Instability of plastic flow in the directions of pure shear: I. Theory
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K. CHAKRABARTI
AND
JOSEPH
W. SPRETNAK
The l o c a l i z a t i o n of p l a s t i c flow (plastic i n s t a b i l i t y ) is a p r e l u d e to f r a c t u r e initiation. P r i o r t r e a t m e n t s a r e b a s e d m a i n l y on the p h e n o m e n o n of n e c k i n g (load m a x i m u m ) which has for its b a s i s f l u c t u a t i o n s in c r o s s - s e c t i o n a l a r e a . A second b a s i c mode of l o c a l i z a t i o n has b e e n c o n f i r m e d e x p e r i m e n t a l l y . This is the l o c a l i z a t i o n of flow within a p l a s t i c zone along c h a r a c t e r i s t i c d i r e c t i o n s (direction of pure s h e a r or zero e x t e n s i o n a l s t r a i n ) . The l o c a l i zation is effected b y the a c t i v a t i o n of t a n g e n t i a l v e l o c i t y r along the c h a r a c t e r i s t i c s . A n e c e s s a r y condition is that the m a t e r i a l s i m u l a t e the ideal p l a s t i c state. The conditions for this second mode of l o c a l i z a t i o n a r e f o r m u l a t e d . A g e n e r a l condition for i n s t a b i l i t y involving both modes is d e r i v e d . In s i t u a t i o n s in which both modes a r e p o s s i b l e , n e c k i n g is always expected to p r e c e d e the l o c a l i z a t i o n along c h a r a c t e r i s t i c s .
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tic s u r f a c e , p r o d u c i n g a ledge at the f r e e s u r f a c e . 3) The f r a c t u r e i n i t i a t i o n o c c u r r e d in the state of plane s t r a i n . This is evidenced b y the c h a r a c t e r i s t i c m a k i n g 45 deg with a f r e e s u r f a c e and the c~ and/3 c h a r a c t e r i s t i c s i n t e r s e c t i n g orthogonally. 4) The d e f o r m a t i o n in plane s t r a i n is r e s t r i c t e d to two d i m e n s i o n s . It follows that f l u c t u a t i o n s in c r o s s s e c t i o n a l a r e a and thus n e c k i n g a r e s u p p r e s s e d in this state of s t r e s s . This is also t r u e for t o r s i o n and for plane s t r e s s for the case e e / a I = 0.5, r e p r e s e n t i n g c o m b i n a t i o n s of plane s t r e s s and plane s t r a i n . The c h a r a c t e r i s t i c s a r e d i r e c t i o n s of z e r o e x t e n s i o n a l s t r a i n (pure s h e a r ) . Only in the state of s t r e s s of plane s t r a i n a r e they also d i r e c t i o n s of m a x i m u m s h e a r s t r e s s , s p e c i f i c a l l y the c r i t i c a l m a x i m u m s h e a r s t r e s s m e e t i n g the flow condition. The t h e o r y of c h a r a c t e r i s t i c s is b a s e d on the ideal p l a s t i c state. The d i r e c t i o n s of the c h a r a c t e r i s t i c s , thus, depend only on the Imposed s t r e s s state. The m o s t f a m i l i a r c h a r a c t e r i s t i c s d i r e c t i o n is that of L u d e r s b a n d s in plane s t r e s s and s i m p l e t e n s i o n , n a m e l y 35~ ' f r o m the h o r i z o n t a l . The c h a r a c t e r i s t i c d i r e c t i o n s a r e found b y s e a r c h i n g out the d i r e c t i o n s of z e r o e x t e n s
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