Elastic stress at a particle-matrix interface
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THE g e n e r a t i o n of i n t e r f a c l a l s t r e s s
between particles and m a t r i x with d i s s i m i l a r t h e r m a l e x p a n s i o n coeffic i e n t s has been r e c e n t l y studied and r e p o r t e d by O l s e n e t a l . 1 The s y s t e m u n d e r i n v e s t i g a t i o n c o n s i s t e d of t h o r i a p a r t i c l e s d i s p e r s e d in a n i c k e l m a t r i x . Since n i c k e l has the higher e x p a n s i o n coefficient the p o s s i b i l i t y a r i s e s that a s u f f i c i e n t l y l a r g e r i s e in t e m p e r a t u r e of the alloy can induce a t e n s i l e s t r e s s which exc e e d s the cohesive s t r e n g t h of the m e t a l - c e r a m i c i n t e r f a c e . The d e v e l o p m e n t of this condition is f a v o r e d by a rapid heating r a t e s i n c e this d i m i n i s h e s the effect of t i m e - d e p e n d e n t s t r e s s r e l a x a t i o n p r o c e s s e s . In t h e i r e x p e r i m e n t s O l s e n et al. employed heating r a t e s of >7000~ p e r sec. F r o m t h e i r e x p e r i m e n t a l technique O l s e n et a l . w e r e a b l e to d e t e r m i n e a m i n i m u m t e m p e r a t u r e r i s e which could be a s s o c i a t e d with decohesion. Using this t e m p e r a t u r e change they c a l c u l a t e d the i n t e r f a c i a l s t r e n g t h of the alloy. It is the p u r p o s e of this note to show that the b a s i s of this c a l c u l a t i o n is i n a c c u r a t e , and l e a d s to a n u n d e r e s t i m a t e of c o h e s i v e s t r e n g t h of a p p r o x i m a t e l y 25 pct. W h e r e ap and a m a r e r e s p e c t i v e l y the l i n e a r t h e r m a l expansion c o e f f i c i e n t s of p a r t i c l e and m a t r i x m a t e r i a l the d i f f e r e n t i a l ' t h e r m a l e x p a n s i o n , 6, which develops on r a i s i n g the t e m p e r a t u r e by a n a m o u n t AT is given by: 5 = (ap--am) AT
In the a b s e n c e of r e l a x a t i o n p r o c e s s e s p r o d u c i n g p l a s t i c s t r a i n s , this d i f f e r e n t i a l t h e r m a l s t r a i n is a c c o m m o d a t e d by e l a s t i c s t r a i n i n g of both the p a r t i c l e a n d m a t r i x . If a (defined a s positive if t e n s i l e ) i s the induced n o r m a l p r e s s u r e at the i n t e r f a c e then a s p h e r i c a l p a r t i c l e of r a d i u s r s u f f e r s a u n i f o r m h y d r o s t a t i c d i l a t i o n (dr~r) given by:
w h e r e Kp is the bulk m o d u l u s of the p a r t i c l e . The e n suing t e n s i l e hoop s t r a i n ~ in the m a t r i x is then:
In the p a p e r by Olsen et al.4 the p r e s e n c e of l a t e r a l s t r a i n s , E00 and Eel, is n e g l e c t e d a s a f i r s t a p p r o x i m a t i o n of the c a l c u l a t i o n for the s t r e s s a c r o s s the i n t e r f a c e . T h i s a p p r o x i m a t i o n leads to an u n d e r e s t i m ation of 25 pct a s c o r r e c t l y shown by Speight and Lobb. s A m o r e a p p r o p r i a t e value for the i n t e r f a c
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