Materials Sciences and Applications, 2011, 2, 497-503
doi:10.4236/msa.2011.25067 Published Online May 2011 (
Copyright © 2011 SciRes. MSA
Experimental and Numerical Analysis of Forming
Limit Diagram (FLD) and Forming Limit Stress
Diagram (FLSD)
Mehdi Safari1*, Seyed Jamal Hosseinipour2, Hamed Deilami Azodi3
1Department of Mechanical Engineering, Islamic Azad University, Aligoodarz Branch, Aligoodarz, Iran; 2Department of Mechanical
Engineering, Babol University of Technology, Babol, Iran; 3Department of Mechanical Engineering, Iran University of Science and
Technology, Tehran, Iran.
Received August 13th, 2010; revised September 25th, 2010; accepted May 11th, 2011.
In this work, forming limit diagram for aluminum alloy 3105 is performed experimentally and forming limit based on
stress (FLSD) calculated from strains that resulted from experimental procedure. In addition, numerical prediction by
ductile fracture criteria using simulation is considered and it is shown that they are well suited with the experimental
results. The strain paths from finite element simulations are found fairly acceptable to represent both sides of the FLD.
Keywords: Forming Limit Diagram, Forming Limit Stress Diagram, Aluminum Alloy 3105, Ductile Fracture Criteria
1. Introduction
Sheet metal formability is generally defined as the ability
of metal to deform into d esired shape without necking or
fracture. Each type of sheet metal can be deformed only
to a certain limit that is usually imposed by the onset of
localized necking, which eventually leads to the ductile
fracture. A well-known method of describing this limit is
the forming limit diagram (FLD), which is a graph of the
major str ain (11
) at the onset of localized necking for all
values of the minor strain (22
).The diagram can be split
into two sides; “left side” and “right side”. At the “right
side”, which was first introduced by Keeler and Back-
ofen [1], only po sitive Major and Minor Strains are plot-
ted. Goodwin [2] completed the FLD by adding the “left
side”, with positive Major and negative Minor Strains.
Various strain paths can be generated in order to create
different combinations of limiting Major and Minor
Strains. The “left side” represents strain paths with strain
ratios (22
) that vary from uniaxial tension
 ) to plain strain (0
). On the “right side”
the strain ratios differ from plain strain to full biaxial
) stretching. Usually FLDs are determined by us-
ing one of the following two types of test methods. The
first one is the Marciniak in-plane test where a sheet
metal sample is strained by a flat-bottomed cylindrical
punch. Between the punch and the metal sheet is a steel
driver with a hole in the centre. This creates a frictionless
in-plane deformation of the sheet. The other test is the
Nakazima (Dome) out-of-plane test, which uses a he-
mispherical punch. Since for this test deformations are
not frictionless, lubricants are used. The necessary strain
paths are obtained by using different lubricants, creating
different friction cond itions, and also with different sam-
ple widths. In this work, Nakazima (Dome) out-of-plane
test is used and sch ematic of this test has sh owed in Fig-
ure 1. The strain-path dependent nature of the FLD
causes the method to become ineffective in the analysis
of complex forming process, especially restrikes, flang-
ing operations, hydroforming, and even first draw dies
with deep pockets or embossments. Experimental evi-
dence for a path-independent stress-based FLD has been
reported in the literature, suggesting that the path depen-
dency of the strain-based approach arises from the path
dependent constitutive laws governing the relationship
between the stress and strain tensors. Kleemola and
Pelkkikangas [3] addressed the limitations of the FLD for
analysis of flanging operations involving copper, brass
and steel alloys following a draw forming operation.
They proposed using a stress-based FLD as an alternative
and gave some experimental results that supported the
Experimental and Numerical Analysis of Forming Limit Diagram (FLD) and Forming Limit Stress Diagram (FLSD)
Copyright © 2011 SciRes. MSA
Figure 1. A schematic of Nakazima out-of-plane test.
path independence of the forming limit in stress space for
these materials. Arrieux et al. [4] rediscovered this phe-
nomenon and proposed using a stress-based criterion for
all secondary forming operations. Despite these earlier
discoveries, the significance of the discovery of a path-
independent stress-based FLD went largely unnoticed in
the literature throughout the 1980’s and 90’s. Stoughton
[5] rediscovered the effect and proposed that it is neces-
sary to use the stress-based criterion in all forming oper-
ations, including the first draw die, in order to get a ro-
bust measure of forming severity. Since 1970s, finite
element theories have been developed for providing the
useful information to the real processes in industries
[6-16]. The FEA usually gives the information of form-
ing process such as the deformed shape, strain and stress
distribution, punching load, and the fracture. Recently,
several researchers [17-20] have used ductile fracture
criteria to determine the limit strains. The limit strains
were determined by substituting the values of stress and
strain histories calculated by the finite element simula-
tions into the ductile fracture criteria.
2. Criteria for Ductile Fracture
Based on various hypotheses, many criteria for ductile
fracture have been proposed empirically as well as theo-
retically [17,21]. It is well known that the forming limit
of sheet metals depends greatly upon the deformation
history. Therefore, the histories of stress and strain may
have to be considered in the criteria.The energy or gene-
ralized plastic work criterion was first given by Freuden-
thal [22]:
Cockcroft and Latham [23] proposed a fracture crite-
rion based on ‘‘true ductility,’’ which states that the
fracture in a ductile material occurs when the following
condition is satisfied:
max 2
The Cockcroft and Latham criterion was modified by
Brozzo et al. [24] to introduce the effect of hydrostatic
stress h
in an explicit form and to correlate their expe-
rimental results.
max 3
Oh et al. [25] modified the Cockcroft and Latham cri-
terion as follows:
max 4
In the above equations
is the equivalent strain at
which the fracture occurs, max
is the maximum normal
stress, h
is the hydrostatic stress,
the equivalent
the equivalent strain, and 123
,,CC C and 4
are material constants. To determine the material con-
tants, destructive tests have to be operated under at least
one or two types of str ess conditio ns. In th e present stud y,
the material constants 14
are determined simply
by uniaxial tension tests as follows: Hosford’s yield cri-
terion for anisotropic materials [26] is expressed as:
11 12
 
 (5)
3. Experimental Procedure
The material used in this investigation was Aluminum
alloy 3105, the chemical composition of it is given in
Table 1. Aluminum alloy was Hot-rolled in 280˚C -
300˚C to receive 7.5 mm thickness. Then it was cold in
environment and finally cold-rolled to 1.2 mm.Uniaxial
tensile test specimens, 50mm long and 12.5 mm wide at
zero degree to the rolling direction, prepared from the
sheets were pulled to fracture at a cross-head speed of 5
mm/min, producing an average strain rate of 31
as the specimen extended. For the measurement of aniso-
tropy coefficient specimens were cut at 0˚, 45˚ and 90˚.
These specimens were stretched in the range of uniform
elongation and the dimensional changes were measured
with the aid of a traveling microscope. R values were
then determined from:
ww w
 
 
 
where w
and l
refer to the transverse and longitu-
dinal strains, respectively. The instantaneous strain-rate
sensitivity index, m, was determined using step changes
in cross-head speed, from 15
mm/min to 250
mm/min which produced strain rates of 31
10 10
. Using extrapolation procedure, two stresses,
Experimental and Numerical Analysis of Forming Limit Diagram (FLD) a nd Forming Limit Stress Diagram (FLSD)
Copyright © 2011 SciRes. MSA
Table 1. Chemical composition of Al 3105.
Material Al Ga V Cr Ti Zn Mg Mn Cu Fe Si
AA3105 0.9695 0.01 0.01 0.02 0.01 0.33 0.61 0.67 0.23 0.81 0.27
and 2
, are compared at the same strain and the m
value was obtained from
21 21
ln lnm
 .
Rectangular strips of various widths, were cut from the
sheets with the long dimension of the rectangle parallel
to rolling direction according to Sadough et al. method
[27]. All specimens were grided with 4.4 mm diameter
circles that the distances between their centers were
maintained 5.4 mm.
The grids were marked on the specimen by a rubber
stamp. In the Figure 2 the specimen with marked grids
of Aluminum alloy 3105 is showed. The deep drawing
machine that was used in this investigation was an auto-
matic hydraulic press (Figure 3). This machine has a ma-
ximum load cap acity of 60 kN, a punching strok e of 250
mm and a variable punch speed up to 200 mm/min.
The strips were clamped firmly at the periphery by a
lock bead and then stretched over a 100 mm diameter
hemispherical punch until they fractured, using polye-
thylene film as a lubricant. The grid circles were de-
formed to elliptic shapes because of stretching strain that
was inserted in the plane of the sheet metal during the
test. For each specimen the major and minor limited
strains were measured from the major and minor axes of
the ellipse that was located at the nearest distance to the
necking zone. The localized necking zone appears as a
groove in the deformed region of the sheet metal. Figure
4 shows the localized necking zone of 200 × 200 speci-
men. In the Figure 5 stretched specimens of Aluminum
alloy 3105 have been showed. Profile projector was used
to measure the major and minor strains in the deformed
circles. The values of the surface strains at the onset of
localized necking constitute the FLDs.
4. Numerical Work
All the geometries in Figure 2 were simulated using
commercially available finite element code ABAQUS/
Standard. In this state, experimental conditions were
duplicated in numerical simulation. During simulation
the analysis of data is done and the ductile fracture crite-
ria used for this purpose. So, when the value of ductile
fracture criteria receive to a critical value obtained from
experimental tensile test, necking has been occurred and
simulation stopped. The punch, die and blank-holder are
modeled as rigid bodies. The Coulomb friction model
with a constant coefficient of friction, μ = 0.11 was ap-
plied. One-quarter of the geometry is used due to sym-
metry for easier to visualize. The appropriate nodal con
strains are applied in the global X, Y directions to impose
symmetry. In Figure 6 simulated 200 × 200 sample is
Figure 2. Speciemen marked with circular grids.
Figure 3. Deep drawing test machine.
Figure 4. Necking zone of 200 × 200 speciemen.
Experimental and Numerical Analysis of Forming Limit Diagram (FLD) and Forming Limit Stress Diagram (FLSD)
Copyright © 2011 SciRes. MSA
Figure 5. Stretched specimens of Aluminum alloy 3105.
Figure 6. Simulated 200 × 200 sample.
5. Evaluation of Forming Limit Stress
Diagram (FLSD)
We obtained from experimental work some data in the
strain form. For accommodating forming limit stress dia-
gram, these data need to be converted first into stress
form. The process of this strain conversion is summa-
rized as the following:
Step 1: Obtain 1
and 2
Step 2: Find Y
from Swift hardening law equation.
In the present work, the effective strain is obtained for
Hosford’s yield function through the corresponding plas-
tic strain rate potential presented by Stoughton in his
paper [5] .
Step 3: Calculate
, a function of material para-
meters derived from the applied yield function. The
strain path is characterized by the strain rate ratio:
And the stress ratio
Experimental and Numerical Analysis of Forming Limit Diagram (FLD) a nd Forming Limit Stress Diagram (FLSD)
Copyright © 2011 SciRes. MSA
For material with in-plane isotropy and for the cases
with non-shear stress in a coordinate system aligned with
the axes of anisotropy, the major and minor true stresses
in the first stage of strain path can be expressed as fol-
lows [5]:
The principal stresses during the second stage of strain
path can be calculated by the following expressions [5]:
121 122
22 11
Yii fifi
fi fi
 
 
 
 (11)
where 1i
& 2i
show the pre-strain state and 1
indicate the final strain state at the second stage of
strain path.
represents the function
defined in Equation (9).
6. Results and Discussions
Typical uniaxial stress–strain curve for alloy AA3105
from uniaxial tests is shown in Figure 7. The strain har-
dening exponent and the strength coefficient k from the
empirical hardening law n
were determined
from plot of lo g
versus log
. Consequently flow
property was summarized as 0.103
MPa. The
tensile strength of the material was found to be 231 MPa.
Measurements of plastic strain ratios (R-values) used in
the prediction of FLDs are given in Table 2. Figure 8
shows the FLD for Al 3105 in terms major and minor
strains, respectively. In out-of-plane tests, the strains in
the sheet vary over the stretched dome from a strain state
close to plane strain near the flange of the sheet to one
approximating balanced biaxial tension at the pole. In
these situations, flow localization and failure site depend
on the factors such as strain hardening, anisotropy beha-
vior of the sheet metal, friction condition and strain gra-
dient. During out-of-plane deformation, geometric and
frictional effect comes into play with regard to their ten-
dency to shift the site of strain localization away from the
pole at which it was first initiated. The shift can also oc-
cur partly because of strain hardening and the deve- lop-
ment of a biaxial stress state in the neck. Using the equa-
tions in section 5, the strain resulted from experimental
work converted to stress. Figure 9 shows the forming
limit stress diagram (FLSD) obtained from experimen-
tally strain values.
In the numerical work, ductile fracture criteria indi-
Figure 7. True stress–True strain curve for alloy AA3105.
Figure 8. Forming limit diagram (FLD) of AA3105.
Figure 9. Forming limit stress diagram (FLSD) of AA3105.
Table 2. Measured R values.
Alloy R0 R45 R90 R*
Al 3105 0.20 0.3 0.24 0.26
cated in Section 2, used to prediction of necking. FLD
and FLSD obtained from these criteria are shown in
Figures 10 and 11.
In Figures 12, 13 experimental and numerical results
compared together and it’s shown that they are in good
agreement. Thus it’s proved ductile fracture criteria is
Experimental and Numerical Analysis of Forming Limit Diagram (FLD) and Forming Limit Stress Diagram (FLSD)
Copyright © 2011 SciRes. MSA
Figure 10. FLD obtained from ductile fracture criteria.
Figure 11. FLSD obtained from ductile fracture criteria.
Figure 12. Comparison between experimental and numeri-
cal results for FLD.
Figure 13. comparison be twe en expe rimental and numer ic al
results for FLSD.
useful selection to predict FLD and FLSD.
7. Conclusions
In this work, forming limit diagram for Aluminum alloy
3105 was obtained experimentally and the forming limit
stress diagram obtained from strain values using Stogh-
toun equations. Also forming limit diagram and forming
limit stress diagram of this alloy predicted by ductile
fracture criteria using commercially available finite ele-
ment code ABAQUS/Standard. It’s showed FLD and
FLSD predicted by ductile fracture criteria are in good
agreement with results obtained from experimental work.
[1] S. P. Keeler and W. A. Backhofen, “Plastic Instability and
Fracture in Sheet Stretched over Rigid Punches,” ASM
Transactions Quarterly, Vol. 56, No. 11, 1963, pp. 25-48.
[2] G. M. Goodwin, “Application of Strain Analysis to Sheet
Metal Forming in the Press Shop,” SAE Paper, No.
680093, 1968.
[3] H. J. Kleemola and M. T. Pelkkikangas, “Efect of Prede-
formation and Strain Path on the Forming Limits of Steel
Copper and Brass,” Sheet Metal Industries, Vol. 64, No. 6,
June 1977, pp. 591-599.
[4] R. Arrieux, C. Bedrin and M. Boivin, “Determination of
an Intrinsic Forming Limit Stress Diagram for Isotropic
Metal Sheets,” Proceedings of the 12th Biennial Con-
gress of the IDDRG, 1982, pp. 61-71.
[5] T. B. Stoughton, “A General Forming Limit Criterion for
Sheet Metal Forming,” International Journal of Mechan-
ical Sciences, Vol. 42, No. 1, January 2000, pp. 1-27.
[6] A. Barata Da Rocha, F. Barlat and J. M. Jalinier, “Predic-
tion of the Forming Limit Diagrams of Anisotropic
Sheets in Linear and Non-Linear Loading,” Materials
Science and Engineering, Vol. 68, No. 2, January 1985,
pp. 151-164. doi:10.1016/0025-5416(85)90404-5
[7] G. Beveridge and R. Schechter, “Optimization Theory
and Practice,” McGraw-Hill, New York, 1970.
[8] M. Gotoh and F. Ishise, “A Finite Element Analysis of
Rigid-Plastic Deformation of the Flange in a Deep Draw-
ing Process Based on a Fourth-Degree Yield Function,”
International Journal of Mechanical Sciences, Vol. 20,
No. 7, 1978, pp. 423-435.
[9] H. Iseki and T. Murota, “Analysis of Deep Drawing of
Non-Axisymmetric Cups by the Finite Element Method,”
Proceedings of the First ICTP, Tokyo, 1984, pp. 678-684.
[10] R. Sowerby, N. Chandrasekaran, X. Chen, M. Rooks and
P. Correa, “The Development of Computer Aids for Sheet
Metal Stampings in CAD/CAM and FEM in Metal
Working,” In: S.K. Ghosh, A.Niku-Lair, Eds., The De-
velopment of Computer Aids for Sheet Metal Stampings in
CAD/CAM and FEM in Metal Working, Pergamon Press,
Oxford, 1988, pp. 187-203.
Experimental and Numerical Analysis of Forming Limit Diagram (FLD) a nd Forming Limit Stress Diagram (FLSD)
Copyright © 2011 SciRes. MSA
[11] R. H. Wagoner, K. S. Chan and S. P. Keeler, “Forming
Limit Diagrams: Concepts, Methods, and Applications,”
The Minerals, Metals & Materials Society, 1989, pp. 153-
[12] R. Sowerby, J. L. Duncan and E. Chu, “The Modeling of
Sheet Metal Stamping,” International Journal of Me-
chanical Sciences, Vol. 28, No. 7, 1986, pp. 415-430.
[13] Y. Q. Guo, J. L. Batoz, J. M. Detraux and P. Duroux,
“Finite Element Procedures for Strain Estimations of
Sheet Metal Forming Parts,” International Journal for
Numerical Methods in Engineering, Vol. 30, No. 8, De-
cember 1990, pp. 1385-1401.
[14] J. H. Vogel, J. Amodo and D. Lee, “An Integration
Process Design Method for Sheet Material Forming,”
Proceedings of the 15th Biennial Conferences on IDDRG,
Dearborn, 1988, p. 265.
[15] K. Lange, M. Herrmann, P. Keck and M. K. Wilhelm,
“Application of an Elasto-Plastic Finite Element Code to
the Simulation of Sheet Metal Forming Processes,”
Journal of Materials Processing Technology, Vol. 27, No.
1-3, August 1991, pp. 239-261.
[16] H.Y. Jiang and D. Lee, “Numerical Simulation of Sheet
Metal Forming Process Based on Large Deformation
Sheet Elements,” Proceedings of the NUMIFORM’92,
1992, p. 485.
[17] S. E. Clift, P. Hartley, C. E. N. Sturgess and G. W. Rowe,
“Fracture Prediction in Plastic deformation Processes”
International Journal of Mechanical Sciences, Vol. 32,
No. 1, 1990, pp. 1-17.
[18] H. Takuda, K. Mori, H. Fujimoto and N. Hatta, “Predic-
tion of Forming Limit in Deep Drawing of Fe/Al Lami-
nated Composite Sheets Using Ductile Fracture Crite-
rion,” Journal of Materials Processing Technology, Vol.
60, No. 1-4, June 1996, pp. 291-296.
[19] H. Takuda, K. Mori and N. Hatta, “The Application of
Some Criteria for Ductile Fracture to the Prediction of the
Forming Limit of Sheet Metals,” Journal of Materials
Processing Technology, Vol. 95, No. 1-3, October 1999,
pp. 116-121. doi:10.1016/S0924-0136(99)00275-7
[20] H. Takuda, K. Mori, N. Takakura and K. Yamaguchi,
“Finite Element Analysis of Limit Strains in Biaxial-
Stretching of Sheet Metals Allowing for Ductile Frac-
ture,” International Journal of Mechanical Sciences, Vol.
42, No. 4, April 2000, pp. 785-798.
[21] A. S. Wifi, A. Adbel-Hamed, N. El-Abbasi and H.
Harmoush, “Finite Element Analysis of Workability of
Some Bulk Forming Processes,” In: T. Abe, T. Tsuta,
Eds., Advances in Engineering Plasticity and its Applica-
tions, Pergamon, New York, 1996, pp. 197-202.
[22] A. M. Freudenthal, “The Inelastic Behavior of Engineer-
ing Materials and Structures,” John Wiley & Sons, New
York, 1950.
[23] M. G. Cockcroft and D. J. Latham, “Ductility and the
Workability of Metals,” Journal Institute of Metals, Vol.
96, 1968, pp. 33-39.
[24] P. W. Bridgman, “Studies in Large Plastic Flow and
Fracture”, McGraw-Hill, New York, 1952.
[25] S. I. Oh, C. C. Chen and S. Kobayashi, “Ductile Fracture
in Axisymmetric Extrusion and Drawing,” Journal of En-
gineering for Industry-Transactions of the ASME, Vol.
101, 1979, pp. 36-44. doi:10.1115/1.3439471
[26] W. F. Hosford, “On Yield Loci of Anisotropic Cubic
Metals,” Proceedings of the Seventh North American
Metalworking Research Conference, SME, Dearborn,
1979, pp. 191-197.
[27] A. Sadough, M. Shakeri and B. M. Dariani, “Theoretical
and Experimental Analysis of Sheet Metal Formability
Limits,” La Revue de Metallurgie-CIT/Science et Genie
des Materiaux Mai, 2000, pp. 663-670.