A numerical analysis of the tensile test for sheet metals
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DURINGtensile tests of engineering sheet materials, the records of load vs elongation are routinely translated into true stress-true strain curves. These records can be utilized up to the maximum load points, since deformation is practically uniform over tile gage length within this load range. In strain hardening metals, true stress-true strain relations are fitted to a power law, a = Ken, and this relationship is assumed to continue beyond maximum load. Beyond the maximum load deformation becomes nonuniform, and gage length elongations can no longer be used to obtain true strain or true stress in the deforming element. In a specimen of strainhardening material containing an imperfection (e.g., variation in cross sectional area) once the maximum load for the imperfection site is attained, a simple onedimensional model suggests that all other material elements in the specimen would stop deforming; since this site can now continue to deform under a falling load. This further suggests negligible post-uniform* exten*In thlspaper,the termuniformelongatioarefers to strainat maximumload, whilepost-urn/ottoexter~smnis the elongationaftermaximumload. Allelongations reportedhere are for a 50.8 mm gagelength, sion o v e r the gage length. H o w e v e r the f a c t s a r e t h a t the i m p e r f e c t i o n s i t e d o e s not d e f o r m by i t s e l f and t h a t the s t r a i n c o n c e n t r a t i o n is s p r e a d o v e r the n e i g h b o r i n g e l e m e n t s . The s t r a i n n o n u n i f o r m i t y s o d e v e l o p e d i s t h u s g r a d u a l and not of a s t e p - f u n c t i o n n a t u r e a s a s s u m e d in m a n y i n v e s t i g a t i o n s . In t h e p r e s e n t p a p e r a p s e u d o t w o - d i m e n s i o n a l a n a l y s i s of p l a s t i c d e f o r m a tion h a s been m a d e with r e a l i s t i c s t r a i n g r a d i e n t s a c c o m p a n y i n g the d e v e l o p m e n t of a neck. It is known that u n i f o r m elongation is d i r e c t l y r e l a t e d to the s t r a i n h a r d e n i n g exponent, n, while the p o s t - u n i f o r m d e f o r m a t i o n is c o n t r o l l e d b y s t r a i n AMIT K. GttOSH, formerly wtth Research Laboratories, General Motors Corporation, Warren, MI is now a Member of the Technical Staff, Science Center, Rockwell International, Thousand Oaks, CA 91360. Manuscnpt submitted September 29~ 1976. METALLURGICAL TRANSACTIONS A
r a t e s e n s i t i v i t y , m (as in the h a r d e n i n g law: g = K~n~rn), as the s t r a i n - r a t e g r a d i e n t s in the s p e c i m e n b e c o m e s i g n i f i c a n t . 1-~ In addition, s i n c e the s t r e s s - s t a t e in the d e v e l o p i n g n e c k d e v i a t e s f r o m u n i a x i a l tension~ the p l a s t i c a n i s o t r o p y p a r a m e t e r , r (the r a t i o of w i d t h - t o - t h i c k n e s s s t r a i n ) which influences the l o c a l flow s t r e s s , a l s o a f f e c t s the p o s t - u n i f o r m e x t e n s i o n . 1'2 The r e l a t i o n s h i p b e t w e e n t h e s e p h
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