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The Journal of Strain Analysis for Engineering

http://sdj.sagepub.com/content/48/6/386

The online version of this article can be found at:

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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Original Article

J Strain Analysis

48(6) 386–394

Ó IMechE 2013

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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

ds

ð4Þ

where dl is the proportionality constant, ds 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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