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Abstract
This paper presents a method for acquiring true stress-strain curves over large
range of strains using engineering stress-strain curves obtained from a tensile test coupled
with a finite element analysis. The results from the tensile test are analyzed using a rigid-
plastic finite element method combined with a perfect analysis model for a simple bar to
provide the deformation information. The reference true stress-strain curve, which predicts
the necking point exactly, is modified iteratively to minimize the difference in the tensile
force between the tensile test and the analyzed results. The validity of the approach is
verified by comparing tensile test results with finite element solutions obtained using a
modified true stress-strain curve.
Keywords : Flow stress; Large strain; Stress-strain curve; Tensile test
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1. Introduction
Metal-forming simulation techniques have become generalized in industry. As a
result, material properties, including the true stress-strain curves, are indispensable for
process design engineers because the accuracy of a simulation depends mainly on that of
the material properties used. This is especially true for true stress-strain curves. A true
stress-strain curve is affected by the manufacturing history, metallurgical treatments, and
chemical composition of the material. Therefore, metal-forming simulation engineers
require true stress-strain curves that reflect the special conditions of their materials.
However, it is difficult to obtain the material properties from experiments and very limited
information about true stress-strain curves can be found in the literature. Most simulation
engineers use the material properties supplied by software companies, which are very
limited and sometimes unproven.
True stress-strain curves can be obtained using tensile (Bridgman, 1952; Cabezas
and Celentano, 2004; Koc and tok, 2004; Komori, 2002; Mirone, 2004; Zhang, 1995;
Zhang et al., 1999), compression (Choi et al., 1997; Gelin and Ghouati, 1995; Haggag et al.
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1990; Lee and Altan, 1972; Michino and Tanaka, 1996; Osakada et al., 1991), ball
indentation (Cheng and Cheng, 1999; Huber and Tsakmakis, 1999a; Huber and Tsakmakis,
1999b; Lee et al., 2005; Nayebi et al., 2002), punch (Campitelli et al., 2004; Husain et al.,
2004; Isselin et al., 2006), torsion (Bressan and Unfer, 2006), and notch tensile
(Springmann and Kuna, 2005) tests. Most of these methods obtain true stress-strain
relations only for strains less than 0.3. However, the maximum strain often exceeds 1.0 in
bulk metal forming, such as in forging, extrusion, and rolling. Sometimes it reaches 3.0 in
multi-stage automatic cold forging, the so-called cold-former forging used to produce
fasteners.
Recently, many researchers have tried to obtain true stress-strain curves using finite
element methods, see e.g. (Cabezas and Celentano, 2004; Campitelli et al., 2004; Choi et al.
1997; Husain et al., 2004; Isselin et al., 2006; Lee et al., 2005; Mirone, 2004; Nayebi et al.,
2002; Springmann and Kuna, 2005). In a tensile test, the true strain reaches its maximum
value at the smallest cross-section in the necked region, and it may exceed 1.5 just before a
ductile material fractures. Therefore, one should be able to obtain the flow stress of
materials at a large strain if finite element methods are used to predict the localized
deformation behavior during a tensile test. A few researchers have attempted to obtain the
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flow stress at a large strain using simulation and experimental approaches, but these
applications have been quite limited, see e.g. (Cabezas and Celentano, 2004; Mirone, 2004)
The first step in obtaining the true stress at a large strain from a tensile test is to
predict the onset of necking exactly using analytical, numerical, or experimental methods.
Many researchers have applied finite element methods to predict the onset of necking (Joun
et al., 2007). However, all researchers who have used simple bar models between gage
marks of a tensile test specimen have included various imperfections or constraints at the
ends to allow necking to take place artificially. Several researchers have used a full tensile
test specimen, including a grip, as the analysis model. A full specimen model causes some
difficulty when matching experimental data with predictions and thereby generalizing the
approach. Dumoulin et al. (2003) satisfied the Considre criterion (Considre, 1885) using
a full model of a sheet specimen. However, they did not discuss the accuracy of their
predictions compared with experiments in a quantitative manner. Joun et al. (2006) were
the first to obtain accurate finite element solutions that satisfied the Considre criterion
exactly in an engineering sense using a perfect tensile test analysis model, that is, a
cylindrical specimen consisting of a simple bar model without any imperfections. They
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recently predicted the exact onset of necking using a rigid-plastic finite element method
(Joun et al., 2007) and Hollomons constitutive law.
This paper presents a new method based on our previous research (Joun et al., 2007)
and an iterative error-reducing scheme to obtain the true stress-strain relationship at a large
strain from the localized deformation behavior in the necked region.
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2. Acquisition of the stress-strain relationship after necking
Figure 1 shows a typical tensile test result selected to illustrate and apply our
approach. When the aspect ratio of the specimen exceeds a certain value, the onset of
necking is dependent on the strain-hardening exponent (Considre, 1885; Joun et al., 2007).
Previously, we predicted the onset of necking exactly in an engineering sense using a rigid-
plastic finite element method and a perfect analysis model. In the analysis, the material was
considered rigid-plastic and isotropically hardened, and its flow stress was described using
Hollomons constitutive law. The analysis model was a simple bar with shear-free ends and
no imperfections, known as a perfect analysis model.
Previously, we defined the reference stress-strain curve as follows (Joun et al.,
2007):
N n N K
=
(1)
where N K is the reference strength coefficient, N n is the reference strain hardening
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exponent, and and are the effective stress and strain, respectively. The reference
strain hardening exponent, denoted as N n , is defined as the true strain at the necking point,
that is,
ln (1 ) N N en = + (2)
where N e is the engineering strain at the necking point. The reference strength coefficient,
denoted as N K , is defined by making the flow stress curve of Equation (1) pass through
the necking point in the true stress-strain curve. Therefore, the reference strength
coefficient can be found from
ln(1 )
(1 )[ln(1 )]
N e
N N e e
N N e
K
++
=
+(3)
where e is the engineering stress at the maximum load point,i.e., the necking point.
Necking starts when the tensile load reaches a maximum value according to conventional
necking theory.
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The reference stress-strain curve shown in Figure 2 is calculated with an emphasis
on the occurrence of necking from the engineering stress-strain relationship shown in
Figure 1. Using the reference stress-strain curve, we previously predicted the elongation
and maximum load exactly at the onset of necking (Joun et al., 2007).
The reference stress-strain curve in Figure 2 must be used to predict the necking
point exactly in an engineering sense. However, problems arise from the fact that the
difference between predictions and experiments increases with the elongation, as shown in
Figure 3, and that the true strain of the material during cold forging sometimes exceeds the
true strain at the necking point by more than a dozen times. Therefore, the reference stress-
strain curve cannot be used to predict the material behavior exactly after necking occurs.
Consequently, an appropriate scheme is necessary to obtain an improved true stress-strain
curve from the reference stress-strain curve. We predict the exact engineering stress-strain
curve using a finite element simulation of a tensile test to obtain an improved true stress-
strain curve.
After necking occurs, the non-uniformity of the true strain increases rapidly in the
longitudinal direction. The maximum strain occurs at the minimum cross-section where the
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shear stress is free due to symmetry and the non-uniformity of the strain distribution is
relatively low. Therefore, it is relatively easy to define the representative strain.
Through finite element analysis, one can trace the minimum cross-section of the
tensile test specimen at a specified or sampled elongationi . The representative strain of
the minimum cross-section at elongationi , denoted as i R , can be calculated from finite
element solutions of the tensile test. The difference between the measured loadit F and the
predicted load ie F at elongationi can be minimized by modifying the true stressi R
corresponding to the representative straini R .
In this paper, the representative straini R is defined using the following average
area scheme:
ii A R i
dA
A
=
(4)
where i A indicates the area of the minimum cross-section of the tensile test specimen at the
sampled elongation i . The current true stress ,i i R old R = at i R is modified to give the
new true stress ,i R new by multiplying the current true stress byi
t i
e
F F
as follows:
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, ,
ii i t
R new i R old e
F F
= (5)
The reference stress-strain curve is used before necking occurs. After necking, the true
stress-strain relationship is interpolated linearly using the sampled points (i R , i R ) defined
at the elongationi
, as shown in Figure 4.
Fig. 1. Experimental results of a tensile test.
Fig. 2. Reference stress-strain curve, defined by N n N K = .
Fig. 3. Comparison of experiments with tensile test predictions.
An iterative algorithm is proposed. The detailed procedure used to calculate the
improved sampled points (i R ,i
R ) at the sampled elongation i is as follows. In the
algorithm, ,i j R and,i j
R are the j-times modified strain and stress, respectively, at the
sampled elongation i .
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Step 1: Calculate the reference strain hardening exponent N n and the reference strength
coefficient N K from tensile test experiments using Equations (2) and (3). Select the
sampled elongations i ( 1,2, ,i M = ) from the experimental data after the necking point.
Step 2: Conduct a finite element analysis of the tensile test using the reference stress-strain
curve and then calculate i R ( 1,2, ,i M = ) at the sampled elongationi from the finite
element solutions of the tensile test.
Step 3: Set j = 1 and ,i j i R R = and then calculate,i j
R from , ,( ) N ni j i j R N R K =
( 1,2, ,i M = ).
Step 4: Replace j with j + 1 and set , , 1i j i j R R
= and , , 1i j i j R R
= ( 1,2, ,i M = ).
Step 5: Conduct a finite element analysis of the tensile test using both the perfect analysis
model and the true stress-strain curve defined by N n , N K and ( ,i j R , ,i j R ) ( 1,2, ,i M = ).
Then check the convergence of the solution at the sampled elongationi ( 1,2, ,i M = ) by
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comparing the measured load it F with the predicted load ie F . If convergence is achieved,
stop the iterations. Otherwise, calculatei R from the finite element solutions and set
, 1i j i R R
+ = .
Step 6: Calculate the stress i R at, 1i j
R += ( 1,2, ,i M = ) by linearly interpolating the
sampled points ( ,i j R ,,i j
R ) ( 1,2, ,i M = ), and calculate the improved stress, 1i j R
+ at
, 1i j R
+= as follows:
, 1i
i j i t R R i
e
F F
+ = (6)
Step 7: Replace j with j + 1 and return to Step 4.
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3. Application example
Figure 4 shows points corresponding to the sampled elongationi on the reference
stress-strain curve as an example, and gives the improved stress-strain curve from the first
iteration. This curve was improved from the reference stress-strain curve using our
approach. Figure 5 compares the predicted load-elongation curve obtained using the
reference, improved, and measured stress-strain curves. The load-elongation curve
predicted from the improved stress-strain curve was considerably more accurate.
Quite accurate results were obtained after a single iteration, although the maximum
elongation was quite large. The maximum error was 474 N or 6.04% of the measured load.
This error could be reduced or minimized through additional iterations. Figures 6 and 7
show the improved stress-strain curves for several iterations and their corresponding
predicted load-elongation curves, respectively.
Fig. 4. Modified true stress-strain curve calculated after first iteration.
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Fig. 5. Comparison of the load-elongation curves.
Fig. 6. Comparison of the stress-strain curves.
Fig. 7. Comparison of the load-elongation curves
Table 1 lists the maximum errors of the predicted loads relative to the measured
loads with the number of iterations. After four iterations, the maximum error was reduced
to less than 0.03 %,i.e., it led to the exact solution in an engineering sense. Therefore, the
convergence characteristics of our scheme are quite good
Table 1 Reduction in error with the number of iterations
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4. Concluding remarks
An approach for acquiring true stress-strain curves at large strains by coupling
experiments with an analysis based on a tensile test and a rigid-plastic finite element
method was presented. The approach uses the reference stress-strain curve before necking
occurs to predict the necking point exactly. An iterative scheme then minimizes the error
between the measured and predicted load-elongation curves after necking occurs by
improving the true stress-strain curve.
Our approach can predict the flow stress at large strains using only the measured
load-elongation curve of a material and a tensile test analysis, yielding exact results from an
engineering viewpoint. This is very important for simulating bulk metal forming. The
approach is simple and systematic, and it can be embedded into commercial metal-forming
simulation software with ease.
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Acknowledgements
This work was supported by grant No. RTI04-01-03 from the Regional Technology
Innovation Program of the Ministry of Commerce, Industry and Energy (MOCIE).
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Figure lists
Fig. 1. Experimental results of a tensile test.
Fig. 2. Reference stress-strain curve, defined by N n
N K = .
Fig. 3. Comparison of experiments with tensile test predictions.
Fig. 4. Modified true stress-strain curve calculated after first iteration.
Fig. 5. Comparison of the elongation-tensile force curves.
Fig. 6. Comparison of the stress-strain curves.
Fig. 7. Comparison of the elongation-tensile force curves.
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Engineering strain
E n g i n e e r i n g s t r e s s ( M P a )
0 0.1 0.2 0.3 0.4 0.50
100
200
300
400
500
Measured
Necking point
Fig. 1. Experimental results of a tensile test.
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True strain
T r u e s t r e s s ( M P a )
0 0.5 1 1.50
250
500
750
Measured and fittedExtrapolated
Necking point
Fig. 2. Reference stress-strain curve, defined by= N
n
N K
.
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Elongation (mm)
L o a d ( N
)
0 2 4 6 8 100
2500
5000
7500
10000
12500
MeasuredReference
Necking point
Fig. 3. Comparison of experiments with tensile test predictions.
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True strain
T r u e s t r e s s ( M P a )
0 0.5 1 1.50
250
500
750
Measured and fittedReferenceFirst improved
Necking point
1098
76
54
321
Fig. 4 Modified true stress-strain curve calculated after first iteration.
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Elongation (mm)
L o a d ( N
)
0 2 4 6 8 100
2500
5000
7500
10000
12500
MeasuredReferenceFirst improved
Necking point
Fig. 5. Comparison of the elongation-tensile force curves.
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True strain
T r u e s t r e s s ( M P a )
0 0.5 1 1.50
250
500
750
Measured and fittedReferenceFirst improvedSecond improvedThird improvedFourth improved
Necking point
Fig. 6. Comparison of the stress-strain curves.
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Elongation (mm)
L o a d ( N
)
0 2 4 6 8 100
2500
5000
7500
10000
12500
MeasuredReferenceFirst improvedSecond improvedThird improvedFourth improved
Necking point
Fig. 7. Comparison of the elongation-tensile force curves.
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Table lists
Table 1
Reduction in error with the number of iterations
Number of iterations 0 1 2 3 4
Maximum error , (%) 30.29 6.04 3.96 0.89 0.28