The effect of a locked screw dislocation on a pile-up of screw dislocations in a two phase system
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P I L E - U P of s c r e w d i s l o c a t i o n s in t w o - p h a s e s y s t e m s c o m p o s e d of two welded e l a s t i c half p l a n e s have b e e n a n a l y z e d in the p a s t u s i n g the method of c o n t i n u o u s l y d i s t r i b u t e d d i s l o c a t i o n s e i t h e r when the p i l e - u p tip is at the i n t e r f a c e 1 or away f r o m the i n t e r f a c e . 2,3 The i n f l u e n c e of a locked s c r e w d i s l o c a t i o n on a p i l e - u p has not b e e n c o n s i d e r e d in the above a n a l y s i s . Using the p o l y n o m i a l method of E s h e l b y et a l . , 4 Chou 5 has c o n s i d e r e d a locked d i s l o c a t i o n in the v i c i n i t y of a p i l e - u p in an infinite h o m o g e n e o u s m e d i u m . However, the p o l y n o m i a l method can be u s e d to a n a l y z e such a s i t u a t i o n in a t w o - p h a s e m e d i u m c o m p o s e d of two welded e l a s t i c half p l a n e s only when one p h a s e is r i g i d . Smith e has given the r e s u l t s u s i n g the method of c o n t i n u o u s l y d i s t r i b u t e d d i s l o c a t i o n s when the p i l e - u p e x i s t s with a locked d i s l o c a t i o n in an i n f i n i t e l y h o m o g e n e o u s m e d i u m . In this p a p e r the method of c o n t i n u o u s l y d i s t r i b u t e d d i s l o c a t i o n s is applied to a t w o - p h a s e m e d i u m so that r e s u l t s a r e o b t a i n e d for the c o m p l e t e r a n g e of s h e a r m o d ulus of one of the p h a s e s . Ashby 7 has given a m o d e l w h e r e i n a r r a y s of s h e a r - l o o p s f o r m a g a i n s t the second p h a s e p a r t i c l e whose b e h a v i o r will depend on the r e l a tive v a l u e of the s h e a r m o d u l u s . B a r n b y a has e x p e r i m e n t a l l y shown that p i l e - u p s of d i s l o c a t i o n s f o r m e d a g a i n s t c a r b i d e p a r t i c l e s in a s t a i n l e s s s t e e l m a t r i x can lead to f r a c t u r e of the second p h a s e p a r t i c l e . In any t w o - p h a s e m a t e r i a l i n the w o r k - h a r d e n e d state, a s in a s i n g l e phase m a t e r i a l , s t r a y d i s l o c a t i o n s m u s t b e c o m e locked as a r e s u l t of v a r i o u s d i s l o c a t i o n i n t e r a c t i o n s , and thus act a s i m p o r t a n t s o u r c e s of h a r d e n i n g . While the effect of t h e s e locked d i s l o c a t i o n s on the d e f o r m a t i o n of s i n g l e p h a s e s has b e e n i n v e s t i g a t e d , the s a m e has not t i l l now b e e n a t t e m p t e d in t w o - p h a s e m a t e r i a l s . In the l a t t e r the p r o b l e m is made c o m p l i c a t e d by the p r e s e n c e of the second phase p a r t i c l e s , which a p a r t f r o m modifying the applied s t r e s s , a l s o i n t r o d u c e i m a g e f o r c e s . T h e s e i m a g e f o r c e s a r e as i m p o r t a n t a s the f o r c e due to the s t r e s s field of a locked d i s l o c a t i o n . In this p a p e r the effect of a loc
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