A systematic investigation of elastic moduli of Wc-Co alloys
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M e a s u r e m e n t s of Young's m o d u l u s , s h e a r m o d u l u s , c o m p r e s s i o n m o d u l u s , and P o i s s o n ' s r a t i o on WC-(1 ~ 30) wt pct Co a l l o y s w e r e c a r r i e d out by d y n am i c r e s o n a n c e method. The e f f e c t s of v o l u m e f r a c t i o n of the WC p h a s e , the c a r b o n content, and the WC p a r t i c l e - s i z e on the e l a s t i c moduli w e r e i n v e s t i g a t e d in s o m e d e t a i l . The r e s u l t shows that the v a r i o u s e l a s t i c m o d u l i of the a l l o y s depend s o l e l y on v o l u m e f r a c t i o n s of the co n st i t u en t p h a s e s and they i n v a r i a b l y fall within Hashin and S h t r i k m a n ' s bounds. M o r e o v e r , the m o d u l i a r e found to be a p p r o x i m a t e d by P a u l ' s " s t r e n g t h of m a t e r i a l s " type f o r m u l a s . It is co n cl u d ed that e l a s t i c b e h a v i o r of the a l loys can be e s s e n t i a l l y p r e d i c t e d on the b a s i s of a s i m p l i f i e d g e o m e t r i c a l m o d e l of WC p a r t i c l e s e m b e d d e d in continuous cobalt m a t r i x . O W I N G to expansion in a p p l i c a t i o n of W C - C o a l l o y s , t h e r e a r i s e s an i n c r e a s i n g d e m a n d f r o m m e c h a n i c a l e n g i n e e r i n g d e s i g n e r s for r e l i a b l e v a l u e s of v a r i o u s e l a s t i c moduli of the a l l o y s . In this r e g a r d , it i s d e s i r a b l e to e s t a b l i s h s o m e g e n e r a l r e l a t i o n s h i p s which will f a c i l i t a t e the p r e d i c t i o n of e l a s t i c moduli of the a l l o y s f r o m e l a s t i c m o d u l i of the c o n s t i t u e n t p h a s e s . Much t h e o r e t i c a l w o r k h a s b e e n c a r r i e d out f o r the d e r i v a t i o n of such relationships.X,2 A c c o r d i n g to t h e s e t h e o r i e s , e l a s t i c m o d u l i of t w o - p h a s e m a t e r i a l s a r e dependent s o l e l y on the v o l u m e f r a c t i o n s of the c o n s t i t u e n t p h a s e s . P a u l z d e r i v e s upper and l o w e r bounds on Yo u n g ' s m o d u lu s u t i l i z i n g e n e r g y t h e o r e m s of e l a s t i c i t y t h e o r y . He a l s o finds an a p p r o x i m a t e f o r m u l a of the Young's modulus of t w o - p h a s e m a t e r i a l s on the b a s i s of s o m e s i m p l e a s s u m p t i o n . The e x p e r i m e n t a l points 4's on W C - C o a l l o y s a r e found to fall w e l l within the bounds. On the b a s i s of v a r i a t i o n a l p r i n c i p l e of e l a s t i c i t y t h e o r y and a s i m p l e g e o m e t r i c a l a s s u m p t i o n of the co n s t i t u en t p h a s e s , Hashin and S h t r i k m a n 6 d e r i v e other m o r e s t r i n g e n t bounds on e l a s t i c m o d u l i which a r e c l o s e r to e a c h o th e r than the ones d e r i v e d by P a u l . K r o c k 7's m e a s u r e d Young's modulus of s e v e r a l t w o - p h a s e or pseudo t w o - p h a s e m a t e r i a l s and found that
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