Strain hardening and instability in biaxially streched sheets
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K. GHOSH
AND
WALTER
A. BACKOFEN
This w o r k is c o n c e r n e d with the o r i g i n s of the two d i f f e r e n t p a t t e r n s of f a i l u r e l i m i t s in b i a x i a l l y s t r e t c h e d s h e e t s which w e r e r e c e n t l y d e s c r i b e d in Ref. 1 : the b r a s s - t y p e in which the l i m i t s t r a i n is i n s e n s i t i v e to s t r a i n s t a t e , and that of f e r r i t i c s t e e l in which the l i m i t s t r a i n i n c r e a s e s a s the i m p o s e d s t r a i n - r a t i o , p = r changes from zero (plane-strain t e n s i o n ) t o w a r d unity (balanced b ia x i a l tension). An e a r l i e r p r o p o s a l that d i f f e r e n t slip m o d e s , i.e. wavy in f e r r i t e v s p l a n a r in b r a s s , m i g h t have c o n t r i b u t e d to t h e s e f a i l u r e l i m i t d i f f e r e n c e s w a s found not to be v a l i d . T h e r e w e r e two p a r t s to the main e x p e r i m e n tal p r o g r a m : the p r e s t r a i n i n g of s m a l l s h e e t s by p r o p o r t i o n a l loading on d i f f e r e n t paths b e t w e e n p = 0 and p = 1, followed by t e n s i o n t e s t i n g , and a m o r e d i r e c t m e a s u r e m e n t of s t r a i n h a r d e n i n g and i n s t a b i l i t y b e t w e e n p ~ - 5 (uniaxial tension) and p = 0. The p r i n c i pal finding was that the o v e r a l l h a r d e n i n g r a t e , e s s e n t i a l l y a s it a p p e a r e d in the m a t e r i a l ' s e f f e c t i v e s t r e s s - s t r a i n c u r v e , changed with the loading path. In a b r a s s it d e c a y e d a s p 1 w a s i n c r e a s e d f r o m ~ - ~ to 1; in f e r r i t i c s t e e l it i n c r e a s e d ; and in a l u m i n u m it w as a f fected v e r y l i t t l e . Such c h a n g e s in h a r d e n i n g r a t e c a u s e s i m i l a r ch an g es in the m a t e r i a l ' s c a p a c i t y for stable flow. The s t a b l e flow, in t u r n , is the b a s e to which a q u a s i s t a b l e - f l o w i n c r e m e n t (when p is >0) is added in r e a c h i n g the o b s e r v e d f a i l u r e l i m i t . Thus a b a s e of p - d e p e n d e n t height can a c c o u n t f o r the f a i l u r e - l i m i t p a t t e r n s . T h e r e is s t i l l no e x p l a n a tion for the p d e p e n d e n c e of the h a r d e n i n g r a t e .
AN e a r l i e r
study of f a i l u r e in b i a x i a l l y s t r e t c h e d s h e e t s showed that the l a r g e s t t e n s i l e s t r a i n b e f o r e cons p i c u o u s l o c a l thinning m a y be l a r g e r than the s t r a i n for necking in p u r e t e n s i o n and m a y even i n c r e a s e a s the d e g r e e of b i a x i a l i t y i n c r e a s e s . ~ T h i s d i s c o v e r y led to d e v e l o p m e n t of the f o r m i n g l i m i t d i a g r a m s in use t o day by many s h eet m e t a l p r o d u c e r s and f a b r i c a t o r s , 3-s and it showed f u r t h e r that the f a i l u r e p r o c e s s under b i a x i a l loading was not w e l l u n d e r s t o o d . The difficult thing to u n d e r s
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