Failure in thin sheets stretched over rigid punches
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It has r e c e n t l y b e e n r e a l i z e d by the p r e s e n t a u t h o r s 11 that t h e r e is no r e a l r e s t r i c t i o n to l o c a l i z e d n e c k i n g in punch s t r e t c h i n g . A s ' p r o p o s e d in Ref. 11, the s t r a i n peaks d e v e l o p i n g f r o m g e o m e t r i c a l and f r i c t i o n a l c o n s t r a i n t s can move t o w a r d the c l a m p e d edge with c o n tinued d e f o r m a t i o n and f i n a l l y s a t i s f y an i n s t a b i l i t y condition followed by a s t a g e of s t r a i n l o c a l i z a t i o n . In the p r e s e n t work, an i n s t a b i l i t y c r i t e r i o n is d e v e l o p e d that is r e a l i s t i c and p h y s i c a l l y c o n s i s t e n t . In o r d e r to do this without i n t r o d u c i n g the u n c e r t a i n t i e s of the c o m p u t e d s t r a i n v a l u e s , the r a d i a l and c i r c u m ferential strains were measured during incremental punch s t r e t c h i n g and u t i l i z e d in the p r e d i c t i o n of i n s t a b i l i t y . Should s a t i s f a c t o r y a n a l y s e s for p r e d i c t i n g t h e s e s t r a i n d i s t r i b u t i o n s b e c o m e a v a i l a b l e , they could be r e a d i l y i n c o r p o r a t e d . The p u r p o s e of such a c r i t e r i o n is, of c o u r s e , to p r o v i d e u n d e r s t a n d i n g of the f a i l u r e p r o c e s s in punch s t r e t c h i n g , and p r e d i c t the s t r a i n l e v e l s for the p o s i t i v e side (e2 > 0) of the f o r m i n g l i m i t d i a g r a m s fl '7 I. E X P E R I M E N T A L Punch S t r e t c h i n g A d e s c r i p t i o n of the m a t e r i a l s t e s t e d and t h e i r m e c h a n i c a l p r o p e r t i e s a r e given in T a b l e I. The four p r i n cipal t e s t m a t e r i a l s w e r e Steel " A " ( a l u m i n u m - k i l l e d ) , B r a s s " A " (70:30 b r a s s ) , T i t a n a l o y ( Z n - T i alloy) and 2 0 3 6 - T 4 A l u m i n u m , which offered a wide r a n g e of m e c h a n i c a l p r o p e r t i e s (n, m , R v a l u e s and f r a c t u r e s t r a i n s ) . The t e s t i n g of such a v a r i e t y of m a t e r i a l s was n e c e s s a r y to e s t a b l i s h g e n e r a l i t y of the p r o p o s e d i n s t a b i l i t y c r i t e r i o n . F o u r o t h e r v a r i e t i e s of low c a r bon s t e e l and a 61:39 b r a s s w e r e used as s u p p o r t i n g m a t e r i a l s in this study. A s c h e m a t i c d i a g r a m of the punch t e s t a p p a r a t u s u s e d is shown in Fig. l(a) and a photograph of a t e s t e d s p e c i m e n (dome) in Fig. l i b ) . A d e t a i l e d d e s c r i p t i o n of the a p p a r a t u s and the t e s t t e c h n i q u e has b e e n p r e s e n t e d e l s e w h e r e . 12 T e s t b l a n k s , 200 m m • 200 m m , p h o t o g r i d d e d with 2.54 m m d i a m c o n t a c t ing c i r c l e s w e r e c l a m p e d f i r m l y b e t w e e n die p l a t e s VOLUME 6A, MAY 1975 1065
r e s p e c t i v e l y ) w e r e m e a s u r e d * along the r o l l i
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