Criteria for ductile fracture and their applications

Journal of Mechanical Working Technology, 4 (1980) 65--81 ©

Elsevier Scientific Publishing Company, Amsterdam

--

65

Printed in The Netherlands

CRITERIA FOR DUCTILE FRACTURE AND THEIR APPLICATIONS

MORIYA OYANE

Department of Mechanical Engineering, Kyoto University, Sakyo-ku, Kyoto (Japan) TEISUKE SATO,

Department of Precision Mechanical Engineering, Tohushima University, Minami-josanjimacho, Tokushima (Japan) KUNIO OKIMOTO

National Industrial Research Institute of Kyushu, Shuku-machi, Tosu, Saga (Japan) and SUSUMU SHIMA

Department of Mechanical Engineering, Kyoto University, Sakyo-ku, Kyoto (Japan) (Received May 10, 1979; accepted September 20, 1979)

Industrial S u m m a r y Criteria for ductile fracture of pore-free materials and porous materials are described. A method of estimating material constants in these criteria is also given. Applications of the criteria to prediction of the fracture strain in several types of metal working processes for pore-free materials and porous materials are described. These processes involve various strain paths and stress paths; in other words, various paths of hydrostatic stress component -- which has a great effect on fracture strain -- are involved. The fracture strain in one process differs from that in another. Although many studies of ductile fracture have already been undertaken, these are not applicable to estimate formability in various metal working processes. In this study, an attempt is made to predict the fracture strain in actual processes using the basic criterion. The calculated fracture strains are in adequate agreement with experimentally measured values.

1 Introduction In t h e s t u d y o f t h e w o r k i n g limits o f the materials in m e t a l w o r k i n g processes, t h e f o l l o w i n g p o i n t s s h o u l d be c o n s i d e r e d : (i) the material m u s t n o t f r a c t u r e in t h e f o r m i n g processes, (ii) t h e p r o d u c t m u s t n o t have d e f e c t s w h i c h lead t o f r a c t u r e in service. I f the material is d e f o r m e d , voids will be initiated at a certain strain, el, with f u r t h e r d e f o r m a t i o n causing the g r o w t h and coalescence o f t h e voids. Even if s o m e small voids are initiated in t h e material, t h e m e c h a n i c a l properties o f the material are n o t necessarily w o r s e n e d . W h e t h e r or n o t f r a c t u r e occurs in service d e p e n d s o n the c o n d i t i o n s in w h i c h the p r o d u c t s are used. F o r these reasons, it is n o t possible t o d e t e r m i n e t h e w o r k i n g limit e x a c t l y . F o r

66

simplicity, however, the fracture strain, el, is determined from noting the point when the crack is observable by the naked eye. A criterion for the ductile fracture of pore-free materials is derived from the equations of plasticity theory for porous materials. In order to apply the criterion to the ductile fracture of porous materials, it is so rearranged that it includes a relative density term. Firstly the criteria are applied here to predict the fracture strain during forming processes and comparisons are made between the calculated and the measured fracture strains. Secondly, the mechanical behaviour of coldworked materials is studied; in particular, the effects of surface cracks due to cold working on the mechanical properties are investigated. A fatiguelife test on torsionally pre~trained specimens is presented and the relation between the fatigue life and pre-strain is described. 2. Criterion for ductile fracture

(a) Pore-free materials For calculation of the strain at fracture, it is desirable that the criterion is expressed in terms of strains. While the voids grow in size and number during plastic deformation, the density of the material decreases; finally the growth and coalescence of voids leads to the fracture of the material. The change in density, or volumetric strain, can thus be a good measure for describing ductile fracture. One of the present authors [1] has derived a criterion for ductile fracture from the equations of plasticity theory for porous materials [2] ; it is assumed that when the volumetric strain reaches a certain value evf, which depends on the particular material, the material fractures. Assuming that after the initiation of fracture the material also obeys the equation for porous metals, the following criterion of ductile fracture is obtained: Cvf

f

f~ O2n- ldev =

0

A

deeq

(1)

eeq.i

where f is a function of the relative density p (defined by the ratio of the apparent density of the porous material to the density of its pore-free matrix), n is a constant, Oeq is the equivalent stress, Om is the hydrostatic component of stress (i.e. (ol + o: + o3)/3), eeq.i is the equivalent strain at which voids are initiated, eeq.f is the equivalent strain at which fracture occurs and A is a material constant. The quantity of the left-hand side of eqn. (1) is dependent only on the material. Therefore eqn. (1) reduces to the following form: eeq.f

Aa e-q deeq=C, eeq.i

(2)

67 where C is a material constant, i.e. e vf

C= /

(f2p:n-llA)dev

0

Osakada et al. [3] have reported that the strain at which voids are initiated depends on the pressure. As a first approximation, however, it can be assumed that the value, eeq.i, is a material constant regardless of the atmospheric pressure. If eeq.i = 0, eqn. (2) reduces to a very simple form:

1+

dE eq = C

(3)

A Oeq 0

(b) Porous materials In order to apply eqn. (3) to the plastic deformation of porous materials, it should be so rearranged that it includes the term of the relative density. The yield criterion for porous materials is given by

pnaeq = X/1 [(o1--02) 2 + (o:--o3) 2+ (o3--al) 2 ] + (Om/f) 2

(4)

where n is a constant and Oeq is again the yield stress, or equivalent stress, referred to the matrix material. Since pnoeq Can be defined as the flow stress of the porous materials, Oeq in eqn. (3) is replaced by pnoeq. Assuming that the value of the right-hand side in eqn. (3) for the porous material is related to the initial relative density of the porous material, it may be written as CpoB (B: a constant, P0: initial relative density). Thus the following ductile fracture criterion can be advanced for porous materials: eeq.f

f 0

1 +

A pnoeq

deeq = Cpo B

(5)

When the initial relative density P0 (and hence p ) is equal to 1.0, eqn. (5) coincides with eqn. (3). 3. Estimation o f material c o n s t a n t s

(a) Pore-free materials Upsetting of a cylindrical specimen with grooved dies -- in which, therefore, sticking takes place at the die--workpiece interfaces -- was employed to estimate the material constants; in this test the bulging of the specimen is large and therefore comparison of the forgeabflity of the materials can be easily made. Axial and circumferential strains were measured at the equator of the bulged surface of the upset cylinders; the stresses were then calculated using the L6vy--Mises equations [4]. Typical results of the upset tests are shown in Figs.l--3.

68

0"6

_

1,0

a Ho/Do= I'0 o H~Do= 1.25

~e

® I..~/Do=I.5

/o

0"3

-

.~.o~/*

Eo.6

uJ

~D

0

~ 1

o HJ~- J-25 • HJD.ffiI'0

0"8

/@

I

~ HI / H o

0"4 0"2

- 0"3

0 -0"6

I

0

I

0.2

I

0"4

.......

0"6

_J

0"8

4H/Ho Fig.1. Variation of principal strains (e 0 ,e z ) with height reduction free surface, in compression with grooved dies. Fig.2. Relationship between height reduction equatorial free surface.

AH/I-t o

AH/H

o

at the equatorial

and equivalent strain eeq at the

cr

~k) !_o.s 0"6

F /,./o..,.o i)

(~"

0"6

0 HJDo:I25

/0////®

H./D..I

0 r//o/

5

0-2 1

~eq

-0.04

I

1

0

I

0'04

I

I

I

0'08

-0"2

Om/ ~q d ~eq -0.4 Fig.3. Relationship between equivalent strain eeq and hydrostatic stress component a m/aeq at the equatorial free surface. eeq.f

Fig.4. Relationship between f (Om/aeq)deeq and eeq.f. o

69 Equation (3) is rewritten as

eeq'f

= C _ ~ 1 j~'q'f "m deeq A

(6)

aeq

o

When a surface crack was observable by the naked eye, the specimen was unloaded and the reduction AH/Ho was measured; this will be called the "limit reduction". The fracture strain, eeq.f , at the equatorial free surface was obtained from this limit reduction (AH/Ho)f and Fig.2. The value, eeq.f

f (am/Oeq ) deeq, was then calculated from the experimentally obtained 0 eeq.f and Fig.3. The experimentally derived relationship between eeq.f

f 0

(o m/Oeq ) deeq and eeq.f for various initial height-to-diameter ratios,

Ho/Do, were plotted; there is a linear relationship, as seen in Fig.4. Material constant A is easily obtained from the slope of the straight line, and material constant C from the intersection of the ordinate and this line.

(b) Porous materials In the case of porous materials, constants A, B and C in eqn. (5) can be estimated from the results of compression tests with grooved dies as for porefree materials (p = 1.0). An atomized iron powder was compacted into a cylindrical billet in a closed-die, and then sintered at l l 0 0 ° C for 90 minutes in vacuum. From the compression tests with grooved dies, the principal strains at the equatorial free surface (co, ez ) and height reduction AH/Ho were obtained; results are shown in Fig.5. The principal stresses (ao, Oz) and equivalent strain eeq were calculated by applying the equations of plasticity theory for porous materials to the results shown in the figure. Figure 6 shows an example of the relationship between height reduction AH/Ho and (aO/aeq, Oz/Oeq, om/aeq ), Fig. 7 shows the relationship between height reduction AH/Ho and the equivalent strain eeq , and Fig.8 shows the relationship b e t w e e n eeq and am/pnaeq. From Fig.7, the equivalent fracture strain eeq.f can be obtained for each limit reduction (AH/Ho)f. Using Fig. 8 eeq.f

f

(Om/pnoeq)deeq can also

be calculated, eeq.~ is plotted against

0 e eq.f

(Om/pnoeq)deeq (see Fig.9). Similarly drawn points in Fig. 9 refer to o data obtained from specimens which are of the same density but having different values of Ho/Do. The double circles refer to the equivalent fracture strains f

I'D 0.1

.

Fig.5. Relationship between height reduction A H / H o and principal strains (eO,e z ) at the equatorial free surface, in compression with grooved dies for (a) various initial relative densities p 0 and (b) various initial height-to.diameter ratios H o / D o.

-,,\

2

0.2

0.3

(b)

~

• Ho/D.=7.13 • H./D~I. 44

0 H,/D~0.94

~--0.69

-0.1 ~ ' ~

0

-0.5 -

I

-0-5

I

0.5 0.6 aH/Ho

-O-4 -

l

0.4

-0-~

(a)

I

0.3

-0.3-

I 0.2

-

-0.3

I

0.1

oO 0

0.3-

-0.2 L

_~

o ?.--0.74 d) .P.=0.69

N

oO

0.4 -

-0.2

-0.1

0

Ol y

uJ 0.2

CO

e ?,=o.a3

o f.=o.aB

O.3 -

• ~'o=1.0o

H o/'0o:1.13

• ~o=0. 93

I

0.4 -

1

0.5 ~H/Ho

0.4

Ee

O

-

q~

I

2.3,rf:F

, ~___

"°°:'"

C=o.83

/

o',q

A

AH/H o

and stress

0.5 "H / Ho

I

o"

0.4,

0.5

0

0

0.1

0.2

-

0 ~ 0

0.1

0.2

0.3

u,) 0.3

u3

0.4 -

0.5

o.

0.1

0-2

o,.~.=o.92

~,o=O.88

I

0.2

I

I

(b)

0.3

0.3

I

///

/

I

0.5 0.6 "H I Ho

S

~,/,, .-

0-4

/

0.4 0.5 "H / Ho

I

, . . , / ~/ J ~ ' " ~cl / ""

(D'~o~0'69

0.1

/~/.

H./Do=I.13

Fig.7. Relationship between height reduction , x H / H o and equivalent strain eeq for (a) various initial relative densities p 0 and (b) various initial height-to-diameter ratios H o l d

Fig.6. Relationship between height reduction ratios (a0/Oeq ~ a z / O e q ' Om/aeq).

-I. 0 -

-0.5

0

~ o.s-

~.o!-

b=4

72

°5f 0-4 o3I

~o=0.88 01-1,/[3,=0.92

@H./O.:1.14 (} HJDo=].39

e ~ D.-~.7~

0.2

/ /

0"1 f-O0 -0.1

I

0,

/0(06

0!2

-

/

-0.2 -0.:

-04

n=Z.5

Fig.8. Relationship between equivalent strain eec1 a n d

am/Priaeq.

which were obtained from torsion tests using specimens of relative densities of Po = 0.88, 0.83 and 0.69. From eqn.(5), the equivalent fracture strain eeq.f is expressed as B

eeq.f

1 ~eq.~ am

= CPo --~

pnoeq

(7)

deeq

0

A straight line can be obtained b y plotting eeq.f against

£ eq.f

f

(°m/pn°eq)deeq

0

(see Fig.9). Except for the initial relative density Po = 0.69, the results obtained from the compression test (for Po = 0.88 and 0.83) agree fairly well with those from the torsion test. Omitting the results for p o = 0.69, material constants A, B and C were determined from Fig.9 as follows: A = 0.424, B = 4.40, C = 0.455.

(8)

Using these constants, the limit reduction for all specimens was calculated, and compared with the experimental results, as shown in Fig.10. There is satisfactory agreement b e t w e e n the calculated and the experimental results.

73

10.6 0.5

kkJ f :

1

"~,~Y.--0.69

(D ~o=0.69

~ 0.2

~?.=o.8o n=2.5

~

I

-0-15

"

-0.10 -f

0.1

ol°

-0.05

O~=o.a8

d~q

-jo"O'eq"

.)o

eeq.f

Fig.9. Relationship b e t w e e n

fy

am/pnaeqdeeq and

1.0

0.5

= 0.8

0.4

..t

equivalent fracture strain 6eq. f-

"T o"

--r 0.6 "1-

U 5 0"3 _

~e

"-'0.4

0.2

// 0.2

/

-

~

0.1 --

Z:o83

o ~:o.e8

/ 0 / 0

--

(9 ~P0=o.74

I

I

I

I

0.2

0.4,

0.6

0.8

(AH/ Ho)~.~,

• • •

1.0

0 0.5

I

I

I

I

0.6

0.7

0.8

0-9

1.0

Fig. 10. Comparison of calculated and experimental values of limit reduction (~ H/H o)f in compression with grooved dies. Fig. 11. Relationship b e t w e e n initial relative density p 0 and equivalent fracture strain eeq.f o b t a i n e d f r o m torsion tests.

74

The ductile fracture criterion for porous materials was then applied to the torsion test. Substituting o m = 0 into eqn. (5), the equivalent fracture strain eeq.f is obt~dned as 6eq -f = C P o B

(9)

Figure 11 shows the relationship between the initial relative density p0 and the equivalent fracture strain. The solid line in Fig. 11 represents the values calculated from eqn. (9); there is thus seen to be satisfactory agreement between the calculated and the experimental results.

4.Application of criterion (a) Pore-free materials Two examples of the application are shown below; further examples are given elsewhere [ 5].

Simple upsetting Upset tests under different frictional conditions were carried out using a pore-free material. The experimentally derived relationship between eeq and Om/Oeq at the equatorial free surface is plotted in Fig.12 from which the empirical expression for a m/Oeq is obtained. Om/Oeq = 1.38 × 1.57~ °.s 6eq ~ - ° ' 2 s - 0 . 3 3 ,

(10)

where ~ is the frictional coefficient, which can be determined by the ring 0.8

1.4 ®

06 0"4

~

/

0"2 •

- 0"4

,N.=0.57 ,U= 0"25 M= 0'15

1"2

ij(=H./o.)=~.o / p=o

I'0

#= 0"055

/

=

(IdJD,= 1-251

k.tO

/

0"8

@

,

0"6

0 -0"2

/

• 0 @

0s ,io

-o.57 1

0'4. '

0-2 L

0 0

0-2

i

0.4.

h

J

0'6

0'8

4 HIHo

Fig. 12. Relationship between equivalent strain eeq and hydrostatic stress component am/aeq.

Fig. 13. Relationship between height reduction A H / H o and equivalent strain 6eq.

I'0

75 compression test, and ~ is the initial height-to
(11)

Substituting the material constants, initial billet size (i.e. ~ = Ho/Do ) and coefficient of friction in eqn. (11), the fracture strain eeq.f in upsetting is estimated. Combining the calculated value of eeq.f and Fig.13, the limit reduction, (hH/Ho)f, is obtained. The comparison between the calculated limit reductions and experimental values is shown in Fig.14. I'0 ,',

S55C 0 ,U=0.57

0"8

X

•

/ --~ /

).Jr- o.15

• l l ~ /

A

o 7" -r-

0"6

0.4

v

/

0"2

0

I

l

0-2

0"4

-0-¢-e-~ I 0"6

/~ : 0-57 ,U = 0.25 ,uz 0.15 JU-"O'05S I 0-8

I'0

(4H/Ho)f.col Fig. 14. Comparison of calculated and experimental values of limit reduction (A H/H o)f.

Simultaneous indirect extrusion Simultaneous indirect extrusion was carried o u t using pore-free materials. Fig.15 (a) shows the schematic figure for indirect extrusion, where Dc is the internal diameter o f the cup, D e is the diameter of the die-exit and t is the thickness of the base; the extrusion ratio is, therefore, R = (Dc/De) 2. The dimensions of the billet and the cup used in this study are as follows: billet height H0 = 10 mm, billet diameter Do = 15 mm, internal diameter of the cup Dc = 15.03 mm and initial height-to-diameter ratio Ho/Do ( = Ho/Dc) = 0.667. The ratio of the thickness of the base to the diameter o f the cup, t/Dc, decreases with the progress of the simultaneous indirect-extrusion process and finally cracks occur at the centre of the product. Even in ductile materials, such as commercially pure copper or aluminium, central cracks occur

76

~ ~ 3"2

(a)

--

2.8 2.4

l

R=4.75 / R=2-88 ®~R-1.98

~2"0

oi/ l i /

LL) i-6 1"2

0"8 0"4 0

~

I

I

1

I , I

0-7 0-6 0"5 0-4 0"3 0"2 0-1 0 t/Dc (b)

Fig.15. Relationship between the ratio of the thickness of the base to the diameter of cup t/Dc and equivalent strain eeq. in simultaneous indirect extrusion, b u t it is difficult to check non
(12)

where R is the extrusion ratio. Substituting eqn. (12) into eqn. (3), (97.7 R -~ eeq.f 3/3A) + (1--0.68/A) eeq.f = C.

(13)

The relationship b e t w e e n eeq at the centre of the specimen and t/De is shown in Fig.15. The fracture strain eeq.f is calculated from eqn. (13) and the working limit (t/Dc)f is evaluated with the calculated fracture strain, eeq.f, and Fig.15. The comparison b e t w e e n the calculated working limits (t/Dc)f and the experimental ones is made in Fig.16, where solid lines show the calculated results. Solid points denote that central cracks have already appeared, and hollow points that cracks have n o t y e t occurred.

77 O 0'1 0.2 O

E3

0"3

4.--

~-)'~,-- 0 , _ ~ .

•

0

0"4 To

0-5

~

,.v[] ~vl$,/~.~ (08

O--I

17S(os

o--e

17S

received)

A - - A CU Mg

0-6

v

-o-

I

I

2

3

0"7

- -e- 6063T5

--÷CHJ I

4

R Fig. 16. Comparison of calculated (solid lines) and experimental values of working limits.

(b ) Porous materials In upsetting with open dies, as shown in Fig.17, fracture occurs along the equatorial free surface, as in compression with grooved dies or simple upsetting. In open-die upsetting, the material at the unconstrained part flows into the constrained part. The density at the constrained part therefore increases during 1.0

o.8'

0.9 -

~

0.7

0.E 0"50

• Lubricated(Mo) ~/

H,/D,-I.24. T,=5.06 I

0.1

I

0.2

I

0.3

0.4

0.5

0.6

=H/Ho Fig.17. Change in density at the constrained part of the specimen during upsetting with open dies.

78

upsetting; this may provide different results from compression with grooved dies or simple upsetting. Fig. 17 shows the change in density at the constrained part of the specimen in open-die upsetting. Experiments were carried out to investigate the effects of the initial relative density P0, the initial height-to-diameter ratio tto/Do, the depth of the constrained part and the lubrication conditions. The relationship between height reduction AH/Ho and (e0, ez ) at the equatorial free surface was determined as in compression with grooved dies. Employing the same method as described in section 3, the relationships between: height reduction AH/Ho and (o0 Heq, az/oeq, Om/aeq); height reduction Att/tto and equivalent strain eeq; and equivalent strain eeq a n d Om/pnoeq, Can also be obtained. Then the relationship between equivalent strain eeq and 1 + a m / A pnOeq cart also be obtained using the material constant as in eqn. (8). Figure 18 shows an example of these relationships. By graphical integration of function (1 + a m/,4 p n o eq) (see eqn. (5)), the equivalent fracture strain eeq.f is obtained. Using the relationship between height reduction AH/Ho and equivalent strain eeq, the limit

2oF

3--~3.06

1.6 ~-

0 H./D,-1.06 H,/D.-I. 24 OH,/D.-1.71

[

1.4

J

1.2

-I .p v--

<

1.0

/

/

0.4

0.5

0.8 0.6 0.4 0.2

ol

0

I

0.1

I

0.2

I

0.3

]

0.6

Fig. 18. Relationship between equivalent strain eeq and 1 + a m/A p n a eq in upsetting with open dies.

79

reduction (AH/Ho)f corresponding to the calculated value of the equivalent fracture strain eeq.f is also obtained. Fig.19 presents a comparison between the calculated values of the limit reduction and the experimentally determined values, where there is seen to be satisfactory agreement. Consequently, the fracture criterion (5) may also be applicable to open-die upsetting. 1.0 0.8 :E

0.6

m

•

0.4

/ 0.2 [-0 V 0

o~o.87

/

e~rO.82 (D9o-'0.73

/

r 0.2

1 0.4

I 0.6

I 0.8

1.0

PH/Ho)r Fig.19. Comparison of calculated and experimental limit reduction with open dies.

(AH/H o)f in upsetting

5. Working limit from the viewpoint of the mechanical properties of coldforged materials To study the effect of surface cracks on the mechanical properties, it is necessary to test the formed specimen without removing the surface layer. Fatigue-life, impact value and quasi-static tensile strength were tested with the torsionally pre-strained specimens. Since the shapes of the specimens should be identical regardless o f the amount of pre-straining, the torsion test was chosen for the pre-straining method. The results of fatigue-life tests are here described. The material for these tests was $55C carbon steel (0.55% C), normalized at 800°C for 90 minutes. The specimens were pre-strained in torsion and then tested on a Schenck-type fatigue testing machine. To evaluate the experimental results, the equivalent stress and the equivalent strain from the Von Mises criterion were introduced. The fatigue life N (number of cycles to fracture) for a constant equivalent stress of a e q = 363MPa is shown in Fig.20. The fatigue life N decreases drastically at an equivalent prestrain of eeq.1 * ~ 0.43, which is less than the ductile fracture strain e e q . f (~0.545) calculated from eqn. (3). Since the surface roughness of a torsion

80

testpiece increases with increasing strain, the fatigue test was then carried out using specimens where the surface layer had been removed after prestraining. Figure 21 shows the relationship between fatigue life and the thickness of the surface layer removed AS O~m). When the pre-strain is less than the ductile fracture strain (eeq. 1 = 0.44, 0.48 and 0.50), the removal of the

10 e

O 0"-

O

0

0

O

Z

10 5

O O

Io" 0

1

I

I

J

0"1

0.2

0"3

0"4

0,5

0-6

(~eq'l

Fig.20. Relationship between equivalent pre-strain eeq. l and fatigue life N.

f

O~ 0~

i 0 6 /

/

(~;/:

0--

.,....-.- 0

• i~eq.I=0.56

•

0 •

" ,, ,,

I

I

I

5

67

Z I 05

104(~

,

I

1

0

I

2

34

I

= 0"50 = 0'48 = 0,44

2(t,S) #m Fig.21. Relationship between' 2AS (~m) and fatigue life N. (AS: thickness of surface layer removed).

81

surface layer of 4--5 pm provides sufficient recovery in fatigue life; the thickness of the surface layer removed corresponds to the surface roughness of the pre-stralned specimen. However, when the pre-strain was larger than the ductile fracture strain (eeq. 1 = 0 . 5 6 ) , the fatigue life did not recover. 6. Conclusions

Criteria for ductile fracture of pore-free materials and porous materials have been derived from the equations of plasticity theory for porous materials, and applied to predict the fracture strain and the working limit during forming processes. The values calculated using these criteria are in good agreement with those measured experimentally. It is thus confirmed that the criteria and the methods proposed in this paper are able to predict the fracture strain in various forming processes. References I M. Oyane, J. Jpn. Soc. Mech. Eng., 75 (1972) 596 (in Japanese). 2 M. Oyane, S. Shima and T. Tabata, J. Mech. Working Technol., 1 (1978) 325. 3 K. Osakada, A. Watadani and H. Sekiguchi, Bull. Jpn. Soc. Mech. Eng., 20 (1977) 1557. 4 H. Kudo and K. Aoi, J. Jpn. Soc. Technol. Plasticity, 8 (1967) 17 (in Japanese). 5 T. Sato and M. Oyane, J. Jpn. Soc. Technol. Plasticity, 15 (1974) 644 (in Japanese).