Elastic interaction energy of two spherical precipitates in an anisotropic matrix
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T H E elastic interaction energy between precipitates is of g r e a t i n t e r e s t in connection with p r e c i p i t a t e a l i g n m e n t d u r i n g c o a r s e n i n g , s t r e s s induced phase t r a n s f o r m a t i o n s , void lattice f o r m a t i o n , and so forth. Although t h e r e have b e e n many s t u d i e s on this s u b ject, most of the p r e v i o u s works have b e e n l i m i t e d e i t h e r to a homogeneous case T M w h e r e i n the e l a s t i c c o n s t a n t s of the p r e c i p i t a t e and m a t r i x phases a r e a s s u m e d to be equal, o r if inhomogeneous to i s o t r o p i c e l a s t i c i t y . 1,S-7 By i n h o m o g e n e o u s it is here m e a n t that the e l a s t i c c o n s t a n t s of the p r e c i p i t a t e s and m a t r i x a r e not the s a m e . However, most p h e n o m e n a caused by e l a s t i c i n t e r a c t i o n s a r e o b s e r v e d in a n i s o t r o p i c m a t e r i a l s with d i f f e r e n t e l a s t i c c o n s t a n t s for the m a t r i x and p r e c i p i t a t e s . U t i l i z i n g E s h e l b y ' s d i f f e r e n c e method s and Yoffe's r e s u l t s on the e l a s t i c s t r a i n field outside a s p h e r i c a l i n c l u s i o n in an a n i s o t r o p i c m a t e r i a l , we i n v e s t i g a t e the e l a s t i c i n t e r a c t i o n e n e r g y b e tween two inhomogeneous s p h e r i c a l p r e c i p i t a t e s e m bedded in an infinite anisotropic m a t r i x o f cubic s y m metry. When the s t r e s s free t r a n s f o r m a t i o n s t r a i n is a unif o r m d i l a t a t i o n for a homogeneous i s o t r o p i c case, the d i l a t a t i o n in the m a t r i x v a n i s h e s . A n i n c l u s i o n , t h e r e fore, e x e r t s no h y d r o s t a t i c p r e s s u r e on a n e i g h b o r i n g i n c l u s i o n and thus t h e r e is no i n t e r a c t i o n between t h e m . Consequently, the e l a s t i c i n t e r a c t i o n between two inhomogeneous s p h e r i c a l p r e c i p i t a t e s in a n i s o t r o p i c m a t r i x o r i g i n a t e s f r o m the i n h o m o g e n e i t y i t s e l f and is of a s h o r t r a n g e n a t u r e s i n c e it is p r o p o r t i o n a l to the sixth power of the d i s t a n c e b e t w e e n p r e c i p i t a t e s .6,7 On the other hand, a s Lee and J o h n s o n 9 have r e c e n t l y shown, t h e r e is a finite d i l a t a t i o n in a n a n i s o t r o p i c m a t r i x , which in t u r n p r o d u c e s an i n t e r a c t i o n of a long r a n g e o r d e r . We s h a l l f i r s t extend the d i f f e r e n c e method 6 to the a n i s o t r o p i c s y s t e m . T h i s method yields the i n t e r a c t i o n e n e r g y b e t w e e n two p r e c i p i t a t e s , by c a n culating the d i f f e r e n c e in t o t a l e n e r g y between two s e p a r a t e s y s t e m s : the f i r s t s y s t e m c o n s i s t i n g of a p a i r of m i s f i t t i n g i n c l u s i o n s and the second of a p a i r of i d e n t i c a l l y m i s f i t t i n g i n h
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