The paper is aimed to present industrial applications of sheet stamping simulation using new finite element formulations developed in the International Center for Numerical Methods in Engineering in Barcelona. Theoretical formulation is briefly reviewed. Both continuum and shell elements have been considered. The new shell elements developed are based on a geometrically exact shell model treating the shell as one-director Cosserat surface. The formulation of the continuum elements employs the multiplicative decomposition of the deformation gradient tensor into its elastic and plastic parts. The new finite element models have been implemented in the in-house explicit dynamic code STAMPACK. A number of practical problems of sheet metalforming have been solved with the program. Some of the problems, namely stamping of a kitchen sink, hydraulic forming of an aeronautical part and stamping of a food can, have been presented in the paper. The examples give an idea of practical information that can be obtained from the computer simulation of a forming process. The results confirm a good behaviour of the formulation and program used in the industrial applications

Jovicevic, J.

Oñate, Eugenio

Rojek, Jerzy

English

Scipedia

ELSEVIER Journal of Materials Processing Technology 60 (1996) 243-247 
Journal of  
Materials 
Processing 
Technology 
Industrial applications of sheet stamping simulation using new finite element models 
J.Rojek, J. Jovicevic, E. Ofiate 
International Center for Numerical Methods in Engineering 
Univ. Politdcnica de Catalu~a 
Gran Capitdn, s/n, 0803,~ Barcelona, Spain 
Abst rac t  
The paper is aimed to present industrial applications of sheet stamping simulation using new finite element formulations 
developed in the International Center for Numerical Methods in Engineering in Barcelona. Theoretical formulation is 
briefly reviewed. Both continuum and shell elements have been considered. The new shell elements developed are based 
on a geometrically exact shell model treating the shell as one-director Cosserat surface. The formulation of the continuum 
elements employs the multiplicative decomposition f the deformation gradient ensor into its elastic and plastic parts. The 
new finite element models have been implemented in the in-house xplicit dynamic ode STAMPACK. A number of practical 
problems of sheet metalforming have been solved with the program. Some of the problems, namely stamping of a kitchen 
sink, hydraulic forming of an aeronautical part and stamping of a food can, have been presented in the paper. The examples 
give an idea of practical information that can be obtained from the computer simulation of a forming process. The results 
confirm a good behaviour of the formulation and program used in the industrial applications. 
Keywords: sheet stamping, industrial applications, finite element simulation, explicit dynamic analysis, 
1. In t roduct ion  
Finite element simulation of stamping processes has 
been successfully applied to the analysis of real parts, it is 
still however an interesting problem for investigators aim- 
ing at the development of enhanced finite element models 
yielding accurate results for complex problems while being 
at the same time computationally effective. In recent years 
new element formulations for the sheet forming simulation 
have been developed in the International Center for Nu- 
merical Methods in Engineering and implemented in the 
numerical code STAMPACK based on the explicit dynamic 
approach [6]. The results of practical industrial applications 
presented in this paper show a good behaviour of the new 
finite element models. 
2. New f in i te  e lement  mode ls  deve loped  
2.1. Continuum element formulation 
In the numerical simulation of sheet stamping the part 
deformed can be discretised with shell or continuum ele- 
ments. Both approaches have their well known advantages 
and disadvantages. In our work both types of elements have 
been considered. The formulation of the continuum ele- 
ments used [3] employs the multiplicative decomposition f 
the deformation gradient tensor F into its elastic and plastic 
0924'0136/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved 
PII0924-0136 (96) 02337-0 
parts, F e and F p, respectively 
F = F e F p (1) 
Using this decomposition the elastic Finger tensor is calcu- 
lated as 
b e-1 = F e -w F e-1 (2) 
This in turn is used in the calculation of the Almansi strain 
tensor 
e e 1 be- l )  = ~(g - (3) 
where g is the spatial metric tensor. Hyperelasticity is as- 
sumed in the constitutive model and the Cauchy stresses 
a are recovered from the free energy function ¢ written in 
terms of Almansi elastic strains e e 
0¢(e e) 
a -  0e ~ (4) 
The potential is adopted in the form 
ce 1 tr(ee)2 + (ee: -~ ~)~ p e e) (5) 
where A and it are the Lamd constants, which is valid with 
the assumption of small elastic deformations fully justified 
for metals. Material yielding is treated using the Von Mises 
criterion with isotropic nonlinear hardening. 
244 J. Rojek  et aL / Journal of Materials Processing Technology 60 (1996) 243-247 
In the numerical implementation a 4-node 2D element 
and 8-node hexahedral 3D element have been employed. 
Both elements are based on the linear approximation of the 
displacement field and a constant pressure is used to avoid 
the volumetric locking. 
2.2. Shell element formulation 
Continuum elements offer the possibility to use a more 
exact 3D description of the deformation and allow us to 
consider better two face contact conditions. However due 
to a greater numerical efficiency a shell model of the blank 
is more often used in the simulation. A family of shell ele- 
ments based on the nonlinear shell theory developed by Simo 
and co-workers [7, 8] have been derived [2], as well as a new 
shell element without rotational degrees of freedom has been 
worked out using a finite volume-like approach [1, 5]. The 
former shell formulation is based on the so-called geomet- 
rically exact shell model treating the shell as one-director 
Cosserat surface. It is assumed that any configuration of 
the shell can be described completely in terms of: 
* the mid-surface of the shell, defined by the mapping ~o: 
.4-+ R a 
• the director field defined by the mapping t: .4 --+ S 2 
where .4 is the set of reference points { = ({1 ~2), { E R ~, 
and S 2 is the unit sphere defined as S 2 := {t E Ral Iltll = 
1}. With these definitions the location of any material point 
of the shell x (x E R 3) can be given by 
x(~,{2,{)  _- ~o({~,~2) + {t({~,{2) (6) 
where { C [ -h/2,  h/2] with h being the thickness of the 
shell. Convective membrane, bending and shear strain mea- 
sures, ¢~,  p~B and am, respectively, are defined at the shell 
midsurface as 
~ = ½(~,~ .~o~ - ~o.  ~,0) 
p~ ~o,~ t,~ t o o = • - ,~ - ~o,~ 
5~ = ~o,~ - t -- t °,~ • ~o °
(7) 
(8) 
(9) 
where a,/3 ---- 1, 2 and index "0" denotes the initial config- 
uration. Having been calculated in the convective base the 
strains are transformed into the Cartesian base defined at 
the initial configuration asthe Total Lagrangian formulation 
is used in the description of large deformations. Two alter- 
native constitutive models have been employed, the stress 
resultant one with the Ilyushin yield criterion and the other 
one make use of continuum constitutive law with the inte- 
gration across the shell thickness. 
Using the formulation presented above a four-node 
isoparametric quadrilateral nd six-node triangular element 
have been derived. In the quadrilateral element both the 
shell mid-surface and the director field are interpolated with 
bilinear functions, in the triangular element quadratic func- 
tions are used. In both cases a mixed formulation with 
assumed shear strain have been used to avoid shear locking. 
The shell elements presented above have been found quite 
efficient in the analysis of sheet stamping problems, never- 
theless till more efficient solution have been looked for. A 
new shell element without rotational degrees of freedom, 
called Basic Shell Triangle (BST) has been derived [5]. The 
BST element is composed of the standard Constant Strain 
Triangle membrane element and a new triangular plate el- 
ement with a single degree of freedom per node [5]. The 
plate element formulation is based on the following formula 
obtained with finite volume-like approach 
/£  ndA: f r  TVwdP (10) 
where { }T 
= {~ ~y,  ~xy} T O2w O2w -2  o2w (11) 
, = Ox  ~ , 0y2  ' 
[ nx 0 1 T= 0 ny 
ny nx 
(12) 
Gq 0 } T 
v = ~'oy  (13) 
with As being an arbitrary subdomain of a plate (shell), F~ 
its boundary, x and y Cartesian coordinates in the plane of 
the plate, w deflection and n ---- {nx , ny} w a unit vector 
normal to the boundary defined in the plane of the plate. In 
the element derived the subdomain As coincides with one 
three-node element. The deflection field is interpolated with 
linear shape functions. The curvature is assumed constant 
in the element and is obtained from Eq. (10) where the 
integration is performed along the element sides with the 
deflection gradient being calculated in terms of the nodal 
deflections of the adjacent elements. The element is adopted 
in the nonlinear analysis using the Updated Lagrangian ap- 
proach. The material yielding is treated in a layer-wise man- 
ner assuming isothropic or anisothropic plastic properties. 
The shell element library is complemented with axisym- 
metric shell elements and strip elements for 2D plane de- 
formation problems. In this case a standard two-node 
Reissner-Mindlin and Kirchhoff elements have been imple- 
mented [4, 10]. 
3. Indust r ia l  p rob lems of sheet meta l fo rming  
3.1. Simulation of stamping of a kitchen sink 
The process of deep drawing of a kitchen sink from a 
stainless teel sheet 0.7 mm thick has been studied using 
the finite element model of a half of the structure. The 
nonuniform esh of 5312 8-node hexahedral continuum ele- 
ments has been used in the discretisation of the sheet. The 
blankholder has been considered eformable and has been 
modelled with 120 elements the same as those used in the 
sheet discretisation. The selection of the continuum ele- 
ments for the model of the sheet and the deformable model 
of the blankholder have been motivated by a better con- 
sideration of an important role played by the nonuniform 
distribution of the pressure between the blankholder and 
sheet which was observed in the real process. Continuum 
elements have enabled us to capture the influence of the 
J Rojek et al/Journal of Materials Processing Technology 60 (1996) 243-247 245 
Fig. 1: Kitchen sink - -  a real part with breakage 
change of sheet thickness on the contact pressure distribu- 
tion. 
The part has been found very difficult to stamp. The 
drawing depth of 140 mm has been found impossible to be 
obtained in one stroke, so a two stage stamping process have 
been adopted in the workshop. This has increased produc- 
tion costs the more that a manual lubricating at some zones 
of the sheet was necessary before each of the two strokes 
since a uniform lubrication did not ensure the proper stamp- 
ing process even with two stage forming. The flow of ma- 
terial was controlled with nonuniform lubrication as well as 
with nonuniform distribution of the blankholder pressure. 
The use of drawbeads was impossible because of the qual- 
ity requirements for the product. With all these means the 
breakage occurred in the part if one stage operation was 
tried, usually at the depth of drawing of about 120 mm. 
The typical breakage observed is shown in Fig. 1. 
The numerical simulation was carried out for one stroke 
stamping for the cases of uniform and nonuniform lubrica- 
tion. The results in the form of the deformed shape with 
the thickness distribution for the case of uniform lubrica- 
tion at the punch stroke of 130 mm are presented in Fig. 
2. There can be seen a significant hinning of the sheet 
on the punch corner coinciding with the place of breakage 
of a real part. The formability of the part is verified us- 
ing the Forming Limit Diagram (Fig. 3). Forming Limit 
Curve has been calculated from the theoretical StSren-Rice 
formula [9]. There can be seen some points above the FLC 
on the diagram, which means that the state of breakage has 
been reached at these places of the sink. The zone repre- 
sented by the points above the FLC on the right hand side 
of the FLD coincides with the zone of the fracture shown 
in Fig. 1. The points above the FLC on the left side of 
the FLD represent the zone which is on the minimum die 
radius, where the stress state is dominated by the bending 
and the maximum deformation occurs on the surface only. 
In some experiments he fracture occurred in this place as 
well. Summing up we can state that the simulation repro- 
duced well the real process of stamping a kitchen sink. 
3.2. Simulation of hydraulic forming with diaphragm 
Sheet metal forming has traditionally involved three 
rigid piece tools. Some new forming techniques which do 
not use the traditional tooling have been developed. In 
many cases they can be used instead of conventional meth- 
ods yielding technological nd tangible benefits. One of 
these methods is hydraulic forming with diaphragm that 
employs hydraulic action of oil on the sheet placed on the 
rigid punch. The sheet is separated from the liquid medium 
by means of a rubber diaphragm. 
A hydraulic forming process of a part from an aeronau- 
tical industry has been simulated with the program STAM- 
PACK. The initial setup of the die and sheet is shown in Fig. 
4. The aluminium sheet 1.6 mm thick has been discretised 
with 1500 quadrilateral shell elements using a nonumiform 
mesh with smaller element size where the greater deforma- 
tion gradients have been expected. One of the features of 
the analysis worth mentioning is the follower type of the 
loading as the oil pressure is always normal to the deformed 
configuration. The material properties have been considered 
rate independent and inertial effects have been considered 
of minor importance so we applied the total pressure of 80 
MPa as a step loading. 
A comparison between the numerical results for the sheet 
deformations and the results obtained in the real process is 
L 
1 i;~) 
Fig. 2: Deformed shape of the kitchen sink with the distribution of the thickness normalised with 
respect o the initial thickness 
246 
0,8 
0,7 
• ~ , 
J. Rojek et al. / Journal of Materials Processing Technology 60 (1996) 243-247 
° t 
o 
-05 -0.4 -0.3 02 O1 0 01 02 o 3 
Minor Logarithmic Strain 
Fig. 3: Forming Limit Diagram for the kitchen sink 
| | | :  I l t l~HS|~| : l ,  l l l l |~ l | | l l l  
' " l i l l l l l l  l |:| t | i::| ] | ] |  i |  I li~ | 
{ ~ I | t  I I I | I ~ | I I | [ I T ~ - ~  
Fig. 4: Initial setup of the die and sheet 
shown in Fig. 5. Excellent agreement between the results 
compared has been found. It can be seen that the simula- 
tion reproduces accurately an important defect due to local 
buckling in the circular flange region. There is a localised 
large wrinkle here. This example shows the possibility of 
verification of a manufacturing process involving hydraulic 
forming technique at the stage of the process design. 
3.3. Simulation of deep drawing of an axisymmetric food 
can 
Deep drawing process of an axisymmetric food can from 
aluminium sheet has been analysed. The initial diameter 
of the flange rim is 116.5 mm and the initial thickness of 
the sheet is 0.22 mm. The initial and deformed configura- 
tions of the tools and sheet at different stages of forming 
are shown in Fig. 6. The sheet was discretised with 206 4- 
node continuum axisymmetric elements using one element 
over thickness. The problem was analyzed for different fric- 
tion coefficients /~ ---- 0, 0.05 and 0.15. The draw-in values 
of the sheet flange obtained in the analysis were compared 
with the values measured in the real process. The values 
Fig. 5: Comparison between the results obtained in the real 
process and numerical results 
obtained 19.2 mm for p = 0, 18.7 mm for # = 0.05 and 
16.9 mm for # -- 0.15 are in a very good agreement with 
the average real value 17.3 mm. The problem was analysed 
later using the discretisation with 103 axisymmetric shell el- 
ements. Three layers were used in each element to consider 
plastification of the section of the shell. The results obtained 
are practically the same as those obtained with continuum 
model. The advantage of the shell model is evident when 
compared to the continuum one as the shell model enables 
us to use much greater time steps and a coarser mesh. 
This examples provides us with a very interesting applica- 
tion of the results of the numerical simulation, namely use of 
the results to design correctly the decoration of the can. The 
decoration of the can is designed for the deformed part (Fig. 
8) however the sheet is painted before forming (Fig. 7). The 
painting pattern of the sheet is traditionally obtained by an 
trial-and-error procedure not always with satisfactory re- 
sults. This process has been automated. The specialised 
software has been developed to transform the painting from 
the flat sheet to the deformed part or in the opposite way 
according to the deformation of the sheet obtained from the 
finite element simulation. Fig. 7 presents the painting of the 
J. Rojek et at/Journal of Materials Processtng Technology 60 (1996) 243-247 247 
sheet obtained with a traditional method and Fig. 8 shows 
the decoration on the deformed can predicted with the new 
methodology after performing transformation of the image 
from Fig. 7. Some possible defects of the decoration can be 
perceived. This shows that the method worked out can be 
used to verify the correctness of the painting pattern. 
Fig. 6: Different stages of stamping a food can Fig. 7: Painting of a flat sheet 
Fig. 8: Painting on the deformed shape 
4. Conc lud ing  remarks  
The results of numerical simulation of the industrial 
sheet metalforming processes presented show a good be- 
haviour of the new finite element models developed and 
implemented in the program STAMPACK. The examples 
show possible application of the finite element simulation 
in the verification of the makeability of the part thanks to 
possibility of predicting part defects such as breakage and 
wrinkling. The numerical simulation can be used to correct 
or optimise the forming process. It can also be underscored 
that the practical applications included reflect a great in- 
terest in the use of numerical simulation existing in small 
companies of the sheet stamping sector. 
References  
[1] P. Cendoya. New Finite Elements for Elastoplastic Dy- 
namic Analysis. PhD thesis, Universitat Politecnica de 
Catalunya, Barcelona, to be published in 1996. (in 
Spanish). 
[2] F. Flores and E. Ofiate. New shell elements for non- 
linear structural analysis. Technical report, CIMNE, 
Barcelona, 1993. 
[3] C. Garcfa Garino. A Numerical Model for the Anal- 
ysis of Large Elasto-plastic Deformations of Solids. 
PhD thesis, Universitat Politecnica de Cataiunya, 
Barcelona, 1993. (in Spanish). 
[4] M.E. Honnor. Non-linear Finite Element Analysis of 
Axisymmetric Shells Applied to Sheet Metal Forming. 
PhD thesis, University of Wales, Swansea, 1985. 
[5] E. Ofiate and M. Cervera. A general procedure for de- 
riving thin plate bending elements with one degree of 
freedom per node. CIMNE, Barcelona, 1993. 
[6] E. Ofiate, J. Rojek, and C. Garcfa Garino. NUMIS- 
TAMP: a research project for assesment of finite ele- 
ment models  for stamping processes. Journal of Ma- 
terials Processing Technology, 50(1-4):17 38, 1995. 
[7] J.C. Simo and J.G. Kennedy. On a stress resultant geo- 
metrically exact shell model. Part V: Nonlinear Plastic- 
ity. Formulations and integration algorithms. Comput. 
Meth. Appl. Mech. Eng., 96:133-170, 1992. 
[8] J.C. Simo, M.S. Rifai, and D.D. Fox. On a stress resul- 
tant geometrically exact shell model. Part IV: Variable 
thickness hells with through-the-thickness stretching. 
Comput. Meth. Appl. Mech. Eng., 81:91-126, 1990. 
[9] S. St6ren, J.R. Rice, Localized necking in thin sheets, J.
Mech. Phys. Solids, 123, 1986. 
[10] O.C. Zienkiewicz. The Finite Element Method in En- 
gineering Science. McGraw-Hill, second edition, 1971. 


Industrial applications of sheet stamping simulation using new finite element models

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Industrial applications of sheet stamping simulation using new finite element models

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