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INTRODUCTION
Forming Limit Curve (FLC) is a map of principal strains that
delineates safe region from failed region for sheet metals and
is a very useful tool to determine formability limits in stampings.
Since the FLC of steel is able to predict the necking during
sheet forming process, it provides a failure criterion for Finite
Element Analysis (FEA) to judge if the part is feasible with the
specied grade in the part design.
Fracture forming limit curve describes the strain limit of
material that can be formed without fracture. Although FLC is
more often used for failure prediction in sheet forming process,
FFLC with limit of fracture strains is also of important input
when fracture prediction is required, such as in crash
simulations.
Several deformation methods exist to acquire FLC, among
which the most often used ones are Nakajima [1] and
Marciniak tests [2]. The Nakajima test uses a spherical shaped
dome, while Marciniak test uses a at topped dome. For both
testing methods, specimens with a series of widths are used to
generate different strain paths. The conventional method of
determining FLC is to measure the strains just before incipient
necking by reading circle grids which were electrochemically
etched on the surface of the sheet. In North America, Nakajima
test is normally used, and the experimental method to catch
the incipient necking is by trial and error. A series of specimens
are formed rst to different heights with different necking
conditions. Thus the test can be controlled such that it is
terminated at the moment when localized necking is initiated.
This process is done by visual observation or hand feel, thus
the test is time consuming (almost one month per steel) and
often carries large error. In Europe, conventionally a position
dependent approach was used to determine incipient necking
strains for sheet metals formed with Nakajima or Marciniak
testing method, where limit strains from different strain paths
were calculated by the extrapolation of strain distribution in the
failure zone before or after fracture occurred [3]. This FLC
determination approach evolved into ISO 12004-2 [4].
In the last several decades, Digital Image Correlation has been
increasingly applied in mechanical characterization of sheet
metal due to the fact that DIC as a non-destructive technique is
able to provide measurement of full-eld strain history in a fast
and accurate fashion compared to most conventional strain
measurement methods. The mechanism of DIC is based on
the assumption that the pattern feature of a certain area of
surface (e.g. paint particles, grids) does not change before and
after deformation, and the displacement and strain eld can be
determined by comparing pixel by pixel between the deformed
Determination of Forming Limit and Fracture Limit Curves Using Digital
Image Correlation
Gang Huang and Sriram Sadagopan
ArcelorMittal USA
Hubert Schreier
Correlated Solutions Inc.
ABSTRACT
Forming limit curve (FLC) and fracture forming limit curve (FFLC) are valuable tools for failure prediction in forming
simulation and die try-out in press shops. In this paper, methods are presented to determine FLC and FFLC for sheets of
advanced high strength steels (AHSS) using digital image correlation (DIC). Dome tests were conducted on AHSS
specimens using DIC system for strain measurement. For generating FLCs, two approaches are introduced to determine
the onset of localized necking by analyzing the strain history at critical locations, one of which has been implemented into
the commercial DIC software Vic-3D (Correlated Solution inc.). For determination of FFLC, a method for measuring
fracture strains based on the strain path evolution is presented. The measured FLCs for several AHSS were compared to
the FLCs using ISO 12004-2, and conventional North American experimental measurements and empirical equations. The
results of comparison revealed that FLCs using DIC are in good correlation with other existing data, thus validating the
presented methodologies.
CITATION: Huang, G., Sadagopan, S., and Schreier, H., "Determination of Forming Limit and Fracture Limit Curves Using
Digital Image Correlation," SAE Int. J. Mater. Manf. 7(3):2014, doi:10.4271/2014-01-0982.
2014-01-0982
Published 04/01/2014
Copyright © 2014 SAE International
doi:10.4271/2014-01-0982
saematman.saejournals.org
574
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and undeformed images of specimens. Figure 1 shows the
schematic of DIC mechanism. During DIC analysis, in the
undeformed image a subset is rst designated around the point
where strain will be measured, then the point of interest on the
deformed images is tracked by matching the subset between
deformed images and reference image based on the
assumption that the feature (e.g. gray scale) does not change
after deformation of specimen. Once the point of interest is
tracked, the displacement and thus the strain can be
determined.
Figure 1. Schematic of tracking process in DIC analysis
With the high popularity of its application in deformation
measurement, DIC has been recently also used in
determination of FLC. There are two categories of incipient
necking criteria to generate FLC. One is the position
dependent approach, which is aligned with the ISO 12004-2 to
determine the limit major/minor strains using an inverse
parabola tting of designated length along cross-section lines
perpendicular to fracture line on both sides of necking zone [4].
The other category is of time-dependent approaches, which
analyzed the evolution of strain(s) at critical location(s) from
DIC to determine the moment of incipient necking. Huang et al.
developed a criterion based on the peak of the second
derivative of strain with respective to time for determination of
the moment of incipient necking [5]. Volk proposed a method of
strain rate frequency diagram from which the onset of necking
is assumed at sudden increased number of measurement
points for higher strain rate inside the necking zone [6].
Feldman et al. presented a criterion for incipient necking based
on the assumption that necking started when the rate of
difference between maximum and average strains in the
necking zone diverged from initial linearity [7].
The main advantage of the position dependent approach is
that only a few images are needed to generate limit strain point
at specic strain path. However, the limit strains are
determined only by postulation using curve tting without direct
access to the real moment of incipient necking. For time-
dependent approaches, the advantage over the time-
dependent approach is that strains are temporally analyzed to
detect the incipient necking moment; however, the varied
methods depend heavily on the mathematical assumption and
process to nd the onset of necking.
It has been reported [7,8] that all the position-dependent and
time-dependent approaches tend to generate FLCs close to
those using the conventional method or ISO 12004-2 practiced
in Europe. Prior to this paper, no approach was reported that
can match the FLC data with North America (NA) experimental
method or the Keeler -Brazier equation. To provide an accurate
and efcient way to generate FLC data which match NA
experimental method/K-B equation, an incipient necking
criterion with mixed time and position dependent approach has
been developed at ArcelorMittal Global R&D, and is presented
in this paper.
The approaches presented in this paper for determination of
FLC are based on strain measurement from DIC. Two criteria
to determine incipient necking are presented, and the resulting
FLCs of several materials are compared with conventional
data. Meanwhile, a method to determine FFLC using DIC is
also presented.
EXPERIMENTAL PROCEDURE
Materials
Seven materials were used in this study. Table 1 shows the list
of materials with tensile properties. The properties were
determined using the ASTM E-8 standard for tensile testing.
Table 1. Steel grades and tensile properties
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In Table 1, CR stands for uncoated steel and GA stands for a
galvannealed coating. DP refers to the dual phase
microstructure and TRIP for transformation induced plasticity
microstructure. TRIP steels have a small percentage of
retained austenite in the microstructure that transforms to
martensite upon deformation. M900 is a fully martensitic
material. The DP980SF steel grade has a higher YS/UTS ratio
and has been developed specically for stretch anging
applications.
Experiment
A dome tester with DIC system was used to form a dome
shape (Figure 2). An MTS servo hydraulic testing system was
employed to conduct LDH tests where Nakajima testing
method was applied. The steel blanks were 177.8 mm long
with the width varying from 25.4 to 177.8 mm to represent
different strain paths. A hemispherical punch with diameter of 4
inches was used. Tests were performed on 3-5 blanks for each
width of each steel grade. The blanks were prepared with
spray paint speckles on the surface. During the forming
process, two cameras oriented with a xed angle were
positioned above the forming equipment, as shown in Figure 2.
Images of the specimen and the speckle pattern were recorded
simultaneously by the two cameras at speed of 15-20 frames/
second during forming until fracture occurred. Then the
dedicated DIC software, Vic 3D (from Correlated Solutions
Inc.), was used to analyze the acquired images to calculate the
major and minor strains of every point of the deformed
specimens at each imaging moment.
Figure 2. Setup of LDH test using DIC system and LDH specimen
For comparison, conventional measurements of FLC based on
NA experimental method and ISO 12004-2 were also
conducted. The experimental procedures of ISO 12004-2 are
detailed in reference [4]. Using the NA experimental method,
the samples (width of 25.4 to 177.8mm and length of
177.8mm) were rst electrochemically gridded with circles of
diameter 2.54mm, as shown in Figure 3. Then the samples
were formed with trial and error to acquire the testing outcome
with incipient necking based on nger touch. Thereafter, the
samples with incipient necking were inspected with a camera
system to measure the strains, and then FLC was drawn as
lower bound to the necking points. The major drawback for this
technique is that it needs considerable amount of time to
generate one experimental FLC, and it is dependent on the
skill and experience of the operator.
Figure 3. Samples formed using the traditional NA technique for
determination of FLC
For determination of FFLC using DIC, the same experimental
procedures for FLC were applied, where the samples with
different widths were formed up to fracture, and pictures
acquired by cameras were analyzed to determine the facture
strain for different strain paths.
INCIPIENT NECKING CRITERIA
Two incipient necking criteria have been developed to
determine FLCs of AHSS at ArcelorMittal Global R&D-East
Chicago. The rst criterion was based on the time-dependent
behavior of major strain history at the critical position where
fracture occurred. The second criterion was based on the strain
history relationship between the critical position (with ultimate
fracture) and other position; therefore it is of a method of mixed
time-dependence and position dependence.
Criterion 1
It is assumed that when the necking occurs inside the sheet
metal through thickness direction, the major strain or minor
strain is subject to some instant change caused by instability.
This should be reected by the change of strain or derivative of
some order.
Figure 4 shows an example dome sample after fracture and its
contour of major strain distribution determined from DIC on the
picture frame right before fracture. The position with maximum
major strain on the contour was selected as the critical point
where the necking and fracture occurred, as shown in Figure 4.
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Figure 4. Point of interest to extract the major strain history. (a)
Specimen after fracture. (b) Major strain distribution with highlighted
position right before fracture occurred
The main task for FLC determination using DIC is to identify
the imaging moment of incipient necking. Within the contour of
strain distribution, the critical point with the eventual maximum
major strain was investigated for incipient necking
identication. It should be noted that, depending on the
specimen geometry and the friction coefcient between punch
and specimen, the position of maximum major strain might not
be xed during the course of forming. Therefore, as shown in
Figure 4, the point with maximum major strain on the image
right before fracture was used to analyze the strain data to
detect the incipient necking moment. The method of analysis is
described in Figure 5.
Figure 5. Histories of strain, strain rate and second derivative
First, the major strain history from beginning up to fracture was
plotted for the critical point. Second, the rst derivative with
respect to time (or strain rate) was calculated numerically; as
shown in Figure 5, the rst derivative was monotonically
increasing up to fracture, and no noticeable instant change or
peak was present. Next, the second derivative of strain with
respect to time was calculated and plotted. As shown in Figure
5, a peak of second derivative was evident before fracture. It
was presumed that the rate of change of strain rate reaches
maximum when local instability occurred due to the strain path
switched to plane strain [9]. Therefore, it was assumed that
incipient necking was at the moment of peak of the second
derivative. Finally, with incipient necking moment identied, the
major and minor strains at the same moment for the specimen
of specic strain path can be determined, which constitutes
one incipient necking data point of FLC. Similarly, by applying
the same procedure the data points for other strain paths can
also be determined, and a complete FLC can be developed.
Criterion 2
It has been reported that Criterion 1 of incipient necking
identication gave rise to consistently conservative FLCs of
AHSS compared to those using NA experimental method or the
K-B equation. To efciently generate FLCs of AHSS consistent
with the NA practice of using K-B equation, so that the data can
be utilized by press shop for die try-out, a new criterion is
needed.
Figure 6. An example of major strain contour with critical point and
offset locations (the circles) right before fracture
In order to abide by the mechanism of localized necking, a
mixed time and position dependent method has been
developed to detect the incipient necking moment. When a
sheet metal is subjected to localized necking during forming
process, physically the major strain within the necking zone
keeps increasing, while for the area away from the necking
zone the major strain reaches a plateau or peak value. Figure
6 shows the contour of major strain with the maximum in the
center for image frame right before fracture; the circles
surrounding the maximum strain represent the offset locations.
Figure 7 delineates the major strain histories of the critical point
and one offset point.
As shown in Figure 7, the peak of major strain was reached
before fracture at the end of the major strain history for the
critical point, which was monotonically increasing. Therefore,
the incipient necking was considered to occur at the moment of
peak strain for the offset point, denoted as tmax. However, due
to the strain gradient tmax might be dependent on the selection
of offset point. On the other hand, tmax might be also affected
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by the fact whether the localized necking or fracture was
initiated in the center or off the center, which was mainly
determined by the tooling condition (Nakajima or Marciniak),
frictional conditions and specimen geometry. To address this
issue, an algorithm was developed to determine the offset point
to extract tmax for two individual situations of critical points. With
tmax acquired, the major and minor strains can then be
extracted at tmax for the sample of the specic strain path.
Similarly pairs of major and minor strains can be determined
for other strain paths as well to form the complete FLC.
Figure 7. Major strain histories at Critical and Offset Points
As an effort to develop an automated system to determine the
FLC using DIC, the incipient necking Criterion 2 has been
implemented into the commercial software Vic-3D (Correlated
Solutions Inc.).
FFLC MEASUREMENT
The fracture forming limit curve provides the limit of strain for
sheet metals before fracture initiation. In a sense similar to
FLC, FFLC composed of limit of major and minor strains under
strain paths varied from being uniaxial to equibiaxial. However,
FFLC might well extend strain limit beyond localized necking till
fracture, especially for ductile materials. With application of
DIC, compared to conventional measurement methods the
local strains at facture zone are relatively easy to measure.
However, for cases of high localization of strain at fracture
zone with indication of steep in-plane strain gradient, the
fracture strain measurement can be highly dependent on the
gauge length (similar to grid size) used in DIC, and
consequently there could be considerable uncertainty in
fracture strain results which constitute FFLC. To circumvent the
associated difculty to reduce the measurement uncertainty, a
hybrid method was developed to measure the fracture strains
for FFLC.
It was documented [8] that, after localized necking starts within
the necking zone the major strain increases rapidly while the
minor strain keeps close to a constant, such that the stress
state switches from original state (drawing or stretch) to plane
strain, as shown in Figure 8 for schematic describing the strain
paths for typical FLC and FFLC of steels. With the minor strain
being close to constant and also based on the fact that prior to
necking there is no localization, the minor strain can be readily
measured using DIC without much uncertainty caused by the
issue of gauge length. While the major strain is drastically
increasing after necking and highly dependent on the selection
of gauge length, the thickness strain can be measured rst
using a point micrometer, and thereafter the major strain can
be calculated using the measured thickness strain and minor
strain with assumption of volume conservation. With the
determined pair of major and minor fracture strains for different
strain paths, a complete FFLC can be generated. It should be
noted that, with the void nucleation and growth inside the
necking zone due to damage accumulation after necking, the
material volume is not strictly a constant; however, with small
volume fraction of voids before fracture for majority of AHSS,
the volume conservation can still be applied for determination
of fracture strain within reasonable error to some extent of
condence.
Figure 8. A schematic of strain paths of FLC and FFLC
RESULTS AND DISCUSSIONS
In this section, the results of FLCs of several steels using two
different incipient necking criteria are presented, followed by
the results of FFLC determined using the measurement
method as previously described.
FLCs Using Criterion 1
The results of FLC using DIC based on incipient necking
Criterion 1 for materials of 1.4mm DP590 CR, 1.2mm DP780
CR and 0.96mm TRIP780 GA are shown in Figures 9, 10 and
11, respectively.
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Figure 9. FLCs of 1.4mm DP590 determined using Criterion 1 and
other methods.
Figure 10. FLCs of 1.2mm DP780 determined using Criterion 1 and
other methods.
Figure 11. FLCs of 0.96mm TRIP780 determined using Criterion 1 and
other methods.
For comparison, also plotted in Figures 9, 10 and 11 are the
FLC data determined using NA experimental method, K-B
equation and ISO 12004-2 for the three materials. Nakajima
tests were also tested on other AHSS to determine FLCs;
however, for the same coils of materials, only DP590 CR and
DP780 and TRIP780 were tested according to ISO 12004-2.
As shown in Figures 9, 10 and 11, the FLCs determined using
incipient necking Criterion 1 are consistently lower than those
from NA experimental method and K-B equation; e.g., the FLC0
using Criterion 1 is lower than FLC0 using NA experimental by
0.055, 0.045 and 0.038 in true strain for DP590, DP780 and
TRIP780 respectively. However, FLCs using Criterion 1 are in
very good agreement with those determined by ISO 12004-2,
the difference of FLC0 between two methods is 0.01, 0.011 and
0.015 for DP590, DP780 and TRIP780, respectively. This
implies that using Criterion 1 will most probably result in FLCs
matched with ISO 12004-2.
FLCs Using Criterion 2
The same images acquired from LDH tests on 1.4mm DP590,
1.2mm DP780 and 0.96mm TRIP780 for FLCs using DIC
based on incipient Criterion 1 were reanalyzed using incipient
necking Criterion 2. In addition, Criterion 2 was applied to
analyze the images with DIC from LDH tests on 0.8mm DQSK,
1.54mm TRIP590, 1.6mm DP980SF and 1.2mm M900 to
determine FLCs.
Figures 12, 13, 14, 15, 16, 17 show the results of FLC using
Criterion 2 for the 7 materials. The FLCs determined using NA
experimental method and K-B equation are also included for
comparison. Although NA experimental method was not
applied to 0.8mm DQSK and the curve was not available for
comparison, it is generally assumed that for conventional
steels of FLC using K-B equation should be very close to that
from NA experimental measurement. It should be noted that
gauge length of 2 mm was used for all FLCs using Criterion 1
and Criterion 2, which is close to the grid size used in
generating traditional NA FLCs.
Figure 12. FLCs of DQSK determined using Criterion 2 and K-B
equation.
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Figure 13. FLCs of DP590 determined using Criterion 2 and other
methods.
Figure 14. FLCs of TRIP590 determined using Criterion 2 and other
methods.
Figure 15. FLCs of TRIP780 determined using Criterion 2 and other
methods.
Figure 16. FLCs of DP780 determined using Criterion 2 and other
methods.
Figure 17. FLCs of DP980SF determined using Criterion 2 and other
methods.
Figure 18. Incipient necking data of M900 determined using Criterion 2
and FLCs using other methods.
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As shown in Figures 12, 13, 14, 15, 16, 17, for all six materials
the FLCs determined using incipient necking Criterion 2 are in
very good agreement with the data generated from NA
experimental method and the traditional NA FLC (K-B
equation) along nearly all strain paths, except for DP980SF of
which the FLC using Criterion 2 is quite different under the
biaxial stretch conditions while it agrees well with data from NA
experimental method. On the other hand, the results imply that
K-B equation well predicted the FLCs for nearly all the 6
materials under nearly all strain paths. However, it was of very
different case for M900. As shown in Figure 18 for FLC data
from DIC using Criterion 2, from NA experimental method and
K-B equation, the K-B equation substantially underestimated
the experimental FLC, with the FLC0 difference of nearly 0.05.
Nonetheless, the incipient necking data points are in very good
agreement with NA experimental data, in spite of the limited
data points that are available.
For a comprehensive comparison between incipient necking
criteria 1 and 2, FLCs generated using the two criteria plotted
in Figures 19 and 20 for DP590 and DP780, as two examples.
As shown in Figures 19 and 20, the FLCs determined using
Criterion 2 is noticeably higher than using Criterion 1 for both
materials. This implies that Criterion 1 detected the necking
earlier, while there might be some small amount of necking
might have occurred prior to the necking moment identied by
Criterion 2. Accordingly, depending on the need and practice of
OEMs or press shops from different regions, the selection
between the two incipient necking criteria can be made. For
example, for the users in North America who follow NA
experimental method or K-B equation, Criterion 2 is
recommended, while for users following ISO 12004-2, Criterion
1 is recommended; or for some users, the FLCs generated
from Criterion 1 and Criterion 2 can be applied as up and low
bounds.
Figure 19. Comparison of DP590 FLCs generated using incipient
necking Criterion 1 and Criterion 2.
Figure 20. Comparison of DP780 FLCs determined using incipient
necking Criterion 1 and Criterion 2.
Results of FFLC
The FFLCs determined using the method with combination of
measurement from DIC and thickness measurement are
shown in Figures 21 and 22 for two materials. Also included in
Figures 21 and 22 are FLCs determined using incipient
necking criteria 1 and 2. Although only limited data of FFLC are
available from this study, demonstrated from Figures 21 and 22
FFLCs of the two materials are linear for varied strain paths,
which is consistent with the traditional theoretical prediction for
steels.
Figure 21. FFLC of 1.4mm DP590
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Figure 22. FFLC of 1.2mm DP780
CONCLUSIONS
Two incipient necking criteria are presented to determine FLC
using DIC. The rst criterion is based on strain evolution at
critical location where localized necking and fracture occurs,
while the second criterion takes into account the strain history
of offset location and its relationship with strain evolution of
critical location. The FLC data determined using the two criteria
are in very good agreement with existing conventional data
using ISO 12004-2 and NA experimental method, indicating
that the two criteria are able to more efciently generate FLCs
that match ISO 12004-2 and NA experimental method,
respectively. Meanwhile, a method to measure FFLC is also
introduced, which is able to provide strain data for fracture
prediction.
REFERENCES
1. Nakajima, K., Kikuma, T., Asaku, K., “Study on the Formability of
Steel Sheet,” Yawata Technical Report 264, 1968.
2. Marciniak, Z. and Kuczynski, K., “Limit strains in the process of
stretch forming sheet metal”, International Journal of Mechanical
Science 9, 1967, 609-620.
3. Bragard, A., Baret, J.-C., Bonnarens, H.: “A simplified Technique
to Determine the FLD at the Onset of Necking”, C.R.M. 33, 1972,
53-63.
4. “Metallic materials -Sheet and strip - Determination of forming limit
curves - Part 2: Determination of forming limit curves in laboratory”,
ISO Standard, ISO-DIS 12004-2, 2006.
5. Huang, G., Sadagopan, S., Yan, B., “Digital Image Correlation
Technique and Its Application in Forming Limit Curve,” IDDRG
2008 International Conference, June 2008, Olofström, Sweden.
6. Volk, V., “New Experimental and Numerical Approach in the
Evaluation of the FLD with the FE Method,” Proceedings of the
FLC Zurich 2006, March 2006, ETH Zurich, Switzerland.
7. Feldmann, P., Schatz, M. and Aswendt, P., “Automatic FLC-Value
Determination from 4D Strain Data,” IDDRG 2009 International
Conference, June 2009, Golden, USA.
8. Sriram, S., Huang, G., Yan, B., and Geoffroy, J., “Comparison of
Forming Limit Curves for Advanced High Strength Steels Using
Different Techniques,” SAE Int. J. Mater. Manuf. 2(1):472-481,
2009, doi:10.4271/2009-01-1173.
9. Lee, Y.-W., “Fracture Prediction in Metal Sheets,” PhD thesis,
Massachusetts Institute of Technology, 2005.
CONTACT INFORMATION
Gang Huang
Applications Technology, Automotive Product Applications
ArcelorMittal Global R&D - East Chicago
3001 E. Columbus Drive, East Chicago, IN 46312
gang.huang@arcelormittal.com
219-399-6561
ACKNOWLEDGMENTS
The authors thank Dr. Benda Yan and Dr. Blake Zuidema for
their support and constructive suggestions.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical,
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Positions and opinions advanced in this paper are those of the author(s) and not necessarily those of SAE International. The author is solely responsible for the content of the
paper.
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