Comparison of the strain-hardening parameters of sheet metals in uniaxial and biaxial tension
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A N D A. J. R A N T A - E S K O L A
T h e s t r a i n - h a r d e n i n g p a r a m e t e r s of s t e e l , copper and b r a s s w e r e d e t e r m i n e d by uniaxi a l and b a l a n c e d b i a x i a l t e n s i l e t e s t s . The s t r e s s - s t r a i n r e l a t i o n s h i p of s t e e l was well d e s c r i b e d by the Hollomon equation w h i l e , the Voce equation was v a l i d for copper and b r a s s . T h u s the use of the n - v a l u e as a m e a s u r e of s t r a i n - h a r d e n i n g a b i l i t y i s j u s t i f i e d f o r s t e e l only and u n i f o r m s t r a i n s have to be used for copper and b r a s s . The u n i a x i a l and b i a x i a l n - v a l u e s of s t e e l w e r e found to be e q u i v a l e n t . However, the s t r e s s - s t r a i n r e l a t i o n s h i p s of copper and e s p e c i a l l y b r a s s w e r e o b s e r v e d to be s t r e s s - s t a t e dependent. F o r m i n g l i m i t d i a g r a m s of the m a t e r i a l s w e r e c o n s t r u c t e d u s i n g h y d r a u l i c bulging and t e n s i l e t e s t s . The F L D s w e r e a l s o c a l c u l a t e d u s i n g the p l a s t i c i n s t a b i l i t y conditions and the s t r a i n - h a r d e n i n g p a r a m e t e r s d e t e r m i n e d . T H E f o r m a b i l i t y of s h e e t m e t a l has m o s t f r e q u e n t l y b e e n e x p r e s s e d by the n - and R - v a l u e s d e t e r m i n e d from uniaxial tension tests. However, formability depends on the s u r f a c e s t r a i n r a t i o applied and thus the p a r a m e t e r s n and R cannot a s s e s s f o r m a b i l i t y u n d e r a l l c o n d i t i o n s . G e n s a m e r ~ suggested that the f o r m i n g l i m i t s of sheet m e t a l can be r e p r e s e n t e d by s u r f a c e s t r a i n s . The e x p e r i m e n t a l m e a s u r e m e n t of t h e s e s t r a i n s , c a r r i e d out by K e e l e r 2 and Goodwin, 3 f o r m e d the b a s i s of the f o r m i n g l i m i t d i a g r a m (FLD). The t h e o r e t i c a l e x a m i n a t i o n of the FLD has been b a s e d on p l a s t i c i n s t a b i l i t y c o n d i t i o n s . Following H i l l , K e e l e r and Backofen 4 d e r i v e d the local and diffuse i n s t a b i l i t y conditions f o r v a r i o u s s t r e s s s t a t e s . V e n t e r and de M a l h e r b e s c a l c u l a t e d a t h e o r e t i c a l e s t i m a t e of the FLD u s i n g local and diffuse i n s t a b i l i t y c r i t e r i a . P a i n t e r and P e a r c e 6 obtained a close a g r e e m e n t b e t w e e n the t h e o r e t i c a l c u r v e and e x p e r i m e n t a l data on the left side of the FLD for s t e e l and a l u m i n u m , but a c o m p l e t e d i s a g r e e m e n t for b r a s s . Backofen and c o w o r k e r s 7,s have shown that the s t r a i n - s t a t e d e p e n d e n c e of the l i m i t s t r a i n s of f e r r i t i c s t e e l type m a t e r i a l s is d i f f e r e n t f r o m that of the b r a s s type. The l a r g e s t s u r f a c e s t r a i n in b i a x i
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