In this work, the finite element method is used to simulate a typical FLD test over tools of different radii. Parameters like the mesh density, element type, numerical determination of the onset of strain localization, limit strain definition etc. have been investigated. Finally, the limit strain for plane strain tension has been determined as a function of the thickness vs. tool radius (t/R) ratio. These simulations confirm that increasing the curvature of the tool increases the value of the limit strains. They also reveal that, as soon as bending becomes important, the practical relevance of the limit strains diminishes - At least with their current definition. The need for new strain localization models is emphasized, together with some of the associated challenges.Projet ANR FORME

ABED-MERAIM, Farid

BALAN, Tudor

English

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Tudor BALAN, Farid ABED-MERAIM - Numerical Investigation of the Limit Strains in Sheet
Forming Involving Bending - In: 9th International Conference on Technology of Plasticity, ICTP
2008, Corée du sud, 2008-09-07 - Proceedings of the 9th International Conference on
Technology of Plasticity, ICTP 2008 - 2008
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 Numerical Investigation of the Limit Strains in Sheet Forming 
Involving Bending 
 
T. Balan1, F. Abed-Meraim1 
1LPMM-UMR7554, ENSAM Metz, Arts et Métiers ParisTech, France 
 
Summary  
In this work, the finite element method is used to simulate a typical FLD test over tools of 
different radii. Parameters like the mesh density, element type, numerical determination of the 
onset of strain localization, limit strain definition etc. have been investigated. Finally, the limit 
strain for plane strain tension has been determined as a function of the thickness vs. tool radius 
(t/R) ratio. These simulations confirm that increasing the curvature of the tool increases the value 
of the limit strains. They also reveal that, as soon as bending becomes important, the practical 
relevance of the limit strains diminishes – at least with their current definition. The need for new 
strain localization models is emphasized, together with some of the associated challenges. 
1. Introduction 
Forming limits represent a key issue in sheet metal forming. The concept of forming limit diagram 
(FLD) has proved to be very useful in describing the forming limits of sheet metals in a general and 
practically relevant manner. Decades of effort have given rise to standardized experimental methods, 
as well as numerous modelling approaches, for the determination / prediction of the FLDs. 
 
The development e.g. of high strength steels and the associated forming technologies revealed several 
limitations of the existing formability models. Among other limitations, FLDs were not aimed for 
application in parts with very small radii, but rather for the study or large panels in areas of reasonable 
curvature. The influence of curvature was already first pointed out in the early work of Gosh and 
Hecker [1]. In an attempt to extend the range of application of the FLD, Col [2,3] suggested the new 
concept of FLS – “Forming Limit Surface” (see Fig. 1). While its beneficial effect is largely 
recognised in the sheet forming press-shops, scarcely any data is available to support the speculations 
about the quantitative impact of the sheet/die curvature on the FLD. However, such information would 
be a prerequisite for the development of more realistic formability models and criteria.  
 
This work is an attempt to quantitatively investigate the impact of curvature on the FLD. The analysis 
is restrained to the “FLD0” point, corresponding to the plane strain mode. The finite element method is 
used to simulate typical FLD tests over dies of different radii. Parameters like the mesh density, 
element type, numerical determination of the onset of strain localization, limit strain definition etc. 
have been investigated. Finally, the limit strain for plane strain tension has been determined as a 
function of the thickness vs. die radius ratio.  
2. Definition of the model problem 
Due to its well known strain-path dependency, the forming limit diagram is formally determined for 
proportional (or linear) strain paths – while an infinite variety of non-linear strain-paths are often 
encountered in practice. Similarly, there is no unique manner to overlap bending to a typical (even 
linear) in-plane loading of a metal sheet. Obviously, the ratio of these two contributions and its history 
of application could have an important role on the limit strains. In this work, a particular choice is 
made that is inspired from the practice of the Nakajima FLD test: metal sheet is supposed clamped on 
its external contour and it is drawn over a punch with a circular profile. Since the plane strain loading 
mode is given particular attention, a two-dimensional, plane-strain configuration is considered (see 
Fig. 2).  
 
 Friction is neglected on both contact interfaces; in order to enforce localization on the top of the 
punch, the die radius is chosen slightly larger than the punch radius. The length of the sheet sample 
and the die radius have been tuned by means of numerical tests, so that their particular values do not 
affect the results.  
ε2
t/R
0,0
0,3
0,6
0,9
0,60,5
0,40,3
0,20,1
0-0,1
-0,2-0,3
-0,4
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
t/R
 
Fig. 1: Forming Limit Surface concept introduced by Col [2]. 
 
In order to address the effect of the thickness vs. punch radius ratio, three values have been tested for 
the punch radius: 50, 25 and 12.5 mm. They correspond to thickness vs. punch radius ratios of 0.06, 
0.12 and 0.24. An arbitrary sheet thickness of 3 mm has been chosen but its particular value should not 
affect the results, provided that the above ratios are respected.  
Sy
m
m
et
ry
 
pl
an
e
R
R+10%
Im
po
se
d
di
sp
la
ce
m
e
n
t
200
3
 
Fig. 2: Geometry and loading considered for the numerical simulations. 
 
Solid finite elements are used for the simulations, in order to describe the strain heterogeneity in the 
pole region, due to bending and contact. Hybrid, displacement – pressure linear elements have been 
used with an enhanced strain definition aiming to accommodate incompatible modes due to bending. 
The simulations are performed with Abaqus/Standard. Fig. 3 gives an outline of the geometry and the 
finite element mesh. In the zone of interest, a number of ten finite elements in the thickness direction 
(twenty integration points) are used to correctly describe the strain gradient and to avoid mesh 
 dependency. 
 
 
Fig. 3: Finite element mesh at the beginning of the simulation and sample results at the end 
of the simulation. Top: detail of the mesh at the pole region. Bottom: the in-plane major 
strain is measured on the top surface of the sheet, along the thick line, starting from the pole. 
 
The strain analysis is made difficult by the presence of a strain gradient through the thickness of the 
sheet, due to bending. Since the experimental strain measurements are always made on the outer 
surface of the sheet, the same surface is conventionally used in the present analysis for the quantitative 
evaluation of strains. In fact, the principal in-plane strain is recorded on the outer surface of the sheet, 
starting from the pole – as indicated in Fig. 3. For large values of the punch radius, the strain value at 
the pole measured in this way should correspond to the so-called FLD0 value – the lowest point of the 
FLD, laying on the vertical axis. The primary aim of this analysis is to quantify the evolution of this 
value when the punch radius is diminishing. 
 
The material is considered isotropic (von Mises yield surface); the evolution of the flow stress Y 
(hardening) is described with a classical Voce law: 0 (1 )bY Q e εσ −∞= + −  with 0 200 MPaσ = , 
400 MPaQ
∞
=  and b = 10. As already underlined, at the current stage the results are not to be 
directly related to an experimental situation – so the material has also been taken into account in this 
simplified manner. 
3. Results of the FE simulations 
Finite element simulations have been performed for the three values of punch radius. Among other 
quantities, the major strain has been recorded on the top surface of the sample. Fig. 4 shows a 
heterogeneous strain distribution corresponding to the zone affected by contact with the punch. 
Globally, the largest values of strain are observed in a small zone close to the pole – although the 
maximum is not necessarily located at the pole. The necking process becomes more and more 
catastrophic and eventually, strain localization occurs. However, this is not an instantaneous 
phenomenon and the definition of the exact moment of strain localization is not unique – as it is also 
the case experimentally. Moreover, the numerical results become less and less reliable as the strain 
localizes – due to pathological mesh sensitivity.  
 
Fig. 4 illustrates the evolution of the strain distribution in the zone of interest at several moments in 
the loading history. The thick line designates a particular simulation increment, when most of the sheet 
is elastically unloaded suddenly, after what the plastic strain occurs only in small areas in the bent 
zones (this is especially true when only the outer surface of the sheet is analyzed). The strain 
 localization relevant for sheet metal forming is often considered to occur somewhat later. However, it 
is clear from Fig. 4 that the subsequent curves can hardly be used for quantitative strain evaluations. 
Indeed, the evolution of strains becomes strongly mesh dependent when most of the mesh is unloaded. 
On the other hand, damage should also be considered in the constitutive model if accurate limit strains 
are searched for at such large values of strain. In contrast, the numerical simulations with different 
mesh densities have shown almost identical strain distributions at the moment of global elastic 
unloading of the sample. 
Major strain
(in-plane)
elastic
plastic
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0 10 20 30 40 50
distance from pole [mm]
m
a
jor
 
st
ra
in
 
Fig. 4: Major strain distribution at different stages of the simulation; punch radius 12.5 mm.  
 
According to these observations, the sudden elastic unloading of more than 80% of the sample is 
formally considered as the localization criterion in this study. Again, this criterion is not aimed to 
mimic the localization criterion adopted experimentally; it mainly allows a consistent quantitative 
comparison of the results for different punch radii. It is however expected that the trends of this 
criterion would correspond to the trends of more realistic localization criteria, in terms of t/R 
sensitivity – and remains representative for applications to sheet metal forming. 
 
The same remarks about the strain distribution stand whatever the value of the die radius, as seen in 
Fig. 5. In order to compare the results to the ideal situation with no bending, a simple, tool-less tensile 
test (plane strain) has been simulated using the same initial mesh. The thick lines from all sub-figures 
in Fig. 5, corresponding to the strain distributions at localization, are overlapped in Fig. 6 (left). In the 
four cases, the strain distribution can be roughly described by two values: the value at the pole – the 
so-called FLD0 value – and the value far from the pole, in the zone unaffected by bending. In real 
situations, this later zone is the largest one for many real parts.  
 
The values of these two representative strains are summarized in Fig. 6 (right) for the different t/R 
ratios. Obviously, the two values are almost identical in the case of in-plane (tool-less) straining – due 
to the particular choice of the localization criterion considered here. When the t/R ratio is increasing, 
the FLD0 strain is increasing, thus confirming the intuitive trend from Fig. 1. By contrast, the strain 
level far from the pole is decreasing with the t/R ratio.  
 0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0 10 20 30 40 50
distance from pole [mm]
m
a
jor
 
st
ra
in
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0 10 20 30 40 50
distance from pole [mm]
m
a
jor
 
st
ra
in
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0 10 20 30 40 50
distance from pole [mm]
m
a
jor
 
st
ra
in
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0 10 20 30 40 50
distance from pole [mm]
m
a
jor
 
st
ra
in
 
Fig. 5: Major strain distribution at different stages of the simulations, for different punch 
radii (from top left to bottom right: no radius; R=50 mm; R=25 mm; R=12.5 mm). 
 
 
0,00
0,05
0,10
0,15
0,20
0,25
0,30
0 10 20 30 40 50
distance from pole [mm]
m
a
jor
 
st
ra
in
Flat
R=50
R=25
R=12.5
    
0
0,1
0,2
0,3
0,00 0,10 0,20 0,30
t/R ratio
m
ajo
r 
st
ra
in at pole
far from pole
 
Fig. 6: Left: Major strain distributions at strain localization. Right: FLD0 limit strain 
dependency to the t/R ratio. 
 4. Discussion and conclusions 
Simple finite element simulations have shown that the impact of bending on the limit strains in plane 
strain loading can be quantified numerically. This effect is beneficial – strain localization is delayed 
by bending. However, the strains attained far from the pole are decreasing when the thickness vs. 
punch radius ratio t/R is increasing. Consequently, the limit strains have a very local meaning and are 
no longer characteristic of the overall straining of the sheet.  
 
In order to relate the numerical results to experiments, one has to carefully adopt the localization 
criterion when limit strains are considered attained. The criterion adopted here was the sudden 
unloading of most of the sample; this choice prevents a too important sensitivity of the results to mesh 
density. On the other side, this criterion is easy to adopt in finite element simulations – while it would 
be more difficult to apply for experiments.  
 
The current analysis has several limitations. First of all, the results depend at some extent on the mesh. 
The mesh density has been tuned by numerical tests to avoid such dependency – however it is far too 
dense to be used in real stamping applications. Eventually, the limit strains at the pole (on the outer 
surface of the sheet) have less practical relevance when bending becomes important: they give no 
information about the underlying strains through the thickness of the sheet – and they are also not 
representative of the strains beyond the tool radius.  
 
The prediction of strain localization appeals for new localization criteria and models, taking into 
account the curvature of the sheet. However, few or no localization criteria currently in use allow a 
straightforward extension to heterogeneous strain fields. Moreover, one can expect that when the 
strains are increasing, damage should not be neglected in the material modelling. Finally, the mesh 
dependency remains a major difficulty that must be taken into consideration. 
 
Numerical simulations like the ones performed in this analysis can bring insight about the required 
features of such new models – and could provide a means of validation. In future work, these 
simulations will be extended to other straining modes (uniaxial tension, balanced biaxial tension etc.) 
in an attempt to simulate the forming limit surface (Fig. 1). This purely numerical study will be 
accompanied by a modelling effort to take curvature into account in strain localization.  
References 
[1] Gosh, A.K., Hecker, S.S.: ***; Met. Trans. 5, 1974, 2161-2164. 
[2] Col, A.: FLCs: Past, present and future. Proceeding of IDDRG 2002, Nagoya, Japan. 
[3] Col, A.: FLCs: are we at a turn? Proceedings of IDDRG 2005, Besançon, France. 


Numerical Investigation of the Limit Strains in Sheet Forming Involving Bending

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Numerical Investigation of the Limit Strains in Sheet Forming Involving Bending

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