Cold and Hot Forging: Fundamentals and Applications / Edited by Taylan Altan, Gracious Ngaile, Gangshu Shen

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76 / Cold and Hot Forging: Fundamentals and Applications

double cup backward extrusion test. The ring

compression test best simulates forging appli-

cations with a moderate amount of deformation,

where the surface expansion induced is on the

order of only 100%, while the double cup back-

ward extrusion test best simulates more severe

forging applications, where the surface expan-

sion and the interface pressure induced are over

500% and 290 ksi (2000 MPa), respectively.

In determining the friction factor, f, or the

shear factor, m, for hot forming, in addition to

lubrication effects, the effects of die chilling or

heat transfer from the hot material to colder dies

must be considered. Therefore, the lubrication

tests used for determining friction factors must

include both lubrication and die-chilling effects.

Consequently, in hot forming, a good test must

satisfy the following requirements [Altan et al.,

1983]:

●

The specimen and die temperatures must be

approximately the same as those encoun-

tered in the actual hot forming operation.

●

The contact time between specimen and

tools under pressure must be approximately

the same as in the forming operation of in-

terest.

●

The ratio of the new generated deformed sur-

face area to the original surface area of the

undeformed specimen (i.e., surface expan-

sion) must be approximately the same as in

the process investigated.

●

The relative velocity between deforming

metal and dies should have approximately

the same magnitude and direction as in the

forming process.

7.8.1 Ring Compression Test

Lubricity, as deﬁned by the friction factor, f,

or the shear factor, m, is commonly measured

by using the ring test [Male et al., 1970] [Doug-

las et al., 1975]. In the ring test, a ﬂat ring-shape

specimen is compressed to a known reduction

(Fig. 7.7). The change in internal and external

diameters of the forged ring is very much de-

Fig. 7.9

Theoretical calibration curves for ring compression test having indicated OD: ID:thickness ratios. (a) 6:3:2 ratio. (b) 6:3:1

ratio. (c) 6:3:0.5 ratio. [Altan et al., 1983]

Friction and Lubrication / 77

pendent on the friction at the die/ring interface.

If friction were equal to zero, the ring would

deform in the same way as a solid disk, with

each element ﬂowing radially outward at a rate

proportional to its distance from the center. With

increasing deformation, the internal diameter of

the ring is reduced if friction is high and is in-

creased if friction is low. Thus, the change in the

internal diameter represents a simple method for

evaluating interface friction.

Simulation of Cold Forging Conditions.

The ring test has an advantage when applied to

the study of friction under cold forging condi-

tions. In order to measure friction with this test,

the force necessary to deform the ring and the

ﬂow stress of the specimen do not have to be

known. Thus, evaluation of test results is greatly

simpliﬁed. To obtain the magnitude of the fric-

tion factor, the internal diameter of the com-

pressed ring must be compared with the values

predicted by using various friction factors, f, or

shear factors, m. Today, these values are most

often predicted by the ﬁnite element method

(FEM). Figure 7.8 shows an example of an FEM

model used for this purpose. The results are plot-

ted in the form of “theoretical calibration

curves,” as can be seen in Fig. 7.9, for rings

having OD:ID:thickness ratios of 6:3:2, 6:3:1,

and 6:3:0.5. The internal diameters used in this

ﬁgure are the diameters at the internal bulge. Un-

der cold forging conditions, these calibration

curves may be considered as “universal” be-

cause changes in material properties (i.e., strain

hardening) have little effect on the curves. In

determining the value of the shear factor, m, for

a given experimental condition, the measured di-

mensions (reduction in height and variation in

internal diameter) are plotted on the appropriate

calibration ﬁgure. From the position of that point

with respect to theoretical curves given for vari-

ous values of “m,” the value of the shear factor,

m, which existed in the experiment is obtained.

Some results obtained from ring compression

tests conducted in a 160-ton hydraulic press with

a ram velocity of 15 mm/s and a ring height

reduction of 50% are shown in Table 7.7.

Simulation of Hot Forging Conditions. In

contrast to the simulation of cold forging con-

ditions, the simulation of hot forging conditions

do not provide for a “universal” set of calibra-

tion curves. The friction calibration curves must

be generated for the speciﬁc ring material under

the speciﬁed ring and die temperatures and the

ram speed conditions. Hence, knowledge of the

ﬂow stress of the material is required [Lee et al.,

1972].

The results from some ring compression tests

conducted under hot forging conditions have

been compiled. The results from ring compres-

sion tests conducted for various materials in a

500 ton mechanical press with a nominal speed

of 90 strokes/min and a total stroke of 10 in.

(255 mm) are shown in Table 7.8.

Fig. 7.10 Metal ﬂow in double cup backward extrusion test

Table 7.7 Values of frictional shear factor, m, obtained from ring compression tests conducted in a

hydraulic press

Specimen/die

temperatures Ring size OD:ID:h(a)

Frictional shear

Material ⬚F ⬚C in. mm factor (m) Lubrication system

AISI 1018 200/200 95/95 1.75:1.13:0.5 44.5:28.7:12.7 0.040 Zinc phosphate coating Ⳮ soap

0.045 Metallic compounds Ⳮ sulfur compounds(b)

0.060 Mineral oil Ⳮ EP additives

Copper 75/75 24/24 2:1:0.67 50.8:25.4:16.9 0.30 Emulsion

0.27 Water-based synthetic

(a) OD, ring outside diameter; ID, ring inside diameter; h, ring height. (b) Environmentally friendly lubrication system developed to replace zinc phosphate coating

based systems. Source: [Gariety et al., 2003] [Hannan et al., 2000]

78 / Cold and Hot Forging: Fundamentals and Applications

7.8.2 Double Cup

Backward Extrusion Test

Lubricity, as deﬁned by the friction factor, f,

or the shear factor, m, is also measured by using

the double cup backward extrusion test. As

shown in Fig. 7.10, the test is a combination of

the single cup forward and single cup backward

extrusion processes. The ratio of the cup heights,

, is very dependent on the friction at the

billet/die and billet/punch interfaces [Buschhau-

sen et al., 1992] [Forcellese et al., 1994]. In par-

Fig. 7.11 Double cup backward extrusion test tooling at the ERC/NSM

Table 7.8 Values of frictional shear factor, m, obtained from ring compression tests conducted in a

mechanical press (die temperatures ⬇ 300 ⬚F, or 150 ⬚C)

Specimen

temperature Ring ratio OD:ID:t

Frictional shear

Material ⬚F ⬚C in. mm factor (m) Contact time, s Lubrication system

6061 Al 800 425 6:3:0.5 150:75:13 0.40 0.038 (a)

6:3:1 150:75:25 0.31 0.047 (a)

6:3:2 150:75:51 0.53 0.079 (a)

Ti-7Al-4Mo 1750 955 3:1.5:0.25 75:38:6.5 0.42 0.033 (b)

3:1.5:0.5 75:38:13 0.42 0.044 (b)

3:1.5:1 75:38:25 0.42 0.056 (b)

403 SS 1800 980 3:1.5:0.25 75:38:6.5 0.23 0.029 (b)

3:1.5:0.5 75:38:13 0.24 0.039 (b)

3:1.5:1 75:38:25 0.34 0.047 (b)

1950 1065 3:1.5:1 75:38:25 0.28 0.06 (b)

2050 1120 3:1.5:1 75:38:25 0.35 0.06 (b)

Waspaloy 2100 1150 3:1.5:1 75:38:25 0.18 0.06 (b)

17-7PH SS 1950 1065 3:1.5:1 75:38:25 0.28 0.06 (b)

2100 1150 3:1.5:1 75:38:25 0.35 0.06 (b)

Ti-6Al-4V 1700 925 3:1.5:1 75:38:25 0.30 0.06 (b)

1750 955 3:1.5:1 75:38:25 0.46 0.06 (b)

Inconel 718 2000 1095 3:1.5:1 75:38:25 0.18 0.06 (b)

2100 1150 3:1.5:1 75:38:25 0.33 0.06 (b)

Ti-8Al-1Mo-1V 1750 955 3:1.5:1 75:38:25 0.27 0.06 (b)

1800 980 3:1.5:1 75:38:25 0.27 0.06 (b)

Udimet 2050 1120 3:1.5:1 75:38:25 0.40 0.06 (b)

7075 Al 700 370 5:3:1 125:75:25 0.37 0.06 (a)

800 425 5:3:1 125:75:25 0.31 0.06 (a)

SS, stainless steel. (a) Caustic precoat Ⳮ graphite coating Dag 137 (Acheson) on the specimens and graphite spray Deltaforge 43 (Acheson) on the dies. (b) Glass-

based coating Deltaforge 347 (Acheson) on the specimens and graphite spray Deltaforge 43 (Acheson) on the dies. Source: [Douglas et al., 1975]

Friction and Lubrication / 79

ticular, it has been found that the ratio of the cup

heights increases as the friction factor, f, or the

shear factor, m, increases. In other words, if

there is no friction, the cup heights will be the

same and the cup height ratio, H

, will be

equal to one. Thus, the ratio of the cup heights

represents a simple method for evaluating inter-

face friction.

Figure 7.11 shows the actual tooling used for

the double cup backward extrusion test. It should

be noted that the lower punch was raised out of

the container for illustration purposes only. Dur-

ing the test, the container and lower punch are

ﬁxed on the bed of the press and held stationary

with the lower punch located completely inside

the container. In addition, the upper punch is

ﬁxed on the ram of the press and moves down-

ward. Thus, there is a relative velocity between

the container and the upper punch, but not be-

tween the container and the lower punch. There-

fore, the material ﬂow to the lower punch is more

restricted in the presence of friction. This ex-

plains why the height of the upper cup is larger

than the height of the lower cup.

To obtain the magnitude of the friction factor,

f, or the shear factor, m, the ratio of the cup

heights, H

, must be compared with the val-

ues predicted by using various friction factors,

f, or shear factors, m. Today, these values are

most often predicted by the ﬁnite element

method (FEM). Figure 7.12 shows an example

of an FEM model used for this purpose. The

results are plotted in the form of “theoretical cal-

ibration curves,” as can be seen in Fig. 7.13. In

determining the value of the shear factor, m, for

a given experimental condition, the measured di-

mensions (cup height ratio and stroke) are plot-

ted on the calibration ﬁgure. Figure 7.14 illus-

trates how the cup height ratio and stroke are

measured. From the position of that point with

respect to the theoretical curves given for vari-

ous values of “m,” the value of the shear factor,

m, which existed in the experiment, is obtained.

It should be noted that in addition to interface

friction, the metal ﬂow in this test is dependent

on the billet material, the billet diameter, and the

Fig. 7.13

Theoretical calibration curves for double cup

backward extrusion test with experimental data

point (shear factor m ⬇ 0.065). [Ngaile et al., 2002]

Fig. 7.12

FEM model of double cup backward extrusion test (shear factor m ⳱ 0.1) (dimensions in millimeters). (a) Initial. (b) Final.

(Ngaile et al., 2002)

Fig. 7.14

Measurement of cup height ratio and stroke. Cup

height ratio: R

⳱ H

. Stroke: S ⳱ initial

height ⳮ (H ⳮ H

ⳮ H

). [Ngaile et al., 2002]

80 / Cold and Hot Forging: Fundamentals and Applications

punch diameter. Thus, the ﬂow stress of the ma-

terial must be known and the appropriate theo-

retical curves should be used to quantify the in-

terface friction. In other words, there is no

“universal” set of calibration curves for this test.

Several double cup backward extrusion tests

have been conducted for various materials in a

160 ton hydraulic press with a ram velocity of

15 mm/s and a punch stroke of 21 mm. The re-

sults of these tests are summarized in Table 7.9.

REFERENCES

[Altan, 1970]: Altan, T., “Heat Generation and

Temperatures in Wire and Rod Drawing,”

Wire J., March 1970, p 54.

[Altan et al., 1983]: Altan, T., Oh, S., Gegel,

H., Metal Forming Fundamentals and Appli-

cations, American Society for Metals, 1983.

[Bay, 1994]: Bay, N., The State of the Art in

Cold Forging Lubrication, J. Mater. Process.

Technol., Vol 46, 1994, p 19–40.

[Bay, 1995]: Bay, N., Aspects of Lubrication in

Cold Forging of Aluminum and Steel, Pro-

ceedings of the 9th International Cold Forg-

ing Congress, Solihull, UK, May 1995, p

135–146.

[Bhushan, 2001]: Bhushan, B., Modern Tribol-

ogy Handbook—Vol 2: Materials, Coatings,

and Industrial Applications, CRC Press,

2000.

[Buschhausen et al., 1992]: Buschhausen, A.,

Lee, J.Y., Weinmann, K., Altan, T., “Evalua-

tion of Lubrication and Friction in Cold Forg-

ing Using Double Backward Extrusion Pro-

cess,” J. Mater. Process. Technol., Vol 33,

1992, p 95–108.

[Doege et al., 1996]: Doege, E., Seidel, R., Ro-

manowski, C., “Increasing Tool Life Quantity

in Die Forging: Chances and Limits of Tri-

bological Measures,” Technical Papers of the

North American Manufacturing Research In-

stitution of SME, 1996, p 89–94.

[Douglas et al., 1975]: Douglas, J.R., Altan, T.,

Flow Stress Determination for Metals at Forg-

ing Rates and Temperatures, Trans. ASME, J.

Eng. Ind., Feb 1975, p 66.

[Forcellese et al., 1994]: Forcellese, A., Ga-

brielli, F., Barcellona, A., Micari, F., “Evalu-

ation of Friction in Cold Metal Forming,” J.

Mater. Process. Technol., Vol 45, 1994, p

619–624.

[Gariety et al., 2002]: Gariety, M., Ngaile, G.,

Altan, T., “Identiﬁcation of Lubricants and

Enhancement of Lubricant Performance for

Cold Heading—Progress Report 1—Identiﬁ-

cation of Lubricants Used for Cold Heading,”

Report No. PF/ERC/NSM-02-R-32A, 2002.

[Gariety et al., 2003]: Gariety, M., Padwad, S.,

Ngaile, G., Altan, T., “Identiﬁcation of Lu-

bricants and Enhancement of Lubricant Per-

formance for Cold Heading—Progress Report

2—Preliminary Lubrication Tests for Cold

Heading,” Report No. PF/ERC/NSM-02-R-

32B, 2003.

[Hannan et al., 2000]: Hannan, D., Ngaile, G.,

Altan, T., “Development of Forming Pro-

cesses for Copper Components for Stanford

Linear Accelerator,” Report No. PF/ERC/

NSM-B-00-20, 2000.

[ICFG, 1996]: International Cold Forging

Group, “Lubrication Aspects in Cold Forging

of Aluminum and Aluminum Alloys,” Doc-

ument No. 10/95, 1996.

[Lazzarotto et al., 1999]: Lazzarotto, L., Ma-

rechal, C., Dubar, L., Dubois, A., Oudin, J.,

“The Effects of Processing Bath Parameters

on the Quality and Performance of Zinc Phos-

phate Stearate Coatings,” Surf. Coat. Tech-

nol., Vol 122, 1999, p 94–100.

[Lee et al., 1972]: Lee, C.H., Altan, T., “Inﬂu-

ence of Flow Stress and Friction Upon Metal

Flow in Upset Forging of Rings and Cylin-

ders,” Trans. ASME, J. Eng. Ind., Aug 1972,

p 775.

[Lenard, 2000]: Lenard, J., “Tribology in Metal

Rolling,” Ann. CIRP, Vol 49, 2000, p 1–24.

[Male et al., 1970]: Male, A.T., DePierre, V.,

“The Validity of Mathematical Solutions for

Determining Friction from the Ring Com-

pression Test,” Trans. ASME, J. Lubr. Tech-

nol., Vol 92, 1970, p 389.

Table 7.9 Values of frictional shear factor, m,

obtained from double cup backward extrusion

tests conducted in a hydraulic press (punch/die

temperatures ⬇ 75 ⬚F, or 24 ⬚C)

Specimen

temperatures

Frictional

shear

factor

Material ⬚F ⬚C (m) Lubrication system

AISI 8610 75 24 0.065 Zinc phosphate coating

Ⳮ soap

0.035 Metallic compounds

Ⳮ sulfur

compounds(a)

0.075 Zinc-based dry ﬁlm(b)

AISI 1038 75 24 0.050 Zinc phosphate coating

Ⳮ soap

(a) Billet size ⳱ 1.25 in. diam ⳯ 1.25 in. height (31.75 mm diameter ⳯ 31.75

mm height). (b) Environmentally friendly lubrication system developed for re-

placement of zinc phosphate coating based systems. Source: [Ngaile et al., 2002]

Friction and Lubrication / 81

[Manji, 1994]: Manji, J., “Die Lubricants,”

Forging, Spring 1994, p 39–44.

[Ngaile et al., 1999]: Ngaile, G., Saiki, H.,

“Cold Forging Tribo-Test Based on Variation

of Deformation Patterns at the Tool-Work-

piece Interface,” Lubr. Eng.: J. Soc. Tribol.

Lubr. Eng., Feb 1999, p 23–31.

[Ngaile et al., 2002]: Ngaile, G., Schumacher,

R., Gariety, M., Altan, T., Kolodziej, J., “De-

velopment of Replacements for Phoscoating

Used in Forging Extrusion and Metal Form-

ing Processes,” 2002, Report No. PF/ERC/

NSM-02-R-85.

[Saiki, 1997]: Saiki, H., “The Role of Tribology

for Improvement of Tool Life in Hot Forg-

ing,” Proceedings of the 1st International

Conference on Tribology in Manufacturing

Processes, Gifu, Japan, 1997, p 22–31.

[Saiki et al., 1999]: Saiki, H., Ngaile, G., Ruan,

L., Marumo, Y., “Evaluation of Cold Forging

Lubricants Under Realistic Forging Tempera-

ture Conditions,” Adv. Technol. Plast.: Ann.

CIRP, Vol 1, p 377–382.

[Schey, 1983]: Schey, J., Tribology in Metal-

working: Lubrication, Friction, and Wear,

American Society for Metals, 1983.

[Schmoeckel et al., 1997]: Schmoeckel, D.,

Rupp, M., ”More Environment Friendly Cold

Massive Forming—Production of Steel with-

out Zinc Phosphate Layer,” Symposium, Lat-

est Developments in Massive Forming, Fell-

bach near Stuttgart, 1997, p 183–200.

[Shen et al., 1992]: Shen, G., Vedhanayagam,

A., Kropp, E., Altan, T., “A Method for Eval-

uation of Friction Using a Backward Extru-

sion Type Forging,” J. Mater. Process. Tech-

nol., Vol 33, 1992, p 109–123.

CHAPTER 8

Inverse Analysis for

Simultaneous Determination of

Flow Stress and Friction

Hyunjoong Cho

8.1 Introduction

The ﬁnite element analysis (FEA) based

simulation of metal forming processes has been

widely used to predict metal ﬂow and to opti-

mize the manufacturing operations. In using

user-friendly commercial FEA software, it is

necessary to assign input parameters for the

simulation. Among those inputs, the parameters

in the ﬂow stress equation, friction factor, and

anisotropy coefﬁcients of a material are usually

obtained from the appropriate tests. The results

of process simulation are extremely sensitive to

the accuracy of ﬂow stress and interface friction

that are input to FEM programs. Therefore, it is

essential that these input values are determined

using (a) reliable material tests and (b) accurate

evaluation methods. A test used to determine

material properties should replicate processing

conditions that exist in practical applications.

A common method for the determination of

the ﬂow stress data for forging simulation is the

cylinder upset test as discussed in Chapter 4 be-

cause (a) during the test the deformation is done

in a state of compressive stress, which represents

well the true stress state of most forging pro-

cesses and (b) the test can be done for a large

strain. However, even in the simplest cylinder

upset test, interface friction leads to an inevitable

bulging of the sample and thereby to an inac-

curate ﬂow stress determination. Thus, the eval-

uation of the test results should be able to over-

come difﬁculties introduced by friction and

inhomogeneous deformation. It is desirable to

consider the unavoidable friction at the tool/

workpiece interface in the test and to identify

the friction together with ﬂow stress using an

appropriate evaluation method. In this chapter,

an inverse analysis technique for the accurate

determination of the input data for FEM simu-

lation is introduced to determine material pa-

rameters in the ﬂow stress model and the friction

at the tool/workpiece interface.

8.2 Inverse Analysis in Metal Forming

8.2.1 Direct and Inverse Problems

An FEA of metal forming process as illus-

trated in Fig. 8.1(a) is regarded as direct prob-

lem. The required input data for direct problem

(i.e., FE simulation) are geometry, process con-

ditions, ﬂow stress, interface friction, etc. In the

direct problem FEA predicts the metal ﬂow,

forming load, and energy by simulating the

forming operation assuming that the ﬂow stress

and friction values are known. Compared with

the direct problem, in the inverse problem the

authors determine one or more of input data of

the direct problem, leading to the best ﬁt be-

tween experimental measurements and FEM

prediction. With experimental measurements

provided to the inverse problem, the input data

are identiﬁed or calibrated (if initial guess is

given). This inverse problem can be applied to

Cold and Hot Forging Fundamentals and Applications

Taylan Altan, Gracious Ngaile, Gangshu Shen, editors, p83-89

DOI:10.1361/chff2005p083

www.asminternational.org

84 / Cold and Hot Forging: Fundamentals and Applications

Fig. 8.1 Direct (a) and inverse (b) problems

any material test in which FEA can be done in-

cluding the cylinder upset and ring compression

tests, provided experimental measurements are

accurate enough, Fig. 8.1(b). Therefore, an in-

verse problem is regarded as a parameter iden-

tiﬁcation problem that can be formulated further

as an optimization problem where the difference

between measurement and FEM prediction is

minimized by adjusting the input parameters.

8.2.2 Procedure for

Parameter Identiﬁcation

The basic concept of an inverse analysis in

ﬂow stress determination consists of a set of un-

known parameters deﬁned in ﬂow stress equa-

tion. First, a ﬁnite element simulation of the se-

lected material test with the assumed parameters

of the ﬂow stress equation is conducted and the

computed load-stroke curve is compared with

the experimentally measured curve. Then, the

assumed parameters of the ﬂow stress equation

are adjusted in such a way that the difference in

the calculated and measured load-stroke curves

is reduced in the next comparison. This proce-

dure is repeated until the difference between ex-

perimental measurements and computed data

disappears. The result of inverse analysis is a set

of the identiﬁed material parameters of the ﬂow

stress equation, which represents material prop-

erties. The procedure used to identify the mate-

rial parameters includes:

1. Guess the material parameters in ﬂow stress

equation.

2. Start FEM simulation of the selected material

test with given ﬂow stress data.

3. Compare the computed forming load with

experimentally measured one.

4. Obtain the amount of adjustments in material

parameters by minimizing the difference be-

tween the computed and measured loads.

5. Improve the material parameters until the dif-

ference becomes within a desired tolerance.

The trial and error procedure is the simplest

way to solve the above inverse problem, i.e., pa-

rameter identiﬁcation. This method may be used

to get some prior information of the parameter

values and get some rough idea what is the most

important parameter for a given problem. How-

ever, this method is time consuming and param-

eters cannot be identiﬁed accurately. Therefore,

it is necessary to use a numerical optimization

technique for robust determination of the param-

eters in the ﬂow stress equation. Generally, the

unknown parameters are determined by mini-

mizing a least-square functional consisting of

experimental data and FEM simulated data. The

FEM is used to analyze the behavior of the ma-

terial during the test, whereas the optimization

technique allows for automatic adjustment of

parameters until the calculated response matches

the measured one within a speciﬁed tolerance.

Derivation of inverse analysis based on rigid-

plastic ﬁnite element formulation was developed

at ERC/NSM [Cho et al., 2003].

8.2.3 Past Studies on the Inverse Analysis

Used for Flow Stress Determination

Chenot et al. formulated an inverse problem

in developing a methodology for automatic

identiﬁcation of rheological parameters. The in-

verse problem was formulated as ﬁnding a set

of rheological parameters starting from a known

constitutive equation. An optimization algo-

rithm was coupled with the ﬁnite element simu-

lation for computing the parameter vector that

minimizes an objective function representing, in

the least-square sense, the difference between

experimental and numerical data. For sensitivity

Inverse Analysis for Simultaneous Determination of Flow Stress and Friction / 85

Table 8.1 Flow stress models and parameters

Forging type Flow stress equation Parameters

Cold

⳱

¯r K¯e

K,n

Warm

⳱

n ˙m

¯r K¯e ¯e

K,n,m

Hot

⳱

¯r K¯e

K,m

analysis of the objective function with respect to

the searching parameters during the optimiza-

tion, Chenot differentiated the FEM code with

respect to the searching parameters [Chenot et

al., 1996].

Boyer and Massoni developed the semianal-

ytical method for sensitivity analysis of inverse

problem in material forming domain. This

method compromises between computation time

and effort of analytical code differentiation. This

technique proved to be a good alternative to the

ﬁnite difference method or analytical differen-

tiation of the FEM code in conducting sensitiv-

ity analysis. As a result of research, the identi-

ﬁcation software CART (Computer Aided

Rheology and Tribology) was introduced [Boyer

& Massoni, 2001].

Pietrzyk et al. used inverse analysis technique

to evaluate the coefﬁcients in the friction and

ﬂow stress model for metal forming processes.

He identiﬁed parameters in the conventional

ﬂow stress equation and in a dislocation density

based internal variable model as well as friction

factor from one set of ring compression test. He

concluded that the determination of both rheo-

logical and frictional parameters from one com-

bined test is ideal because the interpretation of

tests to determine the ﬂow stress depends on an

assumed value of the friction factor [Pietrzyk et

al., 2001].

Zhiliang et al. introduced a new method com-

bining compression tests with FEM simulation

(C-FEM) to determine ﬂow stress from the com-

pression tests where inhomogeneous deforma-

tion is present due to interface friction. In this

method, the ﬂow stress obtained from the com-

pression test is improved by minimizing the tar-

get function deﬁned in load-stroke curves [Zhi-

liang et al., 2002].

8.3 Flow Stress Determination in

Forging by Inverse Analysis

In a large plastic deformation problem, usu-

ally encountered in most forging applications,

the deformation behavior of the material can be

assumed to be rigid-plastic by neglecting the

elastic part. If the material shows strain hard-

ening behavior in cold forging, the power-law

type ﬂow stress equation is used to describe a

stress-strain relationship for plastic range. At el-

evated temperature, ﬂow stress is sensitive to

rate of deformation. Thus, for warm and hot

forgings, ﬂow stress is expressed in function of

strain, strain-rate, and temperature. Examples of

ﬂow stress equations and material parameters

are summarized in Table 8.1.

8.3.1 Material Parameters

In cold forging, K and n are the two material

parameters (i.e., the material strength coefﬁ-

cient, K, and the strain hardening exponent, n)

used to deﬁne the stress-strain relationship dur-

ing plastic deformation. When the experimen-

tally measured load-stroke curve is available,

FEM simulations are made with initial guesses

of K and n values, and then the two material

parameters are to be identiﬁed. Therefore, the

material parameters become design variables in

the optimization problem.

8.3.2 Objective Function

The unknown material parameters are deter-

mined by minimizing an objective function, E,

representing the difference between the experi-

mental and the simulated loads in a least-square

sense:

1Fⳮ F(P)

EXP COM k

E ⳱ (Eq 8.1)

兺

冢冣

i⳱1

EXP

where F

EXP

is the experimental load and F

COM

is the computed load. N is the number of data

sampling points selected from a load-stroke to

construct the objective function. Figure 8.2 il-

lustrates a deﬁnition of the objective function.

The objective function, E, is a nonlinear implicit

function of material parameters P

. Therefore,

the parameter identiﬁcation problem is reduced

to compute a set of the unknown parameter P

⳱ {K,n}, which leads to the best ﬁt between

experimental measurements and corresponding

computed data.

For given material parameters P

, the objec-

tive function E ⳱ E(P

) will be minimum at:

⳵E(P )

⳱ 0 for k ⳱ 1, 2 (Eq 8.2)

⳵P

where P

are the P

⳱ K, P

⳱ n.

86 / Cold and Hot Forging: Fundamentals and Applications

Fig. 8.2

The difference between the computed and the ex-

perimental load

The nonlinear Eq 8.2 is solved with respect

to the parameters P

using Newton-Raphson it-

erative procedure.

⳵ E ⳵E

DP ⳱ⳮ for j, k ⳱ 1, 2 (Eq 8.3)

⳵P ⳵P ⳵P

kj k

The ﬁrst and second gradients of the objective

function with respect to the parameters P

are

evaluated by taking the derivatives of the objec-

tive function E ⳱ E(P

) with respect to P

⳵E2(Fⳮ F)⳵F

EXP COM COM

⳱ⳮ

兺

冦

冧

⳵PN F ⳵P

i⳱1

k EXP k

for j, k ⳱ 1, 2 (Eq 8.4)

⳵ E2

⳱ⳮ

兺

⳵P ⳵PN

i⳱1

1 ⳵F ⳵F(Fⳮ F)

COM COM EXP COM

ⳮⳭ

冦

F ⳵P ⳵PF

EXP k j EXP

⳵ F

COM

for j, k ⳱ 1, 2 (Eq 8.5)

冧

⳵P ⳵P

8.3.3 Advantages

In the inverse analysis, FEM simulation is

used to describe the deformation behavior of

material during the test and the optimization al-

gorithm identiﬁes material parameters using

FEM simulation results. Any material test that

can be simulated by FEM can be selected to de-

termine the material property. Uniform strain-

rate condition, which requires a sophisticated

control of test machine, is not needed in the test

because inverse analysis takes advantage of

FEM simulation where complex stress and strain

states can be handled. Therefore, inverse anal-

ysis technique gives ﬂexibility in selecting ma-

terial tests. For determining ﬂow stress data for

warm and hot forging, only two different test

velocities are required to identify strain-rate sen-

sitivity (m-value) instead of conducting the tests

at several different constant strain-rates.

8.4 Inverse Analysis for

Simultaneous Determination of

Flow Stress and Friction

In the upset test, for determining bulk material

property for a large strain, the main problem is

the existing friction at the die/specimen inter-

face. In order to overcome this problem, friction

is minimized by using a lubricant together with

geometry proposed by Rastegaev [Dahl et al.,

1999]. This treatment will lead to a uniaxial

stress state for a limited reduction in height dur-

ing the test. However, the frictional force at large

compression ratio (a) starts to bulge the sample

(i.e., inhomogeneous deformation) regardless of

the quality of the lubricant and (b) affects the

measured load-stroke curve. In other words, the

measured load-stroke curve has a force contri-

bution from inhomogeneous deformation caused

by frictional force, and this causes an error in

ﬂow stress calculation. Therefore, it is necessary

to consider the inevitable interface friction in the

test and then identify it together with the ﬂow

stress.

For simultaneous identiﬁcation of both the

material parameters P

and the friction factor m

in addition to the measured load-stroke curve,

one more measurable geometrical quantity in the

test, namely the barreling, is used. Barreling re-

ﬂects a degree of inhomogeneous deformation

caused by friction. Therefore, it allows the iden-

tiﬁcation of friction by measuring the barreling

shape of the specimen. During the inverse anal-

ysis, the computed barreling shape is compared

with the measured barreling shape and then the

difference is minimized by adjusting the friction

factor. After several iterations, the difference in

barreling shape disappears and the friction factor

is identiﬁed. Figure 8.3 shows a methodology

for determining ﬂow stress and interface friction

simultaneously.

8.5 Example of Inverse Analysis

8.5.1 Flow Stress Model

The developed inverse analysis algorithm has

been tested by using the real experimental data