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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. 2. Reference stress-strain curve, defined by N n

N K = .


Fig. 4. Modified true stress-strain curve calculated after first iteration.

Fig. 5. 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



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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



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True strain


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



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True strain


0 0.5 1 1.50

250

500

750

Measured and fittedReferenceFirst improvedSecond improvedThird improvedFourth improved

Necking point



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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



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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

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ARTIGO 06 - Bom Artigo