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580 BYotrN .; B, YOON e t a l

s e v e r it y o f t h e f o r m i n g p r o b l e m , i n c h o o s i n g t h e m o s t e c o n o m i c a l m a t e r i a l a n d i n c h o o s i n gt h e s e q u e n c e o f o p e r a t i o n s r e q u i r e d t o f a b r i c a t e t h e p a r t .

O n e o f t h e f i rs t c o m p r e h e n s i v e s t u d i es o f c u p - f o r m i n g w a s t h a t o f C h u n g a n d S w i ft [1 ] .T h e i r a n a l y s e s f o c u se d m a i n l y o n t h e r a d i a l d r a w i n g p r o c e ss , a n d t h e m a t e r i a l s u s e d w e rec o n s i d e r e d t o b e i s o t r o p i c . A l t h o u g h t h e r a d i a l d r a w i n g p r o c e s s w a s s u c c es s fu l ly a n a l y s e d ,t h e a n a l y s i s o f s t r e tc h f o r m i n g o v e r t h e p u n c h h e a d r e m a i n e d u n s o l v e d . In t e n s i v e a n a l y s e s o fs h e e t m e t a l f o r m i n g h a v e b e e n p r e s e n t e d b y m a n y a u t h o r s . E l a s t i c - p l a s t ic [ 2 ] ,e l a s t i c v i s cop las t i c [3 ] o r r ig id -p las t i c [ -4 , 5 ] app roac hes have been us ed s ucces s fu l ly fo rs h e e t m e t a l d e f o r m a t i o n a n a l y s i s . T h e r i g i d - p l a s t i c f o r m u l a t i o n s , i n w h i c h t h e e l a s t i cd e f o r m a t i o n i s a s s u m e d t o b e n e g l ig i b le c o m p a r e d t o t h e l a r g e p la s t ic s t r a in , h a v e g a i n e d aw ide r ange o f app l i ca t ion s des p i t e the f ac t tha t the m e th od f a i l s to p r ed ic t the s t r e s s h i s to ryw h e n e v e r e l a s t i c l o a d i n g o r u n l o a d i n g i s e n c o u n t e r e d .

F r i c t i o n i s a v e r y i m p o r t a n t p a r a m e t e r i n m e t a l f o r m i n g p r o c e ss e s; i t h a s g r e a t i n f lu e n c eo n t h e m e t a l f lo w a n d t h e n e c e s s a r y f o r m i n g l o a d . T h e e x t e n d e d u s e o f m a t h e m a t i c a l a n dn u m e r i c a l a n a l y s i s to s i m u l a t e m e t a l f l o w a n d p r e d i c t f o r m i n g f o rc e s a n d l o c a l p r e s su r ed i s t r i b u t i o n s t h u s r e q u ir e s d e t a i l e d k n o w l e d g e a b o u t t h e f r i c ti o n a l s tr e ss e s a t t h et o o l - w o r k p i e c e i n t er f ac e . T h e s u c c e s s o f t h e o r e t i c a l o r n u m e r i c a l s i m u l a t i o n s o f m e t a lf o r m i n g p r o c e s s e s d e p e n d s o n t h e a b i l i t y t o d e t e r m i n e p r e c i s e l y t h e f r i c t i o n a l p a r a m e t e r sinvo lved .

In th i s r e s ea rch , a t t e n t io n i s f ocus ed on s hee t s t r e t ch ing , w h ich i s m a in ly u s ed to a s s ess thes t r e t c h a b i l i ty o f sh e e t m e t a l s . A c i r c u l a r m e t a l b l a n k i s c l a m p e d f i r m l y a l o n g t h e p e r i p h e r ya n d i s s t r e t c h e d b y a r i g i d h e m i s p h e r i c a l p u n c h . T h e d e p t h o f t h e d e f o r m e d s h e e t w h e n i tf r a c t u re s m a y b e t a k e n a s a m e a s u r e o f d u c t i l it y . I t is a c o m p l i c a t e d p r o b l e m f r o m t h e p o i n to f v ie w o f d e f o r m a t i o n a n a l y s i s, t h e m o s t n o t a b l e d i f f i c u lt y b e i n g t h a t o f q u a n t i f y i n g t h ef r i c ti o n o v e r t h e p u n c h h e a d . A n e l a s t i c - p l a s t i c w o r k - h a r d e n i n g f i n it e e l e m e n t a n a l y s i si n c o r p o r a t i n g a n i s o t r o p i c p r o p e r t i e s o f s h e e t m e t a l s i s v e r if ie d b y c o n d u c t i n g s h e e ts t r e t ch ing exper imen t s . P r e l imina ry r e s u l t s have been p res en ted a t a r ecen t con fe r ence [ -6 ] .In th i s pape r , r e s u l t s a r e d i s cus s ed in de ta i l f o r a cas e tha t u s es the Cou lomb f r i c t ion mode la l o n g t h e p u n c h - s h e e t i n te r fa c e . F u t u r e w o r k w il l i n v o l v e d e t a i l e d u n d e r s t a n d i n g o ff r i c ti o n a l v a r i a ti o n s a l o n g t h e i n t e rf a c e u s i n g a t r a n s p a r e n t p u n c h t e c h n i q u e s i m i l a r to t h a ti n v o l v i n g t he d r a w i n g d i e d e v e l o p e d b y o n e o f t h e p r e s e n t a u t h o r s [ 7 ].

E X P E R I M E N T A L P R O C E D U R E S A N D M A T E R I A L SSheet s t re tching apparatus

A r e c t a n g u l a r d i e se t w i t h fo u r p o s t s f o r m s t h e m a i n p a r t o f t h e s h e e t s t r e t c h i n g a p p a r a t u sa s s h o w n i n F i g . 1. T h e d i e se t w a~ s u p p l i e d b y D a n l e y M a c h i n e C o r p o r a t i o n . P o s t s a n db u s h e s w e r e m a d e f r o m g o o d q u a l i t y s t e e l , h a r d e n e d , c h r o m e p l a t e d a n d g r o u n d t o c l o s et o l e r a n c e s . T h i s a r r a n g e m e n t g i v e s a n e x c e p t i o n a l l y s m o o t h h a r d - w e a r i n g s u r f a c e t h a tr e s is t s co r ro s ion , l e s s ens fr i c t ion and m a in ta in s a c lo s e w o rk in g f it . The d ie s e t w as bo l t ed tot h e b a s e o f a m i c r o p r o c e s s o r - c o n t r o l l e d l n s t r o n t e s t in g m a c h i n e . T h e l n s t r o n i s e q u i p p e dw i t h a l o a d c e l l o f 1 5 0 k N . D e f l e c t i o n c a n e a s i l y b e m o n i t o r e d u s i n g t h e c r o ss h e a d s p e ed .L o a d - d e f l e c t i o n v a ri a t io n s c a n b e m o n i t o r e d a n d r e c o r de d u s i n g t h e H P X Y c h a r tr eco rde r .

A h e m i s p h e r i c a l p u n c h o f 5 0.8 m m r a d i u s a n d a d i e o f o u t s i d e d i a m e t e r 2 06 m m a n di n s id e d i a m e t e r 1 08 m m w e r e m a d e f r o m h i g h - c a r b o n s t ee l. T h e p u n c h w a s f a st e n e d to t h ep u n c h h o l d e r a n d t h e d i e w a s f a s t e n e d t o t h e d i e h o l d e r o f t h e f o u r p o s t d i e se t. A s e r r a te dc i r c u l a r g r o o v e w a s m a d e o n t h e d i e fa c e fo r h o l d i n g t h e b l a n k f i rm l y . A c l o s e - u p v i ew o f th ed i e, b l a n k a n d p u n c h a r r a n g e m e n t is s h o w n i n F i g . 2 . N o l u b r i c a n t w a s u s e d i n a n y o f t h ee x p e r i m e n t s p r e s e n t e d i n t h i s p a p e r . T h e s u r f a c e s w e re n o t c h e m i c a l l y cl e a n e d .Materia l propert ies

A n nea led m i ld s tee l s hee t s o f 0 .762 mm th icknes s , a lum inu m ( I 100-0) s hee t s o f 1 .016 m mth icknes s an d b ra s s (70 :30 ) s hee t s o f 0 .83 m m th icknes s w ere chos en as w ork ma te r i a l s .S h e e t t e n s il e s p e c im e n s o f g a g e l e n g t h 1 65 m m a n d w i d t h 1 2 . 7 m m w e r e s u b j e c t e d t os t a n d a r d t e n si l e t e s ts u n d e r c o n t i n u o u s l o a d i n g u p t o f a i lu r e . T h e y i e ld s tr e n g t h w a sd e t e r m i n e d u s in g t h e 0 . 2 o f fs e t m e t h o d . T r u e s t r e s s -s t r a i n c u rv e s w e r e a p p r o x i m a t e d b y

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Shee t s t re tch ing : a theore t ica l- -exper imenta l com parison 581

FIG. 1 . O vera l l exper im enta l a rrang eme nt o f sheet s t re tch ing .

FIG. 2 . Close up v iew of d ie b lank and punch a rran geme nt .

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Sheet stretching: a theoretical-expe rimen talcomparison 583p o w e r l a w w o r k h a r d e n i n g a s s h o w n b e l o w :

m i l d s te e l: 6 = 5 4 3 . 2 g '2 7 8 ( M P a )a l u m i n u m : 6 = 1 96 .8 g '2 39 6 ( M P a )

b r a s s : ~ = 7 9 7 . 9 g '4 82 5 ( M P a ) .(1)

T h e s t r a i n r a t i o , u s u a l l y r e p r e s e n t e d b y t h e r - v a l u e , is d e f i n e d a s t h e r a t i o o f w i d t h t ot h i c k n e s s s t r a in s . S i n c e t h e t h i c k n e s s s t r a i n c a n n o t b e a c c u r a t e l y m e a s u r e d i n a t h i n s h e e t ,t h i c k n e s s s t ra i n w a s d e d u c e d , u s i n g t h e v o l u m e c o n s t a n c y , f r o m t h e l e n g t h a n d w i d t h s t r a in s .T h e r v a l u e s u s u a l l y v a r y w i t h t h e t e s t d i r e c t i o n , qS, a n d i t i s c o m m o n t o c h a r a c t e r i z e am a t e r i a l b y a n a v e r a g e r v a l u e , ~, w h e r e

ro + 2r45 + rgof = 4 ' (2)t he s ub s c r i p t r e fe r r i n g t o t h e a ng l e ~b ( i f ~b - - 0 , it m e ans t he r o l l i ng d i r ec t i on ) . F ro m t he t en s i l et e s t s , f v a l u e s w e r e c a l c u l a t e d f o r t h e m a t e r i a l s c o n s i d e r e d a s s h o w n b e l o w :

fo r m i l d s t ee l , f= 1 .069f o r a lu m i n u m , ~ = 0 . 9 2 3f o r b r a s s, f = 0 . 9 1 5 .

E V A L U A T I O N O F O V E R A LL C O E F F I C I E N T O F F R I C T IO N F R O M E X P E R IM E N T ST h e m o d e l d e s c r i b e d b y G h o s h [ 8 ] w a s a d o p t e d t o e v a l u a t e th e o v e r a l l c o e f fi c ie n t o f

f r i c t i o n i n s t r e t c h i n g f r o m l o a d - d e f l e c t i o n p l o t s . T h e r e a r e t h r e e a s s u m p t i o n s . F i r s t , t h a ti n t e rf a c e p r e ss u r e i s u n i f o r m a t a n y s ta g e o f d e f o r m a t i o n . S e c o n d , t h a t A m o n t o n ' s l a w h o ld s :z = p p , w h e r e z is t h e s h e a r s t re s s , p i s t h e c o e f f i c i e n t o f f r i c t i o n a n d p i s t h e n o r m a l p r e s s u r e .T h i r d , t h a t t h e i n f le c t io n p o i n t o n t h e l o a d ~ l e f l e c t i o n p l o t c o r r e s p o n d s t o t h e a t t a i n m e n t o fm a x i m u m p r e s su r e . T h e s e f r ic t io n a l b o u n d a r y c o n d i t i o n s w e r e u s e d a s in p u t s t o t h e f i ni tee l e m e n t c o d e a s d e s c r i b e d i n t h e n e x t s e c t i o n .

F i g u r e 3 s h o w s a d e f o r m i n g s h e e t in c o n t a c t w i th t h e p u n c h w i t h u n i f o r m n o r m a l p r e s s u r ep , p u n c h l o a d P , p u n c h r a d i u s p , ra d i u s o f c o n t a c t b o u n d a r y r c a n d a f r ic t i o n a l f o r c e p p e ru n i t a r e a a t t h e c o n t a c t s u rf a c e. T h e v e r t ic a l c o m p o n e n t , d V , o f th e f o r c e a c t in g o n a na n n u l a r e l e m e n t o f t h e s h e e t w h i c h s u b t e n d s a n a n g l e d O a t t h e p u n c h c e n t e r is g iv e n b y

d V = (pcosO + pp s in O)(2~p 2 s in OdO). (3)I n t e g r a t i o n o f d V o v e r t h e e n t i r e p u n c h c o n t a c t a r e a ( 0 = O c) g i v e s t h e v e r t i c a l f o r c e , V , a s

V = 2 ~ p 2 p s in O c o s O + s in 2 0)d0 , (4)w h i c h e q u a l s t h e p u n c h l o a d , P . I n t e g r a t i o n o f e q u a t i o n ( 4) l e a ds t o

p = ~p2 p [sin 2 0c + p(0~ - s in 0~ co s 0 c)]. (5)R e c o g n i z i n g t h a t r c = p s i n 0 ~, t h e d i f f e r e n t i a t i o n o f P w i t h r e s p e c t t o r c a n d s u b s t i t u t i o n o fd p / d r ~ ) = 0 c o r r e s p o n d i n g t o t h e p r e s s u r e m a x i m u m y i e ld s

d P / d r o ) ] , . = 2~ pp s i n 0o (1 + ~ t an 0o ), (6)w h e r e r c = r o a n d 0~ = 0 o a t m a x i m u m p r e s s u r e . T h e r e f o r e , a t r = r o , w e o b t a i n

[ d P / d r ~ ) l 2 r 1 + p t a n 0 1A , o = ~ [ _ s i n Oo + p { O o / s i n O o ) _ c o s O o . (7)

R e p l a c i n g s in 0 o b y f r o ~ P ) a n d c o s O b y p x / - ~ - r ~ / p ) , e q u a t i o n ( 7) w o u l d n o w l e a d t o t h ef o l lo w i n g e x p r e s s i o n f o r d e t e r m i n i n g p :

2 - X r o- X { p E / r o s i n - l r o / p ) - ~ } - [ 2 r o /x / p 2 - r g ) ] ' ( 8 )

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584 BYOUNG B YOON e t a lPunch Load, P

dV = Ve r t i c a l C o m p o n e n t

/ Fr ict ion Force / un it areaElementa l Su rface rea HP

2Kp ~e ae

FIG 3 A schematic representation of the sheet stretching process

w here X = [ d P / d r c ) / P ] , o , r c = p s i n 0 c, r o = p s i n 0 c a n d 0 , = 0 o a t m a x i m u m p r e s su r e . T h ee x p e r i m e n t a l l o a d - d e f l e c t i o n c u r v e s s h o w n i n F i g s 5 - 7 h a v e b e e n u s e d t o e v a l u a t e i ne q u a t i o n (8 ). T h e e x p e r i m e n t a l l y c a l c u l a t e d o v e r a l l c o e f f ic i e n t o f f ri c t i o n o f e a c h m a t e r i a l a tt h e p u n c h - s h e e t i n t e r f a c e i s s h o w n b e l o w :

f o r a l u m i n u m , / ~ = 0 . 2 3 5f o r m i l d s te e l, = 0 . 1 9 1for bras s , /~ = 0 .293.

T H E O R E T I C A L A N D C O M P U T A T I O N A L P R O C E D U R E ST h e a n a l y s e s p r e s e n t e d h e r e a r e b a s e d o n t h e fi n it e e l e m e n t p r o g r a m E P A F R I C I- 9] ,

t o g e t h e r w i t h c e r t a i n m o d i f i c a t i o n s t o i t w h i c h a l l o w t h e s o l u t i o n o f s h e e t s t re t c h i n g . T h et h e o r e t i c a l b a s is o f th e p r o g r a m i s a r a t e f o r m u l a t i o n o f i s o t r o p i c e l a s t ic - p l a s t i c m a t e r i a lb e h a v i o r b y e m p l o y i n g th e J a u m a n n s t re s s r a te s t o e n s u r e m a t e r i a l f r a m e i nd i f fe r e n c e. T h eo n s e t o f p l a s t ic y i e l d i n g i s c h a r a c t e r i z e d b y t h e v o n M i s e s c ri t e r i o n a n d t h e P r a n d t l - R e u s se q u a t i o n s. T h e p r o g r a m u s es a n i n c r e m e n t a l f in i te e le m e n t a n a l y s is e m p l o y i n g c o n s t a n ts t r a in t r i ang les .

O n e s p e c i a l f e a t u r e o f t h i s p r o g r a m i s t h e c o n s t r u c t i o n o f a n i n c r e m e n t a l c o n s t i t u t iv er e l a t i o n f o r f r i c t i o n c o n t a c t i n a n a l o g y w i t h t h e p l a s t i c i t y t h e o r y [ 1 0 ] . T h i s f e a t u r ea c c o m m o d a t e s a p p r o p r i a t e v a r i a t i o n s i n t h e i n t e r fa c i a l c o n d i t i o n s , s u c h a s c o n t a c t p r e s s u rea n d t h e h a r d e n i n g o r s o f t e n i n g r e s p o n s e i n t h e t r a n s i t i o n f r o m s t i c k i n g t o s l i d i n g m o t i o n .T h i s i s a n a t t r a c t i v e f e a t u r e , b e c a u s e t h e b r o a d g o a l o f t h i s r e se a r c h i s to g a i n a b e t t e ru n d e r s t a n d i n g o f f r i c t io n a l o n g t h e i n t e r f a c e i n s h e e t f o r m i n g p r o c e s se s :

T h e o r i g i n a l p r o g r a m h a s b e e n m o d i f i e d to i n c o r p o r a t e a n i s o t r o p i c p r o p e r t ie s o f s h e e tm e t a l s a n d G r e e n s t r e s s ra t e s. T h e y i e l d c r i t e r i o n p r o p o s e d b y H i l l h a s b e en u s e d t oc h a r a c t e r i z e t h e a n i s o t r o p i c e f fe c t r l 1 ]. i t i s k n o w n t h a t t h e u s e o f t h e m o s t c o m m o no b j e c t iv e s t r es s r a t e , t h e J a u m a n n r a t e , in t h e g e n e r a l i z a t i o n o f t h e i n f in i t e s i m a l t h e o r y l e a d st o a n i n s t a b i l i ty i n t h e e v o l u t i o n o f th e y i e l d s u r fa c e w h i c h i s e x p l a i n e d o n t h e b a s i s o f t h ee l a s ti c s o l u t i o n [ 1 2 ]. A n y i n s t a b i l i ty t h a t a p p e a r s i n t h e e l a s ti c s o l u t i o n o f t h e s i m p l e s h e a rp r o b l e m c a n a l s o a p p e a r i n t h e s o l u t i o n o f a n e l a s t o - p l a s ti c p r o b l e m i n w h i c h k i n e m a t i ch a r d e n i n g i s u s e d . I n o r d e r t o a v o i d t h i s p r o b l e m , G r e e n s t r e s s r a t e s [ 1 3 ] b a s e d o n n o t i o n s o fi n v a r i a n c e u n d e r s u p e r i m p o s e d r i g i d b o d y m o t i o n h a v e b e e n a d o p t e d .

A m a t e r i a l i s sa i d t o b e h y p o e l a s t i c i f t h e c o m p o n e n t s o f st r es s r a t e a r e l i n e a r f u n c t i o n s o ft h e c o m p o n e n t s o f t h e r a t e o f d e f o r m a t i o n [ 14 -1:a V C i j k l D i j (9)

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Sheet stretching: a theoretical-expe rimen tal omp arison 585w h e r e t rv r e p r e s e n t s a n y m e a s u r e o f th e c h a n g e o f C a u c h y s tr e ss t h a t i s u n c h a n g e d a p a r tf r o m o r i e n t a t io n u n d e r s u p e r p o s e d r i gi d b o d y m o t i o n s. T h e r a t e o f d e f o r m a t i o n t e n s o r D i j isa w e l l d e f i n e d q u a n t i ty ; i t v a n i s h e s w h e n t h e b o d y p e r f o r m s a r i gi d b o d y m o t i o n . H o w e v e r ,f o r a s tr e s se d b o d y p e r f o r m i n g r ig i d b o d y r o t a t i o n , n e i t h e r t h e t i m e d e r i v a t iv e O t r i j / c ~ tn o r t h em a t e r i a l d e r i v a t iv e D a i J D t v a n i s h e s i d e n t i c a l l y . A r i g i d b o d y r o t a t i o n c h a n g e s t h e s t r e s st e n s o r , e v e n t h o u g h t h e s t a te o f s t re s s in t h e b o d y r e m a i n s u n c h a n g e d . T h u s n e i t h e r c~ t r i j / c~ tn o r D t r ~ j / D t c a n s e r v e a s a n a p p r o p r i a t e s t r e s s r a t e m e a s u r e t o b e r e l a t e d s i m p l y t o t h ed e f o r m a t i o n r a te D i ~ T h e s t r es s ra t e t e n s o r m u s t c o n t a i n a r o t a r y t e r m t h a t c o m p e n s a t e s f o rt h e f a c t t h a t t h e s t r e ss c o m p o n e n t s w i t h r e s p e c t t o a fi x e d c o o r d i n a t e s y s t e m x~ c h a n g e , e v e nw h e n t h e r e is n o c h a n g e i n t h e s tr e ss c o m p o n e n t s w i t h r e s p e c t to a c o o r d i n a t e s y s t e m x ) t h a tp a r t i c ip a t e s i n t h e i n s t a n t a n e o u s r o t a t i o n o f t h e n e i g h b o r h o o d o f th e c o n s i d e r e d p a r ti c le .T h i s i s w h y a n e w r a t e d e r i v a t i v e s u c h a s t rv i s i n t r o d u c e d . T h e d e s i r e d s t r e s s r a t e d e r i v a t i v em u s t b e in v a r i a n t w i th r e s p e c t t o r i gi d b o d y r o t a t i o n , b u t t h is r e q u i r e m e n t d e m a n d s n ou n i q u e a n s w e r .

G r e e n s d e f i n i t i o n , d e n o t e d b y a v , is o b t a i n e d b y d i f f e r e n t i a t i n g t h e e x p r e s s i o n f o r cr i nt e rm s o f o r* , g i v i ng

~rv = /r + ~r~ - f~a = R~r* R t , (10)wh ere a * i s a ro t a t ed C a uc hy s t r e s s , a* = R t c rR , f~ i s a r a t e o f ro t a t i on t ens o r , f~ = I~ R , an d Ris a n o r t h o g o n a l r o t a t i o n t e n s o r t h a t c o m e s f r o m t h e p o l a r d e c o m p o s i t i o n o f t h ed e f o r m a t i o n g r a d ie n t d x ~ / O X j W e s e e t h a t t h i s s t r es s r a t e r e p r e s e n t s t h e d e r i v a t i v e o f a ni n v a r i a n t m e a s u r e o f s tr e s s, a * , w h o s e p r i n c i p a l i n v a r i a n t s h a v e t h e s a m e v a l u e s a s th o s e f o rt he C au ch y s t re s s . Thu s , i f ~rv is z e r o , P r a g e r s c o n d i t i o n t h a t t h e i n v a r i a n t s o f C a u c h y s tr e ssb e s t a t i o n a r y i s s a t is f ie d . T h e a d v a n t a g e o f t h i s r a t e m e a s u r e o v e r J a u m a n n s is t h a t i t isd e r i v a b l e f r o m a p h y s i c a l l y m e a n i n g f u l s t re s s , a * , a n d i t is t r u l y a m e a s u r e o f th e r a t e o fc h a n g e o f s t re s s in t h e d e f o r m e d c o n f i g u r a t i o n . T h i s f o r m s t h e t h e o r e t i c a l b a s i s f o r t h e f i n it ee l e m e n t m e t h o d .

C o m p u t a t i o n s w e r e ca r r ie d o u t u n t il p u n c h d e p t h r e a c h e d 4 0 m m a f te r 2 31 t i m ei n c r e m e n t s . T h e p u n c h s p e e d w a s 1 m m p e r s e c o n d a n d t h e d a t a w e r e o b t a i n e d a t e a c h 5 m mp u n c h t r av e l. M e s h c o n f i g u r a t i o n s a re s h o w n i n F ig . 4 f o r e a c h 5 m m p u n c h t ra v e l. T h e t o t a lC P U t im e c o n s u m e d f o r 52 7 n o d e s a n d 2 3 2 4 - C S T e l e m e n ts w a s a b o u t 2 0 m in . T h e p r o g r a mw a s r u n o n a n I B M 3 0 9 0 -4 0 0 c o m p u t e r .

D I S C U S S I O N O F R E S U L T SD u r i n g t h e d e f o r m a t i o n p r o c e s s w e a s s u m e t h a t t h e i n t e r f a c e p r e s s u re i s u n i f o r m , t h a t t h e

c o e f f i c ie n t o f f r i c t i o n i s c o n s t a n t a n d t h a t t h e i n f l e c t io n p o i n t o n t h e l o a d ~ t e f l e c t i o n p l o tc o r r e s p o n d s t o t h e a t t a i n m e n t o f m a x i m u m p r e s su r e . T h e a v e r a g e v a l u e o f th e c o e f fi c ie n t o f

FIG. 4. M esh configuration at d ifferent stages of shee t stretching.

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S h e e t s t r e tc h i n g : a t h e o r e t i c a l e x p e r i m e n t a l c o m p a r i s o n 5 8 7

~ o

g

P u n c h T r a v e l ( ra m )

Experiment /

: m m

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( b ) T h e o r yF i G . 8 . R a d i a l t h i c k n e s s v a r i a t i o n f o r s te e l .

- 1 0 r a m-m- 15 mm--o- 20 mm- s - 2 5 r a m- o - 30 m m

8 0

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588 BYOUNG B YOON e t a l

The overall coefficient of friction evaluated from experiments represents only an estimateof the average value. It is quite likely that the friction will change from point to point on thepunch-sheet interface and also as a function of deformation. Such variation would bedifficult to determine and if determined would change from one application to another.

Experimentally and theoretically obtained punch-load deflection plots for mild steelaluminum and brass specimens are shown in Figs 5-7. Experimental and theoretical valuesare favorably matched. The small differences between experiment and theory shown in Figs5-7 are mainly due to limited element numbers and stepwise contact conditions in numericalcalculations. The maximum punch depth in this case is about 35 ram.

The experimental measurements of thickness variations for each incremental stretching ofmild steel aluminum and brass specimens over a hemispherical punch are shown in Figures8-10. As the punch travels the friction forces increasingly retard the metal movement overthe punch head thus causing less deformat ion there and more in the unsupported areas.Thus the strain peak gradually shifts towards the edge of the hemispherical dome.

The thickness variations indicate that the stress and strain localization zones are severe atlater stages in stretching. This implies that at a certain stage the sheet will have a surfacedepression or a neck. All specimens--steel aluminum and brass- -have been successfully

1.1

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\ I . 1 m m~ - ~ - 1 5 m m\ / 20 ~r~

- ~ 2 5 m m- o - 3 0 m m

2 0 4 0 6 0 8 0R a d i a l C o o r d i n a t e ( r a m )

( a ) E ~ p e r l m e n t

~ 1 0 m m~ - 1 5 m m

FIG 9 Radial thickness variation for alumin um

- o - 2 0 m r n= - 2 5 r a m

- c - 3 0 m mi i j

2 0 4 0 6 0 8 0R a d i a l C o o r d i n a t e r a m )

( b ) T h e o r y

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Sheet s t re t ch ing: a t heore t i ca l expe r ime nt a l com par i s on 589

0 . 9

0 . 8

0 . 7

0 . 6

0 . 9

0 8

0 . 7

. -o- 20 m m

. -B- 25 m m- o - 3 0 m m

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R a d i a l C o o r d i n a t e ( r a m )( a ) E x p e r i m e n t

0 . 6

~ -' - lO, , , r, ,t-t3:].~- '-~ -m- 15 m m

o . - 20 r n m4 - 2 5 m m 0 - 30 m m

i i m2 0 4 0 6 0 8 0

R a d i a l C o o r d l n a t a ( r a m )( b ) T h e o r y

FIG t0 Radia l t h i cknes s var ia t ion f or bras s .

s tretched with no fracture unt i l the f inal s tage for 35 mm punc h travel. T hen we can expectthat as s tretching proceeds thinning at the root of the neck wil l be enhanced and w il l occurmu ch faster than the deform ation o f the rest of the sheet ow ing to the stress and s trainlocal izat ion. Init ia l ly cold rol led aluminu m sheet was used in our experiments but itss tretchabil ity was l imited by i ts prior cold w ork. Alum inum sheets were annealed to im proveducti li ty . Hen ce the m easured V values we re clo se to unity .

Fro m Figs 8-1 0 w e can see that the overal l shapes are wel l matched. H owe ver the Contactregions along the pu nch -shee t an d die-she et interfaces are not wel l matched. This suggeststhat our experimen tal ly calculated overal l coeff ic ient of fr ict ion is not g oo d eno ugh to m odelthe real forming processes . I t seems that the actual value o f the coeff ic ient of fr ict ion alongthe punch-sheet interface is somewhat less than the prescribed value in the analys is asobtained from experim ents This s i tuat ion is reversed in the contact region alon g thedie-sh eet interface. Th is mean s that the coeff ic ient of fr ict ion varies from point to point andwe need a better understanding of such local variat ions in frict ion along the punch-sheetinterface.

Althou gh the theoret ical results com pare favorably w ith the experimental results i t hasbeen realized that a more realistic frictional variation along the punc.h-sheet interface isneeded. In fact research is in progress to invest igate such frict ional boun dary cond it ions

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590 BYOUNG B. YOON et al.u s i n g a n e w t r a n s p a r e n t p u n c h t e c h n i q u e . I n t h is e x p e r i m e n t a l t e c h n i q u e , t h e a s p e r it i e s o fs h e e t m e t a l a l o n g t h e p u n c h - s h e e t i n t e rf a c e a r e o b s e r v e d a n d r e c o r d e d . A n e w f r ic t i o n m o d e lw i ll b e d e v e l o p e d o n t h e b a s i s o f t h e m i c r o s c o p i c a s p e r i t y a n a l y s is o f t h e p u n c h - s h e e ti n t e rf a c e . S u c h a c c u r a t e d e s c r i p t i o n o f f r i c t i o n a l v a r i a t i o n s a l o n g t h e i n t e r fa c e w i l l f u r t h e re n h a n c e t h e p r e d i c t i v e c a p a b i l i t i e s of t h e n u m e r i c a l m e t h o d s .

C O N C L U S I O N SE x p e r i m e n t s a n d t h e o r e t i c a l c o m p u t a t i o n s o n s h e et s tr e t c h in g h a v e b e en p e r f o r m e d a n d

t h e re s u l ts c o m p a r e d . E x p e r i m e n t s h a v e b e e n c o n d u c t e d o n c o m m e r c i a l m a t e r i a l s s u c h asm i l d s te e l, a l u m i n u m a n d b r a s s w i t h a h e m i s p h e r i c a l s t e el p u n c h u n d e r d r y c o n d i t i o n s . T h et h e o r e t i c a l c o m p u t a t i o n s e m p l o y e d a l a r g e - s t r a i n f in i te e l e m e n t t e c h n i q u e b a s e d o n a ne l a s t i c / w o r k - h a r d e n i n g - p l a s t i c m a t e r i a l m o d e l i n c o r p o r a t i n g a n i s o t r o p i c p r o p e r ti e s . As p e c i a l f e a t u r e o f t h i s re s e a r c h i s t h e l i n k i n g o f th e a n a l y s i s to t h e e x p e r i m e n t b y d e t e r m i n i n gt h e o v e r a l l c o e ff i ci e n t o f f r i ct i o n a l o n g t h e p u n c h - s h e e t i n t er f ac e . S u c h m e a s u r e d c o e f fi c ie n t so f f r i c ti o n h a v e b e e n u s e d a s i n p u t s t o t h e n u m e r i c a l a n a l y s i s . I n t h i s w a y , v e r y g o o da g r e e m e n t h a s b e e n a c h i e v e d b e t w e e n e x p e r i m e n t a l a n d t h e o r e t ic a l r e su l ts s u ch a s lo a d s ,d e f l e c t i o n s a n d s t r a i n d i s t r i b u t i o n s a t v a r i o u s s t a g e s o f t h e s h e e t s t r e t c h i n g p r o c e s s . F i n a l l y ,r e s e a r c h i s i n p r o g r e s s t o r e fi n e t h e f r i c t i o n a l b o u n d a r y c o n d i t i o n s f u r t h e r b y a c t u a l l yd e v e l o p i n g a n e w f r i c t i o n m o d e l b a s e d o n t h e m i c r o s c o p i c a s p e r i t y a n a l y si s .

R E F E R E N C E S1. S. Y. CHUNG an d M . W . SWIFTCup-drawing from a flat blank. Proc . Ins t . Mech . Engng 165 , 199 1951).2. N. M. WANGand B. BUDIANSKY,Analysis of sheet metal stam ping by a f inite element method. J. appl . Mech.

T ra ns . A S M E 45, 73 1978).3. N. M. WANG and M. L. WENNER Elastic viscoplastic analysis of simple stretch forming p roblems. InMechanics of Sheet M etal Form ing, Symposium held at the G eneral M oters Research Lab orator ies , Warren,Michigan, p. 367 1977).4. C. H . LEE an d S. KOBAYASHI,New solut ions to r igid-plas t ic deformation problems using a m atr ix method. J .E n g n g I n d . T r a n s . A S M E 95, 865 1973).5. S. KOBAYASHIand J. H. K IM , Deformation analysis of axisymmetric sheet metal forming processes by therigid -plas tic f inite element method. In Mechanics of Sheet Metal Fo rmin g, Symposium held at the GeneralM oters Research Lab oratorie s, W arren, Michigan, p. 345 1977).6. B. B. YOON, R. S. RAO an d N. KIKU CHI,Experimental an d num erical comparisons of sheet stretching. In Proc.16th N. Am. Manu facturing Res. Con f., University o f Illinois, Ur ban a, p. 159 1988).7. R. S. RAO an d S. D. K . BARBAT, Fric tion al interactions in forming processes: interfacial observations an dpho toelastic studies with transp aren t dies. Proc. M anufacturing Processes, Machines and Systems, 14th Conf.on Prod . Res. and Tech. held at The U niversity o f Michigan, p. 105 1987).8. A. K. GHOSH, A m ethod for determining the coefficient of friction in punch stretching of sheet metals. In t . J .Mech. Sci . 19, 457 1977).9. J.-H. CHENG an d N. KIKUC HI,An analysis o f meta l forming processes using large deformation elastic plasticformulations. Comput. Meth . app l . Mech . En gng 49, 71 1984).I0. J.-H. CHENG and N. KIKUCHI, An incremental constitutive relation o f unilateral con tact fr iction for largedeformation analysis. J . app l. Mech . Trans . A S M E 52, 639 1985).11. R. HILL T h e M a t h e m a t i c a l T h e o r y o f P l a s t ic i t y. Oxford University Press, Lond on 1950).12. G . C. JOHNSONand D. J. BAUMMANA discussion of stress rates in f inite deformation problems. Sandia ReportSand 82-8821 1982).13. A . E. GREEN an d P. M . NAGHDIA general theory of an elas t ic-plas t ic continuum. Arch. Rat . Mech. Anal . 18,251 1965).14. W. PRAGER ntroduc t ion to M echanics o f Cont inua . Gin n and Company, Boston 1961).15. F. P. BOWDENan d D . TABOR,The Fr ic t ion and Lubr ica t ion o f So l ids par t I I . Oxford University Press, Lon don1 9 6 4 ) .