The flow stress of carbon steel in warm metalworking

dot,~rnal of Mechanical Working Technology,.2 (1978) 241- -253

241

© Elsevier Scientific Publishing Company, Amsterdam - - Printed in The Netherlands

THE FLOW STRESS OF CARBON

STEEL IN WARM METALWORKING*

IL OSAKADA

Department of Mechanical Engineering, Kobe University, Rokko, Nada, Kobe (Japan) and S. FUJII

Department of Systems Engineering, Kobe University, Rokko, Nada, Kobe (Japan) (Received May 5, 1978)

Industrial Summary For predicting the load in the warm metalworking of carbon steel, it is necessary to obtain the flow-stress data of the material under the specified conditions. Although quite a few experimental results of flow stress have been reported, they are not always effectively utilized. This is because (i) the combinations of the related factors are too many, and (ii)the amount of data at medium strain rate - - of the order of 10/s - - is not large, although metalworking is usually carried out in this strain-rate range. In this report, stress--strain curves of seven carbon steels with carbon contents of from 0.033 to 0.63% are obtained at temperatures of from 0 to 700°C using a mechanical press (strain rate of about 10/s). Two models of flow stress of carbon steel are proposed, considering strain, strain rate, initial temperature and carbon content as variables. The constants in the models are estimated by the non-linear regression method using 56 stress--strain curves obtained in the present experiments and 128 curves collected from the literature. The maximum pressure in forward extrusion is calculated using the proposed model of flow stress and the upper bound method. The calculated results show fairly good agreement with experimental results reported in the literature, except for large extrusion ratios.

1. I n t r o d u c t i o n I n d e s i g n i n g a w a r m m e t a l w o r k i n g o p e r a t i o n , i t is n e c e s s a r y t o k n o w t h e maximum load, the die pressure, and the energy required for the operation. T o d e t e r m i n e t h e s e v a l u e s , t h e f l o w s t r e s s o f t h e m a t e r i a l t o b e d e f o r m e d is u s e d as t h e b a s i s f o r c a l c u l a t i o n . S i n c e t h e s t r a i n a r i s i n g w i t h i n t h e m a t e r i a l is usually large, compression tests with good lubrication have been widely a d o p t e d t o m e a s u r e t h e f l o w stress. C a r b o n s t e e l is o n e o f t h e m o s t c o m m o n m e t a l s d e f o r m e d in w a r m w o r k i n g , and quite a few compression tests have been conducted at warm working temperatures. Krause [1], Hashizume [2] and Samanta [3] have examined the e f f e c t o f s t r a i n r a t e o n t h e f l o w s t r e s s o f l o w - c a r b o n s t e e l s , a n d O y a n e e t al. [ 4 ]

*Paper presented at the International Conference on W a r m Working, Sunderland, U.K., September 11--12, 1978.

242

have measured the flow stress of five carbon steels at a high strain rate. Many other experiments have been carried out using tensile testing machines and Hopkinson-bar equipment to study the fundamental properties of metals, rather than to obtain practical flow-stress data. In spite of this extensive work, it is n o t easy to find the required stress-strain curve for given working conditions from published literature, because (i) the combinations o f the related factors are t o o many, and (ii) the a m o u n t of data for carbon steels with various carbon contents in the strain-rate range of the order of 10/s is n o t sufficiently large, although metal-working opera-, tions are carried o u t frequently in this range. It is considered important for practical purposes to be able to evaluate the flow stress of carbon steels systematically at a strain rate of the order of 10/s, and further, to be able to express the flow stress, by a mathematical model, in terms of strain, strain rate, temperature, and carbon content. 2. Measurement of flow stress

2.1 Experimental procedure Figure 1 shows the outline of the experimental equipment. A mechanical press with a load capacity of 45 tons (450 kN) and a speed of 75 strokes per minute was used for compression tests. The size of a specimen was i 5 mm in height and 10 mm in diameter. The working load was measured by a straingauged load-cell. The ram travel was measured by a displacement transducer in which a strain-gauged plate-spring was fastened to the ram and the end of the spring deflected as it slid along an inclined guide surface attached to the bolster. As shown in Fig. 2, signals were stored by a digital signal m e m o r y (1024 words in each channel) through a dynamic strain amplifier (0--10 kHz). After each experiment, they were recalled and recorded by an X--Y recorder. The initial temperature of the specimen was from 0°C to 700°C. For experiments at elevated temperatures, the specimen was heated by a furnace controlled by a transistor temperature controller. The temperature drop in the specimen while it was being carried from the furnace to the press, was measured prior to carrying o u t the experiments, and was allowed for when setting the furnace temperature. The side surface of the specimen was covered with asbestos cloth, and the specimen was supported by steel wires at a b o u t 1 m m above the lower platen prior to compression. Thin P.T.F.E. sheets were stuck onto the platens to provide effective lubrication at all temperatures.

2.2 Experimental results The materials tested were seven carbon steels with different carbon contents. Their chemical compositions and heat treatments are listed in Table 1. The m a x i m u m logarithmic strain achieved in the tests was 0.65, and the strain rate was varied as shown in Fig. 3, irrespective of the test material and

243

Fig. 1. Photograph of the experimental equipment.

ML:~nanicalPres~_ Displacement Transducer J Specimen o

~5 ~o cc c ~2 5

~ L o a d Cell

o

__J

0

X-Y Recorder DigitalSignal Strain J Memory Amplifier

L

Fig. 2. Diagram of the measuring and recording system. Fig. 3. Strain-rate history in the compression test.

0.2

0.4 Strain

0.6

244

TABLE 1 Chemical compositions (wt.%) and heat treatments of the steels used Steel

C

Si

Mn

P

S

Heat treatment

Designation in Fig. 4

0.20 0.24 0.22 0.26 0.27 0.24 0.26

0.18 0.40 0.49 0.77 0.71 0.74 0.77

0.063 0.016 0.023 0.020 0.014 0.021 0.034

0.010 0.016 0.011 0.014 0.014 0.022 0.030

950°C 880°C 850°C 810°C 810°C 800°C 790°C

(a) (b) (c) (d) (e) (f) (g)

(dlS) CH 1 0.033 S 15 C 0.17 S 20 C 0.20 S 45 C (1) 0.49 S 45 C (2) 0.47 S 50 C 0.50 SK 7 0.63

× × × × × x x

60 min 90 rain 90 rain 90rain 90 rain 90 rain 90 rain

the test temperature. The initial strain rate was 12/s, and the strain rate was kept within (10+3)/s up to a strain o f 0.5. The stress--strain curves obtained for the materials in Table 1 are displayed in Fig. 4. It is observed that the stress--strain cur~es at 300°C show marked work-hardening and those at 500, 600, and 700°C work-softening, for all the materials tested. The flow-stress data of the materials at a strain of 0.1 are p l o t t e d against temperature, in Fig. 5. The peaks of flow stress due to blue brittleness, or dynamic strain-ageing, appear at a b o u t 500°C. 3. Mathematical modelling of flow stress

3.1 Collection of data To establish an appropriate model o f the flow stress of carbon steel under warm working conditions, flow-stress data were collected:, from published literature in addition to using the data from the present experiments. The selection of the data was carried o u t on the following basis: (i) The material tested is annealed plain carbon steel with a k n o w n chemical composition. (ii) The strain (logarithmic) is greater than 0.3. (iii) The strain rate is greater than 10 -2/s, and the initial temperature is lower than 700°C. (iv) More than three stress--strain curves are available for the material. The list of the literature sources used is given in Table 2, in which the experimental conditions and the n u m b e r of stress--strain curves referred to are illustrated. 184 stress--strain curves (comprising 128 from the literature and 56 from the present experiments) were used as the basis for model building. The values of flow stress were read from the stress--strain curves at every 0.1 strain (logarithmic). Overall, 1056 units of data were obtained. The a m o u n t of data at each strain is given in Table 3. Since the temperature and strain rate in the billet during deformation were n o t measured for almost all the data employed, reference was made to only the initial values. Approximate

245

~E I000

"£ 1 0 0 0 z

/ / L°ffc 3oo~¢ 400°C 500~C

:£ 800 o

600 tD

)=

~ /

400

o a.

~

~

~- 800

oot 600 -- ~ o ~

400

. . . . . . . . ,C~Oo,c

o~ 200

200

L~

/-

Z

800

800

60o 40O

7oo'~ 02 i

(c)

z

0.3 ' 0.4 ' 05 : Strain

j_ 20o

o'1 o12 ob o,',- &

o

Q6 '

"_.£lO0O

L2OOt

z I

ot

BOO

zlOOt

..... ~q:~ ....

2oo~c

600

6oo

•

--------__~o.-¢

,~

400

oo 400

o

200

o

(e)

o'~

Strain

(d)

ot

~

7o~----

o ~

ta

o

I 0.6

z

---~600C

/ 0.1 '

"E 1000

i 05

1000

100~C

~_i----~:~_.~~ - ..... ":-2oo°c

soo

~o 400 [1_

I O.z,

Strain

o ~ o o ' c

800

200

I 0.3

%

~E lOOO

o

L 02

(b)

Strain

(o)

®

i 0.1

o(

0I

~oo'c

/

7OO13

200 I

ol 1

0.2'

0 3'

0.4'

0

0.6 '

o.' 5

I

OJ 1 0.2

(f)

Strain

%

I

I

03 • 04 Strain

I

I

OS

08

Ot 1000

z :£

~.~-_:::~

800

4oooc

600 4OO o h-

200 0(~

(g)

0t

G2

05

06

Strain

Fig. 4. Stress--strain curves f o r d i f f e r e n t c a r b o n steels at a strain rate o f a b o u t 10/s: for details o f t h e steels r e f e r t o Table 1.

246

Z

~: 1000

~I CH1 (0033~IoC)

[] 5z.5C(2)(0.47o2~c)

(~) SlSC(O.17"IoC)

II 550C(050~&C)

II S20C(0.20QI,C)

0

SK7(O63'hC)

A 545C(I)('0/.9Olo C )

800 600

o~

z,O0

LL

20O I

0

I

t

300 400 Temperature

600 °C

Fig. 5. F l o w stress--initial temperature curves for carbon steels at a strain of 0.1 (~ = lO/s).

TABLE2 List of literature sources used for data collection N 1 = number of steels, 7 = carbon content, e = strain rate, 0 = temperature, N 2 = number of collected stress--strain curves. Author

N,

Krause [1] Hashizume [2] Samanta [3] Oyane [4] Ohmori [5] Shida [6] Present experiments

1 1 1 5 1 5 7

(%)

e

0

(s-')

(°c)

0.11 0.15 0:10 0.01- -0.52 0.14 0.04--0.42 0.033--0.63

0.09--20 0.2--650 0.66--430 450 0.012--110 0.01--9 12

20--450 0--600 20--630 0--700 -78--20 20 0--700

7

N2

Total

13 21 14 46 5 29 56 184

TABLE 3 Number of items of data collected for each strain Strain

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Total

Items of data

184

184

184

172

124

107

46

21

21

13

1056

t e m p e r a t u r e r i s es m a y b e c a l c u l a t e d b y a s s u m i n g a d i a b a t i c d e f o r m a t i o n . A s f o r c h e m i c a l c o m p o s i t i o n s , o n l y c a r b o n c o n t e n t is r e f e r r e d t o as a v ar i a b l e , b e c a u s e i t is n o t u s u a l t o k n o w t h e d e t a i l s o f c h e m i c a l c o m p o s i t i o n s o t h e r t h a n t h a t o f t h e c a r b o n c o n t e n t in p r a c t i c a l s i t u a t i o n s .

247

3. 2 P r e p a r a t i o n f o r m o d e l b u i l d i n g

The flow stress a is expressed in MN/m 2 . The variables considered are logarithmic strain e, strain rate ~ in s - ' , temperature 0 in K, and carbon c o n t e n t ~ in %. To express the effect of strain on the flow stress, the following equation is often used. o

=Ae n ,

(1)

where A and n are constants, which depend on the strain rate, temperature and carbon content. The effect of strain rate may be expressed by o

(2)

= B~ m .

Where the ffrow stress decreases m o n o t o n o u s l y with temperature, the following expression is useful. o

= C exp (- D / O ) .

(3)

MacGregor and Fisher [7] proposed the concept of "velocity-modified temperature", 0m, in which the effect of strain rate is converted into the effect of temperature as 0m = 0(1

-

k

In ~/~0),

(4)

where ~0 is the strain rate at which the temperature 0m is defined. These authors indicated that the flow stress of a carbon steel at warm-working temperatures can be expressed by a single curve using this method. However, because the characteristic feature of the flow stress of carbon steel in warm working is the existence of blue brittleness, or dynamic strain-ageing, the effect of temperature on the flow stress may n o t be expressed by a simple expression. The flow stress increases m o n o t o n o u s l y with carbon c o n t e n t for hypoeutectoid steel, and the following expression may be applied. o = E + F7

(5)

where E and F may be affected by strain, strain rate and temperature. Based on the above considerations, the flow stress may be represented by a function of e, ~, 0, and 7 with constants Q (Q1, Q2 . . . . . Qy, J: number of constants), o

= f ( e , e, O, 7; Q)

of which the form is to be established. 3. 3 N o n - l i n e a r r e g r e s s i o n

Since equation (6) may be a non-linear function, the m e t h o d which has

(6)

248

been used in non-linear systems to estimate the values o f constants m ay be applied. Suppose t h a t there are N observations (data) available o f the f o r m Ou, eu, eu, 0u, ~/u, where u = 1, 2 , . . . , N. F o r any values of t he constants, the m odel can be written in the form,

Ou = f(eu, eu, Ou, 7u; Q) + 5u

(7)

5u is th e error f o r the uth data. The sum of squares of errors S f o r t he model is N S = ~ {au.- f(eu, eu, Ou, 7u; Q)} 2 (8) u=l

By using appropriate metl)ods, t he values of constants which minimize the sum o f squares, or give the least squares, can be obtained. F r o m among several algorithms p r o p o s e d for non-linear regression, Marquardt's m e t h o d [8] was used in the present$1iudy. The standard deviation o f the error s is also estimated as follows, s = v~N-

j).

(9)

3.3 Mathematical models Model 1 A relationship be t w een flow stress and t e m p e r a t u r e for a material at a given strain and strain rate is assumed t o be as indicated by curve A in Fig. 6. The peak o f the curve is due to blue brittleness, and moves t o a higher t e m p e r a t u r e as the strain rate increases. This curve is considered to consist of the following three terms, as shown in the figure: (i) a h y p o t h e t i c a l curve B for ferrite -- ignoring blue brittleness -- which is expressed b y eqn. (3), (ii) a peak C due t o blue brittleness, which m ay be a p p r o x i m a t e d by a funct i o n similar to th e density f u n c t i o n of normal distribution, and (iii) a curve D representing the e f f e c t o f carbon c o n t e n t . The ef f ect o f strain rate is taken into consideration by using the velocity m o d if ied t e m p e r a t u r e , as given in eqn. (4). The equation obtained is: o=a~ exp

0(l-a31n~)

I

+ -a5

]

+ as 7 exp

.

(10)

Estimation of the constants was carried out i n d e p e n d e n t l y for e = 0.1 and e = 0.5. To increase the a m o u n t o f data, 20 results f r o m Hopkinson-bar tests [9] and 6 results f r o m tension tests [10] were added for e = 0.1. T h e stress--strain curves listed in Table 2 were e x t r a p o l a t e d from e = 0.3 or 0.4 t o e = 0.5,

249

a n d 60 a d d i t i o n a l i t e m s o f d a t a w e r e o b t a i n e d f o r e = 0.5. T h u s 2 1 0 i t e m s o f d a t a f o r e = 0.1 a n d 184 f o r e = 0.5 w e r e used; t h e e s t i m a t e d c o n s t a n t s are p r e s e n t e d in T a b l e 4. T h e s t a n d a r d d e v i a t i o n o f the e r r o r increases as t h e strain increases, p o s s i b l y b e c a u s e the w o r k - h a r d e n i n g c h a r a c t e r i s t i c s are aff e c t e d b y f a c t o r s in t h e c h e m i c a l c o m p o s i t i o n s o t h e r t h a n t h a t o f t h e c a r b o n content. In Fig. 7, t h e calculated f l o w stress f o r 0 . 1 7 % c a r b o n steel at strains o f e = 0.1 a n d e =- 0.5 are c o m p a r e d w i t h t h e e x p e r i m e n t a l l y o b s e r v e d values o f Fig. 4(5). By using this m o d e l , the f l o w stress at e = 0.1 (e0.1) a n d e = 0.5 (e0.s) can b e calculated. T h e stress--strain c u r v e m a y b e a p p r o x i m a t e d b y eqn. (1) using 0-0.1 a n d O0.s as, 1200 ~1000

8=0.1 E=0.5 Cmlculated . . . . ExperimentQ[ o •

800 e 6oo

B

\

U3

°~

o L~

Loo

LT_ 200

200 400 600 TemperQture °C

Temperoture

Fig. 6. A m o d e l o f t h e f l o w s t r e s s - - t e m p e r a t u r e c u r v e f o r c a r b o n steel. Fig. 7. C o m p a r i s o n o f e x p e r i m e n t a l a n d c a l c u l a t e d f l o w s t r e s s e s f o r 0 . 1 7 % C steel at a s t r a i n r a t e o f a b o u t 10/s. TABLE 4 Estimated constants for Model 1 s -- s t a n d a r d d e v i a t i o n o f e r r o r . e = 0.1 al a2 a3 a4 a5 a6 a7 a8 a9

e = 0.5

2.08 x 1.43 X 7.06 X 1.18 X -4.59 X 6.51 x 4.26x 4.74 x 1.73 X

102 102 10 -2 102 10 102 10 -2 102 102

3.18 x 1.29 x 8.81 x 1.27 x -2.89 x 6.54 X 4.54x 2.31 x 6.41 x

102 102 10 -2 102 10 l0 s 10 -2 102 10 ~

S

(MN/m 2)

66

111

250

o

= exp [1.43 In (o0.s) - 0.43 In (o0.1)] e °'62 In (%.,l-o.,)

(11)

Model 2 Since the above model does n o t explicitly include the strain term, various models were examined to a c c o m m o d a t e the strain, based on Model 1. The following model gave the best fit from amongst those examined.

a = ( b i e b2 + b 3 ) ( b 4 +(bse+b6)"/~ X exp t ~U' l (- bb7 es ' ")

1+

+bexpE+ ° tln0'l +11 +11n+'I21 Estimation of the constants was carried out utilizing all the data in Table 3, and the estimated values are given in Table 5. The standard deviation of the error drops b e t w e e n that for e = 0.1 and that for e = 0.5, of Model 1. TABLE 5

Estimated c o n s t a n t s f o r M o d e l 2 s = standard deviation of error b1 b~ b3 b4 b~ b~

2.94 7.04 6.94 6.18 -3.34 2.62

X X × ×

102 1 0 -1 10 10 -I

b7 b, b9 blo bll b12

2.84 4.50 2.88 -2.28 4.97 4.67

× 102 X 1 0 -2 X 102 × 102 × 1 0 -2

s = 87 MN/m 2

3.4 Discussion of the mathematical models The models proposed above are valid only in the range given for most of the experimental data used for the estimation of the constants. The appropriate values are: strain up to a b o u t 0.8, strain rate between 10 -2 and 5 × 102/s, temperature b e t w e e n 0 and 700°C, and carbon c o n t e n t up to 0.6%. Since the data are collected from various sources, they do n o t distribute uniformly, and further, their reliability is n o t known. Thus, the accuracy of the estimated constants is limited. To increase the accuracy, it is necessary to carry o u t the experiments systematically, and using statistical methods. Although it is n o t possible to detect the causes of the difference between the calculated and the experimental result for each case, these may be classifted as follows: (i) Errors associated with the material: effect of chemical compositions other than carbon content, grain size, structure, deformation history etc. which are neglected in the model. (ii) Errors associated with the experiment: error of stress and strain measure-

251

ments, effect of friction -- which is inherently involved in the compression test -- and effect of variation of temperature and strain rate during deformation. (iii) Errors associated with insufficient modelling: lack of knowledge of constitutive equations of commercis.1 metals at elevated temperature, and e m p l o y m e n t of a restricted number of constants for convenience of calculation. To compensate for the error due to (i), which may be systematic to some extent, an approximate m e t h o d is proposed. The basic concept is that a material which shows much lower (or higher) flow stress than that calculated by the proposed models under a certain condition, will probably show lower (or higher) flow stress under other conditions. Thus a modification factor may be introduced, defined as, = experimental flow stress/calculated flow stress.

(13)

By measuring one or more stress--strain curves of the material by a trial experiment, the average value of ~ can be calculated and the flow stress under various conditions may be estimated, multiplying the modification factor by the calculated values obtained from the models. From simulation with the collected data, it was confirmed t h a t the standard deviation of the error could be reduced to about 2/3 of the original value, on average, depending on the conditions of the trial experiment. 4. Prediction of working load To demonstrate a usage of the mathematical model, the extrusion pressure in warm extrusion was predicted. In the case of a rigid-perfectly plastic material (no work-hardening) of flow stress U, the extrusion pressure p can be calculated by various methods as: P = q~

(14)

where q is a function of extrusion ratio, die angle and frictional condition. Because the average strain ~-in the deformed material is approximated by q for zero friction [11], the mean flow stress om in the deformation zone may be calculated by: 1 ~ 1 Om----- f o(e)de = e q 0

q f

a(e)de.

(15)

0

By substituting om into ~ in eqn. (14), the extrusion pressure can be calculated. The approximate strain rate is calculated, assuming a deformation zone, by the following equation,

252

(16)

= ~ud 2vl4V

where V is the volume of the deformation zone, v is the extrusion speed and d is the diameter of the billet. In the present calculation, the upper-bound method for axisymmetric extrusion [12] is used, neglecting the effect of friction, and Model 2 is used ~or a(c). In Fig. 8, calculated extrusion pressures are compared with the m a x i m u m extrusion pressures obtained experimentally by Ishii [ 13], Yuasa [ 14], and Geiger [15]. The experimental conditions are listed in Table 6. The calculated values of extrusion pressure agree fairly weU with the experimental values at low extrusion pressures, but the scatter increases as the extrusion pressure increases. The high extrusion pressures are mainly for high extrusion ratios, i.e., for large strains. It may be inevitable that the accuracy of calculation deteriorates as the strain increases, because a very limited amount of flow stress data is available at large strains. 4 0 0 0 -E Z

o

Ishii

•

Yuasa

A Geiger

3ooc -

2

•

eo Oon

eee

2

~

g

~

oO~g

•" . /

.~ 2 0 0 0 m

~/

%#o o~o o

o o

~000

I

oo

I

1000 Calculated

Fig. 8. C o m p a r i s o n

I

2000 Extrusion

I

3000

4000

Pressure

MN/m

of experimental

2

a n d c a l c u l a t e d e x t r u s i o n pressures.

TABLE 6 E x p e r i m e n t a l c o n d i t i o n s for forward e x t r u s i o n Author

Material C %

Ishii [ 1 3 ]

0.02 0.11 0.17

Y u a s a [14]

0.25 0.53

Temperature (o C) 20--700

0.2 0 . 4 3

0.15 20--700 0.45

2.2 3.8 6.0

Die angle 2 ~ (deg)

Diameter d (mm)

Velocity (m/s)

Heat treatment

60

30

6.6

Annealed

120

36

0.I

Hot-roned

24.7

0.25

Annealed

1.7 300--700

0.35 Geiger [15]

Extrusion ratio

2.5 5.0 1.7 2.5 3.6

90

253

5. Conclusions (1) Stress--strain curves were obtained for seven carbon steels with carbon contents of from 0.033 to 0.63%, at temperatures of from 0 to 700°C. The strain rate is a b o u t 10/s, and the strain is up to 0.6. The stress--strain curves at 300°C show marked work-hardening and those at 500, 600 and 700°C worksoftening. (2) Two models of flow stress under warm working conditions are proposed, considering strain, strain rate, initial temperature and carbon content as variables. The constants in the models are estimated by the non-linear regression method, using stress--strain curves determined in the present work and those collected from the literature. It is observed that the difference b e t w e e n actual and calculated flow stresses increases as the strain increases. (3) The maximum extrusion pressure in warm forward-extrusion was predicted, using one of the presently reported models for the flow stress and employing the upper-bound method. Calculated results show fairly good agreement with experimental results reported elsewhere, except for large extrusion ratios. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

U. Krause, Arch. Eisenhfittenwes., 34 (1963) 745. S. Hashizume, J. Japan Soc. Tech. Plasticity (in Japanese), 4 (1963) 733. S.K. Samanta, Int. J. Mech. Sci., 10 (1968) 613. M. Oyane, F. Takashima, K. Osakada and H. Tanaka, Proc. 10th Japan Congr. Testing Materials, 1967, p. 72. M. Ohmori and Y. Yoshinaga, Proc. 11th Japan Congr. Testing Materials, 1968, p. 95. S. Shida, J. J~')an Soc. Tech. Plasticity (in Japanese), 13 (1972) 935. C.W. MacGregor and J.C. Fisher, J. Appl. Mech., 13 (1946) 11. N.R. Draper and H. Smith, Applied Regression Analysis, John Wiley & Sons, Chichester, 1966, p. 263. K. Tanaka, J. Japan Soc. Mech. Eng. (in Japanese), 69 (1966) 48. M. Fujino, T. Nitta, M. Yamamoto and S. Taira, J. Soc. Mat. Sci. Japan (in Japanese), 20 (1971) 467. K. Osakada and Y. Niimi, Int. J. Mech. Sci., 17 (1975) 241. J. Halling and L.A. Mitchell, Int. J. Mech. Sci., 7 (1965) 277. M. Ishii, J. Japan Inst. Met. (in Japanese), 32 (1968) 154. K. Yuasa, Metall. Met. Form., 41 (1974) 164. R. Geiger, Load and Work Required in Warm Extrusion of Steel, 9th Int. Cold Forging Group Plenary Meeting, Paris, August 1976.