A Theoretical examination of the plastic deformation of ionic crystals: II. analysis of uniaxial deformation and axisymm
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IN a p r e v i o u s
p a p e r , 1 h e n c e f o r t h r e f e r r e d to a s I, the s t r e s s s t a t e s r e q u i r e d f o r the s i m u l t a n e o u s a c t i v a t i o n of five i n d e p e n d e n t s l i p s y s t e m s have been d e r i v e d for s l i p on {110} (110> p l u s {100} (110) s y s t e m s . T h e s e s t r e s s s t a t e s w e r e u s e d to a n a l y z e d e f o r m a t i o n by a x i s y m m e t r i c flow using the m a x i m u m w o r k m e t h o d of B i s h o p and Hill. 2 In the p r e s e n t p a p e r the p r o b l e m of a x i s y m m e t r i c flow i s a n a l y z e d in g r e a t e r d e t a i l . Num e r i c a l s o l u t i o n s of the T a y l o r f a c t o r M a r e o b t a i n e d for s e v e r a l v a l u e s of a , the r a t i o of the c r i t i c a l r e s o l v e d s h e a r s t r e s s (CRSS) for {100} (110> s l i p v s {110} (110> s l i p . The l a t t i c e r o t a t i o n d e v e l o p e d by t e n s i o n o r c o m p r e s s i o n h a s a l s o been c a l c u l a t e d . The p r e s e n t r e s u l t s w e r e a c h i e v e d using the p r e v i o u s l y d e v e l o p e d l i n e a r p r o g r a m m i n g technique 3 of o b t a i n i n g s o l u t i o n s of T a y l o r ' s m i n i m u m w o r k a n a l y s i s , 4 which i s m a t h e m a t i c a l l y e q u i v a l e n t to the B i s h o p and Hill a n a l y s i s . ~ In a d dition, a n a l y s i s w a s m a d e for the c a s e of u n i a x i a l d e f o r m a t i o n , which r e q u i r e s the a c t i v a t i o n of a single s l i p s y s t e m only.
3//2 = x / 2 a / ( c o s O c o s ~p sin r + c o s 0 sin O cos 2 q~)[1] for (100)[011]. In Eq. [1], 0 and ~ a r e the a n g l e s defining the s p e c i m e n a x e s a s in F i g . 1 of I, and a = rloo/rlio is the r a t i o of the CRSS for the two s l i p m o d e s . Both Mt and M2 a r e n o r m a l i z e d with r e s p e c t to {110} (110) s l i p . It m a y be noted that M, h a s a m i n i m u m v a l u e for a x i a l o r i e n t a t i o n at [100] and that M2 i s l o w e s t n e a r [111]. Hence the {110} and {100} m o d e s a r e e x p e c t e d to p r e d o m i n a t e n e a r [100] and [111] r e s p e c t i v e l y . The equation of the b o u n d a r y s e p a r a t i n g the a c t i v i t y of the two s l i p m o d e s can b e o b t a i n e d by equating M1 to Ms. A plot is shown in F i g . 1 for s e v e r a l v a l u e s of a . F i g s . 2 to 4 show p l o t s of c o n s t a n t M c o n t o u r s for
[111] ~=~, 51 0 ~
L30
UNIAXIAL DEFORMATION If the a x i a l d i r e c t i o n x is l o c a t e d within the s t a n d a r d 100-110-111 s t e r e o g r a p h i c t r i a n g l e a s shown in F i g . 1 of I, the p r i m a r y {110} (110) s l i p s y s t e m s a r e (101) [101] and (101) [101]. B e c a u s e of the r e c i p r o c a l r e l a t i o n s h i p b e t w e e n the s l i p p l a n e and d i r e c t i o n , both s l i p s y s t e m s s u s t a i n the s a m e r e s o l v e d s h e a r s t r e s s . F o r the {100} (110) s l i p m o d e , the p r i m
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