Inhomogeneity of plastic flow in constrained deformation
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l e a s t amount of s l i p d T , a s s u m i n g that the c r i t i c a l s h e a r s t r e s s T is the s a m e f o r a l l s y s t e m s . F o r plane s t r a i n c o m p r e s s i o n of a (111)[1i2] c r y s t a l , s l i p i s e x p e c t e d on the ( i i l ) plane in both the [011] and [101] d i r e c t i o n s , r e f e r r e d to a s the c o p l a n a r p a i r , c o m b i n e d with s l i p in the [110] d i r e c t i o n on both the ( l i i ) and (111) p l a n e s , the c o l i n e a r p a i r , F i g . 4. T h e s e r e s u l t s w e r e o b t a i n e d p r e v i o u s l y 3'4 b y the m e t h o d of Bishop and Hill, which has r e c e n t l y b e e n shown s to be c o m p l e t e l y e q u i v a l e n t to the T a y l o r t h e o r y . T h i s o r i e n t a t i o n has b e e n a n a l y z e d in m o r e d e t a i l by D i l l a m o r e , B u t l e r and G r e e n 6 and by us 7 f o r p l a n e s t r a i n d e f o r m a t i o n with no s h e a r s t r a i n s p e r m i t t e d , i.e., exx =-ezz, Cyy = e y z = ~ z x = e x y = 0, such a s m a y be e n c o u n t e r e d in a p o l y c r y s t a l l i n e a g g r e g a t e . The (x, y , z) d i r e c t i o n s a r e taken a s the c o m p r e s s i o n plane n o r m a l [ 111], the c o n s t r a i n t d i r e c t i o n [ i10] and the e x t e n s i o n d i r e c t i o n [112], r e s p e c t i v e l y . In t h i s
ANALYSIS F r o m the T a y l o r a n a l y s i s , s l i p i s e x p e c t e d on s y s t e m s which will p r o d u c e the d e s i r e d shape change with the l e a s t a m o u n t of w o r k ~-d7 and, t h e r e f o r e , the B. C. WONSIEWICZand G. Y. CHIN are Members of Technical Staff, Bell Telephone Laboratories, Murray Hill, N. J. Manuscript submitted June 30, 1969. METALLURGICAL TRANSACTIONS
Fig. 1--View of the compression plane of a Cu-6 wt pct A1 crystal initially compressed on (111) with expansion limited to [112]. Regions A and B are the result of inhomogeneous deformation. Cx x = -0.48. VOLUME 1, JANUARY 1970-57
c a s e , if t h e r e w e r e a c o n s t r a i n t ezx = 0, it c o u l d be s a t i s f i e d only if b o t h the c o l i n e a r and c o p l a n a r p a i r s o p e r a t e t o g e t h e r by d e f i n i t e a m o u n t s of s h e a r . A s the c r y s t a l d e f o r m e d h o m o g e n e o u s l y , it w o u l d r o t a t e toward and past (112)[1il], achieving a stable orientat i o n about 8 d e g f r o m (112). H o w e v e r , in the p r e s e n t e x p e r i m e n t a l s e t u p , t h e r e i s no c o n s t r a i n t o n ezx o r eyz. The c r y s t a l i s t h e n p e r m i t t e d to d e f o r m by e i t h e r p a i r of s l i p s y s t e m s a l o n e . As the f o l l o w i n g a n a l y s i s s h o w s , t h e s e two p a i r s a r e m u t u a l l y e x c l u s i v e in that the o p e r a t i o n of one r a i s e s i t s r e s o l v e d s h e a r s t r e s s r e l a t i v e to the o t h e r . With the (x, y, z) d i r e c t i o n s d e f i n e d a s a b o v e , the r e s o l v e d s h e a r s t r e s s on e a c h of the f o
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