Hydro-Mechanical Deep-Drawing

Hydro- Mechanical Deep-Drawing J. Reissner and P. Hora, lnstitut fur Umformtechnik, ETH-Zurich - Submitted by E. Matthias

An a n a l y s i s , b a s e d on t h e e n e r g e t i c s t a b i l i t y c r i t e r i o n combined w i t h n u m e r i L a ? c d l c u l a t i o n s . f o r t h e h y d r a u l i c d r a w i n g o f t a p e r e d c y l i n d r i c a l c u p s i s d e v e l o p e d i n t h i s J a p e r . As t h e c a l c u l a t i o n shows, t h e l i m i t i n g d r a w i n g r a t i o s t r o n g l y deoends o n t h e 2 r e s s u r e d u r i n g h y d r a u l i c d r a w i n g . The t h s p r e t i c a l and e x p e r i m e n t a l r e s u l c s agree d e l l .

1.

Introduction

3.

The d i s t r i b u t i o n o f s t r e s s and d e f o r m a t i o n

H y d r o m e c h a n i c a l d e e p - d r a w i n g i s a p a r t i c u l a r l y eCOnOmic a l process f o r t h e manufacture o f t h i n - w a l l e d paraboli c o r c o n i c a l h o l l o w b o d i e s . T h i s process i s a l s o emin e n t l y s u i t a b l e f o r t h e p r o d u c t i o n o f work p i e c e s w h i c h have s e v e r a l p r o f i l e shapes and a d d i t i o n a l c o n t o u r s . W h i l e p u b l i c a t i o n s i n r e c e n t y e a r s have q u a n t i f i e d t h e m a i n f a c t o r s i n f l u e n c i n g t h e f o r m a b i l i t y lim i t s i n c o n v e n t i o n a l d e e p - d r a w i n g , t h e r e i s a l a c k of s u c h i n f o r m a t i o n f o r t h e Hydromec p r o c e s s .

= o r t h e c a l c u l a t i o n t h e c u p is s u b - d i v i d e d i n t o

Thus i n t h e p r e s e n t w o r k a method i s i n d i c a t e d i n w h i c h f o r m a b i l i t y l i m i t s a r e c a l c u l a t e d u s i n g t h e ene r g e t i c s t a b i l i t y c r i t e r i o n [1.2.3,4,5.6;in connection w i t h t h e n u m e r i c d e t e r m i n a t i o n l7.6; o f s t r e s s - and d e f o r m a t i o n d i s t r i b u t i o n s . i f s t r e s s - s t r a i n c u r v e s dnd y i e l d l o c u s c u r v e s o f t h e m a t e r i a l a r e a v a i l a b l e . The a n a l y s i s i s done f o r t h e h y d r o - m e c h a n i c a l deep d r a w i n g o f c o n i c a l cups w i t h c i r c u l a r c r o s s s e c t i o n from s t a b i l i s e d austenitic steel.

3b.

7 regions (Fig. 2): 1.

Region of

the flange w i t h blank holder contact

( r -rbl 2.

Region o f t h e f l a n g e w i t h o u t b l a n k - h o l d e r c n n t o c t ( rb-rc I

95.

4.

Region o f c u r v a t u r e o f t h e drawing p l a t e ( r - r ' I c g Region o f c u r v a t u r e o f t h e blank h o l d e r ( r - r 1 c g R e g i o n of t h e b u l g e ( r ' - r 1

5.

Region o f t h e c o n i c a l r i m

6.

Region of

7.

R e g i o n o f t h e cup b o t t o m ( r A - 0 )

g

g

(r

g

-r'-pSt.cosOStl

g

c u r v a t u r e o f t h e punch ( r ' + p S t c o s O

st-%]

I f i t is assumed t h a t t h e m a t e r i a l behaves w i t h n o r m a l anisotropy, then t h e flow c o n d i t i o n i s

2 . The h y d r o - m e c h a n i c a l d e e p - d r a w i n g p r o c e s s I n deep-drawing w i t h r i g i d d i e s , t h e r e q u i r e d drawing f o r c e i s i n t r o d u c e d i n t h e r e g i o n o f t h e punch c u r v a t u r e and t r a n s m i t t e d v i a t h e c o n i c a l r i m t o t h e f l a n g e . The d r a w i n g f o r c e i s l i m i t e d by t h e s t r e s s w h i c h t h e w o r k - p i e c e m a t e r i a l c a n s u p p o r t a t t h e p o i n t where t h e p u n c h c u r v a t u r e ceases. The deep d r a w i n g o f c o n i c a l shapes i s made m o r e d i f f i c u l t by t h e f a c t t h a t a t t h e s t a r t o f t h e procees, t h e e f f e c t i v e t r a n s m i t t i n g zone i s r e l a t i v e l y s m a l l w h i c h c a n l e a d t o l a r g e l o c a l def o r m a t i o n s and f a i l u r e . I n a d d i t i o n , f o l d s f o r m i n t h e a r e a between t h e p o i n t o f a p p l i c a t i o n of t h e punch and t h e d i e p l a t e w h i c h is s u p p o r t e d by t h e b l a n k h o l d e r . I n hydro-mechanical deep-drawing t h e r e i s a pressure c u s h i o n b e l o w t h e d i e - p l a t e ( F i g . 1 ) . The i m m e r s i n g d r a w i n g punch d i s p l a c e s a p a r t o f t h e h y d r a u l i c f l u i d . The r e s u i t i n g p r e s s u r e a c t s o n t h e p a r t o f t h e workp i e c e o n t h e punch and i n t h e d r a w i n g gap and must be s u f f i c i e n t l y h i g h t h a t t h e c o n i c a l r i m behaves q u a s i r i g i d l y a s a consequence o f f r i c t i o n between t h e w o r k p i e c e and punch a f t e r t h e f i r s t punch c o n t a c t . The m a g n i t u d e of t h e p r e s s u r e must however be chosen s u c h t h a t i t l e a d s t o a b u l g e shaped f o r m a t i o n o f t h e u n s u p p o r t e d d e f o r m a t i o n zone. O n l y i n t h i s c a s e can f o l d s o f f i r s t and second o r d e r b e s a f e l y a v o i d e d b a s e d on t h e s t r e s s d i s t r i b u t i o n . The l i m i t i n g d r a w i n g r a t i o i s s i m u l t a n e o u s l y s i g n i f i c a n t l y improved, s i n c e t h e c r i t i c a l zone ha5 been t r a n s l a t e d f r o m t h e p o i n t where t h e punch c u r v a t u r e c e a s e s t o t h e t r a n s i t i o n b e t ween t h e c o n i c a l r i m and punch. A t o o h i g h c u s h i o n pressure leads t o premature f a i l u r e . The mode of o p e r a t i o n o f a H y d r o n e c m a c h i n e c a n b e d i v i d e d i n t o two t i m e p e r i o d s .

A t the s t a r t o f the working cycle the pressure f l u i d i s pumped o u t o f t h e s u p p l y c o n t a i n e r i n t o t h e p r e s s u r e c u s h i o n . I n t h e second t i m e p e r i o d t h e i r m e r s i n g punch d i s p l a c e s t h e w a t e r e m u l s i o n o v e r an a d j u s t a b l e gap p r e s s u r e c o n t r o l v a l v e ( F i g . 11. The p r e s s u r e d i s t r i b u t i o n i n t h e w a t e r compartment depends o n t h e p o s i t i o n o f t h e p r e s s u r e c o n t r o l v a l v e , t h e punch g e o m e t r y 6nd t h e punch speed. The f l o w i n g o u t o f t h e p r e s s u r e f l u i d u n d e r t h e b l a n k h o l d e r i s p r e v e n t e d by p r e s s i n g t h e p l a t e o f t h e b l a n k h o l d e r a g a i n s t a s e a l i n g r i n g . Water i s m o s t l y used as t h e p r e s s u r e f l u i d , e m u l s i f i e d w i t h m a c h i n i n g o i l f o r c o r r o s i o n reasons.

111 and t h e f l o w r u l e d e r i v e d f r o m i t i s

I f t h e 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 i s v a l i d , ( 2 ) can be r e w r i t t e n i n a f o r m more s u i t a b l e f o r t h e n u m e r i c a l calculation:

P r o v i d i n g t h a t work h a r d e n i n g o c c u r s , t h e e q u i v a l e n t s t r a i n 'p i s d e f i n e d b y t h e i n t e g r a l o f dlp o v e r t h e def o r m a t i o n p a t h o f an e l e m e n t 'p=Jdlp=&J[ ZR+1

R (dc -ds

r

8

1

+

( der-RdEnl

+

( dsO-RdEn)

1

"',5) (

while the hardening c h a r a c t e r i s t i c o f the m a t e r i a l i s given by the e m p i r i c a l f u n c t i o n kf

= A(lp+B)"-C

.

161

The e o m e t r i c a l r e l a t i o n s h i p between t h e l o c a t i o n o f a p o i n f on t h e s t a r t i n g d i s c and t h e c o r r e s p o n d i n g p o s i t i o n on t h e c u p a s a f u n c t i o n of t h e punch d i s p l a c e ment are f o u n d w i t h t h e h e l p o f t h e c o n t i n u i t y c o n d i t i o n ( F i g . 3). The d e f o r m a t i o n i n t h e f l a n g e p a r t o t t h e drawn p a r t o c c u r s u n d e r t h e a c t i o n o f r a d i a l t e n s i l e and t a n g e n t i a l compressive s t r e s s e s . F o l d f o r m a t i o n caused by t h e s e i s p r e v e n t e d by t h e b l a n k h o l d e r . As a c o n s e quence o f t h i s b l a n k h o l d e r p r e s s u r e , a m a c r o - c o n t a c t a r e a w i t h c o n s t a n t s h e e t t h i c k n e s s w i l l form, a t t h e edge o f t h e f l a n g e ( F i g . 2 ra-rbl. A t e a c h d r a w i n g s t a g e n f r o m t h e assumed t h i c k n e s s s t r a i n s en t h e t a n g e n t i a l s t r a i n s €0 c a n b e f o u n d w i t h t h e h e l p o f t h e c o n t i n u i t y e q u a t i o n , and t h u s ben. he0 and Alp can be c a l c u l a t e d . The r e f e r e n c e s t r a i n l p = X A q ~ d f t e r (51 gives w i t h (61 the reference s t r e s s kf. W h i l e ( 3 1 dnd ( 4 ) p r o v i d e t h e d i f f e r e n c e s o f t h e s t r e s s e s [ c r ~ - U r l and ( a n - o r ) , t h e strclsses t ' l e m s e l v e s

Annals of the ClRP Vol. 30/1/1981

201

are o b t a i n e d w i t h t h e h e l p of t h e e q u i l i b r i u m c o n d i tion. This can be numerically integrated using t h e t r a o e z o i d a l r u l e and t h e n w r i t t e n i n t h e form

* <.- /, , I

si

T h e l i m i t i n g r a d i u s i s t h e n r e a c h e d when t h e i n t e g r a l o f don i n t h e r e g i o n ra-rb z o r r e s p o n d s t o t h e e x t e r n a l l y a p p l i e d b l a n k h o l d e r f o r c e F ~ r eJ d u~c e d b y t h e h y d r o static pressure; F

= FN

-

.

~ r ~ -g r ” I . p ~

NH

i n w h i c h S i k is t h e t e n s o r a n d t h e v e c t u r (If t h e n o m i n a l s t r e s s v e l o c i t y a n d v i t h e v e l o c i t y v e c t o r . :n o r d e r t o formulate t h e c h a r a c t e r i s t i c bounddry v a l J e problem, t h e v e l o c i t y f i e l d v i must oe given t o p a r t o f t h e s u r f a c e A, a n d t h e s u r f a c e s t r e s s s i or i t s v e l o c i t y s i t o t h e r e m a i n i n g s u r f a c e A,. Additionally a p r e s s u r e p c h a n g i n g w i t h t i m e p a n d a c t i n g o n A p is v e r y i m p o r t a n t i n t h e p r e s e n t case. Taking i n t o account t h e occurring pressure one aventua l l y o b t a i n s from (12) a f t e r s e v e r a l computational s t e p s and t h e a p p l i c a t i o n o f t h e p r i n c i p l e o f v i r t u a l work t h e f o l l o w i n g s t a b i l i t y f u n c t i o n

< av.

S i n c e t h e b l a n k h o l d e r p r e s s u r e a c t s o n l y up t o r b , t h e r e is n o f r i c t i o n f o r c e i n t h e r e g i o n b e t w e e n r b a n d rc, o n l y d e f o r m a t i o n f o r c e s . After assuming a n a v e r a g e t h i c k n e s s 5, t h e t a n g e n t i a l a n d t h i c k n e s s s t r a i n s arm o b t a i n e d w i t h t h e h e l p o f t h e c o n t i n u i t y c o n d i t i o n . The c h o i c e of t h i c k n e s s ( s ) i + lr e s u l t s f r o m t h e r e q u i r e m e n t t h a t t h e v a l u e ( ~ ~ . s )g iiv e+n ~by t h e e q u i l i b r i u m c o n d i t i o n a g r e e s a s d e l l a s p o s s i b l e w i t h t h e v a l u e c a l c u l a t e d from ( 4 ) . I n t h e r e g i o n s of c u r v a t u r e ( F i g . 2 rc-rd) t h e s h e e t is t a n g e n t i a l l y c o r n p r e s s e d a n d b e n t twice. A c c o r d i n g

Iiik V

av

dV

- I aij V

av

$P 2J

dV

-

p 1%

vjnkdA (131

+

i, IA

vinidA

which is v a l i d f o r i n c o m p r e s s i b l e m a t e r i a l s r e l a t e d t o t h e actual configuration xi. T h e t h e o r y o f t h e p l a s t i c p o t e n t i a l w i t h t h e v n n MISES c r i t e r i o n leads t o t h e m a t e r i a l law f o r r i g i d p l a s t i c bodies with hardening

t o CHUNG a n d SWIFT [ e l t h e i n f l u e n c e of t h e d o u b l e b e n d i n g on t h e f o r c e - d i s p l a c e m e n t c u r v e c a n b e n e g l e c t e d f o r p z / s o > 6.

The d e t e r m i n a t i o n of ( B I i . 1 a n d t h u s [ r ) i f l is d o n e w i t h t h e h e l p of t h e c o n t i n u i t y c o n d i t i o n . The s t r a i n s c a l c u l a t e d from t h i s a r e r e g a r d e d a s b e i n g s u f f i c i e n t l y a c c u r a t e i f t h e value [ar.s)i+l from ( 3 ) and (4) a g r e e s w e l l with t h a t c a l c u l a t e d from t h e equilibrium condition

+

)1

I [p+ars

lpde

.

which forTs t h e connection between t h g shape change v e l o c i t y c i j and t h e s t r e s s v e l o c i t y akk. n i j is t h e e x t e r n a l u n i t normal and h a p o s i t i v e s c a l a r f u n c t i o n which c h a r a c t e r i s e s t h e hardening behaviour and a l s o t a k e s i n t o a c c o u n t t h e p r e d e f o r m a t i o n . h is d e t e r m i n e d w i t h t h e h e l p of t h e s t r e s s - s t r a i n c u r v e s k f ( q ) f r o m (6) a n d t h e y i e l d l o c u s c u r v e s $ ( a i j ) from ( 1 1

If o n e p u t s ( 1 4 1 i n ( 1 3 1 , t h e n a f t e r a p p l y i n g t h e i n t e g r a l c r i t e r i o n o f GAUSS o n e o b t a i n s t h e f o l l o w i n g s t a b i l i t y condition

On t h e o c c u r r e n c e o f a b u l g e , t h e f r i c t i o n f o r c e i s i n c r e a s e d b y t h e h y d r o s t a t i c p r e s s u r e . W h e t h e r or n o t a b u l g e f o r m s can b e checked by u s i n g t h e c r i t e r i o n [arsli.

2rr [ r I i . s i n ( e l i < n . p .

[r;-(r)2]

(101

A s s u m i n g t h a t t h e b u l g e i s c i r c u l a r , t h e c u r v a t u r e is c a l c u l a t e d up t o a g i v e n l i m i t i n g c o n t a c t r a d i u s r S u b s e q u e n t l y t h e f o r c e e q u i l i b r i u m i s c h e c k e d i n g’ t h e d i r e c t i o n of t h e punch a x i s , a n d i f t h i s c o n d i t i o n is n o t f u l f i l l e d a l s o t h e b u l g e r a d i u s c o r r e c t e d . A s a c o n s e q u e n c e of t h e f r i c t i o n b e t w e e n t h e w o r k p i e c e and punch, t h e requirement (degIj

0

(111

must a l s o be s a t i s f i e d . The s h a p e o f t h e b u l g e d e p e n d s o n t h e p r e s s u r e , t h e p o s i t i o n o f t h e p u n c h , t h e t o o l g e o m e t r y a n d on t h e m a t e r i a l p a r a m e t e r s . F i g . 4 s h o w s b u l g e s h a p e s of a s t a b i l i s e d a u s t e n i t i c steel, which were formed i n t h e drawing gap with a l i n e a r l y increasing pressure. 4.

Formability l i m i t

The f o r m a b i l i t y o f t h e w o r k - p i e c e i n t h e Hydromec process is s u b j e c t e d t o l i m i t s . Provided t h a t t h e c u s h i o n p r e s s u r e a l l o w s t h e s h e e t t o l i e on t h e p u n c h , a neck which i n i t i a t e s f r a c t u r e w i l l occur a t t h e end o f t h e d r a w i n g p l a t e c u r v a t u r e or o f t h e b u l g e . T h e neck w i l l be prevented during forming a s long as t h e d i f f e r e n c e between t h e d i s s i p a t i v e and e x t e r n a l work is p o s i t i v e a n d f i n i t e . T h e w o r k d i f f e r e n c e r e l a t e d t o t h e reference configuration X i can be e x p l i c i t l y w r i t t e n f o r s u f f i c i e n t l y smalI t i m e s T a s

208

which can a l s o be w r i t t e n i n t h e form

T h e l i m i t i n g t h i c k n e s s of t h e c r i t i c a l z o n e , c a l c u l a t ed with t h e help of (17) and taking i n t o account t h e predeforrnation and t h e inhomogeneous d e f o r m a t i o n d i s t r i b u t i o n of t h e b u l g e d e t e r m i n e s t h e l i m i t i n g d r a w i n g ratio

0’

31

RO

r C

f o r a g i v e n geometry, f r i c t i o n c o n d i t i o n and deforrnat i o n c o n d i t i o n ( F i g . 5 ) . T h e m a g n i t u d e of 6’ is d e c i s i v e l y i n f l u e n c e d b y t h e c u s h i o n p r e s s u r e or t h e v a r i a t i o n of p r e s s u r e w i t h d i s p l a c e m e n t . T h e p r e s e n t c a l c u l a t i o n s t a r t e d w i t h a minimum p r e s s u r e w h i c h , from s t a r t i n g t o draw t h e d i s c t o t h e t e c h n o l o g i c a l l y necessary remaining flange, increased l i n e a r l y t o t h e g i v e n maximum p r e s s u r e . T h e m e a s u r e d v a l u e s o f 0. a g r e e w e l l w i t h t h o s e d e t e r m i n e d t h e o r e t i c a l l y . On t h e o t h e r h a n d , t h e l i m i t i n g r a t i o s w h i c h were c a l c u l a t e d d i t h o u t t a k i n g i n t o a c c o u n t t h e inhomogeneous deformat i o n d i s t r i b u t i o n i n t h e b u l g e were c l e a r l y t o o l o w .

5 . ;urnr,ary

5 . References

As w e i l a s the p r o i s s s - and natsria:-technical

ques-

tions. an i.nportart parameter infiuencing the hydromechanical C E B P drawing of conical hollow bodies is t n e cushion D r E s s u r e . The process Y a s analysed to q L a n t i f y this infiuence and the s t r e s s e s and deforqac i on s u c c u r r i n g t h r o u g h o u t t h e a e e p - draw i n g p r c c e s s determinsd 3 y iteraticn. T h e determination o f the formability limit is done 3 y applying the energetic stability critericn ta
Fig. 1

Hydronec machine hydraulics

Fie. 2

with

a diagrann

Regions of

3

of

L11

FlILL,

i?.

J . M~cl.P7ya.Soiids 4 ( 1 9 5 G 1 2 4 7

[21

gILL,

R.

[31

qILL.

R.

J. Mo-h.Phys.So:ids 3 . NEch.PhyS.Solids

[41

;ILL,

R.

[51

RAUTCR, A.

D.M.

[61

dOD,

[71

WISSJER,

[O]

;-!UHG, S.Y.and SWIFT, P.W.

J.

5 (19561 1 5 (19571153

J. Mech.Phys.Solids E 1 1 9 5 7 ) 1 Diss. ETh-ZIrich 6 2 2 1 , 1 9 7 8 J. Msch.Eng.Sc. 6 ( 1 9 6 4 ) 116 Sericht Institut f i r Umformtecbnik ETH-ZGrish 1980 Proc.1nst.mech.Engrs.London (:351) 1 9 9

165

the

conical c u p

209

0

100 "0

'k

c

1% I

Fig. 3

Geometrical relationships

Fig. 5

210

Influence o f t h e pressure path on t h e limiting ratio -calculated with inhomogeneous deformation i n bulge - - - c e i c u l a t e d f o r t h e c o n t a c t point o n l y