Dislocation behavior at tilt boundaries of infinite extent
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(A - a) w h e r e A a n d a a r e the d i m e n s i o n s of the l a r g e and s m a l l c r y s t a l s , respectively.2'3 W h e r e a s the b o u n d a r y shown in Fig. l(a) has no long r a n g e s t r e s s e s a s s o c i a t e d with it, 4 the b o u n d a r i e s
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r F i g . 1--Various types of i n t e r n a l b o u n d a r i e s . {a) S y m m e t r i c a l
tilt boundary; (b) plastically deformed symmetrical tilt boundary; (c) coherent interface between crystals of differing size. VOLUME 1,DECEMBER 1970-3397
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Fig. 2--Removal of the long-range stress fields associated with the boundaries in Fig. l(b) and l(c) by the introduction of crystal dislocations into the boundaries.
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0.4 shown in F i g . l(b) and 1(c) do h a v e such long r a n g e s t r e s s f i e l d s . The long r a n g e s t r e s s f i e l d s shown in F i g s . l(b) and l(c) can h o w e v e r b e e l i m i n a t e d by the i n t r o d u c t i o n of c r y s t a l d i s l o c a t i o n s into the b o u n d a r y a s shown in F i g s . 2(a) and 2(b), r e s p e c t i v e l y . The r e m a i n d e r of the p r e s e n t study w i l l deal with the n a t u r e of the s t r e s s f i e l d s a s s o c i a t e d w i t h the b o u n d a r i e s in F i g . 1 and t h e i r a b i l i t y to n u c l e a t e c r y s t a l d i s l o c a t i o n s so a s to r e d u c e t h e s e s t r e s s f i e l d s in the m a n n e r shown by F i g . 2. In the p r e s e n t a n a l y s i s , the c r y s t a l s t r u c t u r e s a r e a s s u m e d to be s i m p l e cubic and the s l i p s y s t e m i s c h o s e n a s (100) {100} so that the B u r g e r s v e c t o r of the c r y s t a l d i s l o c a t i o n s is ao (100), w h e r e a o is the d i m e n s i o n at the unit c e l l edge. In a bcc l a t t i c e on the o t h e r band, the c r y s t a l d i s l o c a t i o n s would h a v e B u r g e r s v e c t o r s e q u a l to ~-ao(111), ~ and so forth. N A T U R E O F THE STRESS F I E L D S A T I N T E R N A L BOUNDARIES The c o m p o n e n t s of s t r e s s a s s o c i a t e d with an infinite v e r t i c a l a r r a y of edge d i s l o c a t i o n s h a v e b e e n d e r i v e d by Li 4 and a r e given a s f o l l o w s :
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Fig. 3--Equistress contours for Oxx in units of~b/2h (1 - v) about an infinite vertical wall of edge dislocations corresponding to (a) & = 0 deg, (b) 4, = 45 deg, and (c) & = 90 Oeg
a x x = - c o s ~b sin 2k(cosh 2o~ - c o s 2k + 2a sinh 2a) -
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[la]
a y y = c o s ~ sin 2k(2ot sinh 2ol - c o s h 2ol + cos 2k)
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[lb]
a z z = 2v(sin 9 sinh 2a - c o s ~ sin 2k)(cosh 2or -
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