The elastic strain energy of coherent ellipsoidal precipitates in anisotropic crystalline solids
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U N D E R S T A N D I N G of the e l a s t i c s t r a i n e n e r g y a s s o ciated with a c o h e r e n t e l l i p s o i d a l p r e c i p i t a t e has p r o g r e s s e d r a p i d l y d u r i n g the l a s t two d e c a d e s p r i n c i p a l l y due to the c l a s s i c a l work of Eshelby.l-3 He has shown that under the a s s u m p t i o n of i s o t r o p i c e l a s t i c i t y , the s t r e s s field i n s i d e an e l l i p s o i d a l p r e c i p i t a t e is c o n s t a n t when the s t r e s s - f r e e t r a n s f o r m a t i o n s t r a i n is constant. Subsequently Walpole, 4 Willis, S Kinoshita and Mura, 6 and Asaro and B a r n e t t 7 have proved or extended this t h e o r e m for the a n i s o t r o p i c case. To date, however, n u m e r i c a l v a l u e s of the s t r a i n e n e r g y have been obt a i n e d only for the i s o t r o p i c case by Robinson, 8 B a r n e t t et al 9 and Shibata and Ono, ~~d e s p i t e the need for c a l c u l a t i o n s b a s e d on a n i s o t r o p i c e l a s t i c i t y in m a n y alloy systems. Recently, Wert ~ has c a l c u l a t e d a n i s o t r o p i c s t r a i n e n e r g i e s a s s o c i a t e d with d i s c - s h a p e d p r e c i p i t a t e s u s ing the a p p r o a c h of K h a c h a t u r y a n , ~ under the a s s u m p tion that the p r e c i p i t a t e and m a t r i x have the s a m e e l a s tic c o n s t a n t s . However, the method of E s h e l b y does not r e q u i r e t h i s r e s t r i c t i v e a s s u m p t i o n p r o v i d e d that the p r e c i p i t a t e is e l l i p s o i d a l and the s t r e s s - f r e e t r a n s f o r m a t i o n s t r a i n is a p o l y n o m i a l function of position c o o r dinates. 7 We shall f i r s t c a l c u l a t e the a n i s o t r o p i c e l a s t i c s t r a i n e n e r g y for p r e c i p i t a t e s which a r e all e l l i p s o i d s of r e v o l u t i o n as a function of e l l i p s o i d a l a s p e c t r a t i o and the o r i e n t a t i o n r e l a t i o n s h i p b e t w e e n p r e c i p i t a t e and m a t r i x , u s i n g the method of Eshelby. The e l a s t i c c o n s t a n t s of JONG K. LEE and H. I. AARONSON are Assistant Professor and Professor, Department of MetallurgicalEngineering, MichiganTechnological University, Houghton, MI 49931 and D. M. BARNETT is Associate Professor, Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305. Manuscript submitted October 4, 1976. METALLURGICALTRANSACTIONSA
t h i r t e e n e l e m e n t s or compounds a r e used for the p r e cipitate and those of A1 and Cu a r e used for the m a t r i x . T h e s e r e s u l t s will be c o n s i d e r e d in t e r m s of c r y s t a l s t r u c t u r e , e l a s t i c c o n s t a n t s and p r e c i p i t a t e morphology, and c o m p a r e d with the few a p p l i c a b l e e x p e r i m e n t a l o b s e r v a t i o n s a v a i l a b l e . F i n a l l y , we will i n c o r p o r a t e e l a s tic s t r a i n e n e r g y into the m i n i m i z a t i o n p
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