Journal of Mechanical Working Technology, 4 (1980) 65--81 65 Elsevier Scientific Publishing Company, Amsterdam - - Printed in The Netherlands

CR ITER IA FOR DUCTILE FRACTURE AND THEIR APPL ICATIONS

MORIYA OYANE

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

TEISUKE SATO,

Department of Precision Mechanical Engineering, Tohushima University, Minami-josanjima- cho, 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 Summary

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 compo- nent -- 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 Int roduct ion

In the study of the working limits of the materials in metal working pro- cesses, the fol lowing points should be considered:

(i) the material must not fracture in the forming processes, (ii) the product must not have defects which lead to fracture in service. I f the material is deformed, voids will be initiated at a certain strain, el,

with further deformat ion causing the growth and coalescence of the voids. Even if some small voids are initiated in the material, the mechanical proper- ties o f the material are not necessarily worsened. Whether or not fracture oc- curs in service depends on the condit ions in which the products are used. For these reasons, it is not possible to determine the working limit exactly. For

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 cold- worked materials is studied; in particular, the effects of surface cracks due to cold working on the mechanical properties are investigated. A fatigue- life 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 duc- tile fracture. One of the present authors [1] has derived a criterion for duc- tile 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 = A deeq (1) 0 eeq.i

where f is a function of the relative density p (defined by the ratio of the ap- parent 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, (2) eeq.i

67

where C is a material constant, i.e.

e v f

C = / (f2p:n-l lA)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 pres- sure. 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, re- ferred 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 Cpo B (B: a constant, P0: initial relative density). Thus the following ductile fracture criterion can be advanced for porous materials:

eeq.f

f 1 + deeq = Cpo B (5) A pnoeq

0

When the initial relative density P0 (and hence p ) is equal to 1.0, eqn. (5) co- incides with eqn. (3).

3. Estimation of material constants

(a) Pore-free materials Upsetting of a cylindrical specimen with grooved dies -- in which, there-

fore, 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

~D

0"6

0"3

0

- 0"3

-0"6

_ a Ho/Do= I '0 o H~Do= 1.25 ~e I..~/Do=I.5 /o

/@

- .~ .o~/*

~ 1 I ~H/Ho I

1,0

0"8

Eo.6 uJ

0"4

0"2

0 0

o HJ~- J-25 HJD.ffi I'0

I I I . . . . . . . _ J

0.2 0"4 0"6 0"8

4H/Ho

Fig.1. Variation of principal strains (e 0 ,e z ) with height reduction AH/H o at the equatorial free surface, in compression with grooved dies.

Fig.2. Relationship between height reduction AH/ I - t o and equivalent strain eeq at the equatorial free surface.

0"6

F /,./o..,.o i ) 0 H JDo: I25

(~" /0 / / / / H. /D . . I 5

0 r / /o / ~eq

-0"2

-0 .4

cr

~k) !_o. s

0"6

0-2

1 I 1 I I I I

-0.04 0 0'04 0'08

O m / ~q d ~eq

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)dee q and eeq.f.

o

69

Equation (3) is rewritten as

= C_~ 1 j~'q'f "m deeq (6) eeq ' f A aeq

o

When a surface crack was observable by the naked eye, the specimen was un- loaded 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 ob- tained 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 (o m/Oeq ) deeq and eeq.f for various initial height-to-diameter ratios, 0

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 pore- free materials (p = 1.0). An atomized iron powder was compacted into a cylin- drical billet in a closed-die, and then sintered at l l00C 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 ob- tained; results are shown in Fig.5. The principal stresses (ao, Oz) and equi- valent 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, o m/aeq ), Fig. 7 shows the relationship between height re- duction AH/Ho and the equivalent strain eeq , and Fig.8 shows the relationship between 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

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 differ- ent values of Ho/Do. The double circles refer to the equivalent fracture strains

CO

uJ

I H o

/'0o:

1.13

0.

4 -

~

'o=1.0

o ~

o=0. 93

o f.

=o.

aB

O. 3

- e

?,=o.

a3

o ?.

--0.7

4 d)

.P.=

0.69

0.2 O ly

0

I I

_~

0.2

-0.1

-0.2

-0.3

-0-~

-0-5

I l

I I

0.3

0.4

0.5

0.6

aH/H

o

(a)

N

oO

oO

0.4

-

~--

0.6

9

0 H

,/D~

0.94

Ho/

D.=

7.13

0

.3-

H

./D

~I.

44

0.

2~

-

Ee

0.1 0

1 I'D

0.

1 0.

2 0.

3 0.

4 0.

5 -0

.1 ~

'~

~

H/H

o

L -,

,\

-0.2

-0.3

-

-O-4

-

-0.5

-

(b)

O

Fig

.5. R

ela

tio

nsh

ip

be

twe

en

h

eig

ht re

du

cti

on

A

H/H

o a

nd

pri

ncip

al st

rain

s (eO

,e z ) a

t th

e

eq

ua

tori

al fre

e su

rfa

ce

, in c

om

pre

ssio

n

wit

h g

roo

ve

d die

s fo

r (a

) va

rio

us init

ial re

lati

ve

d

en

siti

es p 0

an

d (b

) va

rio

us in

itia

l he

igh

t-to

.dia

me

ter

rati

os Ho

/D o

.

~.o

!-

C=

o.83

":

'"

A

~ o

.s-

, ~

___

2.3,

rf:F

/

0 I

I

-0.5

-

q ~

o'

,q

-I. 0

-

0.5

"H /

Ho

Fig

.6. R

ela

tio

nsh

ip

be

twe

en

h

eig

ht re

du

ctio

n

AH

/H o

an

d st

ress

ra

tio

s (a0

/Oeq

~

az/O

eq

' O

m/a

eq

).

o"

u3

0.5

0.4

0.3

0.2

0.1 0

0.5

0.4

,

u,) 0

.3

0.2

0.1 0 0

///

- H

./D

o=I.1

3 /~

/.

(D'~

o~0'

69

,..,

/~

~cl"

/J~

'"/"

~

I I

I I

I 0

0.

1 0.2

0.3

0

.4

0.5

"H

/ H

o

- ~,

o=O.

88

/ S

o,.~

.=o.

92

/ ~

,/,,

.-

0.1

0-2

0

.3

0-4

0.5

"H

I H

o Fi

g.7

. Re

lati

on

ship

be

twe

en

he

igh

t re

du

ctio

n ,x

H/H

o a

nd

eq

uiv

ale

nt st

rain

eeq f

or (

a)

vari

ou

s init

ial r

ela

tive

de

nsi

tie

s p 0

an

d (b

) va

rio

us in

itia

l he

igh

t-to

-dia

me

ter

rati

os H

old

o.

(b)

I 0

.6

b=4

72

5 f 0-4 o3 I 0.2

0"1 f-O 0

-0 .1 -

-0.2

~o=0.88 01-1,/[3,=0.92 @H./O.:1.14 (} HJ Do=].39 e ~ D.-~.7 ~

I 0, 0!2

/ /

/0(0 6 /

-0.:

-04 n=Z.5 Fig.8. Relationship between equivalent strain eec 1 and 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 1 ~eq.~ am eeq.f = CPo --~ pnoe q deeq

0

A straight line can be obtained by plotting eeq.f against

(7)

eq . f

f (m/pneq)deeq 0

(see Fig.9). Except for the initial relative density Po = 0.69, the results ob- tained 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 between the calculated and the experimental results.

I -0-15

10.6 0.5

f : 1 "~,~Y. - -0 .69 ~ 0.2

n=2.5 ~ " 0.1

ol -0.10 -0.05 -f

.)o -jo"O'eq" d~q

kkJ

(D ~o=0.69

~?.=o.8o

O~=o.a8

eeq.f

Fig.9. Relationship between f am/p naeqdee q and equivalent fracture strain 6eq. f- y

73

1.0 0.5

= 0.8

. .t

--r 0.6 "1-

" - '0 .4

0.2

0.4

_ ~e

// (9 ~P0=o .74

- / ~ Z:o83 / o ~:o.e8

/ I I I I 0.2 0.4, 0.6 0.8 1.0

(AH / Ho)~.~,

"T o"

U5 0"3

0.2 --

0.1 --

0 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 between initial relative density p 0 and equivalent fracture strain eeq.f obtained from 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 = CPoB (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 be- tween 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 ~-'2s -0.33, (10)

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

0.8 1.4

06

0"4

~ 0"2

0

-0"2

- 0"4

,N.= 0.57 ,U= 0"25 0 M= 0'15 @ #= 0"055 / = (IdJD,= 1-251 k.tO /

0s ,io '

1"2

I '0

0"8

0"6

0'4.

0-2

0 0

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

/ /

@ ,

-o.57 1

L i h J

0-2 0.4. 0'6 0'8 I'0

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.

75

compression test, and ~ is the initial height-to

76

~ ~ - (a)

3"2 - -

2.8 2.4

~2"0

LL) i-6 1"2

0"8 0"4

0 0-7

R=4.75 l / R=2-88 ~R-1.98

oi / l i /

~ I I 1 I , I 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, but it is difficult to check non

77

O

E3 4.--

O

0'1

0.2

0"3

0"4

0-5

0-6

0"7

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

O-- I 17S(os received) To ~ o - -e 17S

,.v[] ~vl$,/~.~ (08 A- -A CU

Mg

v -o- - -e- 6063T5

--CHJ I I I

2 3 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 con- strained part. The density at the constrained part therefore increases during

1.0

0.9 - ~ o.8' 0.7

Lubricated(Mo) 0.E H,/D,-I.24. ~/

T,= 5.06 I I I

0"50 0.1 0.2 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 con- strained 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 de- scribed 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 and Om/pnoeq, Can also be obtained. Then the relation- ship between equivalent strain eeq and 1 + am/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 rela- tionship between height reduction AH/Ho and equivalent strain eeq, the limit

-I < .p v - -

2oF 3--~3.06 0 H./D,-1.06

H,/D.-I. 24 1.6 ~- OH,/D.-1.71 [

1.4

1.2

1.0

0.8

J / /

0.6

0.4

0.2

o l I I I ] 0 0.1 0.2 0.3 0.4 0.5 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 deter- mined values, where there is seen to be satisfactory agreement. Consequent- ly, the fracture criterion (5) may also be applicable to open-die upsetting.

:E

1.0

0.8

0.6

0.4

0.2

0 0

m

/ o~o.87 / e~rO.82

[-- / (D 9o-'0.73

V r 1 I I 0.2 0.4 0.6 0.8 1.0

PH/Ho)r Fig.19. Comparison of calculated and experimental limit reduction (AH/H o)f in upsetting with open dies.

5. Working limit from the viewpoint of the mechanical properties of cold- forged 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 of 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 800C for 90 minutes. The specimens were pre-strained in torsion and then tested on a Schenck-type fatigue testing machine. To evalu- ate 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 aeq = 363MPa is shown in Fig.20. The fatigue life N decreases drastically at an equivalent pre- strain of eeq.1 * ~ 0.43, which is less than the ductile fracture strain eeq.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 pre- straining. Figure 21 shows the relationship between fatigue life and the thick- ness 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

Z

10 e O

O 0"- 0

- 0

10 5

O

O

O

I o " 1 I I J

0 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~ 0- - i 0 6 / / .,....-.- 0

(~; / : i~eq. I = 0.56 " = 0"50

Z 0 ,, = 0 '48 05 ,, = 0,44 I

104(~ , I 1 I I I I

0 I 2 34 5 67

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 thick- ness 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 .56) , 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).