Distribution of plastic strain and negative pressure in necked steel and copper bars
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s t u d i e s of i n c l u s i o n s e p a r a t i o n and p l a s t i c hole growth. It was t h e r e f o r e c o n s i d e r e d useful to obtain a n u m b e r of solutions for the d i s t r i b u t i o n of the p l a s t i c s t r a i n and negative p r e s s u r e along the axis of a b a r with a v a r i e t y of i n i t i a l neck c o n t o u r s and made of m a t e r i a l having p l a s t i c s t r e s s s t r a i n c u r v e s of some c o m m o n m e t a l s such as s t e e l and copper. Here we d e s c r i b e the r e s u l t s of a set of such n u m e r i c a l c o m p u t a t i o n s u s i n g the app r o a c h of N e e d l e m a n . 8 However, we develop f i r s t an a p p r o x i m a t e solution by extending the B r i d g m a n d e v e l opment. 1. BOUNDS FOR THE STRESS AND STRAIN DISTRIBUTION IN THE NECK REGION OF A BAR Rhee and McClintock ~~have a r g u e d that useful bounds to the d i s t r i b u t i o n of s t r a i n s (and s o m e t i m e s s t r e s s e s ) in a p r o b l e m involving p l a s t i c flow of a s t r a i n h a r d e n i n g m a t e r i a l can be obtained f r o m two solutions for l i m i t i n g b e h a v i o r . In the f i r s t s o l u t i o n the m a t e r i a l is m o d e l l e d as a p l a s t i c r i g i d , n o n - h a r d e n i n g one with a yield s t r e s s r e s u l t i n g in the e x p a n s i o n of the s a m e amount of p l a s t i c work as in the r e a l s t r e s s s t r a i n c u r v e over the r a n g e of p l a s t i c s t r a i n s of i n t e r e s t . In the second solution the m a t e r i a l is m o d e l l e d as a l i n e a r l y h a r d e n i n g one with zero yield s t r e s s and a P o i s s o n ' s r a t i o of 0.5 to s i m u late no change of v o l u m e . T h e s e s o l u t i o n s for the necked b a r are r e a d i l y o b t a i n a b l e for given neck d i m e n s i o n s and f u r n i s h bounds for c a s e s in which the n u m e r i c a l solutions of the type to be p r e s e n t e d below a r e u n a v a i l a b l e . 1.1 Non Strain Hardening, Rigid Plastic Material The d i s t r i b u t i o n of the n e g a t i v e p r e s s u r e along the axis of the b a r in a n o n - h a r d e n i n g r i g i d p l a s t i c m a t e r i a l with yield s t r e s s Y0 is r e a d i l y obtained b y the method of B r i d g m a n in which it is a s s u m e d that the s u r f a c e s of c o n s t a n t p r i n c i p a l s t r e s s in the r e g i o n of the neck have u n i f o r m c u r v a t u r e as shown in Fig. 1. The r e s u l t i n g d i s t r i b u t i o n of n e g a t i v e p r e s s u r e aT, along the axis of the s p e c i m e n a d i s t a n c e z away f r o m the p l a n e of the neck is then given by (see Appendix) Yo - 3 + l n
1+ ~-
-2
In
1 + a(a+~
(r = 0 )
[1] VOLUME 6A, APRIL 1975-815
tioned above~ The s a m e r e s t r i c t i o n d i s c u s s e d above in c o n n e c t i o n with Eq. [1] also applies to Eq. [3]. 1.2 L i n e a r l y H a r d e n i n g M a t e r i a l with Zero Yield S t r e s s The d i s t r i b u t i o n of t r i a
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