity; Friction; Explicit method; Parametric study; Modelling;
Optimization; Clusters

Reference to this paper should be given in the following way:

F. Ayari, E. Bayraktar, Parametric Finite Element Analysis for a square cup deep drawing process, Journal
of Achievements in Materials and Manufacturing Engineering 48/1 (2011) 64-86.

64 Research paper © Copyright by International OCSCO World Press. All rights reserved. 2011
 Analysis and modelling

1. Introduction
1. Introduction is the material chosen to be studied in this work). The rigid die
is a flat surface with a square hole 84 mm by 84 mm, rounded
at the edges with a radius of 8 mm. The rigid punch measures
In this study we are analysing the forming of three
70 mm by 70 mm and is rounded at the edges with the same
dimensional shapes by deep drawing process. Different numerical
10 mm radius. The blank holder can be considered a flat plate,
process can be used as it is mentioned in the literature [1-5].
since the blank never comes close to its edges. The geometry
The most efficient way to analyse this type of problem is to
of these parts is illustrated by Figure 1 and Figure 2. The rigid
analyse the forming step with a FEM code that allows both
surfaces are offset from the blank by half the thickness of the
dynamic and static analysis. In this study, Abaqus Explicit [6]
blank to account for the shell thickness.
is used to carry FE analysis. Since the forming process is essentially
a quasi-static problem, computations with Abaqus /Explicit are
a)
performed over a sufficiently long time period to render inertial
effects negligible.
Forming processes are generally expensive, for this reason there
is a great amount of researches studies related to their optimizations.
Indeed, the coupling of simulation software’s with mathematical
algorithms for optimizing the process parameters is widely
increasing in various fields of forming. It was demonstrated that this
kind of coupling reduces and improves the products’ cost [7].
Optimization of process parameters such as die radius, blank
holder force, friction coefficient, etc., can be accomplished based
on their degree of importance on the sheet metal forming
characteristics. In this investigation, a statistical approach based
on computing with categorical array technique was adopted
to determine the degree of importance of some geometric design
parameters on the thickness distribution of deep drawn
rectangular cup. Then a mono objective optimization method
scheme has been applied in forming study to design the process
providing guidance how to choose the best fixed geometric
parameter which leads to the selected minimum failure criterion. b)

2. Description
2. Description of theof themodel
initial initial
model
The material of the blank will form the base of the cup which
is in contact with the face of the punch, the die and the holder.
This material can stretch and slides over the surface of the punch;
however, minimal variation in thickness of this material is
expected (Figure 1).
During a deep drawing operation, the blank is subjected
to radial stresses due to the blank being pulled into the die cavity
and there is also a compressive stress normal to the element which
is due to the blank-holder pressure. Radial tensile stresses lead to
compressive hoop stresses because of the reduction in the Fig. 1. a) 3D key dimensions of the FE assembly model;
circumferential direction. b) principal geometric parameters of the FE model
In fact, the load applied on the blank is modelled as
a distributed load on the contact surface holder-blank. The wall While Abaqus/Explicit automatically takes the shell thickness
of the cup is primarily encountering a longitudinal tensile stress, into account during the contact calculation. A mass of 0.65 kg
as the punch transmits the drawing force through walls of the cup is attached to the blank holder, and a concentrated load of 19.6 kN
and through the holder as it is drawn into the die cavity. There is applied to the contact surface blank - holder. The blank holder
is also a tensile hoop stress caused by the cup being held tightly is then allowed to move only in the vertical direction to
over the punch. accommodate changes in the blank thickness. The coefficient
The choice of the different geometric dimensions and material of friction between the sheet and the punch is taken to be variable
properties was conformed to experimental previous data. In fact, from (0.01 to 0.125), and that between the Blank and the Punch.
before starting the parametric FE study, we have performed It is (from 0.01 to 0.25). In fact, in previous studies it was
a comparative study with experimental previous work and we confirmed that the coefficient of friction between contact surfaces
have used it as a validation study of this model. All the initial has an important effect in the forming process [1].
dimensions are chosen to be identical to those used in the The simulated punch velocity is kept constant and equal
experimental previous study [7]. The blank is initially square, to 1.66 mm/sec while the considered minimum nodal distance
150 mm by 150 mm, and is 0.78 mm for the (mild steel Ms, that is less than the blank thickness.

READING DIRECT: www.journalamme.org 65

 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

a)

b)

Fig. 3. The stress-strain curve used for the numerical simulations

The computer time involved in running the simulation using

explicit time integration with a given mesh is directly proportional
to the time period of the event, since the stable time increment
size is a function of the mesh size (length) and the material
stiffness. Thus, it is usually desirable to run the simulation at an
artificially high speed compared to the physical process. If the
speed in the simulation is increased too much, the solution does
not correspond to the low-speed physical problem; i.e., inertial
effects begin to dominate. In a typical forming process the punch
may move at speeds on the order of 1 m/sec, which is extremely
slow compared to typical wave speeds in the materials to be
formed. (The wave speed in steel is approximately 5000 m/sec.)
Fig. 2. a) FEA model of the DDP of a square cup; b) FEA model In general, inertia forces will not play a dominant role for forming
of the DDP of a square cup; blank mesh rates that are considerably higher than the nominal 1 m/sec rates
found in the physical problem.
The blank is made of Mild steel Ms. The relation between true In the results presented here, the drawing process is simulated
stress and logarithmic strain, stress strain of this materials are by moving the reference node for the punch downward through
done with the following expressions [7]. a total distance of 11- 15- 30 and 40 mm (6.626506, 9.036145,
18.072289 and 24.096386). In this analysis we used the technique
(1) of mass scaling to adjust the effective punch velocity without
where ı is the equivalent tensile stress, İP the equivalent plastic altering the material properties.
strain and the other material parameters are identified by mechanical
tests (Table 1). Figure 3 shows the equivalent stress vs. the plastic
deformation of the MS with different test directions.
3. The FEM
3. The FEM model
model
Table 1.
Material properties Finite element simulations of deep drawing process provide
Mild Steel an effective means to investigate the interaction between the
VY (Mpa) 173.1 process parameters and the material response. They provide useful
E (GPa) 206 information for fine-tuning the production processes. In this study
VUS (MPa) 311.4 the deep drawing process of rectangular cups is modelled using
U (kg/m3) 7800 FEA. The general purpose commercial FEA code Abaqus/Explicit
is used for the simulations.
The stress-strain behaviour is defined by piecewise linear The movement of the punch was defined using a pilot node.
segments matching the Ramberg-Osgood curve up to a total This node was also employed to obtain the drawing force during
(logarithmic) strain level of 107%, with Mises yield, isotropic the simulation. After applying appropriate boundary conditions
hardening, and no rate dependence. to the models of sheet, punch, die and blank holder, the numerical
Given the symmetry of the problem, it is sufficient to model simulation of the process was performed.
only a one-quarter sector of the box. However, we have employed Figure 4 shows the simulation with 51 elements. The distribution
a one-quarter model to make it easier to visualize. We use 4-node, of the von Mises stresses is illustrated in Figure 4a for the
three-dimensional rigid surface elements (type R3D4) to model the numerical analysis with 106 solid elements. To facilitate doing
die, the punch, and the blank holder. The blank is modelled with a comparison between various results, the remaining of the FE
8-node, linear finite-strain shell elements (type SC8R). findings are presented and discussed in the next section.

66 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

a) b) Table 2 below illustrates results obtained for a square blank

with a punch stroke of 15 mm and 40 mm for Mild steel material.
Earring profile shows asymmetric flow. In fact, in deep drawing
it derives from planar anisotropy. Thus, the plastic flow of
anisotropic sheet can be regarded as the sum of two superimposed
deformation processes occurring simultaneously; normal flow
controlled by normal anisotropy, and asymmetric flow due to planar
anisotropy. In fact, before the parametric study, a blank sheet
mesh size sensitivity study has been built and as it is shown by the
Figure 5 above a mesh size of 55, the earring profiles dimensions
DX, DY and DD are insensitive to the mesh size variation. That is
the reason for which we have adopted the 55 mesh size as the
optimal value in the model.

c) Table 2.
Comparative results of experimental study
Material Ms Ms
Travel 15 mm 40 mm
DX_FEA 7.07 28.6
DX_EXP 7.0 28.1
DX_ERROR 1.0% 1.8%
DD_FEA 3.7 14.89
DD_EXP 3.9 15.1
DD_ERROR 5.1% 1.4%
DY_FEA 7.06 28.6
DY_EXP 7.1 28.5
DY_ERROR 0.6% 0.4%

Fig. 4. a) Earing profile of the blank at the end of the DDP;

b) direction used for FE validation study; c) X-Y plane projection
of the DD, DX and DY earing profile measurements distances and
AB Strain sampling path
Fig. 5. Curve that demonstrates the mesh sensitivity
4. Validation
4. Validation study
study The numerical results presented in the Table 1 are obtained
with the following coefficient of friction 0.02 for the Blank
To check the validity of results computed by deep drawing Holder contact, 0.02 for the Blank Die contact, 0.25 for the Punch
simulations, two numerical comparative studies were investigated. Blank contact and then 0.034 for the Blank Holder contact, 0.04
Results from these two studies were compared to experimental for the Blank for the global contact surfaces for MS material and
results and a good correlation was deduced. These parameters 0.0 Die contact, 0.16 for the Punch Blank contact, then 0.04 for
refer to mild steel sheets. the global contact surfaces for the Al material. These coefficients
are chosen to simulate the real contact surfaces of the experimental
conditions; as before each experiment both sides of the Blank
4.1. First validation with general
4.1 First validation with general displacements
displacements
sheet surface were wiped with a paper towel dipped in the
lubricant and they were kept in a vertical position for 30 minutes [7].
Thus, we have conducted several finite element simulations
The first validation study consists on a comparison between by varying the Blank- Holder coefficient of friction, the Blank
numerical and experimental displacements in the following Die coefficient of friction and the Blank Punch coefficient of
directions: DX-called 'rolling direction', DD (diagonal direction) friction in the range from 0 to 0.25. We have measured at each
and DY (transverse direction) as shown in Figure 4. time the edge displacements which are describing the earring

Parametric Finite Element Analysis for a square cup deep drawing process 67
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

profile in the X direction, the Y direction and the Diagonal ultimate punch stroke of 40 mm. Yet, the simulation results
direction. These displacements values were then compared to those produced the trend in deformation behaviour of the MS blank.
obtained experimentally for the Ms Material. Results from Table 1 In all the cases, the elements in the corner area do not reach
show that experimental and FE earring profile displacements were a state of plastic instability even within a draw depth of 40 mm.
being close so that the error is varying from 1% to 5.1%.

5. Parametric
5. Parametric FEMFEM
study study
4.2. Strain comparative
4.2 Strain comparative study. study
Finite element simulations of deep drawing process provide
The second validation to be considered in this study, consists an effective means to investigate the interaction between the
of the comparison between the experimental and the numerical process parameters and the material response. They provide useful
principle strains (H1, H2, H3) in the diagonal direction of the blank information for fine-tuning the production processes. In this study
denoted by the path AB (Figure 4), for a punch travel of the deep drawing process of rectangular cups is modelled using
respectively 15 mm and 40 mm with the MS material (Figs. 6, 7). FEA. The general purpose commercial FEA code Abaqus/Explicit
The experimental strain profiles trend, along the diagonal path, 6.7 is used for the simulations.
was closely replicated by the FEA model. In particular, Figure 7d.
shows a slight gap in the plot between thickness strain and
distance along diagonal displacement of the Mild steel with 40 mm
5.1. 
5.1 Model
Model geometry
geometry
travel punch. The difference between experimental and numerical
strains in this case is due to an important out of plan strain
deformation for x = 75 mm (at the vicinity of the corner cup for All parts are modelled as rigid bodies except for the blank
which a maximum thinning is denoted). In general, the numerical which is modelled as an elastic-plastic material with metal
values overestimate the biaxial state flow strain. As far as we are plasticity. The DDP simulation is accomplished in two phases:
dealing with high anisotropic materials, the Von Mises criterion the blank holder applies a predetermined force on the blank then
used for numerical strains predictions overestimates the a displacement equal to the desired depth is applied to the punch.
experimental strain values, but this discrepancy can be allowed as All geometric dimensions of the parts may vary and the model
the maximum error between numerical and experimental geometry can be easily changed. Parts used in the DDP are shown
thickness strains does not exceed the 18% except for the MS with by Figure 1b, principal dimensions of a cup shown therein may be

Fig. 6. Comparison between FEA and experimental results for Mild steel with punch travel of 15 mm: a) Mises stress distribution and
deformed shape of the square cup; b) Principal strain H1 along diagonal path AB; c) Principal strain H2 along diagonal path AB; d) Principal
strain H3 along diagonal path AB

68 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

Fig. 7. Comparison between FEA and experimental results for Mild steel with punch travel of 40 mm: a) Mises stress distribution and
deformed shape of the square cup; b) Principal strain H1 along diagonal path AB; c) Principal strain H2 along diagonal path AB; d) Principal
strain H3 along diagonal path AB

varied in the FE model. Thus, the Figure 1b describes the reduced integration and one element through the thickness.
geometric parameters of the deep cup model. In fact, tb is the The punch, die and blank holder are all modelled as 3 dimensional
blank thickness, WP is the punch width section, WB is the blank discrete rigid surfaces using four-node rigid surface elements
width section, WD the width of the die cavity, and respectively; (R3D4). In addition, the geometry has been variable, mesh
RsP section normalized radius of the punch this radius is measured parameters such as element size and mesh density may also be
in the xy plane, RsD section normalized radius of the die measured varied. This feature is needed to ‘tune’ the model in order to get
in the xy plane, RfD fillet normalized radius of the die measured mesh independent converged results.
in the xz plane, RfP fillet normalized radius of the punch measured
in the xz plane and SP normalize punch travel (stroke). When we
are dealing with rectangular cup, additional geometric parameters 5.2. Boundary conditions
5.2 Boundary conditions
are considered, such as tLP total length of the punch section,
tLB total length of the blank and tLD total length of the die cavity. As described in section 5.1 the DDP consist of two steps.
The parameters tLP, tLB and tLD are similar to WP, WB and WD used During the second step, the punch moves at a constant speed
in the case of square cup but they measure the dimensions in the VP for a travel distance SP with blank holder pressure P still
xz plane for rectangular cup geometry. applied. The die is fixed while the punch and blank holder are free
The blank is assumed to have frictional contact with the to move in a direction normal to the blank plane. The double
remaining parts. Due to the anisotropy of material behaviour, symmetry of DDP configuration is exploited and only a one-fourth
a 3D analysis has been considered modelling only a quarter of the of the assembly was modelled with symmetry boundary conditions
deep drawing test is achieved. Adequate boundary conditions applied at symmetry planes.
must be imposed at the symmetry axes. These symmetry axes are
defined as the global X and Y axes in the FE mesh; the global
Z axis is parallel to the punch displacement direction. The geometry 5.3. 
Numerical
5.3 Numerical considerations
considerations
of the FE model is that of the experimental process shown
in Figure 1. The tools (punch, die and holder) are considered to be Mass scaling
perfectly rigid and are modelled by rigid elements. At low punch speeds (and constant speeds) DDP is essentially
The structure is modelled using both 2D and 3D elements a quasi-static process. Therefore, inertia forces do not play a major
available in Abaqus. The blank sheet metal was modelled using role and it is possible to speed up the convergence of the
eight-node continuum shell elements in Abaqus (SC8R) with numerical solution using mass scaling. This approach requires

Parametric Finite Element Analysis for a square cup deep drawing process 69
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

increasing the density of the material artificially in order to increase (such as wrinkling), (ii) defects due to asymmetrical flow (caring)
the stable time increment for the numerical integration. This (iii) surface defects, and (iv) distorted geometry in the
procedure is termed “mass scaling”. In the same manner as unconstrained state (such as thinning).
load-factoring this is an attractive method in instances where The most primary defects that occurs in deep drawing
inertia plays a comparatively small part in the structural behaviour. operations are the over thinning and wrinkling of sheet metal
As far as, in our case a blank forming analysis is considered, materials. Generally, the thinning is produced in the blank wall
where a large proportion of the deformed structure is constrained compressed between the punch and the die; while the wrinkling
by rigid surfaces and the kinetic energy of the blank itself is a small is occurred in the flange of the blank. Those major defects are
component of the overall energy balance of the problem, the mass preventable if the deep drawing systems are designed properly.
scaling is applied to the blank only. The greater the die cavity depth, the more blank material has to
Thus, this is done by multiplying the density of blank material be pulled down into the die cavity and the greater the risk of
by a factor. This factor is chosen so that the solution time is reduced thinning and wrinkling in the blank. The maximum die cavity depth
considerably all the while keeping the ratio of kinetic energy is a balance between the onset of wrinkling and the onset excessive
to strain energy of the blank low. A ratio of 0.05 is recommended thinning or fracture, neither of which is desirable. This balance is
[8] for best performance. Since the contributions of the die, punch described with limit drown values commonly fixed by the industrial
and holder to inertia forces are negligible (the punch is moving exigency, for example, in the automotive industry 20% of thinning
at a constant speed) they were all assigned unit point masses. in the sheet thickness is the maximum tolerable value.
We have varied this factor from 500 to 60000 and several In this parametric study, several parameters have been
numerical examples were conducted to adjust this factor, so that considered to be variable. For all the following cases, when ultimate
we obtained converged solution while the ratio of kinetic energy thinning riches more than 30% of the initial blank thickness,
to strain energy is kept less than 5% of the strain energy; a value the process is considered as failed. The simulations presented here
of 10000 of the mass scaling have been used. are run with the following considerations: the material properties
are those of MS material; the coefficient of friction are 0.02
Contact parameters for the Blank Holder contact, 0.02 for the Blank Die contact, 0.25
The formulation of contact parameters between rigid surfaces for the Punch Blank contact and then 0.03 for the global contact
(all the DDP tools except the blank) and a deformable body surfaces as follows, the punch speed is of 1.66 mm/s and the
(which is the blank) is modelled with surface-to-surface model holder force is maintained at a value of 19.6 kN.
that is less sensitive to master and slave surface designations. In order to assess the effect of parameters such as RfD, RfP, RsD,
In this formulation the finite sliding is undergo. However, the RsP, Vb,SP, lD and tb on the DDP, we have adopted a simplified
finite-sliding, surface-to-surface formulation with the path-based notation all the geometric parameters were done as a ratio to the
tracking algorithm do allow for double-sided surfaces used in this final cross section width WD; so that they can be dimensionless.
study. As far as, surface-to surface contact discretization has more The following ratios are then introduced:
continuous behaviour upon sliding, contact conditions with finite- lD= tLD/ WD, sP= SP/WD, rsD = RsD / WD,
sliding contact tend to converge in less iteration with surface-to rsP = RsP / WD, rfP = RfP / WD, rfD = RfD / WD.
surface contact discretization. In addition during all the parametric With the purpose of accomplishing this task, four parametric
simulations, the analytical rigid surfaces are simulated by the studies have been considered. In the first parametric study the effect
master surfaces, slave surfaces are attached to deformable blank. of the aspect ratio (lD) and the blank thickness on the limit
of drawability of the DDP are considered. The second parametric
Friction coefficient parameter study treats the effect of aspect ratio (lD), the punch section radius
Friction is one of the most important parameters affecting the (rsP) and die fillet radius (rfD) on drawability, wrinkling and percent
material flow and the required load in forming process [9, 10]. thinning of the formed cup. The third parametric study examines
FE investigations treating this parameter as a single parameter the effect of the cup aspect ratio (lD), die section radius (rsD) and
of DDP and drawing forming limit in sheet forming processes, punch fillet radius (rfP) on DDP. The fourth and last study
are well précised in references [11] and [12]. is dealing with the effect of the aspect ratio (lD) and punch travel
In fact, friction has both positive and negative roles in metal distance (sP) on the DDP A total of 136 finite element analyses
forming. There are numerous instances where friction opposes the cases have been used to carry out the described parametric study.
flow of metal in forming processes, and there are also several
cases where the forming process is made possible by friction. Effect of rectangular cup aspect ratio and blank thickness
A high value of friction between the blank and the matrix causes on drawability of the DDP
a significant thinning of the blank. On the opposite side by In this section, we are interested to attempt a correlation that
removing the lubricant, there appeared signs of damage at the describes the limits of drawability between the aspect ratios lD,
contact surfaces materials. and the blank thickness tb. In fact, to involve the relation between
the initial blank thickness and the final aspect ratio of the
rectangular cup; it is important to derive the trend of thickness
5.4. 
Parametric
5.4 Parametric simulation
simulation of the DDP variation along the particular path (the diagonal path highlighted
of the DDP with a red dashed line in Figure 4a. In this section study, thickness
of the blank, tb is varying in the range of 0.8 to 1.6 mm and the
In DDP defects imply that the drawing of a cup has been aspect ratios lD is varying in the range of 1.2 to 1.6. The most
completed but that the finished shape has some undesirable important geometric parameters (rsD, rfD, rsP, rfP and wD) are
features in terms of geometry and/or mechanical properties. simultaneously; 0.3, 0.25, 0.1,0.2, 40. Figures 8 to 10 are
The defects fall into four main categories: (i) defects due to buckling, describing the thickness evolution vs. the diagonal distance for the

70 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

two limit aspect ratios. The variation of the thickness profile of the thickness) otherwise exchange by value of the initial thickness.
formed cup is measured in two ways. The first method is to evaluate The same work is repeated for all the aspect ratio lD. as it is shown
the maximum thinning value which is expressed as: by Figure 12.
t b  min t b
pt * 100 (2)
tb
In this case the minimum is sampled over the diagonal path.
The second method is to calculate the deviation of the final profile
from the initial thickness this is expressed as:
N

Gt ¦
2
tb  t k (3)
k 1

where N is the number of sampling points along the diagonal

direction and tk is the thickness at the kth sampling point.
The x-axis of Figure 11a describes the different initial
thickness of the blank, and the vertical axes define the values
of the blank thicknesses after deformation along the diagonal path.
From this graph, it is shown that the difference between the
maximum and minimum thicknesses of the deformed blank
is minimized for low initial thickness of the blank for the same
geometric shape. The Figure 11b represent the group summery Fig. 10. Variation of the blank thickness along the diagonal path
statistic table of the results for the ratios lD=1.2. for various aspect ratios lD and initial blank thickness tb=1.6

a)

Fig. 8. A square rectangular cup with highlighted diagonal path

b)

Fig. 11. a) Variation of the blank thickness along of the deformed

sheet for ratios lD=1.2 and tb=0.8, 1.2, 1.4, 1.6; b) summary
of statistics results

Graphs in Figure 12 show that the variation of the thickness

profile pt and the deviation of the final profile Gt increases when
Fig. 9. Variation of the blank thickness along the diagonal path blank thickness tb increases and also when aspect ratio lD increases.
for various aspect ratios lD and initial blank thickness tb=0.8 In the graphs represented by Figure 12, for the spectrum of lD
and tb the trends of the thinning deviation curves measured by the
Indeed, the dispersion indicator (standard deviation) varies Gt parameter and those of the thinning variation are quite similar.
from one sample to another depending on the group (the initial In fact, the results show that the values of the maximum and

Parametric Finite Element Analysis for a square cup deep drawing process 71
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

minimum of the thickness changes are minimized for smallest the maximum thinning trends to decrease. Figure 14 represents
blank thicknesses and they are largest for highest aspect ratios lD. the thinning deviation along the diagonal path, for various aspect
From Figure 12, it is established that the rate thinning defined ratios lD and punch section radius (rsP).We note that the trend
as the aspect of the maximum thinning to the blank thickness of thinning deviation is closely similar to the trend of maximum
tb is depending on the aspect ratio lD. It seems that as much as the thinning vs rsP. In fact, the increase of the final cup aspect ratio
aspect ratio lD is low, the maximum thinning rate is upper. Thus, lD induces decrease of the maximum thinning independently from
the rate of maximum thinning is slowed down by the increasing (rsP). Figure 15, show that above rsP =0.4, the increase of lD ratio
of the final cup aspect ratio lD. Results show that, varying the leads almost to increase of the maximum thinning. It is thus
aspect ratio lD from 1.2 to 1.6 has reduced the thinning of the noticed that we have to avoid high rfD parameters combined with
blank thickness by 30% for the same blank thickness of 1.6 mm. high rsP parameters, especially with thick sheet blanks, because
it leads to greatest maximum thinning.

Fig. 12. Thinning deviation of the thickness distribution along

diagonal path versus the blank thickness tb and aspect ratio lD
Fig. 14. Variation of the thinning deviation along the diagonal
Effect of die fillet and punch section radii on drawability path for various rsP and lD
of the DDP
In this section we are dealing with the study of the geometric
parameters such as the punch section radius (rsP), the die fillet
radius (rfD), and their interaction with the aspect ratio lD, on the
drawability of the DDP. In fact, several finite element analyses were
performed, 36 FEA experiments are done with lD, which rises
between 1 to 1.5, rsP is varying between 0.2 to 0.8 and rfD between
0.4 to 0.8. For the entire calculations in this section the normalized
punch stroke sP is fixed to 0.75, the blank thickness tb is fixed to 1.2.

Fig. 15. Variation of the maximum thinning along the diagonal

path for various rsP, rfD and lD

Effect of punch fillet and die section radii on drawability

of the DDP
In some literature review [13], [14] and [15], the effect of the
punch fillet radius was considered as effective parameter which
can influence the drawability of the DDP. Nevertheless, interaction
between punch fillet radius (rfP) and die section radius (rsD) has
not been well discussed. In this paragraph we will elucidate
Fig. 13. Variation of the thinning along the diagonal path for the effect of interaction between those parameters. We will focus
various aspect ratios rsP and rfD on the impact of this interaction on DDP and highlight the change
in the behaviour of the maximum thinning according to variation
From Figure 13, it seems that when rsP takes minim values; of the parameters; aspect ratio lD and tb.
the growth of maximum thinning is well emphasized. By the way, A series of 64 FEA experiments is done by varying the blank
in Figure 13 it is noticed that as much as rfD and rsP are declines; thickness in the range from 1.2 to 1.6 mm, the aspect ratio lD from

72 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

1.2 to 1.6, the die section radius from 0.3 to 0.8 and the punch thinning deviation versus rsD for different values of rfP. The thinning
fillet radius from 0.1 to 0.22, the punch stroke sP was kept constant profile is generally growing, according to increasing of the rfP for
and equals to 0.75. the same aspect ratios lD and rsD. Results confirm that the lower rfP
is, the more severe maximum thinning is. In fact, it is observed that
interaction between rfP and rsD is well emphasized for a relatively
thick blank (tb=1.6 mm). In addition, values of rsD larger than 0.6
improve the critical thinning, when the punch fillet radius is smaller
than 0.1. This result is very significant; in fact a high value of the
fillet punch radius > 0.1, combined with section die radius < 0.6,
leads to a minimum thinning Figures 18, 19 and 20.
Figure 20 shows the large amount of thinning localized in the
critical zone of the diagonal path when we move from tb=1.2
to tb=1.6.

Fig. 16. Variation of the thickness along the diagonal path for
various rsD for lD=1.6 and tb=1.4

Fig. 19. Variation of the maximum thinning % along the diagonal

path for various rsD, and rfP for lD=1.6 and tb=1.6

Fig. 17. Variation of the thinning deviation along the diagonal

path for various rsD, and rfP. For lD=1.4 and tb=1.6

Fig. 20. Variation of the thickness along the diagonal path for
various rfP for lD=1.4 and tb=1.4; 1.2
The effect of interaction between rsD, rfP, lD and tb is presented
by Figures 21 to 23. A significant sensitivity to the interaction
between the rsD ,rfP, with lD and tb, is noticed on the thinning profile.
We kept the initial blank thickness constant and equal to
tb=1.6 mm, then we have varied the final cup aspect ratio lD from
1.2 to lD=1.6. According to Figure 21, the thinning deviation for
Fig. 18. Variation of the maximum thinning % along the diagonal the different values of rsD has been distinguished with a significant
path for various rsD, and rfP. For lD=1.4 and tb=1.6 reduction. Figure 22 confirm that when the final cup aspect ratio
lD increases, the maximum thinning with the same rsD, rfP,
According to Figure 16 it is observed that, the maximum is slowed down. The growth of rfP, parameter is accompanied
thinning for the different rsD ratios is located at the same zone which with the decrease of maximum thinning, but this maximum
corresponds to the corner of the cup. Figure 17 introduces the thinning has to grow for increasing rsD values.

Parametric Finite Element Analysis for a square cup deep drawing process 73
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

In the cases with thin initial blank tb=1.2, it appears that for From Figures 22 and 23, it appears that for larger aspect ratio
the entire studied cases from Figures 20 and 21, the maximum lD, the thinning deviation and the maximum thinning are being
thinning profile tends to take a rapid rate reduction within the decreasing. We note at least, that above rsD =0.6, even though the
increase of rsD parameter. This fact is particularly noted for rsD maximum thinning is slowed down, the increase of rfP ratio leads
above 0.6. to a small growth of the maximum thinning instead of decreasing.

Sensitivity to the Punch travel

In order to take advantage of this parametric FEA study, the
interaction between the lD aspect ratio and the punch travel sP ratio
are emphasized in Figures 24 to 28. Four values of parameter sP are
experimented at constant value of lD; then the analysis is repeated
for a different value of lD. It appears that for a square cup the
growth of the sP doesn’t lead to the increase of the maximum
thinning. In fact, as it is mentioned by Figure 28, beyond sP=0.6
the increase of the punch travel slow down the maximum thinning
of the square cup. Conversely, under 0.6 of the sP value, the growth
of the ratio sP leads to a rapid increase of the maximum thinning.
This paradoxes result can be attributed to the fact that a friction
of the sheet between the blank holder and the die is almost
important so that a higher thickening is observed for the flange
of any rectangular cup characterized with a sP above 60% of the
blank cross section width. Therefore, when the punch travel
Fig. 21. Variation of the thinning deviation along the diagonal path increases, a uniform distribution of the thickness can be transferred
for various rsD, and rfP with lD=1.4 and tb=1.2 from the thickening amount of material. The critical thinning
is being slowed at the end of the process.

Fig. 22. Variation of the maximum thinning % along the diagonal Fig. 24. Variation of the thinning along the diagonal path for
path for various rsD, and rfP for lD=1.4 and tb=1.2 various aspect sP and lD=1

Fig. 23. Variation of the thinning deviation along the diagonal path Fig. 25. Variation of the thinning along the diagonal path for
for various rsD, and rfP for lD=1.6 and tb=1.2 various aspect sP and lD=1.5

74 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

Figures 26 to 28 show that maximum thinning increased with 5.5. Discussion

5.5. Discussion
sP for cups with lD larger than unity. Beyond a punch travel equals
to the blank width of the cross section, some wrinkling can be FE analysis results show that localized deformation and
observed at the vicinity of the corner section as highlighted wrinkling occur along the major axis of diagonal DD when
by Figure 27, this fact is considered as major failure. Figure 28, interaction between some of the geometric parameters is hold
illustrate two important facts; the larger the aspect ratio lD is, the as detailed in section 5. This is attributed to the non-uniform
smaller the maximum thinning for the same sP. Indeed, as well as contact in the cross-section between the mid-blank and the punch
the aspect ratio lD is higher, the maximum thinning trends to grow during the forming process. With compared results between the
linearly. various couple of geometric parameters (radius of the punch and
the die), it was confirmed that for a selected value of the final
geometry of the blank (lD), we can associate particular values
of punch section and fillet radius to avoid wrinkling and tearing
of the blank. In fact, if the choice of the following parameters
rsP=RsP / WD, rfP=RfP / WD, rfD=RfD / WD is not compatible
different an excessive thinning can occurred and lead to a tearing.
As it is known, tearing in the drawing mode occurs when the tensile
flow stress at a local neck exceeds the ultimate stress. In such
location the strain takes also its ultimate value and then it could
be considered as a forming limit. This important consideration for
general rectangular cross sections, has been incorporated by FE
DDP forming sequence, in which we maintain a punch speed
constant for the entire cases and we change the contact section
between punch, blank and die with a sensitivity analysis for
rectangular cups to the various combination of fillet and sections
radius which in turn minimizes the maximum thinning.
Fig. 26. Variation of the thinning along the diagonal path for In section 5, a limit of drawability according to aspect ratio
various aspect sP and lD =2 lD blank thickness tb is considered by controlling the material flow
and avoiding necking at the bottom corners of rectangular cup.
If rsD and rfP, are too small, sheet material does stick to the die and
cannot flow easily to the die cavity. This could be associated to a
failure of the process; (wrinkling and excessive amount of thinning).
As it was indicated, for larger aspect ratios lD, smaller initial
sheet blank thicknesses, the maximum thinning are being decreasing.
But, we note that above rsD=0.6, even though the maximum
thinning is slowed down with conditions above, the increase of rfP
ratio leads to a small growth of the maximum thinning instead of
decreasing. Indeed, great value of the rfP ratio above 0.7 can lead
to the development of local wrinkling phenomenon.

6. Optimisation
6. Optimization FEMFEM
study study
Fig. 27. Plastic deformation out in the thickness direction for the
following cases sP=1 and aspect ratio, lD =1.5 and 2 respectively Simulations of forming processes are increasingly used
by large and small companies in the early stage of a product
design. These simulations have become indispensable for the
development of products and the automation of this process itself.
Optimization of parameters such as die radius, blank holder
force, friction coefficient, etc., can be accomplished based on their
degree of importance on the sheet metal forming. In this investiga-
tion, a statistical approach called optimization mon-objective method
has been applied to design the process providing the best geometric
parameters which lead to the selected minimum failure criterion.

6.1. Summery of some

6.1 Summery of some parametric
parametric FE results
FE results
and discussion and discussion

The aim of this section is to outline the most important results

Fig. 28. Variation of the thinning versus the sP for various aspect presented in section 5, in order to summarize the principle
ratio lD geometric parameters that were the mostly affecting the drawability

Parametric Finite Element Analysis for a square cup deep drawing process 75
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

of the final product. In fact, after studying the effect of different statistical parameters such as: the mean, variance, standard
geometric parameters (rsP, rfD, rfP, rsD, lD, sP and tb) on the forming deviation, and dispersion. In fact, the interest of statistical
process and more specifically on the thinning phenomenon and representations lies on the fact that the lecture and interpretation
the thickness distribution along the critical diagonal path as of those results according to statistical variables give a best
mentioned by Figure 8a the following conclusions were underlined: meaning and enhance enlightenment.
x within a fixed value of the final geometry of the deep drawn
rectangular cup (wD, lD), it is possible to associate particular Cases study of the forming problem
values of section and fillet radius of the punch rsP to avoid The function chosen to perform the statistical study is called
wrinkling and tearing of the blank; "vartestn" via to the statistical Matlab toolbox. It allows the
x if rsD and rfP are too small, the material of the sheet sticks calculation of certain statistical parameters, comparison of variance
to the die and cannot flow easily into the cavity of the die of multiple samples using Bartlett's test with a graphical
which leads to the possible existence of wrinkles and excessive representation.
thinning, leading to the failure of the forming process; The synthax parameters are as follow, vartestn (X), vartestn
x for large values of lD, and low values of initial blank thickness (X, group), P = vartestn (...) are defined such as:
tb, the maximum thinning decreases. But beyond rsD=0.6, and x vartestn (X): using the Bartlett test to check the equal variance
with the increase of the rfP it is noted that we have an increase for the columns of the matrix X. Indeed, this is a test of the
of the maximum thinning. In fact, within a ratio rfP greater than null hypothesis H0 that postulate that the columns of the
0.7, local development of wrinkling phenomenon is developed; matrix X are of a normal distribution with the same variance,
x increasing of rsD usually causes the decrease of maximum against the alternative hypothesis Ha with columns of the
thinning, but a low value of punch fillet radius rfP associated matrix X of even distribution which have different variances.
with a high value of blank thickness tb leads to the thinning The result is displayed in graphical form with a table that
increase, so that local wrinkles can appear on the blank sheet; contains the values of statistical parameters;
x increase of punch stroke sP for rectangular initial blank sheet x vartestn (X, group) requires a vector argument X and a group
causes the existence of thinning. For values of punch stroke which can be variable, a vector row of character with one row
above the section width of the blank, some wrinkles may for each element of the matrix X. The values of the matrix
appear at the corner of the final geometry of the sheet metal. X are in the same group. This function tests the homogeneity
of variance in each group;
x P = vartestn (...) returns the p-value, i.e. the probability
6.2. Statistical analysis
6.2 Statistical analysis results ofresults
the FE model of observing the given result when the null hypothesis
of the FE model of homogeneity of variances is true. In cases where this value
is very small, there is a doubt on the validity of this hypothesis.
The FE numerical simulation of forming process such
as drawing process can provide a large amount of final drown Effect of the blank geometric shape and thickness tb on the
configurations based on multiple combinations of the different forming process
variable parameters of the model. To optimize the model behaviour In the following example we set lD =1.2 mm and it has
and also to save cost of the big amount of time calculation, it is changed the value of the initial thickness of the blank tb in the
almost versatile to hold up with statistical techniques. In fact, rage of (0.8, 1.2, 1.4, 1.6). Taking into account all these data,
statistical computing is essential when seeking optimized solution we obtain the results shown by Figures 29a and b.
to reduce costs and manufacturing time. This kind of calculation The graphic representation illustrated by Figure 29b consists
is actually done using the well known statistical Matlab tool box. on a schematic rectangular representation called “box plot”. This
The statistical study is based on the calculation of certain representation is one way to approach the statistics summery
statistical parameters such as: (mean, variance, standard deviation, concepts. In fact, it can summarize data in a very visual outcome
median, correlation.) and summaries results with standard graphics see Figure 29 and easily compare various statistical variables.
(histogram, box plot, chart points ...). This representation is located in a two landmark axes; the samples
Indeed, the interest of statistical representations lies in the fact group, axis and the axis containing all values of the samples. For
of presenting the influence of several variables in a well extended each group, a “box plot” which presents some statistical
spectrum of values; against a limited one as in it was described parameters such as:
in the parametric study for described in the previous section. x the middle – it divides the data into two equal sets;
x quartile – the quartiles of statistical series are the three values
Q1, Q2 and Q3 of character who share the population into four
6.3. Statistics problem
6.3 Statistics problem parts of the same size;
x the inter-quartile range – the difference between the upper and
The graphs of variations in thickness and thinning rates lower quartiles (Q3 - Q1) and also indicates the dispersion
presented in the previous section 5, gives a general idea on the of a dataset;
deep drawing model behaviour within variation of various x group – group of diffrent tb variables;
parameters. Indeed, the interpretations were based on the decrease x count – number of values in the vector tb: (thickness values
or increase of the thickness and rate of thinning of the blank along the diagonal path), this number is also the length of the
without calculating the limit deviations of such variations. Using vector thickness tb;
mathematical tools such statistical appropriate functions, we have
xi  m ;
2
x STd – standard deviation: S
¦
N
interpreted statistically numerical results defined in the section 5.
The statistical study is based on the calculation of certain i 1
N 1

76 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

x mean – mean value of each group that is defined with the than the risk of error Į (0.05), this fact confirms that the
N xi hypothesis H0 is well justified. Certainly, the indicator of dispersion
folowing equation:; m
¦i 1 N (standard deviation) varies from one sample to another depending
x Barletts statistic: statistical test of Barlett on the group, so according to the value of the initial thickness
of the blank tb. Therefore, the samples presented in Figure 29b
( N  k ) ln S p2  ¦i 1 ( N i  1) ln Si2
k
haven’t the same variance. It is concluded that for rectangular
T
1  (1 / 3(k  1))(¦i 1 (1 / N i  1)  1 /( N  k )) final geometry of the blank with lD > 1, and with initial law
k

thickness tb, there is important thinning of the deformed blank.

with: Si² – the variance of ith Group, N – total length of the
discretized and weighted simples, Ni – length of the vector Effect of the section radius of the punch rsP parameter on the
corresponding to the group i, k – group’s number, blank thinning rate
Sp² – pooled variance: mean weighted of the group variance In order to visualize the influence of rsP parameter on the
¦
k
i 1
( N i  1) Si2 thickness distribution of a blank, we have considered different
Variances are considered unequal if T> X²(Į,k-1) with X²(Į,k-1) values rsP with for the tb varying in the range from 0.2 to 0.8 and
is the biggest critical distribution of Chi square (Į=significant an aspect ratio equals to 1.2. Results are then obtained with
threshold, k-1: degree of freedom, and k is the number Matlab as shown it in Figures 30a and b.
of samples).
x P-value – is related to Į (for great values of P, the hypothesis a)
H0 is true, otherwise this hypothesis is false).
a)

b)

b)

Fig. 30. a) Thickness variation of the blank with lD=1, tb=1.2 and
rsP=0.2, 0.4, 0.6 and 0.8; b) summary table for the same
conditions as in Figure 30a

The results presented in Figure 30a show that the 4 samples of

the parameter rsP have almost the same inter-quartile; we can say
Fig. 29. a) Thickness variation of the blank with lD=1.2 and tb=0.8; that they are compatible for a certain range of thickness. We have
1.2; 1.4 and 1.6 mm; b) summary table thickness variation of the noticed from the value p-value (Figure 30b) that also the same
blank with lD =1.2 and tb=0.8; 1.2; 1.4 and 1.6 mm samples have the same variances. We also note from these figures
that as far as the parameter rsP decreases the thinning of the final
In Figure 29a, the x-axis which represents the different groups, blank sheet increases. In conclusion, the thinning rate is high for
defines the initial blank thickness tb, and the y-axis values low values of rsP in the case of a square plate and initial thickness
represent the different thickness ranges of the blank along the tb=1.2 mm.
diagonal path during the forming process. From this graph, we see
that the difference between the maximum and minimum thickness Effect of the fillet radius of the die rfD parameter on the blank
of the blank at the end of the process along the diagonal path thinning rate
is almost obtained with law thicknesses of the blank. Indeed, for In this example, we will study the influence of the rfD variation
tb=0.8 mm, we notice that the stamped sheet metal has undergone on the thinning rate of the blank for a square initial sheet blank with
less thinning much lower than it is for the other tb cases. From tb=1.2 mm. In this case rsp=0.4, lD=1 and tb=1.2 mm, rfD is varying
Figure 29b we notice that the p-value is equal to 0, which is lower from 0.4 to 0.8. We obtained the following results (Figure 31).

Parametric Finite Element Analysis for a square cup deep drawing process 77
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

a) a)

b) b)

Fig. 31. a) Blank thickness for rfD=0.4, 0.6 and 0.8 with rsP=0.4,
lD=1 and tb=1.2 mm; b) group summary table of data related to the
Figure 31a
Fig. 32. a) Blank thickness for rsD=0.4, 0.6 and 0.8 with rfP=0.4,
It is concluded from Figure 31, that the lowest mean value lD=1 and tb=1.2 mm; b) group summary table with blank thickness
is (1.1656) that is corresponding to the 1st group (rfD=0.4). for rsD=0.2, 0.25, 0.4, rfP=0.1, lD=1 and tb=1.2 mm
We notice that according to this statistical parameter, the highest
thinning is found for rfD=0.4, we can also notice that as far as this a)
parameter increases the thinning is spectacularly diminishing.
We can conclude that with a square DDP with initial blank
thickness of 1.2 mm more the parameter rfD increases, more the
thinning is decreasing.

Effect of the section radius of the die rsD parameter on the

blank thinning rate
From Figure 32a, we can see that the formed metal reaches
a maximum rate of thinning (thickness <1 mm) for rsD=0.4.
In conclusion, for a square plate, more the parameter rsD increases,
there is a chance to get important thinning.
Figure 33a shows that the minimum variations in thickness
corresponds to rfD=0.2. We can conclude that a high value of rfD
minimizes the risk of excessive thinning leading to failure.

7. Description
7. Description of optimisation
of optimization problem b)
problem
The improvement and the cost reduction in forming process
products has been always a major objective in automotive industry.
In a forming process, the sheet metal is subjected to mechanical
tools action; punch, die and blank holder. These tools are generally
considered as rigid bodies, causing contact actions, the deformation
of the sheet along a well defined kinematic. The normal and
tangential interactions due to contact between tools and sheet metal
are taken into account. The coefficients of friction blank-tools
have a great influence on the process development and its quality. Fig. 33. a) Variation of rfP=0.1, 0.2, 0.4 with rsD=0.25, lD=1 and
Taking into account all these considerations and from finite element tb=1.2 mm; b) group summary table

78 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

calculations performed by Abaqus, we used the values of blank

thickening in deep drawing along the diagonal path of sheet metal
to judge the quality and the acceptability of the final formed
product. In this study, a criterion of maximum thinning tolerance
of 20% is adopted. We chose this criterion because the risk of
developing structural defects resulting from thinning as, wrinkles,
breaks, tears, is mostly high when thinning reaches 20%.
However, according to formatting examples presented in the
previous section, the numerical predictions are far from
experimental realities. For this reason, we chose an approximation
method for optimization of geometrical parameters of the drawing
process such as the different radii of the die and the punch.

7.1. Results
7.1 and discussion
Results and discussion
After studying the effects of different geometric parameters Fig. 34. DDP of a rectangular profile
such as (rSP, rFD, rfp, rsD, lD, and SP, tb) on the forming process,
specifically on the thinning phenomenon and the thickness
distribution along the critical diagonal path; the following
conclusions are considered:
x according to a final geometry dimension of the clank lD,
we can associate particular values of the tools radii to avoid
wrinkling and tearing of the blank. In fact; several
Remarque’s are underlined;
x if rfp is too small, the material of the sheet sticks to the die
matrix and cannot flow easily into the of matrix cavity, which
leads to the appearance of wrinkles and excessive thinning;
x for large values of lD, and low values of initial blank thickness
tb, the maximum thinning decreases. But for rsD=0.6, and with
the increase of rFP thinning was growing up instead
of decreasing. A radius rfp smaller than 0.7 leads to the
development of local wrinkles;
x the fact of increasing rsD usually causes the decrease of Fig. 35. High thinning with rfD=0.6; rsP=0.7; tb=1.2, lD=1.2
maximum thinning;
x a low value of the fillet punch radius rfp associated with a high
value of blank thickness leads to increased thinning and the
possible appearance of wrinkles;
x increase of Sp for rectangular plates causes the appearance
of thinning. For values of punch travel above the section
width of the blank, some wrinkles may appear at the corner
of the sheet metal after forming.
On the light of these interpretations we have reviewed the
following examples to show the thinning distribution of the final
formed product according to highlight the particular combinations of
geometric parameters that can lead to excessive thing and wrinkling.

Effect of rsD and rfP on the forming process

rfp=0.2 ; rsD=0.3; tb=1.2; lD=1.4: law values with rfP and rsD,
we have reported that for rfP>0.1 and rsD<0.6 we have less
thinning and les wrinkling as shown by Figure 34 with rfP=0.2; Fig. 36. Thinning reductions in the critic path; rsP=0.4, rfD=0.6,
rsD=0.3; tb=1.2; lD=1.4. tb=1.2 and lD=1

Effect of rfD and rsP on the forming process 8. Optimisation problem

In this case, we have considered the simulation of the following 8. Optimisation
of the DDP problem of the DDP
parameters: rfD=0.6; rsP=0.7; tb=1.2, lD=1.2, with high values
of parameters rsP and rfD, it is shown that the thinning is more 8.1. Optimisation
8.1 Optimization method
method
important in this case (Figure 35).
rsP=0.4, rfD=0.6, tb=1.2 and lD=1, for a square blank sheet with Basic concept of optimization techniques
law values of the parameters rsP and rfD on a reduction of thinning The mathematical concept of optimization is presented by
according to Figure 35 is shown in Figure 36. Figure 37. It is composed mainly of two key phases: modelling

Parametric Finite Element Analysis for a square cup deep drawing process 79
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

and solving the optimization problem. The modelling phase The algorithm therefore consists in the following expression:
consists of: xk +1 = xk-Į kǻf (xk)
1. selecting a number of variables where the user is authorized with Į, the step taken in the direction of the highest slope.
to adjust, The stopping criterion may include a tolerance on the variation
2. choosing an objective function, of the cost function, a tolerance on the variation of x, a tolerance
3. taking into account the possible constraints. value of the gradient, a maximum number of iterations or
a maximum number of evaluations.
 The effectiveness of this method is low. It can be shown that
two consecutive directions will be orthogonal and that this feature
may cause oscillations and lead to the divergence of the algorithm.

Choice of used method

According to previous sections, it was concluded that
by taking some basic precautions into account, we are able to finally
reproduce fairly well with a finite element simulation a forming
process operation. thinning phenomenon is one of the most
Fig. 37. The concept of basic mathematical optimization [16] difficult to control during the development of stamping operations,
because many parameters such as the geometry of tools, stamping
According to [15], most research has focused on solving some speed, lubrication tools and so influence the geometry and
optimization problems, where the selection and application residual stress state of the final product. This methodology
of an optimization algorithm can be adapted. Accordingly, is devoted primarily to the geometric configuration of the forming
the application of optimization techniques to the forming process equipment and determining target parameters during optimization.
of a particular metal requires a large expertise. However, most The parameters are chosen to optimize the rays of the die and the
professionals in the process of forming lack this expertise, which punch as those factors are most sensitive in the forming process.
is an obstacle to fully exploit the potential of forming process In general for a problem of single-objective optimization,
optimization. To overcome this obstacle, it is necessary for an we define an objective function (response function), we seek
optimization strategy, in forming process to adopt a structured to optimize with respect to involved parameters. The objective
approach that can solve major problems of metal forming. function chosen in our case is the difference between the reference
The subsequent sections are devoted to describe the optimization curve of the initial thickness of the plate and thickening of the
approach used in this work to enhance the metal forming process curve obtained after numerical simulation via Abaqus explicit,
defined in previous sections. In the following scheme we are this function is called monobjective function.
describing the different steps of resolution adopted in this work. Let f (x) the objective function described as:
f(x) = e-y(x)
with: e – initial thickness of the blank; y(x) – thickness function,
8.2. Optimisation
8.2 Optimization methods
methods in forming process x – x coordination along the diagonal path.
in forming process We have applied optimization of mild steel deep drawing
process, with initial blank thickness of 1.2 mm.
There are several methods to optimize forming processes such
as: Newton method, genetic algorithms, design of experiments Looking for the thickening function
and Tagauchy techniques. These methods are used directly to deal In the previous chapter, the FE numerical simulation using
with problems mathematically modelled based on mechanical Abaqus software has allowed us to extract the thickening curves
models, or indirectly by example in learning sequences through for different cases. However, it was not possible to extract the
artificial neural networks. equations for these curves via Abaqus software. For this reason,
we have chosen to use a mathematical curve fitting program
Classic method included in Matlab software to provide an approximate equation
of the curve thickening.
Principle
The minimization of a function called cost function or objective
function is the most used for the optimization of forming process.
This function depends on several parameters that affect the
calculation of first and second derivatives of the function. The number
of function parameters to minimize also affects the number
of iterations to obtain a solution that solves the whole problem [16].
Among the conventional common methods using the gradient
of the function we are explaining briefly in the following
paragraph the principle of this technique.
Considering f(x) the cost function to be minimized and ǻf the
gradient of this function. The algorithm will therefore seek
to construct a sequence of points x1 x2 ... xk x3, such as indicated
by [16]:
f (xk +1) <f (xk) Fig. 38. Thickening vs. x distance along the diagonal path

80 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

Example: Here above is a thickening curve shown in Figure 38 x R-square: the square of the correlation between input values and
obtained with: rsD=0.25; rfP=0.1; tb=1.2 and lD=1. predicted values after adjustment. A value closer to 1 indicates
To know the equation describing this curve we extracted first a good correlation between the actual and fitted values;
the coordinate values of points forming the curve. x adjusted R-square: it is the degree of freedom of R-square.
The values are then saved in a text file. After finishing with A value closer to 1 indicates a better fit;
the numerical simulation by Abaqus, then data are imported to the x RMSE: the root of the mean errors. A value closer to 0 indicates
Matlab workspace for example. Coordinate values of the thickness a good fit with less error.
curve as shown in Figure 39. Because our goal is to study the rate of thinning in a formed
blank, we choose the values of thickness variations which are less
than the initial thickness of the blank (1.2 mm). Indeed, areas
at the final product whose thickness exceeds the value of the initial
thickness of the blank, and undergo high thickening.
Then, we obtained for these values of thickness, the curve shown
by Figure 41. In this case, the corresponding equation proposed by
Matlab is an order 7 Gaussian curve with corresponding coefficients:
Adjusted R-square and ESS are respectively closer to 1 and 0.

Fig. 39. Thickness distributions via Matlab software Fig. 41. Thinning curve
The curve presented by Figure 39 is built from the coordinates The best equation corresponding to the represented curve is as
of the thickness distribution extracted at first from FE results. follows:
Indeed, in the next step we will choose the most suitable
equation that overlies the shape of the curve formed by point list y(x)=a1*exp(-((x-b1)/c1)^2) + ... + a7*exp(-((x-b7)/c7)^2
previously defined among all equations proposed by Matlab.
Below are presented the equation coefficients with
a confidence equal to 95%.
a1=0.4232 (-3.359, 4.206); b1=56.95 (26.06, 87.85); c1=19.18
(-58.1, 96.47)
a2=1.105 (1.067, 1.144); b2 = 122 (77.07, 167); c2=103 (-501.1, 707.2)
a3=0.04211 (-0.7084, 0.7926); b3=97.21 (22.51, 171.9); c3=12.88
(-57.95, 83.71)
a4=0.04122 (-0.3036, 0.386); b4=87.12 (74.04, 100.2); c4=8.848
(-10.05, 27.75)
a5=-0.0006871 (-0.06063, 0.05926); b5=88.92 (16.41, 161.4);
c5=0.6008 (-47.75, 48.96)
a6=-0.01663 (-0.06769, 0.03444); b6=75.42 (74.6, 76.24);
c6=2.121 (-0.2897, 4.532)
a7=-0.006177 (-0.06756, 0.05521); b7=78.68 (58.73, 98.64);
c7=3.954 (-23.07, 30.98)
Fig. 40. Fitting results and statistics parameters The best fit of this curve is obtained with the following equation

To verify the correct choice of the equation, there are y(x)=a1*exp(-((x-b1)/c1)^2) + ... + a7*exp(-((x-b7)/c7)^2
statistical parameters that help us to take the right decision. These
sets are listed in the fitting results window (Figure 40). They are Below are giving the equation coefficients with a confidence
represented as follows [16-18]: equal to 95%.
x SSE (Sum of Square due to Error) is the sum of squared errors a1=0.4232 (-3.359, 4.206); b1=56.95 (26.06, 87.85); c1=19.18
of adjustment. A value closer to zero indicates a successful (-58.1, 96.47)
adjustment; a2=1.105 (1.067, 1.144); b2=122 (77.07, 167); c2=103 (-501.1, 707.2)

Parametric Finite Element Analysis for a square cup deep drawing process 81
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

a3=0.04211 (-0.7084, 0.7926); b3=97.21 (22.51, 171.9); c3=12.88 a4=-0.04269 (-0.3768, 0.2914); b4=81.04 (65.63, 96.45);
(-57.95, 83.71) c4=7.37(-16.51, 31.25)
a4=0.04122 (-0.3036, 0.386); b4=87.12 (74.04, 100.2); c4=8.848 a5=-0.04958 (-0.2447, 0.1455); b5=75.1 (72.75, 77.45); c5=4.8
(-10.05, 27.75) (1.245, 8.356)
a5=-0.0006871 (-0.06063, 0.05926); b5=88.92 (16.41, 161.4); a6=-0.00895 (-0.1945, 0.1766); b6=89.27 (-74.38, 252.9);
c5=0.6008 (-47.75, 48.96) c6=10.83 (-96.06, 117.7)
a6=-0.01663 (-0.06769, 0.03444); b6=75.42 (74.6, 76.24); with the following statistics parameters:
c6=2.121 (-0.2897, 4.532) x SSE: 3.76exp5
a7=-0.006177 (-0.06756, 0.05521); b7=78.68 (58.73, 98.64); x R-square: 0.9992
c7=3.954 (-23.07, 30.98) x Adjusted R-square: 0.9989
x RMSE: 0.001008
Optimization function
After determining the equation of the thickness curve, the f(x) is minimal for x=113.5970 and its value is equal to:
objective function becomes equal to
f(x)=1.2- y(x) =a1*exp(-((x-b1)/c1)^2) + ... +
f(x)=1.2-a1*exp(-((x-b1)/c1)^2) + ... + a7*exp(-((x-b7)/c7)^2 a6*exp(-((x-b6)/c6)^2 = 0.0972.

To optimize this objective function, we have used the function rsD=0.25 and rfP=0.4:
"fminbnd" performed in Matlab optimization toolbox which The expression of the equation of thickness curve is done by:
allows the minimization of a function in one variable in a fixed y(x) = a1*exp(-((x-b1)/c1)^2) + ... + a8*exp(-((x-b8)/c8)^2
interval. The minimization function is then saved as an M-file and
it is done with the following variables: with the following equation coefficients verifying a confidence
x=fminbnd (@(x) Abaqus (x,a1,b1,c1, a2,b2,c2, a3,b3,c3,a4,b4,c4, interval of 95%:
a5,b5,c5, a6,b6,c6 ,a7,b7,c7), 61.3441,120) a1=0.08261 (0.002547, 0.1627); b1=59.03 (57.8, 60.26);
when we calculate the value of this minimized function we find: c1=9.908 (7.012, 12.8)
f=0.0938 is thus obtained for x=107.1561 mm, f (x)=0.0938 a2=0.2163 (-1.124, 1.557); b2=49.75 (25.33, 74.17); c2=27.33
The value found for f (x) becomes minimal. Otherwise for (-24.02, 78.69)
a distance 107.1561 mm we have less thinning. a3=1.181 (1.006, 1.356); b3=-2.879 (-37.2, 31.44); c3=83.14
By following the same steps as in the previous example, we (-143, 309.2)
calculate the functions minimized for different values of geometric a4=1.011 (-0.4632, 2.484); b4=127.5 (59.19, 195.7); c4 = 59.7
parameters rsD, RfP and RsP, RfD. By comparing the values of f(x) (22.14, 97.26)
obtained for different cases, we choose the lowest. Indeed, the a5=0.02191(-6.857*1013,6.857*1013);
objective function is minimal in our case a low rate of thinning. b5=92.17(-5.312*1014, 5.312*1014);
Therefore, the risk of occurrence of these defects decreases for c5=0.1333(-3.626*1014, 3.626*1014)
selected parameters. It then derives the optimal values of these a6=0.04702 (0.03598, 0.05806); b6=69.88 (69.45, 70.31);
geometric parameters: radii of punch and die. c6=4.827 (4.127, 5.526)
a7=0.03335 (-0.02731, 0.09401); b7=85.62 (84.82, 86.42);
c7=1.104 (-0.1832, 2.391)
9. Search of optimal
9. Search of optimal geometric a8=0.05877 (-0.1257, 0.2432); b8=93.79 (90.54, 97.05); c8=13.04
geometric
parameters parameters (2.518, 23.56)
and with the following statistics parameters:
x SSE: 6.083 exp (-5)
9.1. Search
9.1 Search of theofoptimal
the optimal
RsD and rfP values x R-square: 0.9988
RsD and rfP values x Adjusted R-square: 0.9978
x RMSE: 0.001424
Taking into account the previous example, we'll continue to
look for other values and RsD rfP equations of thickness and objective f(x) is minimum for x=116.6013 mm and its value in this case
functions to determine the optimal values of these parameters. is equal to:
rsD=0.4 and rfP=0.1
f(x)=1.2- y(x) =a1*exp (-((x-b1)/c1)^2) + ... +
by adopting a similar methodology as in the previous example, we
a8*exp(-((x-b8)/c8)^2 = 0.0701
have determined the thickness equation as follows.
rsD=0.4 and rfP=0.4
y(x)=a1*exp (-((x-b1)/c1)^2) + ... + a6*exp (-((x-b6)/c6)^2
The expression of the equation of thickness curve is done by:
with the following coefficients verifying a confidence value of y(x) = a1*exp(-((x-b1)/c1)^2) + ... + a8*exp(-((x-b8)/c8)^2
95%:
a1=0.1233 (-0.96, 1.207); b1=47.88 (-119.8, 215.6); c1=23.35 with the following equation coefficients verifying a confidence
(-114, 160.7) interval of 95%:
a2=1.102 (1.092, 1.113); b2=111.1 (-76.69, 298.9); c2=409.2 a1=0.2897 (-0.9158, 1.495); b1=65.51 (57.54, 73.48); c1=9.022
(-4854, 5672) (-4.807, 22.85)
a3=-0.01318(-0.02148, -0.004891); b3=80.97 (80.73, 81.21); a2=1.123 (1.091, 1.154); b2=121.5 (89.44, 153.6); c2=109
c3=1.791(0.9269, 2.656) (-161.5, 379.5)

82 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

a3=0.04296 (0.03036, 0.05557); b3=85.75 (85.45, 86.04); a1=3.561exp5 (-1.926exp10, 1.926exp10); b1=-23.88 (-3.246exp5,
c3=1.51(0.7563, 2.263) 3.245exp5)
a4=0.008928 (0.003119, 0.01474); b4=92.54 (92.04, 93.04); c1=21.45 (-4.56exp4, 4.565exp4); a2=1.306 (-8.893exp4,
c4=1.22(0.1885, 2.251) 8.893exp4)
a5=0.002508 (-0.002666, 0.007682); b5=95.7 (93.4, 98); b2=223.6 (-2.548exp7, 2.548exp7); c2=330.5 (-1.509exp7,
c5=2.061 (-1.371, 5.493) 1.509exp7)
a6=0.02983 (-0.03026, 0.08993); b6=80.13 (78.93, 81.33); a3=-0.01705 (-0.02054, -0.01356); b3=75.5 (75.38, 75.63);
c6=3.19 (0.268, 6.113) c3=1.28 (0.9817, 1.578)
a7=0.05673 (-0.5539, 0.6673); b7=90.55 (33.45, 147.6); c7=16.27 a4=0.02843 (-52.13, 52.18); b4=107.2 (-2887, 3101); c4=17.34
(-37.41, 69.95) (-6722, 6756)
a8=0.01087 (-0.006902, 0.02865); b8=76.48 (75.28, 77.67); a5=0.145 (-45.07, 45.36); b5=64.37 (-703.4, 832.1); c5=10.47
c8=1.868 (0.5105, 3.225) (-523, 543.9)
The statistics parameters: a6=0.01094 (-0.009594, 0.03148); b6=86.76 (85.79, 87.73);
x SSE: 3.68 exp (-5) c6=3.395 (1.021, 5.768)
x R-square: 0.9992 a7=0.02533 (-1.735, 1.785); b7=92.57 (42.49, 142.7); c7=9.187
x Adjusted R-square: 0.9985 (-103.8, 122.2)
x RMSE: 0.001127 a8=-0.1288 (-5.514exp4, 5.514exp4); b8=167.3 (-1.126exp7,
1.127exp7)
f (x) is minimum for x=117.6832 mm and its value is done by: c8=84.72 (-1.043exp7, 1.043exp7)
f(x)=1.2- y(x) = a1*exp(-((x-b1)/c1)^2) + ... + The statistics parameters are:
a8*exp(-((x-b8)/c8)^2 = 0.0749 x SSE: 0.0001832
x R-square: 0.9982
rsD=0.25 and rfP=0.2 x Adjusted R-square: 0.9978
The expression of the thickness curve in this case is done by: x RMSE: 0.001427
y(x)=a1*exp(-((x-b1)/c1)^2) + ... + a8*exp(-((x-b8)/c8)^2 f (x) is minimal for x=113.0403 and its value in this case is equal to:
with the following coefficients verifying a confidence interval
of 95%: f(x)=1.2- y(x) =a1*exp(-((x-b1)/c1)^2) + ... +
a1=0.184 (-9.013, 9.381); b1=65.78 (-249.3, 380.9); c1=10.08 a8*exp(-((x-b8)/c8)^2 = 0.0922
(-127.5, 147.6) The previous results for the objective minimized functions f(x)
a2=1.109 (1.109, 1.11); b2=112.5 (110.7, 114.2); c2=174.7 (84.74, are resumed in the following comparative Table 2.
264.7)
a3=0.01317 (-0.1128, 0.1392); b3=87.55 (85.46, 89.64); c3=2.182 Table 2.
(-3.562, 7.926) Comparison between values of optimization functions
a4=0.05056 (-0.8239, 0.9251); b4=78.72 (72.03, 85.4); c4=3.801 rsD 0.25 0.4 0.25 0.4 0.25 0.2
(-13.13, 20.74) rfP 0.1 0.1 0.4 0.4 0.2 0.1
a5=-0.1224 (-9.81, 9.565); b5=75.22 (-335.7, 486.1); c5=9.535 f(x) 0.0938 0.0972 0.0701 0.0749 0.0910 0.0922
(-174.1, 193.2)
a6=0.04234 (-0.2425, 0.3272); b6=72.55 (70.14, 74.96); c6=2.922 According to this table, it is shown that the lowest value of f(x)
(-0.5512, 6.396) is equal to 0.0701 corresponding to RsD=0.25 and RfP=0.4. For
a7=0.005245 (-0.1151, 0.1256); b7=91.09 (80.96, 101.2); these two values of the radii of die and punch there is less
c7=2.504 (-10.01, 15.02) thinning and high risk of defect occurrence within the end of the
a8=0.002789 (-0.06102, 0.0666); b8=96.13 (42.23, 150); forming process. We can say that to optimize the forming process
c8=4.521 (-27.41, 36.46) for a predefined material, it is preferable to choose the following
The statistics parameters are: values of RsD rfP, to minimize the problems of thinning as follows
x SSE: 2.636exp-005 rsD=0.25 and rfP=0.4
x R-square: 0.9996 In conclusion, at this stage of optimization, mathematical
x Adjusted R-square: 0.9993 modeling show a decrease in the rate of thinning during forming
x RMSE: 0.0009373 process for rfP > 0.1 and RsD <0.6, according to initial geometric
f(x) is minimal for x=112.5001 mm and its value in this case is parameters considered in this problem.
done by:
f(x)=1.2- y(x) =a1*exp(-((x-b1)/c1)^2) + ... + 9.2. Search
9.2 Search of theofoptimal
the optimal
rsP and rfD values.
a8*exp(-((x-b8)/c8)^2 = 0.0910 rsP and rfD values
rsD=0.2 and rfP=0.1
The equation of thickness curve is done by: rsP=0.4 and rfD=0.25:
Following the same steps as it was defined in the previous
y(x) = a1*exp(-((x-b1)/c1)^2) + ... + a8*exp(-((x-b8)/c8)^2 section. We have looked for the equation of y(x), and it was
defined as:
with the following coefficients corresponding to a confidence
interval of 95%: y(x)=a1*exp(-((x-b1)/c1)^2) + ... + a6*exp(-((x-b6)/c6)^2

Parametric Finite Element Analysis for a square cup deep drawing process 83
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

with the following coefficients that are done with a confidence with the following coefficients that are done with a confidence
interval of 95%: interval of 95%:
a1=0.5184 (-1.462, 2.499); b1=62.29 (54.02, 70.56); c1=10.08 a1=0.2302 (-2.65, 3.11); b1=62.05 (16.38, 107.7); c1=16.41
(-3.152, 23.31) (-21.98, 54.79)
a2=1.102 (1.021, 1.183); b2=120.4 (85.83, 155); c2=77 (-150.5, a2=0.467 (-7.568, 8.502); b2=45.89 (-13.71, 105.5); c2=30.61
304.5) (-136.9, 198.1)
a3=0.1308 (-1.291, 1.553); b3=87.15 (32.2, 142.1); c3=17.65 a3=1.166 (0.1066, 2.226); b3=-8.196 (-136.6, 120.2); c3=62.93
(-38.06, 73.37) (-418.4, 544.3)
a4=-0.0149 (-0.01998, -0.009813); b4=78.87 (78.72, 79.03); a4=0.1311 (-1.149, 1.411); b4=75.2 (55.89, 94.5); c4=8.671
c4=1.698 (1.266, 2.131) (-6.359, 23.7)
a5=0.02207 (-0.0006604, 0.0448); b5=88.25 (87.27, 89.23); a5=1.098 (-0.06019, 2.257); b5=122 (52.14, 191.8); c5=50.6
c5=4.342 (2.762, 5.921) (-245.7, 346.9)
a6=0.141 (-0.0693, 0.3514); b6=74.45 (72.48, 76.42); c6=6.463 a6=-1.61 (-3848, 3844); b6=90.09 (-12.66, 192.8); c6=6.076
(3.651, 9.275) (-131.1, 143.2)
The statistics parameters are done by: a7=0.1149 (-2.247, 2.477); b7=97.72 (35.94, 159.5); c7=12.14
x SSE: 2.208e-005 (-58.56, 82.84)
a8=1.721 (-3844, 3847); b8=90 (-25.39, 205.4); c8=6.195 (-132.3,
x R-square: 0.9997
144.7)
x Adjusted R-square: 0.9995 The statistics parameters are done by:
x RMSE: 0.0007832 x SSE: 6.142 exp(-5)
The function f(x) is minimized for x=116.1692 mm, this x R-square: 0.998
means that there was minimum thinning after the sheet forming x Adjusted R-square: 0.9965
at the distance x=116.1692 mm and at this distance: x RMSE: 0.001408

f(x)=1.2- y(x) =a1*exp(-((x-b1)/c1)^2) + ... + The function f(x) is minimized for x=114.5862 mm and f(x)
a6*exp(-((x-b6)/c6)^2 = 0.0928 is done with:
f(x)=1.2- y(x) =1.2-a1*exp(-((x-b1)/c1)^2) + ... +
rsP=0.4 and rfD=0.4:
a8*exp(-((x-b8)/c8)^2=0.0608
The equation of thickness curve is done by:
y(x)=a1*exp(-((x-b1)/c1)^2) + ... + a7*exp(-((x-b7)/c7)^2 rsP=0.4 and rfD=0.8:
The equation of thickness curve is done by:
with the following coefficients that are done with a confidence
y(x)=a1*exp(-((x-b1)/c1)^2) + ... + a8*exp(-((x-b8)/c8)^2
interval of 95%:
a1=0.2302 (-2.65, 3.11); b1=62.05 (16.38, 107.7); c1=16.41 with the following coefficients that are done with a confidence
(-21.98, 54.79) interval of 95%:
a2=0.467 (-7.568, 8.502); b2=45.89 (-13.71, 105.5); c2=30.61 a1=0.08399 (-0.1323, 0.3002); b1=54.02 (36.61, 71.43); c1=24.35
(-136.9, 198.1) (6.695, 42.01)
a3=1.166 (0.1066, 2.226); b3=-8.196 (-136.6, 120.2); c3=62.93 a2=1.198 (1.193, 1.204); b2=1.391 (-7.908, 10.69); c2=214.9
(-418.4, 544.3) (-122, 551.8)
a4=0.1311 (-1.149, 1.411); b4=75.2 (55.89, 94.5); c4=8.671 a3=0.2938 (-0.674, 1.262); b3=129.9 (111.4, 148.4); c3=33.79
(-6.359, 23.7) (-23.17, 90.76)
a5=1.098 (-0.06019, 2.257); b5=122 (52.14, 191.8); c5=50.6 a4=0.01307 (-0.008136, 0.03427); b4=75.81 (61.58, 90.04);
(-245.7, 346.9) c4=6.714 (-2.881, 16.31)
a6=-1.61 (-3848, 3844); b6=90.09 (-12.66, 192.8); c6=6.076 a5=-0.03196 (-0.06004, -0.00388); b5=80.54 (80.02, 81.06);
(-131.1, 143.2) c5=3.92 (2.943, 4.897)
a7=0.1149 (-2.247, 2.477); b7=97.72 (35.94, 159.5); c7=12.14 a6=0.05901 (-0.2239, 0.3419); b6=93 (85.41, 100.6); c6=17.5
(-58.56, 82.84) (-3.022, 38.02)
The statistics parameters are done by: a7=0.006954 (-0.01532, 0.02923); b7=67.47(58.15, 76.79);
x SSE: 2.369e-005 c7=5.481 (-0.4445, 11.41)
x R-square: 0.9995 a8=0.04047 (-0.7717, 0.8526); b8=72.65 (69.85, 75.45); c8=6.073
x Adjusted R-square: 0.9992 (-23.08, 35.23)
The statistics parameters are done by:
x RMSE: 0.0008472
x SSE: 1.541e-005
The function f(x) is minimized for x=115.00 mm x R-square: 0.9991
x Adjusted R-square: 0.9985
f(x)=1.2-a1*exp(-((x-b1)/c1)^2) + ... + a7*exp(-((x-b7)/c7)^2 = x RMSE: 0.0006732
0.0790
The function f(x) is minimized for x=114.6336 mm, with f(x)
rsP=0.4 and rfD=0.6: is done by the following
The equation of thickness curve is done by:
f(x)=1.2- y(x) =1.2-a1*exp(-((x-b1)/c1)^2) + ... +
y(x)=a1*exp(-((x-b1)/c1)^2) + ... + a8*exp(-((x-b8)/c8)^2 a8*exp(-((x-b8)/c8)^2=0.0395

84 Research paper F. Ayari, E. Bayraktar

 Analysis and modelling

rsP=0.25 and rfD=0.25: f(x)=1.2- y(x) =1.2-a1*exp(-((x-b1)/c1)^2) + ... +

The equation of thickness curve is done by: a7*exp(-((x-b7)/c7)^2=0.0950
y(x)=a1*exp(-((x-b1)/c1)^2) + ... + a7*exp(-((x-b7)/c7)^2 for the different cases considered in this section, we will extract
optimal values of rsP and rfD parameters according to the
with the following coefficients that are done with a confidence comparative values in the Table 3 below.
interval of 95%:
a1=0.1444 (-0.05489, 0.3437); b1=59.18 (56.11, 62.25); c1=11.05 Table 3.
(7.398, 14.71) Comparison of optimization results
a2=0.2571 (-1.065, 1.579); b2=48.42 (23.3, 73.54); c2=24.93 rsP 0.4 0.4 0.4 0.4 0.25 0.6
(-14.48, 64.34) rfD 0.25 0.4 0.6 0.8 0.25 0.25
a3=1.19 (1.096, 1.285); b3=0.05335 (-27.16, 27.27); c3=82.72 F(x) 0.0928 0.0790 0.0608 0.0395 0.0905 0.0950
(-111.5, 277)
a4=0.9655 (-0.4893, 2.42); b4=124.1 (89.38, 158.7); c4=46.87 Comparing the results presented in the table, we notice that
(-14.58, 108.3) f(x) is minimal for rsP=RfD=0.4 and 0.8. We can conclude then the
a5=0.09067 (0.03569, 0.1457); b5=70.36 (69.59, 71.13); following optimal values:
c5=6.174 (5.017, 7.33) rsP=RfD=0.4 and 0.8
a6=0.1376 (-0.4407, 0.7159); b6=89.13 (80.58, 97.68); c6=15.42 We checked in the previous chapter for a square plate with
(-2.594, 33.44) thin rsP < RfD (more precisely rsP=RfD=0.4 and 0.6) there is less
a7=0.02248 (0.01368, 0.03129); b7=79.17 (78.78, 79.55); risk of developing thinning process during formatting. Therefore,
c7=2.233 (1.411, 3.055) with the optimal values of RsP and RfD, you get a quality product.
The statistics parameters are done by:
x SSE: 2.685 exp (-5)
x R-square: 0.9996 10. Conclusions
10. Conclusion
x Adjusted R-square: 0.9993
x RMSE: 0.0008887 In order to improve the comprehension of some experimental
The function f(x) is minimized for x=116.6566 mm with f(x) results that we may face in the industrial deep drawing process,
is done by the following a larger parametric analysis is needed. Analysis using FEM
parametric study is therefore indispensable. Nevertheless, a best
f(x)=1.2- y(x) =1.2-a1*exp(-((x-b1)/c1)^2) + ... + FEM model has to be at the same time the lower in the CPU cost
a7*exp(-((x-b7)/c7)^2=0.0905 time and closely the most representative to experimental cases.
In this study we have conceived a FEM model of parametric
rsP=0.6 and rfD=0.25: deep drawing process analysis, a spectrum consisting of 136
The equation of thickness curve is done by: geometries is used to assess a sheet metal DDP. As a first step, the
numerical simulation using dynamic explicit finite element
y(x) =a1*exp(-((x-b1)/c1)^2) + ... + a7*exp(-((x-b7)/c7)^2 analysis has been validated within existing experimental data.
This validation has concerned (MS material), different punch
with the following coefficients that are done with a confidence
travels, various blank thicknesses; a good correlation has been
interval of 95%:
noticed between experimental and simulated results.
a1 = 0.155 (-0.9685, 1.279); b1 = 58.28 (55.85, 60.7); c1 =
The second step which is the major objective of this study
10.51 (2.373, 18.65)
consists on a geometric parametric FEA study. In fact, even
a2 = 0.8499 (-1.344, 3.044); b2 = 40.92 (-14.37, 96.2); c2
though we have a large amount of numerical investigations in the
= 26.09 (-141, 193.2)
literature review through this last decade; a specific detailed
a3 = 1.096 (0.06832, 2.124); b3 = -6.846 (-41.14, 27.45);
parametric study that involves the most dominating deep drawing
c3 =28.57 (-166.7, 223.8)
parameters is well suitable at this stage. To emphasize interaction
a4 = 0.2629 (-2.187, 2.713); b4 = 74.28 (57.65, 90.9); c4
between the most fluctuant of this parameters we have choose
=12.88 (-11.7, 37.45)
to deal with the following.
a5 = 1.104 (1.085, 1.123); b5 =117.3 (106.8, 127.8); c5 =
In this paper, effect of the following geometric parameters
63.62 (-54.77, 182) (rsP, rfD, rfP, rsD, lD, sP and tb) and their interaction on drawability
a6 = 0.04622 (-0.03824, 0.1307); b6 = 90.81 (89.95, 91.66); c6 = of DDP are well analyzed. In particular, their sensitivity
5.024 (2.523, 7.525) to thinning phenomena and thickness distribution along critical
a7 =0.07014 (-0.2923, 0.4326); b7 = 97.29 (84.34, 110.2); c7 paths were obtained according to this trail;
= 9.485 (-7.38, 26.35) As it is known the increase of rsD parameter leads
The statistics parameters are done by: to diminishing of the maximum thinning in general, but it was
x SSE: 5.271e-005 observed that a law value of the fillet punch radius associated
x R-square: 0.999 with a high blank thickness leads to a growth of maximum
x Adjusted R-square: 0.9985 thinning and sometimes to wrinkling. In addition, the larger the
x RMSE: 0.001264 aspect ratio lD is, the smaller the maximum thinning particularly
The function f(x) is minimized for x=111.2032 mm, with f(x) for deep drawing punch travels. These results could be effectively
is done by the following applied to produce successful DDP.

Parametric Finite Element Analysis for a square cup deep drawing process 85
 Journal of Achievements in Materials and Manufacturing Engineering Volume 48 Issue 1 September 2011

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