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PREVIOUSLYH o s f o r d

and Chin ~ have obtained a l i s t of s t r e s s s t a t e s for the s i m u l t a n e o u s a c t i v a t i o n of f i v e or m o r e {111}(112) slip or twin s y s t e m s . M o r e r e c e n t l y , Chin and W o n s i e w i c z 2 have obtained a s i m i l a r l i s t f o r {123}(111) slip. T h e s e s t r e s s s t a t e s w e r e obt a i n e d by s o l v i n g five or m o r e s i m u l t a n e o u s e q u a t i o n s e x p r e s s i n g 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 f o r s l i p (or twin) in t e r m s of the a p p l i e d s t r e s s e s , with the r e q u i r e m e n t that the c r i t i c a l s t r e s s is not e x c e e d e d in the r e m a i n i n g s y s t e m s . T h e p r o c e d u r e was f a c i l i t a t e d by the l i s t of a c t i v e s y s t e m s f o r d e f o r m a t i o n in a x i s y m m e t r i c flow, obtained f r o m our c o m p u t e r 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 . 3'4 A h y p o t h e s i s had b e e n d i s c o v e r e d e m p i r i c a l l y by one of us 5 that in the c a s e of {111}(110) s l i p , the l i s t of a c t i v e s y s t e m s for e n f o r c i n g a x i s y m m e t r i c flow i s i d e n t i c a l to the l i s t p r e v i o u s l y obtained by B i s h o p and Hill 6-8 for e n f o r c i n g an a r b i t r a r y shape change. Since the g e n e r a l a p p l i c a b i l i t y of this h y p o t h e s i s is u n c e r t a i n , it is i m p o r t a n t that a s y s t e m a t i c s e a r c h be m a d e for a l l p e r m i s s i b l e s t r e s s s t a t e s , such a s that done by Bishop and H i l l , b e f o r e the l i s t s for a x i s y m m e t r i c flow can be a p p l i e d f or a r b i t r a r y d e f o r m a t i o n . T h e p r e s e n t p a p e r d e a l s with such a s e a r c h in the c a s e s of {111}(112) slip and twinning. ANALYSIS O F { 1 1 1 } ( 1 1 2 ) TWINNING U s i n g the p r e v i o u s l y a d a p t e d n o t a t io n s 1 for the t w e l v e {111}(112) twin s y s t e m s , s e e T a b l e I, and s l i g h t l y a l t e r i n g the r e s o l v e d s h e a r s t r e s s e q u a t i o n s , e x p r e s s i o n s for yielding on t h e s e s y s t e m s a r e : System

Yield E x p r e s s i o n

al

2F - G - H = $1

bl cl dl

-2F +G-H=SI 2F+G+H=SI -2FG + H = S1

a2 b2

- F + 2 G - H = S2 F - 2G - H = $2

c2 d2

-F-2G

+H

=$2

F + 2 G + H =$2

Constraints -~SI St, F i g . 5(e), t h e r e i s no i n t e r s e c t i o n of the horizontal edges. c) DIAGONAL EDGES T h e d i a g o n a l e d g e s lie at fixed v a l u e s of H. T h e s e a r e , f r o m T a b l e II.

G

-5

T3

3) E d g e s As m a y be noted in F i g . 2 and deduced f r o m T a b l e II, the four s i d e s of the s o l i d outline of the r e c t a n g l e (or s q u a r e ) f o r m e d on the F , G plane r e p r e s e n t the l o c i of p r o j e c t i o n s of f o u r of the six e d g e s of a t e t r a h e d r o n . The other two e d g e s p r o j e c t a s d i a g o n a l s ; they a r e the top and b