Prediction of entire forming limit diagram from simple tensile material properties

Abstract

The purpose of this study is to develop a phenomenological model for prediction of the entire forming limit diagram from simple tensile material properties. The phenomenological model is based on the necking and ductile damage theories. In the proposed model, void nucleation is described as a function of the equivalent plastic strain, and void growth is a function of the stress triaxiality. The forming limit curves calculated from the proposed phenomenological model matched reasonably well in the region of uniaxial tension to balance biaxial tension with the experimental forming limit curves generated on C-Mn 440 steel, interstitial-free 340 steel, and interstitial-free steel sheets.

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The Journal of Strain Analysis for Engineering
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DOI: 10.1177/0309324713488886
2013 48: 386 originally published online 9 July 2013The Journal of Strain Analysis for Engineering Design
Surajit K Paul, G Manikandan and Rahul K Verma
Prediction of entire forming limit diagram from simple tensile material properties
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DOI: 10.1177/0309324713488886
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Prediction of entire forming limit
diagram from simple tensile material
properties
Surajit K Paul, G Manikandan and Rahul K Verma
Abstract
The purpose of this study is to develop a phenomenological model for prediction of the entire forming limit diagram
from simple tensile material properties. The phenomenological model is based on the necking and ductile damage the-
ories. In the proposed model, void nucleation is described as a function of the equivalent plastic strain, and void growth
is a function of the stress triaxiality. The forming limit curves calculated from the proposed phenomenological model
matched reasonably well in the region of uniaxial tension to balance biaxial tension with the experimental forming limit
curves generated on C-Mn 440 steel, interstitial-free 340 steel, and interstitial-free steel sheets.
Keywords
Forming limit diagram, tensile test, sheet metal, ductile damage, stress-based forming limit diagram
Date received: 22 November 2012; accepted: 9 April 2013
Introduction
In order to design the shape of products and process
parameters of thin sheet forming processes such as
stamping, it is essential to use a failure limit criterion
up to which material can be safely deformed without
necking or failure. The forming limit diagram (FLD) is
the most commonly used failure limit criterion in sheet
metal forming industries. FLD is a diagram of major
and minor strains at the onset of local necking, schema-
tically shown in Figure 1. Figure 2 schematically repre-
sents FLD as a deep drawn part and its individual
loading zones that are located in different positions of
the FLD. Two types of neck become visible during sim-
ple tensile test: diffuse and localize types of necks. The
diffuse neck appears during tensile test when the maxi-
mum force is reached; it is generally observed in the
width direction of the tensile specimen, whereas the
localized neck is formed in the thickness direction
(within the diffuse neck region) and it is very close to
the fracture point. The forming limit curve (FLC) gen-
erally represents localized necking at various strain
ratios. Localized neck is highly influenced by strain rate
sensitivity exponent (m) of the material. The strain rate
in the neck region increases once the neck is formed.
As flow stress of the metals and alloys increases with
the increasing strain rate, high strain rate sensitivity
exponent (m) of the material means more resistance for
deformation in the neck region and high postuniform
elongation.
1,2
The concept of FLD was first introduced by Keeler
and Backofen
3
and Goodwin.
4
The FLC can be split
into two branches: ‘‘left branch’’ and ‘‘right branch.’’
Keeler and Backofen
3
first introduced the ‘‘right
branch’’ of FLC, which is valid for positive major and
minor strains. Goodwin
4
completed the FLC by intro-
ducing the ‘‘left branch’’ of FLC, which is applicable
for positive major and negative minor strains. After
that, many theoretical models were developed to calcu-
late FLD. Three different types of models are available
to compute FLD: (a) the bifurcation method: bifurca-
tion analysis was first introduced by Hill
5
to predict dif-
fuse necking on metal sheet, Sto
¨
ren and Rice
6
introduced a pointed vertex on yield surface to
compute FLC, and Hutchinson et al.
7,8
carried out
localized-band bifurcation analysis to predict FLC; (b)
geometrical imperfection: Marciniak and Kuczynski
(M–K)
9
approach that predicts instability of sheet by
considering geometrical imperfection in terms of
R&D, Tata Steel Limited, Jamshedpur, India
Corresponding author:
Surajit K Paul, R&D, Tata Steel Limited, Jamshedpur 831001, Jharkhand,
India.
Email: paulsurajit@yahoo.co.in; surajit.paul@tatasteel.com
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thickness deference; and (c) damage mechanics-based
approach: Chow et al.
10
conducted computer simula-
tion to analyze the effects of plastic damage on the
formability of very ductile interstitial-free (IF) steel
under both proportional and nonproportional loading
conditions. Paul
11
conducted a comparative study and
showed that the FLD prediction capabilities among
available various analytical and bifurcation models.
The M–K model considered geometrical inhomogene-
ity as a variation of the sheet thickness directed along
the minimum principal stress axis. It was assumed that
during the biaxial straining, the strain localization
occurs in the region of geometric inhomogeneity of the
sheet. The M–K model computes limit strain using von
Mises yield criteria and is underestimated in the
domain of plane strain and overestimated in the
domain of biaxial straining. Use of the yield criterion
in the M–K model has strong influence on the shape
and position of FLC. The shape and size of geometrical
imperfection (thickness ratio, width, and angel of
imperfect region) also have great influence on the shape
and position of FLC. As a result, for different thickness
ratios, different FLCs can be achieved. Rather a band
of FLC can be obtained from M–K model for a partic-
ular material. Ghazanfari and Assempour
12
calibrated
FLC obtained from M–K model with an experimental
point to find a unique FLC. The analytical models
mentioned above are applied to construct FLC, which
is based on thickness reduction or necking. Recently,
few fracture-based models are published in the litera-
ture, which are used to predict fracture-forming limit
diagram (FFLD). The fracture-forming limit curve
(FFLC) lies above the classical FLC, as FFLC is based
on fracture and FLC is based on necking of sheet.
Takuda et al.
13
used ductile fracture criterion in a finite
element platform to compute forming limit. Lou
et al.
14
introduced void nucleation, growth, and shear
coalescence-based ductile damage criteria in uncoupled
manner to calculate FFLD of DP780 steel sheet. Lou
et al.
14
reported that the FFLD is also valid for low or
negative triaxiality (pure shear to uniaxial tension),
where negligible or no thickness reduction takes place.
However, to characterize material constants for these
fracture models, many advanced tests are required; for
example, in the Lou et al.
14
model, uniaxial tensile,
plane strain tensile, balanced biaxial tensile, and pure
shear tests are recommended in their original article to
characterize the material constants. In the present
work, a phenomenological model is formulated to pre-
dict FLC and the material constants for the proposed
model, which can be calculated solely from the simple
uniaxial tensile test.
Experimental procedure
Three different strength level steels such as carbon man-
ganese (C-Mn 440), high-strength IF 340, and IF steels
are selected for present investigation. Depending upon
the strength level of those steels, applications are also
different; IF 340 and IF steels are used for car outer
and inner panels, whereas C-Mn 440 steel is used as
internal structural member of car body. IF 340 and IF
steels are ultralow carbon steels, carbon level main-
tained up to 0.0035% (wet %). Microalloying elements
Mn (0.65%) and Nb (0.04%) are present in IF 340 steel
as strengthening elements. In C-Mn 440 steel, around
0.09% carbon and 1.53% manganese are present.
Tensile test of the steel sheets is carried out in a ser-
vomechanical (Instron made) tensile testing system.
The flat specimen of 50 mm gauge length is tested at a
fixed strain rate of 0.001/s (i.e. cross head velocity of 3
mm/min). The longitudinal and transverse strains are
measured simultaneously through a video extensometer
(Instron made) during tensile testing. Load is measured
by inbuilt load cell attached to servomechanical tensile
testing system. Engineering stress is calculated by divid-
ing the load with initial cross-sectional area. True stress
is calculated from engineering stress by assuming the
volume consistency condition.
FLDs are generated by following normal Nakazima
et al.
15
procedure. In this process, different states of
strains are generated by varying the width of the sam-
ple. All the samples have fixed length of 200 mm and
width varied between 25 and 200 mm in steps of 25
Figure 1. Schematic of forming limit diagram (FLD).
Figure 2. Schematic of individual loading zones of a deep
drawn part.
Paul et al. 387
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mm. Specimens are prepared by shear cutting and side
edges are polished to remove extra edges. Here, the
length 200 mm represents the rolling direction. For
each blank width, at least three specimens are tested to
get maximum number of data points. Specimens of var-
ious sample width after testing are shown in Figure 3.
The experimental procedure to determine FLD involves
three stages: grid marking on the sheet specimens,
punch-stretching the grid-marked samples to failure or
onset of localized necking, and measurement of strains.
The circles on the sheet samples became ellipses after
deformation, falling into safe (forming is done and
without any difficulty), necked (near the crack region),
wrinkled (located on the left side of the diagram), and
failed (right side is related to initial fracture in punch
radius zone and the left side is related to final fracture
in flange zone) zones. Finally, FLD is drawn by plot-
ting the minor strain along the abscissa and the corre-
sponding major strain along the ordinate and by
drawing a curve that separates the safe region from the
unsafe region.
16,17
In all the samples, grid patterns
(small dots) are printed by screen printing method. The
distance from center to center of dots is 3.0 mm in both
the rolling and perpendicular to the rolling directions.
After experimentation, center-to-center distances of the
dots become altered in the rolling and perpendicular to
the rolling directions, one direction becoming higher in
comparison with the other. This center-to-center dis-
tance of the dots is used to calculate major and minor
strains. The experiments are carried out on a 1300-kN
capacity double-action servo hydraulic forming press.
The main ram (drive punch) capacity is 600 kN, and
the blank holder capacity is 700 kN. For the current
work, hemispherical punch with diameter 101.6 mm is
used. The speed of the punch 1.0 mm/s is maintained
for all the experiments. The test end criterion (onset of
necking) is automatically detected by initiation of load
drop and stops the punch drive. Teflon with rubber
pad is used to ensure the localize necking at the center
of the specimen. After forming, premarked circles
became ellipses. In this work, the major and minor
strains are measured using automatic strain analyzer
(ARGUS). Finally, the FLC is then drawn clearly
demarcating the safe limiting strains from the unsafe
zone containing the necked and fractured zones.
Empirical equations for FLC
FLC-zero or FLC
0
is the most important parameter in
the FLC. FLC
0
is the major principal strain at the onset
of necking at plane strain condition. The parameter
FLC
0
is often defined from empirical formula or makes
approximation from similar metals. An empirical for-
mula is used to determine FLC
0
in a wide range of steels
as proposed by Keeler and Brazier
18
FLC
0
=ln 1+
13:3+14:13t
100

n
0:21

for n40:21
ð1Þ
where t is the sheet thickness measured in millimeters
and n is the strain hardening exponent in power law
expression. In the above formula, FLC
0
is represented
in true strain. The full FLC can be calculated from
equation (2) by considering isotropic von Mises yield
criteria
e
2
 0, e
1
= FLC
0
 e
2
e
2
= 0, FLC
0
=ln1+
13:3+14:13t
100

n
0:21

e
2
 0, e
1
= 1 + FLC
0
ðÞ1+e
2
ðÞ
0:5
 1
8
<
:
for n40:21
ð2Þ
where e
2
and e
1
are the true minor strain and major
strain, respectively.
Plasticity relations
All describing mathematical formulations in this article
are derived from isotropic plasticity theory. The von
Mises yield function and associated flow rule are used
for all analysis.
The von Mises yield function can be defined as
f = s
eq
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s
2
1
+ s
2
2
 s
1
s
2
q
ð3Þ
where s
1
and s
2
are the major and minor principal true
stresses, respectively, and plane stress (s
3
= 0) condi-
tion is adopted. s
eq
is the equivalent true stress.
Associated flow rule can be described as
d

e
p
= dl
df
ds
ð4Þ
where dl is the proportionality constant, ds is the stress
rate, and d

e
p
is the plastic strain rate.
The ratio of the minor true strain rate (de
2
)tothe
major true strain rate (de
1
) is defined by the parameter
Figure 3. Specimens of various sample widths after
experimentation.
388 Journal of Strain Analysis 48(6)
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a =
de
2
de
1
ð5Þ
Similarly, the ratio of the minor true stress (s
2
)to
the major true stress (s
1
) is defined by the parameter
b =
s
2
s
1
ð6Þ
This flow rule (equation (4)) leads to a relation
between a and b, which can be expressed as
19,20
a =
2b  1
2  b
ð7Þ
b =
1+2a
a +2
ð8Þ
The rate of change of the yield function can be
defined as
df
d

e
p
dl =
∂f
∂s
1
ds
1
+
∂f
∂s
2
ds
2
ð9Þ
The most commonly used representation of stress–
strain relation is the power law
s
eq
= Ke
n
teq
ð10Þ
where K and n are material constants, and e
teq
is the
total equivalent strain.
By rearranging equations (3) and (6), equivalent
stress (s
eq
) can be written as
s
eq
= s
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1  b + b
2
q
ð11Þ
Similarly, equivalent plastic strain increment (de
eq
)
can be written as
de
eq
=
2
ffiffiffi
3
p
de
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1+a + a
2
p
ð12Þ
Triaxiality (T
r
) is the ratio of mean stress (s
m
) and
equivalent stress (s
eq
). s
m
=(s
1
+ s
2
)/3 for plane stress
condition, and combining equations (11) and (8) results
in equation (13)
T
r
=
s
m
s
eq
=
1+a
ffiffiffi
3
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1+a + a
2
p
ð13Þ
where mean stress (s
m
)is
s
m
=
1
3
s
1
+ s
2
+ s
3
ðÞ ð14Þ
Triaxiality can be written also in terms of stress ratio
(b)as
T
r
=
1+b
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1  b + b
2
p
ð15Þ
All these relationships are used in the next section to
deduce the simulated FLC.
Phenomenological model
The FLD is very useful in finite element method (FEM)
analysis, die design optimization, die tryout, and qual-
ity control during production.
20
In recent years, many
techniques have been developed to evaluate FLD
experimentally.
16,21
Large numbers of experimentation
are required to determine FLD accurately. For that
reason, with the increase of computational techniques,
several researchers have proposed numerical models to
predict the FLDs. In the current work, a damage
mechanism–based FLD determination technique is pro-
posed. FLD is generated by plotting major and minor
strains at the necked area of specimens with different
widths. Causes of necking in the ductile material were
studied extensively in the last seven decades. Initiation
and growth of voids cause necking, and coalescence of
voids results in final fracture.
22–24
In the proposed
model, the parameters related to initiation and growth
of voids are selected as variables. Function of those
variables can be defined as necking of the material at
different strain ratios. The material constants of that
function are determined from uniaxial tensile test. The
final strategy is to verify the proposed model with
experimental FLD of steels with different strength
levels.
Under tensile loading, ductile material shows neck-
ing phenomenon before final fracture. Microscopic
analysis is evident that nucleation, growth, and coales-
cence of voids is the prime cause of the necking and
finally ductile failure.
22,23
A number of models have
been proposed to explain the ductile failure, for exam-
ple, the Gurson,
24
Tvergaard, and Needleman (GTN)
model.
25
Bandstra et al.
22
reported that deformation
instability on HY-100 steel is triggered by the growth
of MnS inclusion-nucleated voids. Bonora et al.
23
experimentally studied the ductile damage evolution by
nucleation, growth, and coalescence of voids under
triaxial stress state and simulated the ductile damage
by continuum damage mechanics model. Steels usually
contain inclusions and secondary phase particles
because of their alloy concepts and production process.
Due to plastic deformation in ductile steels, primary
voids are normally formed from inclusions and second-
ary phase particles. The mechanism of void nucleation
has been explained by a number of stress- and strain-
based nucleation models. Gurson
24
proposed a strain-
based nucleation model that describes the rate of void
nucleation as a function of equivalent plastic strain.
Thus, in the proposed phenomenological model, void
nucleation can be expressed as a function of equivalent
plastic strain as below
D
n
= f e
eq

ð16Þ
The primary voids in material cause a local notch
effect, which also affects the local multiaxial stress
concentration (i.e. triaxiality). Mean stress is mainly
influenced by the void growth, observed by experimen-
tation.
26,27
McClintock
26
studied the growth and
Paul et al. 389
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coalescence of cylindrical holes under different stress
states. Li et al.
27
showed the effect of stress triaxiality
created by different sample size and geometry on void
growth. Triaxiality can be defined as mean stress nor-
malized by equivalent stress. High-tensile mean stress
accelerates void growth, while negative mean stress
suppresses void growth, and hence delaying failure was
shown by Hosford and Caddell.
1
Therefore, normally,
it can be said that void growth is a function of triaxial-
ity. In the current model, necking strain is considered
as a function of triaxiality as below
D
g
= fT
r
ðÞ ð17Þ
Leo
´
n-Garcı
´
a et al.
28
and Narayanasamy and Sathiya
Narayanan
29
studied ductile damage (void initiation
and growth) on IF steels.
Weck and Wilkinson
30
reported two coalescence
mechanisms in their material model: the necking of the
ligaments between voids caused by the highest principal
stress and shear-linking up of voids along the direction
of the maximal shear stress. These two mechanisms
were also microscopically observed by Bao and
Wierzbicki
31
in different sets of tests, that is, upsetting
tests, shear tests, tensile tests of smooth, and notched
round bars. Li et al.
27
reported that the necking of the
ligaments between voids is referred as dimple-dominant
fracture, while the linking up of voids is named as shear
fracture. Ghosh
32
showed that the dimple-dominant
fracture is uncommon in the sheet metals and shear
fracture is common for sheet metals. Bressan and
Williams
33
and Li et al.
34
proposed models to describe
shear fracture. Bressan and Williams
33
proposed a
shear instability criterion to calculate local necking in
sheet metal forming under biaxial stretching. Li et al.
34
proposed a modified Mohr-Coulomb (MMC) fracture
criterion, which can predict ductile fracture including
shear fracture. Normally, void nucleation and growth
take place during necking and void coalescence leads
necked region to fracture. Therefore, void coalescence
effect (i.e. shear fracture) is not considered in the cur-
rent phenomenological model.
According to Tresca’s theory,
35
acting shear stresses
are (s
1
–s
2
)/2, s
1
/2, and s
2
/2 for plane stress condition.
In the region of uniaxial tension to equi-biaxial tension,
maximum shear stress will be s
1
/2. In the region of uni-
axial tension to plane strain tension, second maximum
shear stress will be (s
1
–s
2
)/2, while in the region of
plane strain tension to equi-biaxial tension, it will be
s
2
/2. It is assumed in the current work that the first
maximum and the second maximum shear stress have
an effect on material strain hardening behavior and
hence necking. As maximum shear stress is constant in
the region of uniaxial tension to plane strain tension, it
is assumed that the second maximum shear stress (t
s
)
has a role on necking. Therefore, the damage during
necking is modeled by the t
s
, which is normalized by
equivalent stress in a form of
D = f
2t
s
s
eq

ð18Þ
For the region of uniaxial tension to plane strain ten-
sion, 2t
s

s
eq
can be expressed by
2t
s
s
eq
=
1  b
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1  b + b
2
p
for 04b40:5 ð19Þ
For the region of plane strain tension to equi-biaxial
tension, 2t
s

s
eq
can be expressed by
2t
s
s
eq
=
b
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1  b + b
2
p
for 0:54b41:0 ð20Þ
A phenomenological model is proposed in the cur-
rent work based on ductile damage criteria (initiation
and growth of voids) to predict necking at different
loading conditions. The proposed model can be
expressed as
1+3T
r
2

2t
s
s
eq

8
+
3
2
T
r

1
=
3n
2
4
3
5
C = e
eqn
ð21Þ
where e
eqn
is the equivalent plastic strain at necking.
Only two material constants n and C are presented in
equation (21). They can be determined from simple ten-
sile test. n is the strain hardening exponent in the true
tensile stress–strain curve. For plane strain condition,
2t
s

s
eq
=2

ffiffiffi
3
p
and T
r
=1

ffiffiffi
3
p
. C can be determined
from equation (22)
C =
e
ul
1+nðÞ
1+
ffiffi
3
p
2

2
ffiffi
3
p

8
+
ffiffi
3
p
2

1
=
3n
ð22Þ
where e
ul
is the uniform elongation o the material.
Therefore, to draw the complete FLD, only tensile
properties of the material, namely, uniform elongation
(e
ul
) and strain hardening exponent (n) are required.
From equations (21) and (22), for different values of
strain ratio (a), the equivalent plastic strain at necking
(e
eqn
) can be calculated. From equations (12) and (5)
and from the equivalent plastic strain at necking (e
eqn
)
for different values of strain ratio (a), the major and
minor strains at the time of necking can be calculated.
Results and discussion
C-Mn 440 steel, IF 340 steel, and IF steel are consid-
ered in the current work for this model validation.
Engineering tensile stress–strain curves of these three
different steels is depicted in Figure 4(a). Tensile mate-
rial properties that are used as material constants in
and material constant ‘‘C’’ used in the proposed model
are tabulated in Table 1 for these three different steels.
Determination of strain hardening coefficient (K) and
exponent (n) is straightforward for isotropic hardening.
K and n are the only two unknowns, and they are
390 Journal of Strain Analysis 48(6)
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determined by fitting the theoretical stress–strain curve
s = Ke
n
teq
to the experimental true tensile stress–strain
curve. Power law–fitted true tensile stress–strain curves
of three different steels are shown in Figure 4(b). C can
be determined from equation (22) for these three differ-
ent steels.
Strain-based and stress-based FLDs are commonly
used to predict failure in industrial sheet metal forming
operations. Plane stress loading condition is normally
present in industrial sheet metal forming operations
except hydro forming, where through thickness stress
also exists. For three-dimensional loading conditions,
that is, nonplane stress condition, Simha et al.
36
intro-
duced extended stress-based FLD (plot of mean stress
vs equivalent stress) to describe necking. In this current
study, for plane stress forming operations, only strain-
based and stress-based FLDs are examined. All experi-
mental strain-based FLDs are generated by direct
experimental measurement, whereas experimental
stress-based FLDs are generated by mapping experi-
mental data points from strain space to stress space
with the help of basic plasticity theories, which are dis-
cussed in the section ‘‘Plasticity relations.’’
In order to validate this proposed phenomenological
model, the FLDs of C-Mn 440 steel, IF 340 steel, and
IF steel sheets are calculated, and the predicted FLDs
are compared with experimental FLDs, which are
shown in Figure 5. For C-Mn 440 steel shown in
Figure 5(a), the right side of the FLC, that is, the
stretching zone of the proposed model matches well
with the experimental results, whereas the left side of
the FLC, that is, the drawing zone of the proposed
model is slightly underpredicted when compared with
the experimental results. Left-hand side dotted line in
FLD (Figure 5(a)–(c)) represents pure uniaxial tensile
loading (a = 20.5), whereas right-hand side dotted
line represents equi-biaxial stretching (a = 1), while
for IF 340 and IF steel shown in Figure 5(b) and 5 (c),
respectively, FLC predicted by the proposed model
matches reasonably well with the experimental results
on both sides of the FLC, that is, stretching and draw-
ing zones. Similarly, predicted stress-based FLD by
proposed model for C-Mn 440 steel, IF 340 steel, and
IF steel sheets is matched reasonably well with the
experimental results, which is shown in Figure 6. First
dotted line (left-hand side) in stress-based FLD (Figure
6(a)–(c)) represents plane strain loading (b = 0.5),
whereas second dotted line (right-hand side) represents
equi-biaxial stretching (b = 1).
Conclusion
A phenomenological model is proposed for the predic-
tion of FLD from simply tensile material properties.
Table 1. Material constants of different steels.
Materials e
ul
CnK(MPa) Thickness (mm)
IF steel 0.3 0.29042 0.3 555 0.8
IF 340 steel 0.26 0.38611 0.26 675 0.7
C-Mn 440 steel 0.187 0.44871 0.147 695 1.2
IF: interstitial-free; C-Mn: carbon manganese.
Figure 4. Tensile stress–strain diagram of different steels: (a) engineering tensile stress–strain diagrams and (b) true tensile stress–
strain diagrams and the power law–fitted curves, power law coefficients, and exponents as tabulated in Table 1.
IF: interstitial-free; C-Mn: carbon manganese.
Paul et al. 391
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Figure 5. Comparison of predicted FLD with experimental results for different steels (a) C-Mn 440 steel, (b) IF 340 steel, and (c)
IF steel.
IF: interstitial-free; C-Mn: carbon manganese.
Figure 6. Comparison of predicted stress-based FLD with experimental results for different steels (a) C-Mn 440 steel, (b) IF 340
steel, and (c) IF steel.
IF: interstitial-free; C-Mn: carbon manganese.
392 Journal of Strain Analysis 48(6)
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The model is constructed with consideration of damage
accumulation induced by nucleation and growth of
voids. Coalescence of voids is not considered in the
model formulation because FLD is a necking criteria
and coalescence of voids will be minimum during neck-
ing. These two processes of nucleation and growth of
voids are described as functions of the equivalent plas-
tic strain and the stress triaxiality to be multiplied to
represent a phenomenological model. The all material
constants of the proposed model are determined from
solely uniaxial tensile experiment. To check the validity
of the proposed model, three steels are chosen with dif-
ferent strength levels; they are C-Mn 440 steel, IF 340
steel, and IF steel. Comparison with the experimental
data shows that the calculated FLD and the forming
limit stress diagram (FLSD) can predict the sheet metal
forming limits accurately, especially for the right-hand
side of FLD and FLSD.
Acknowledgements
Authors would like to thank Dr. Saurabh Kundu,
Head Product Research Group, R&D, Tata Steel
Limited, Jamshedpur, India, for his kind cooperation.
Declaration of conflicting interests
The authors declare that they have no conflict of
interest.
Funding
This research received no specific grant from any fund-
ing agency in the public, commercial, or not-for-profit
sectors.
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