Plastic Flow, Instability and Ductile Fracture in Upsetting of Solid Cylinders

Plastic Flow, Instability and Ductile Fracture in Upsetting of Solid Cylinders J. C. Gelin, IUT Le Creusot, J. Oudin, Y. R a v a l a r d , Universitede Valenciennes - Submitted by A. Moisan (1). ENSAM Paris/F

____ SUt'dVRY

: The knowledge o f metal f l o w and s t r a i n a t f r a c t u r e i s o f most importance t o p r e d i c t t h e p l a s t i c damace i n metal forming processes. I n t h i s paper, we analyse t h e metal f l o w i n u p s e t t i n g of s o l i d c y l i n d e r s o f h i c h t t o diameter r a t i o equal t o 1.5, w i t h t h e veiahted r e s i d u a l method. The r e s u l t s about t h e e q u i v a l e n t s t r a i n oatterns a r e compared w i t h those obtained by Vickers hardness i n d e n t a t i o n t e s t s , f o r t h r e e d i f f e r e n t cases o f f r i c t i o n : l u b r i c a t e d v i t h o i l , d r y f r i c t i o n , and rough platens. It shows t h a t t h e experimental and c a l c u l a t e d s t r a i n s are i n good ayeement. A t h e o r e t i c a l method i s prooosed t o determine t h e e q u a t o r i a l surface s t r a i n paths from the maximum diameter t o minimum diameter r a t i o o f c y l i n d r i c a l specimens. using a l o c a l v e l o c i t y f i e l d and a phenomenological r e l a t i o n . The maximum t e n s i l e s t r e s s t o f l o w s t r e s s r a t i o , and t h e h y d r o s t a t i c s t r e s s component t o f l o w s t r e s s r a t i o a r e deduced from t h e s t r a i n paths and LEVY-WISES r e l a t i o n s . The a o p l i c a t i o n o f COCKCROFT f r a c t u r e c r i t e r i a , OYP.IIE f r a c t u r e c r i t e r i a and HILL c r i t e r i a f o r i n s t a b i l i t y orovide s a t i s f a c t o r y canparison w i t h v i s u a l observations i n experimental i n v e s t i g a t i o n . I t seem t h a t t h e values obtained fran the OYANE c r i t e r i a a r e i n good agreement w i t h exnerimentaly measured values. Furthermore, an i d e n t i f i c a t i o n method f o r d e t e n i n a t i o n o f m a t e r i a l c h a r a c t e r i s t i c s i n d u c t i l e f r a c t u r e i s presented.

1. INTRODUCTIOH The p r e d i c t i o n of defects o c c u r r e n c e i n forming processes r e q u i r e s t h e knowledge o f t h e stress, s t r a i n and s t r a i n r a t e t h a t take place d u r i n g deformation i n t o the r!orkpiece. The f l o w o f metal i s mainly a f f e c t e d by t h e i n i t i a l geometry o f t h e workpiece, t h e geanetry o f t h e t o o l s , t h e f r i c t i o n forces a t t h e t o o l -workpiece i n t e r f a c e . The present i n v e s t i g a t i o n i n v o l v e d t h e u p s e t t i n g o f s o l i d c y l i n d e r s o f carbon s t e e l w i t h i n i t i a l height-diameter r a t i o equal t o 1.5, according t o t h e standard procedure f o r t h e Kudo from cooperative works supervised c o l d u p s e t t a b i l i t y t e s t [ 1, by t h e "F" Group o f t h e C I R P . Kudo and Aoi [3J , i n t h e i r i n v e s t i g a t i o n o f t h e w o r k a b i l i t y o f carbon s t e e l , r e l a t e d t h e compression t e s t c o n d i t i o n s t o t h e mode o f f r a c t u r e , t h e f r i c t i o n c o n d i t i o n a t t h e i n t e r f a c e was c o n t r o l l e d q u a l i t a t i v e l y by usina qrooved d i e s and conical dies, w i t h and without l u b r i c a t i o n . An important experimental study on t h e change i n geometry O f t h e f r e e surface i n u p s e t t i n g o f s o l i d c y l i n d e r s was c a r r i e d out i n 1969 by Kulkarni and K a l p a k j i a n [E.] Thej have specified parameters c h a r a c t e r i z i n g b a r r ? l i n g under d i f f e r e n t cases of l u b r i c a t i o n . b a r r e l i n g i s s t u d i e d as an exemple o f f r e e deformat i o n i n p l a s t i c working. consisted of s e r i e s The experiments r e a l i z e d by Kobayashi o f t e s t s i n v o l v i n g t h e u p s e t t i n g o f s o l i d c y l i n d e r s and r i n g s o f SAE 1040 s t e e l under annealed c o n d i t i o n s a t room temperature. t!easurments o f t h e displacement d i s t r i b u t i o n s a t t h e tool-workpiece i n t e r f a c e , bulge p r o f i l e s , s t r a i n s a t t h e f r e e surface, and observations of crack formation f o r various f r i c t i o n condit i o n s and specimen dimensions have been r e a l i z e d . The observat i o n s concerning s t r a i n and s t r e s s h i s t o r i e s , and t h e i r r e l a t i o n s h i p s t o t h e mode o f f r a c t u r e c o n f i r m t h e r e s u l t s of Kudo and Aoi f o r u p s e t t i n g o f s o l i d c y l i n d e r s .

21

.

153

!lore r e c e n t l y . K i v i v u o r i and Sulonen [ 6 ] analysed f o r m a b i l i t y l i m i t s and f r a c t u r i n g modes o f u n i a x i a l compression specimens. The experimental procedure a p p l i e d i s t h e same as t h a t preconiThey assume t h e existence o f sed by Lagasse and De Meester [2] t h r e e modes of f r a c t u r e :

.

W o r k a b i l i t y o f metals have been studied by many authors. Cockcroft and Latham [9] proposed a d u c t i l e f r a c t u r e c r i t e r i a based on t h e t e n s i l e s t r a i n energy density, f r a c t u r e occurs when a maximum t e n s i l e s t r a i n energy i s reached. Oyane [lo] proposed a c r i t e r i a derived fran t h e equations o f p l a s t i c i t y f o r porous m a t e r i a l s [la , f r a c t u r e occurs when t h e volumetric s t r a i n reaches a c e r t a i n value. t h a t depends on m a t e r i a l . s t r a i n h i s t o ry and s t r a i n r a t e . H i l l 0 2 1 , l o c a l i s a t i o n o f deformation o f a defined an i n s t a b i l i t y c r i t e r i a , H i l l analysis o f i n s t a b i l i t y t o I n t h e present work, we have used and compared these d i f f e r e n t approachs t o p r e d i c t t h e defects OccurRnce on t h e e q u a t o r i a l f r e e surface, fran t h e c a l c u l u s o f t e n s i l e s t r e s s f a c t o r and hydros t a t i c s t r e s s f a c t o r a t t h e e q u a t o r i a l c y l i n d e r circumference.

2. _EXPERIMENTAL PROCEDURE Canpression t e s t s o n 14 mn diameter-21 mn h e i g h t c y l i n d e r s were c a r r i e d o u t using a 5 MN h y d r a u l i c press w i t h means o f measuring and recording loads and die-displacements. Two s o r t s o f d i e s have been used :

-

smooth d i e s : Standard Roughness Ra ; 0.2pm M a t e r i a l AFNOR reference 2 200 C 13 (SAE D 3) Hardness : 58-62 Rockwell C.

Before t e s t s t h e d i e s were cleaned and polished w i t h 8u diamond paste. These smooth d i e s have been used w i t h o u t l u b r i cant o r with Mobilube O i l SPE 80/90 l u b r i c a n t .

- grooved d i e s

: Height o f standard scratch : 1 mn M a t e r i a l P.FNOR reference 2 200 C 13 (SP.E D 3 )

Recessed and g r o w e d d i e s were a l s o used. w i t h no l u b r i c a n t . We used two t e s t m a t e r i a l s : AFNOR reference XC 38 (SAE 1038 s t e e l ) , and AFNOR reference 30 ElCC 16 (no SAE equivalence). I n order t o present a normalized s t r u c t u r e . these two m a t e r i a l s have been annealed :

- l o n g i t u d i n a l cracks, - oblique cracks. - double o b l i q u e cracks. Theoccurrence o f two f i r s t modes have been explained by Kudo and Aoi ; l o n g i t u d i n a l cracks occurs when t h e a x i a l component o f t h e e q u a t o r i a l surface s t r e s s i s p o s i t i v e , oblique cracks occurs when t h e a x i a l canponent o f t h e e q u a t o r i a l surface s t r e s s i s negative.

-- aa tt 850" 660"

C C

for for

The chemical canpositior! t a b l e 1.

one hour f o r XC 38. one hour f o r 30 NCD 16. o f t h e two s t e e l s a r e given i n

The aim o f t h e present i n v e s t i g a t i o n i s t o study t h e i n f l u e n ce o f various l e v e l s o f f r i c t i o n a t t h e workpiece-dies i n t e r f a c e on t h e i n t e r n a l metal flow, a t any g i v e n p o i n t a t any i n s t a n t . A procedure and a canputer package. based on t h e weighted r e s i dual method c7.83, has been developped f o r t h e axisymmetric u p s e t t i n g o f s o l i d c y l i n d e r s , which permits t h e determination o f load, a x i a l and r a d i a l displacements, f l o w patterns, s t r e s s and s t r a i n , f o r various l e v e l s o f i n t e r f a c e shear stress. Hardness d i s t r i b u t i o n w i t h i n t h e b i l l e t , c a l c u l a t e d from t h e weight e d r e s i d u a l method have been compared w i t h those obtained from Vickers hardness i n d e n t a t i o n t e s t .

A p r a c t i c a l problem i s t h e r e l a t i o n s h i p between t h e geometric a l change o f f r e e surface and t h e d e f e c t Occurrence on t h e f r e e surface. H i s t o r i e s o f t h e s t r e s s and s t r a i n a t t h e e q u a t o r i a l c i r c u n f e r e n c e where an eventual f r a c t u r e i s expected a r e t h e major concern o f t h i s problem. Kudo and Aoi [3] , i n t h e i r i n v e s t i g a t i o n o f t h e w o r k a b i l i t y o f carbon s t e e l , measured d i r e c t ly, by means o f intended marks, t h e s t r a i n v a r i a t i o n s a t t h e e q u a t o r i a l f r e e surface, t h e i n v e s t i g a t i o n s o f Kobayashi a r e i n t h e same way. I n t h i s study. we propose t h e determination o f t h e s t a t e o f s t r a i n a t t h e e q u a t o r i a l surface f r a n t h e measure o f maximum diameter- minimum diameter r a t i o o f t h e compressed c y l i n d e r .

Lq

Annals of the CIRP Vol. 30/1/1981

Table 1

-

Chemical composition o f t e s t

materials

The c y l i n d e r s were c a r e f u l l y machined i n bars o f diameter 18 mn. The machining c o n d i t i o n s were ( w i t h carbide t o o l ) : C u t t i n g speed : 6 h / m i n , : 0.2 mn, Depth Longitudinal feed : 0. 1 mn/min.

193

I n order t o realize three d i f f e r e n t conditions o f lubrication a t t h e d i e - c y l i n d e r s i n t e r f a c e , t h e c y l i n d e r s were c l a s s i f i e d i n t h r e e grouo, according t o t h e rouqhness o f thP faces.

x O < Ra< 1 . 2 ~ corresponds t o smooth d i e i w i t h l u b r i c a n t llobi lube o i 1 SAE 80/90 x 1.24 Ra< 2um corresponds t o smooth d i e s w i t h no l u b r i cant ( d r y f r i c t i o n ) , x 2 4 Ra 4311171 corresponds t o grooved d i e s .

3. WEIGHTED RESIDUAL IlETHOD AN0 INTERNAL FLOW OF METAL

3.1 - Ihep-e-jca!-a_feep;f We have used t h e weighted r e s i d u a l method [8] t o p r e d i c t deformation patternand equivalent s t r a i n over the diametral s e c t i o n o f workpiece.

Vickers hardness t e s t s have been r e a l i z e d on t h e median c i r cumference o f each c y l i n d e r , a f t e r p o l i s h i n g w i t h special paper. The charge used was equal t o 300 g, the f o l l o w i n g values have been obtained :

The p l a s t i c f l o w i n t h e c y l i n d e r , Figure 2, can be described by using t h e f l o w f u n c t i o n :

HV300 = 162

3 f o r XC 38 s t e e l ,

vo being t h e v e l o c i t y of the d i e and A . . t h e unknown parameters.

HV300 = 260

4 f o r 30 NCD 16 s t e e l .

The f u n c t i o n 0 ( r , z ) i s continuous with f i r s t order d e r i v a t ives. The components o f t h e v e l o c i t y a r e obtained from t h e p a r t i a l d e r i v a t i v e s o f t h e f u n c t i o n 0 (r, z ) :

1J

Marks were i’tlented by Vickers hardness t e s t s a t two l o c a t i o n s along t h e equatocial f r e e surface o f t h e specimen. Distance between t h e marks i n t h e a x i a l d i r e c t i o n a t each l o c a t i o n was 1 mn. A f t e r each increment o f deformation, t h e h e i g h t o f t h e specimen. t h e minimum diameter o f t h e c y l i n d e r and t h e maximum diamet e r o f t h e f r e e surface were measured w i t h a p r o f i l e p r o j e c t o r (accuracy : 0.01 nm). The procedure was repeated f o r each increment o f deformation, u n t i l cracks Were observed w i t h a naked eye ( x 10). Ue decided t h a t f r a c t u r e occured when i t was possible t o observe a t l e a s t one crack o f 0.3 nm length. The mechanical p r o p e r t i e s o f t h e two metals have been d e t e r mined from compression t e s t o f s o l i d c y l i n d e r s . Figure 1 shows t h e s t r e s s - s t r a i n r e l a t i o n u s i n g an exponenti a1 smoothing f o r SAE 1038 s t e e l . The paaameters o f t h e exponential s t r a i n hardening law, ao = AE. , a r e i n t a b l e 2 ; (10 i s t h e flow-stress, A i s t h e f l o w - s t r e s s f o r a u n i t s t r a i n and n strain-hardening coefficient

.

The boundary c o n d i t i o n s a r e : Vz = 0 f o r z = 0 , V r = 0 f o r r = 0, Vz = Vo f o r z = h.

-

To c a l c u l a t e t h e s t a t e o f s t r e s s i n t h e c y l i n d e r , two shape f u n c t i o n s 01 and 02 a r e used ; t h e s t r e s s components a r e :

,FlcI=r 1

a02

;Ti-

(3)

The f o l l o w i n g boundary c o n d i t i o n s a r e required : t h e shear stress i s equal t o zero along r and z axis. t h e normal component o f t h e stress i s equal t o zero on t h e f r e e surface, ( i i i ) a t t h e d i e - c y l i n d e r i n t e r f a c e , t h e shear s t r e s s i s d e f i ned by the f o l l o w i n g r e l a t i o n :

(i) (ii)

ME

%

a,

(T ( ~ = h ) = - % rz vhere ii i s t h e mean f r i c t i o n c o e f f i c i e n t (sublayer model).

The f u n c t i o n s 01 and 07 a r e defined as f o l l o w s :

500

The r e s i d u a l f u n c t i o n i s formed by t h e f o l l o w i n g q u a n t i t y :

2 51 t h a t includes t h e boundary c o n d i t i o n ( i i i ) The weighting f u n c t i o n s a r e :

C

I0

050

Fig. 1

I

E

1.00

- P e l a t i o n s h i v between 7flow-stress

00

and mean

equivalent s t r a i n

-

We o b t a i n L = I x J + (P 1 ) x N + P x Q l i n e a r equations and f o r t h e c a l c u l a t i o n o f t h e unknown parameters F.... 1 J , ,B cPq *

where k €[l L, 1 , and ak i s i d e n t i f i e d t o Aij Bmn o r C 3.2

Table 2

-

I d e n t i f i c a t i o n parameters f o r t e s t m a t e r i a l s

-

w). Nurller!E?!_a_nrr_ex~~~~en~?!-~e~~~;~

The i n f l u e n c e o f f r i c t i o n on t h e i n t e r n a l metal f l o w i s pred i c t e d by t h e g r i d d i s t o r s i o n s i l l u s t r a t e d i n F i g u r e 3. When t h e i n t e r f a c e f r i c t i o n i s h i g h (compression w i t h grooved o r rough d i e s ) , t h e r a d i a l and a x i a l displacements increase and a small f o l d i n g occurs, i n t h e experimental r e s u l t s , an amount o f f o l d i n g i s observed when t h e r e d u c t i o n o f h e i g h t increases strong 1y

.

194

(respectively

The equivalent strain distribution is also a reflection of the internal flow patterns. Figure 4 indicates clearly the areas of high and low deformation obtained from Yickers hardness measunnents and calculated fran the weighted residual method (at 50 % of height reduction for SAE 1038 steel. with dry friction).

D r y friction

I

- ?ii,O.36

Experimental equivalent strain distribution for SPE 1038 steel. at 80 X of height reduction with dry friction, is presented on Figure 5. The maximwi equivalent strains are inside a zone extending from the middle r axis to the corner of the specimen . The maximum hardening zone extends when friction increases.

Initial

heiq ht -

diomeler rafio

= 1.5

-//A””

21

0.4

i

S

4

.

( b ) calculated

Fig. 2

0

values

/

I /I 0.4

Fig. 4

- Canparison

Fig. 5

- Experimental equivalent strain distributions, SAE 1038 steel, dry friction, 80 X of heigth reduc-

between experimental and calculated equivalent strain distributions.

- Upsetting of solid cylinder between two flat platens tion.

4. VARIATION OF STRAINS AND STRESSES AT THE EQUATORIAL FREE SURFACE IN UPSETTING OF SOLID CYLINDER? 4.1

- bnalur!cal_consldnratlons

The state of stresses at the equatorial free surface of a canpressed cylinder is a plane state. indeed the radial stress and the shear are equal to zero. The levy-Mises equations relate plastic strain increments to stresses.

I

a)

The increment of local equivalent strain i s given by the following relation : 2 (1 t B t 8’)‘ 1 Id .cZZ; I dT = -

Dry friction

(8)

6

where B

= d80

;

(9)

dEZZ

-5 0%

8 is the local slope of the strain paths.

Fran Levy-tlises equations and above relations, we obtain the expressions o f the maximum tensile stress to flow stress ratio and the hydrostatic stress to flow stress ratio :

b)

Compression

with

rough dies

The aim of the present investigation is to give a method for the calculus of B. The strain path can be represented by the following phenomenological relation :

-

=

Fig. 3 Grid distorsions at 50 % reduction in height for two different conditions of friction.

ba,:

-

7

5 2

(12)

Where b i s a parameter that depends on the severity of friction at the die-cylinder interface. 4.2

- h~~l-y-locj~y-f~fld 195

and The p r e d i c t i o n o f the e q u a t o r i a l s t r a i n s E , f i g u p 8 6. r e a l i z e from a l o c a l v e l o c i t y f i e l d

;51

rZZ

The s t r a i n and s t r e s s paths a r e mainly influenced by t h e s e v e r i t y o f f r i c t i o n a t the d i e - c y l i n d e r i n t e r f a c e , when t h e shear s t r e s s increases, t h e d i s t r i b u t i o n o f s t r a i n s become more inhanogeneous.

is

However, t h e c i r c u m f e r e n t i a l s t r a i n s obtained fran t h e d i f f e r e n t c o n d i t i o n s o f l u b r i c a t i o n (smooth dies, grooved o r rough d i e s and l u b r i c a t e d d i e s w i t h llobilube o i l SAE 80/90) a t t h e s a w h e i g h t reduction, present very s i m i l a r values. t h i s i s a t t r i b u t e d t o t h e f a c t t h a t t h e r a d i a l disolacement o f t h e e q u a t o r i a l f r e e surface i s n o t a s i g n i f i c a n t parameter f o r t h e occurence o f b a r r e l i n g . Stress paths f o r c i r c u m f e r e n t i a l and h y d r o s t a t i c stresses are h i g h l y s i g n i f i c a n t o f t h e s t a t e o f s t r e s s a t e q u a t o r i a l surface o f compressed c y l i n d e r w i t h b a r r e l i n g . For 30 NCD 16 s t e e l . we can note t h a t f r a c t u r e occurs when h y d r o s t a t i c s t r e s s f l o w s t r e s s r a t i o i s p o s i t i v e , i t i s i n agreement w i t h the observations reported by Kudo and Aoi

w Fig. 6

- Meridian s e c t i o n o f a compressed barreling.

.

The r a d i a l displacement o f t h e p o i n t

(1

dur = alr

M

131.

cylinder with

i s choosen l i k e t h a t :

- u z 3 t a2 r 3

(13)

where : al and a2 a r e constants. u i s a parameter t h a t depends o f f r i c t i o n .

The i n c o m p r e s s i b i l i t y c o n d i t i o n involves : du,

=

-

2al

(z

-

z3)

-

4 a2 r 3L

(14)

The constants a and a a r e c a l c u l a t e d from t h e boundary c o n d i t i o n s a t t h e eAuatori%l f r e e surface and we o b t a i n :

=zz = 4b

- ( 3 + a RB P) -( 1 t 4 # - 2n RE n) B

(15)

p i s t h e r a d i u s o f curvature o f f r e e surface a t t h e p o i n t 8.

If we assume t h a t t h e t r a c e o f t h e f r e e surface i n meridian s e c t i o n i s approximated by a p a r a b o l i c curve, t h e radius o f curvature i s given by :

Fig. 7a

-

S t r a i n paths f o r t h r e e c o n d i t i o n s o f l u b r i c a t i o n .

H2

(16)

P =

Parameters b and c a r e evaluated by minimizing t h e f o l l o w i n g quantity : (171

( i ) is the ith step o f upsetting. n i s t h e nunber o f steps ; R i s equal t o Log ( ’). and (czz)i i s replaced by equation

EL:)

$’

(15). Thus, s t r e s s and s t r a i n DathS are evaluated fran maximum diameterminimum diameter r a t i o using t h e equation ( l o ) , (11). (12). (15) and (16). 4.3

-

hpgrjFgw~a!-r?+:

F i g . 7b

-

Stress paths f o r t h r e e c o n d i t i o n s o f l u b r i c a t i o n .

5. DUCTILE FRACTURE

The e q u a t o r i a l s t r a i n s and stresses versus h e i g h t r e d u c t i o n curves a r e i l l u s t r a t e d i n F i g u r e 7, f o r 30 NCO 16 s t e e l ( t h r e e conditions o f lubrication).

5 - 1 - neelicatlo?_of_Fncrnscoeic_fra_ctu_re_c uelett~ns-of_rolid-cY!~~dgr:

The continuous curves correspond t o t h e c a l c u l a t e d values, and t h e d i s c r e t e p o i n t s correspond t o t h e experimental s t r a i n s measured w i t h t h e above-mentioned procedure.

Cockcroft and Latham [9] c r i t e r i m s t a t e s t h a t f r a c t u r e occurs i n a m a t e r i a l when t h e q u a n t i t y :

The t h e o r e t i c a l and experimental r e s u l t s a r e i n good agreement, t h e maximum r e l a t i v e d i f f e r e n c e i s l e s s than 5 X czz exp.

- cZZ cal. Q

sZZ exp.

5::

(f

& o o d r > C

(18)

Where CJ* i s t h e maximum t e n s i l e stress, dE i s t h e increment o f e q u i v a l e n t s t r a i n and C i s a c h a r a c t e r i s t i c constant o f a m a t e r i a l a t a given s t r a i n r a t e and temperature. A t t h e e q u a t o r i a l f r e e surface o f a s o l i d c y l i n d e r

equal t o

2

(equation 10) and

d Zis

given from t h e equation (8).

The constant C i s evaluated from a t e n s i l e t e s t :

196

*

$is

-ntl C=A'f,

( E 4

,

c a h l b Oyane crit eria

n + l

o

Nhere A and n a r e t h e parameters o f exponential s t r a i n hardening law, Tf i s t h e equivalent s t r a i n reaches a t t h e f r a c ture.

0

Oyane and a l . [lo] c r i t e r i a s t a t e s t h a t f r a c t u r e occurs when t h e volumetric s t r a i n reaches a c r i t i c a l value :

f

'f ( 1 t

o

:

Hill criteria Cockcroft criteria

0.50

k) c' dF = C CiO

i s t h e h y d r o s t a t i c s t r e s s factor, C ' and C are constants of

t h e m a t e r i a l a t a given s t r a i n - r a t e and temperature. 'm

.

i s given from t h e equation (11) and t h e constants C and C '

a00

a r e evaluated w i t h an i d e n t i f i c a t i o n method.

H i l l c r i t e r i a f o r i n s t a b i l i t y [12; f o l l o w i n g form :

a50

can be expressed on t h e Fig. 8b

1 doo "40 ' 'z __ uo <- 2no dF

-

Comparison o f c a l c u l a t e d and experimental circumference s t r a i n s i n upsetting.

I

From t h e equations (8). ( 9 ) and (1l);the becanes :

1 & - - .d 3 E

2

l t P (1 + 6

(21)

I

t

6. CONCLUSION

above r e l a t i o n

6')'

m i m p o r t a n t remark i s deduced from t h i s r e l a t i o n ; t h e f r a c t u r e due t o i n s t a b i l i t y occurs when t h e s l o w o f l o c a l s t r a i n path i s smaller than 1.

-

5.2

- ve!i~!~y_of-fra_du_r~-~~~~~~~~~~ -

F i g u r e s 8 (a, b) shows t h e s t r a i n paths l c z z ] ) for t h r e e d i f f e r e n t c o n d i t i o n s o f l u b r i c a t i o n and a c o r r e l a t i v e c u r ve between experimental and c a l c u l a t e d values o f c i r c u m f e r e n t i a l s t r a i n s . It seems t h a t t h e Oyane c r i t e r i a presents a good correl a t i o n w i t h t h e experimental values o f s t r a i n s when t h e f i r s t crack occurs. The Cockcroft c r i t e r i o n g i v e s overestimated values, however t h e c h a r a c t e r i s t i c constant o f t h e m a t e r i a l determined from t e n s i l e t e s t i s n o t very s i g n i f i c a n t i n upsetting. H i l l c r i t e r i c a p r e s e n t s an important d i s p e r s i o n comparatively t o Oyane c r i t e r i o s a n d Cockcroft c r i t e r i o n .

The present i n v e s t i g a t i o n about i n t e r n a l metal f l o w i s very s i g n i f i c a n t , t h e weighted r e s i d u a l method i s a poverful t o o l t o t h e l o c a l analysis(f1ow pattern, p r e d i c t i o n o f equivalent s t r a i n s p r e d i c t i o n o f b a r r e l i n g ) . The r e s u l t s r e l a t i v e t o t h e a n a l y t i c a l determination o f s t r a i n and s t r e s s paths a r e i n good agreement w i t h t h e experimental values, t h e maximum r e l a t i v e d i f f e r e n c e i s smaller than 5 X . Furthemore, t h e measure o f maximum diameterminimum diameter r a t i o i s more accurate than t h e measure of s t r a i n s a t t h e e q u a t o r i a l f r e e surface. Maximum diameterminimum diameter r a t i o i s an important f a c t o r t o A r a c t e r i z e t h e s e v e r i t y o f f r i c t i o n and t h e amount o f b a r r e l i n g . The a n a l y t i c a l determination o f t e n s i l e stress-flow s t r e s s r a t i o and h y d r o s t a t i c s t r e s s - f l o w s t r e s s r a t i o i s o f a major importance t o p r e d i c t d u c t i l e f r a c t u r e with macroscopic c r i t e rions. The Oyane c r i t e r i a i s verv s i g n i f i c a n t , and t h e f a c t t h a t volumetric s t r a i n reaches a c r i t i c a l value i s a p r o o f o f the i n t e r n a l damage o f t h e m a t e r i a l .

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