Minimization of shears for pencil glide in body-centered cubic crystals
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P. PENNING is Professor of Metal Physics, Interd,sciphnary Department of Metals Science and Technology, Delft, University of Technology, The Netherlands. Manuscript submitted November 19, 1974, METALLURGICAL TRANSACTIONS A
p e n c i l glide to compute n u m e r i c a l l y the f o u r m a g n i t u d e s of s l i p . In t h i s p a p e r an a n a l y t i c a l s o l u t i o n f o r the p r o b l e m of (111> p e n c i l glide is given by u s i n g T a y l o r ' s m i n i m u m p r i n c i p l e . In view of the r e s u l t s of Chin and M a m m e l 3 t h i s s o l u t i o n m u s t be, and i n d e e d i s , i d e n t i c a l with the one o b t a i n e d by u s i n g B i s h o p and H i l l ' s a p p r o a c h . In a d d i t i o n the c r y s t a l r o t a t i o n due to s l i p has b e e n d e t e r m i n e d .
1. D E F O R M A T I O N AND ROTATION DUE TO SLIP Throughout this paper a crystal-based co-ordinate s y s t e m i s u s e d . T h e s l i p d i r e c t i o n i s (111). It is c o n venient to i n t r o d u c e a s y m m e t r i c m a t r i x ( a m n ) in the following way:
11,)
1 (~mn) =
1
1
-i
1
-1
-I
-I
1 -1
1
"
[1]
1
The four possible slip directions are: Dn = [~2n, ann, C q n ] / ' ~ 3 : , 7 = 1, 2, 3, 4. P e r p e n d i c u l a r to Dn two o r t h o g o n a l a x e s Mnl and Mn2 a r e c h o s e n to fix the o r i e n t a t i o n of the s l i p p l a n e n o r m a l Nn. Mnl = [ 0 , - ~ 3 n , ~ 4 n ] / ~ , Mn2 = [2~en, - ct~n, - c%n V v~, N n = Mnl c o s Cn + Mn2 sin ~n" A d i s t i n c t i o n i s m a d e b e t w e e n ~n and ~n + 7r. T h e a m o u n t of s l i p , F n , i s t a k e n to b e n o n n e g a t i v e . R e v e r s a l of s l i p i s e x p r e s s e d by adding n to ~n and h e n c e by a r e v e r s a l of s i g n in N n. The d i s p l a c e m e n t v e c t o r a r i s i n g f r o m t h i s s l i p i s e q u a l to: V. = F n (R "Nn)Dn,
[2]
w h e r e R d e n o t e s the r a d i u s v e c t o r of the c o n s i d e r e d l o c a t i o n . T h e s t r a i n c o m p o n e n t s can be d e r i v e d i m m e d i a t e l y f r o m t h i s value of Vn and s u m m i n g the cont r i b u t i o n of a l l s l i p s y s t e m s : En = -- ~ ~n rn sinSn, 3 n VOLUME 7A, JULY 1976-1021
+
E~ = -~- ~n axn r n sin (~bn - 2~r/3),
~2
E33 = " ~ ~
n
+L~3{~3 ~z~ 2 E ~ , - ~ n F n s i n ( ~ n + 27r/3)}
Otln r n s i n (~n + 2~r/3),
[3] + x~ {3 ~ E ~
+ ~ a ~ n r ~ sin
~.}
2E~a = - ---~ ~ Oten F n s i n ~n, 3
+ X~a{3 ~'2E~a + ~-~c%nr n s i n ( ~ n - 27r/3)} n
+ X~e{3 ~/'2E~2 + Y~ot~nF n s i n ( ~ n + 2~r/31}.
2Ex~ = - -~- L~ a~n F n sin(~bn - 2~r/3), n
2E~z = - ~~,/-22 " ~n ~ n Fn s i n (~0n +
S e t t i n g the d e r i v a t i v e with r e s p e c t to F n and ~n e q u a l to z e r o l e a d s to:
2'm/3).
It m u s t b e noted that the s u m of the f i r s t t h r e e s t r a i n s i s equal to z e r o (no change in v o l u m e ) so that t h e r e a r e five independent s t r a i n s . The r o t a t i o n of the c r y s t a l d u r i n g p l a s t i c flow cons i s t s of two p a r t
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