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Plastic Re laxat ion of Th erm oe last icS t r e s s i n A l u mi n u m/ C e r a mi c C o mp o s i t e sC . T . K I M , J . K . L E E , an d M . R . P L I C H T AThe d i s loca t ion gene r a t ion due to a the r m oe las t i c s t r e s s in 2024 Al /ce r am ic ( S iC or T iC) com pos i te sw a s s t u d ie d u s i n g t r a n s m i s si o n e l e c t r o n m i c r o s c o p y ( T E M ) . C o m p o s i t e s c o n t a i n i n g d i ff e r e n tc e r a m i c p a r t i c u l a t e s , c e r a m i c v o l u m e f r a c t i o n , a n d p a r t i c l e s i z e w e r e i n v e s t i g a t e d . D i s l o c a t i o nd e n s i t y p r o f i l e s w e r e m e a s u r e d a s a f u n c t i o n o f th e d i s t a n c e f r o m a n A 1 / c e r a m i c i n t e r f a c e a n dc o m p a r e d w i t h t h o s e c a l c u l at e d f r o m a n e l a s t o p l a s ti c i ty m o d e l w h i c h a c c o u n t s f o r t h e v o l u m ef r a c t i o n o f t h e c e r a m i c p a r t i c l e s . T h e i n t e n s i t y o f d i s l o c a t i o n g e n e r a t i o n s h o w e d a s t r o n g p a r t i c l es i z e d e p e n d e n c e : a s t h e c e r a m i c p a r t i c l e s iz e b e c a m e o f t h e o r d e r o f a m i c r o n , t h e i n t e n s i t y o fd i s l o c a t i o n g e n e r a t i o n i n c r e a s e d s i g n i f i c a n t l y . W i t h a n i n c r e a s e i n t h e v o l u m e f r a c t i o n o f t h ec e r a m i c p a r t i c l e s , t h e d i s l o c a t i o n d e n s i t y a l s o i n c r e a s e d , a n d t h e d i s l o c a t i o n s t r u c t u r e b e c a m ea m o r e t a n g l e d a r r a n g e m e n t . I f h e a t d i s s i p a t i o n w a s t a k e n i n t o a c c o u n t a s p a r t o f t h e p l a s t icw o r k , t h e p r e d i c t e d d i s l o c a t i o n d e n s i t i e s o f t h e e l a s t o p l a s t i c i ty m o d e l w e r e f o u n d t o b e i n r e a -s o n a b l e a g r e e m e n t w i t h t h e m e a s u r e d d i s l o c a t i o n d e n s i t i e s o f 1 09 t o 1 0 1 ~ c m - 2 .

I . I N T R O D U C T I O NP L A S T I C r e l a x a ti o n o f t h e e la s t ic s t r es s a n d s t r a in e n -e r g y a s s o c i a t e d w i t h b o t h t h e r m a l m i s f i t t i n g i n c l u s i o n sa n d l a t t i c e m i s m a t c h i n g p r e c i p i t a t e s h a s b e e n l o n g r e c -o g n i z e d . E x a m p l e s o f d i s lo c a t i o n g e n e r a t io n d u e t o at h e r m a l m i s f i t i n t h e v i c i n i t y o f a n i n c l u s i o n i n c l u d e t h eo b s e r v a t i o n s b y V o g e l s a n g e t a l . t~ l a n d A r s e n a u l t a n dF i s h e d 2] o n d i s l o c a t i o n g e n e r a t i o n i n t h e a l u m i n u m a l l o ym a t r i x s u r r o u n d i n g S i C w h i s k e r s o r p l a t e l e t s a n d t h ef i n d i n g b y C h a w l a a n d M e t z g e r ]3~ o f h i g h e r d i s l o c a t i o nd e n s it i e s at t h e C u / W c o m p o s i t e in t e r fa c e . T h e m e -c h a n i c a l p r o p e rt i e s o f t h e c o m p o s i t e , t h e r e f o r e , s h o u l db e a f f e c t e d b y t h e m a g n i t u d e a n d e x t e n t o f t h e d i s l o -ca t ion gene r a t ion tha t takes p lace in the so f t m a t r ix a r oundh a r d p a r t i c l e s a s a r e s u l t o f t h e r e l a x a t i o n o f t h e r m a l m i s -f i t s tr e s s i n t h e i n t e r f a c e r e g i o n .

T h e p l a s t i c d e f o r m a t i o n o f a m i s f i t t i n g i n c l u s i o n i n a ni n f i n i t e m a t r i x h a s b e e n a n a l y z e d v i a a c o n t i n u u mm e cha nic s m o de l t4,5,61 a s we l l a s a d i s loc a t ion loop -pun chin g m o de l . 17,sl Le e e t a l . t41 we r e ab le to d ed uce ap a r t i c l e s i z e - d e p e n d e n t y i e l d s t r e s s f r o m t h e A s h b y -J o h n s o n m o d e l is] f o r t h e n u c l e a t i o n o f a d i s l o c a t i o n f r o ma p a r t i c l e / m a t r i x i n t e r f a c e . T h e y s h o w e d t w o e f f e c t i v ey i e l d s tr e ss e s w h i c h d e p e n d u p o n t h e c o h e r e n c y o f t h ep a r t i c l e / m a t r i x i n t e r f a c e . O n e i s e s s e n t i a l l y i n d e p e n d e n to f t h e p a r t i c l e s i z e a n d e q u a l t o t h e t h e o r e t i c a l y i e l ds t r e n g t h , / z / 2 z r , w h e r e / x i s t h e m a t r i x s h e a r m o d u l u s .T h e o t h e r e f f e c t i v e y i e l d s t r e ss a p p l i e s t o i n c o h e r e n t p a r -t i c le s a n d i s a s t r o n g f u n c t i o n o f p a r t i c l e s i z e . W h e n t h ep a r t i c l e is v e r y s m a l l , s a y o f t h e o r d e r o f 1 0 t o 2 0 n m ,th i s s t r e s s appr oaches the theor e t i ca l y ie ld s t r eng th . Whent h e p a r ti c l e s iz e b e c o m e s o f t h e o r d e r o f a m i c r o n , t h ee f f e c t i v e y i e l d s t re s s a p p r o a c h e s a n a v e r a g e ( l o w -t e m p e r a t u r e ) m a c r o s c o p i c y i e l d s t r es s .S i n c e i n m e t a l / m a t r i x c e r a m i c c o m p o s i t e s ( M M C ' s )t h e p a r t i c l e / m a t r i x i n t e r f a c e i s c o n s i d e r e d t o b e i n c o -

C.T. KIM, Gra dua te S tude nt , J .K . LEE, Profe s sor , a nd M.R.PLICHTA, Assoc ia te Profe s sor , a r e w i th the De pa r tm e nt ofMe ta l lurg ic a l Engine e r ing , Mic higa n Te c hnologic a l Unive r s i ty ,Houghton , MI 49931.Manuscript submitted June 7, 1989.

h e r e n t , i t i s e x p e c t e d t h a t t h e d i s l o c a t i o n g e n e r a t i o n d u et o a t h e r m a l e x p a n s i o n m i s m a t c h w o u l d d e p e n d s t r o n g l yo n t h e s i ze o f t h e c e r a m i c p a r ti c l es i n M M C ' s . A l t h o u g ht h e c o n t i n u u m m e c h a n i c s a n d d i s l o c a t i o n l o o p - p u n c h i n gm o d e l s a p p e a r t o y i e l d f a i r l y r e a s o n a b l e p r e d i c t i o n s f o rt h e e x p e r i m e n t a l l y o b s e r v e d d i s l o c a t i o n g e n e r a t i o n , t od a t e t h e r e h a v e b e e n n o s y s t e m a t ic e x p e r i m e n t a l s t u d ie so n t h e p a r t i c l e s i z e - d e p e n d e n t d i s l o c a t i o n g e n e r a t i o n .A d d i t i o n a l l y , t h e p r e v i o u s t h e o r e ti c a l m o d e l s w e r e b a s e don a s ing le inc lus ion em bedded in an in f in i t e m a t r ix , thusn e g l e c t i n g t h e e f f e c t o f a m u l t i p a r t i c le d i s t r i b u t i o n .T h e r e f o r e , a th e o r e t i c a l m o d e l f o r t h e p l a s t i c r e l a x a t i o ni s d e s ir e d t o a c c o u n t f o r t h e v o l u m e f r a c t i o n o f c e r a m i cp a r t i c l e s i n c o m p o s i t e s .T h e p u r p o s e o f t h i s w o r k w a s t o e x a m i n e e x p e r i m e n -t a l l y t h e d i s l o c a t i o n g e n e r a t i o n d u e t o t h e d i f f e r e n t t h e r -m a l c o n t r a c t i o n s i n A 1 / c e r a m i c (S i C o r T i C ) c o m p o s i t e s .M a t e r i a l s c o n t a i n i n g d i f f e r e n t c e r a m i c p a r t i c u l a t e s , v o l -u m e f r ac t i o n s o f t h e c e r a m i c , a n d s i ze s o f c e r a m i c w e r ei n v e s ti g a t e d . A n e f f o r t w a s a l s o m a d e t o d e v e l o p a t h e -o r e t i c a l m o d e l f o r t h e p l a s t i c r e l a x a t i o n a r o u n d a s p h e r -i c a l p a r t i c l e , w h i c h i n c o r p o r a t e s t h e v o l u m e f r a c t i o n o fc e r a m i c p a r t i c l e s .

I I. T H E O R YF o r m a t h e m a t i c a l s i m p l i c i t y , w e a s s u m e t h a t a l l o f th ec e r a m i c p a r t i c l e s a r e o f s p h e r i c a l s h a p e w i t h r a d i u s a ,a n d f u r t h e r , t h a t a g i v e n p a r t i c l e i s s u r r o u n d e d b y u n i -f o r m l y d i s t r i b u t e d n e i g h b o r i n g p a r t i c l e s. T h i s a s s u m p -t i o n a l l o w s u s t o u s e s p h e r i c a l c o o r d i n a t e s w h o s e o r i g i ni s at t h e c e n t e r o f a c e r a m i c p a r t i c l e w i t h r a d i a l s y m -m e t r y , a s s h o w n s c h e m a t i c a l l y i n F i g u r e 1 . I n t h e m a n -n e r o f L e e e t a l . ,l a ] p e r f e c t p l as t ic b e h a v i o r i s a s s u m e d

f o r t h e m a t r i x p h a s e , a n d t h e c r y s t a l l o g r a p h i c n a t u r e o fp l a s t i c f l o w i s n e g l e c t e d . H o w e v e r , t h e p r e s e n t m o d e la c c o u n t s f o r t h e v o l u m e f r a c t i o n o f c e r a m i c p a r t i c l e s .T h u s , t h e m a t r i x i s c o n s i d e r e d t o y i e l d u n d e r t h e c o n -d i t i o n o f a c o n s t a n t y i e l d s t r e s s, a n d t h e y i e l d i n g i s c o n -s ide r ed independ ent o f the s t r e s s ax i s . W e sha l l f i r s t ob ta int h e s t r e ss a n d s t r a in a s s o c i a t e d w i t h a m i s f i t ti n g s p h e r i c a lh a r d p a r t i c l e i n t h e a b s e n c e o f p l a s t i c r e l a x a t i o n , i . e . ,

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@ @Fig . 1 - - A sc he m a t ic d iagra m for a n ide a l i z e d c e ra m ic c onf igura tionand plas t ic zone .u n d e r p u r e l y e l as t ic c o n d i t io n s . T h e n , e m p l o y i n g t h ef o r e g o i n g a s s u m p t i o n s a n d c o n t i n u u m p l a s t i c i t y t h e -o r y , [9'101 t he so l u t i on f o r t he e l a s t o p l a s t i c d e f o r m a t i o n i sd e t e r m i n e d w h e n p l a s t i c r e l a x a t i o n t a k e s p l a c e i n t h em a t r i x .A . P u r e E l a s t i c S t a t e

C o n s i d e r a t h i c k h o l l o w s p h e r e w h o s e i n t e r n a l s u r f a c eo f r a d i u s a i s s u b j e c t e d t o a p r e s s u r e , P 1 , b y t h e t h e r m a lm i s f i t ti n g c e r a m i c p a r t ic l e a n d w h o s e e x t e r n a l s u r f a c e o fr a d i u s b i s s u b j e c t e d t o a p r e s s u r e , P 2 , b y t h e s u r r o u n d -i n g c e r a m i c p a r t i c l e s , a s s h o w n i n F i g u r e 1 . B e c a u s e o ft h e r a d i a l s y m m e t r y a p p r o x i m a t i o n , t h e t a n g e n t ia l d i s -p l acement s , a s we l l a s t he shea r s t r e s ses and shea r s t r a i ns ,a r e a ll z e r o , a n d t h e r a d i a l d i s p l a c e m e n t , u , i s a f u n c t i o no f th e r a d i a l d i s t a n c e , r . F u r t h e r , t h e e q u i l ib r i u m e q u a -t io n s i n th e a b s e n c e o f b o d y f o r c e s r e d u c e t o

d o t 2 ( o r - o '0 )- - + - 0 [11d r rw h e r e o r a n d o 0 a r e r a d i a l a n d t a n g e n t i a l s t r e s s c o m -p o n e n t s , r e s p e c t i v e l y . T h e s t r a i n s a r e r e l a t e d t o t h e r a -

d i a l d i s p l a c e m e n t , u , b yd u ue r - d r ' eo r [21

F o r t h e t h i r d d i a g o n a l c o m p o n e n t s , o 6 = o '0 a n d e 6 = e 0a r e im p l i ci tl y a s s u m e d . H o o k e ' s l a w p r o v i d e s1er = E (Or - - 2 v ~ 1 7 6 a " A T [ 3a ]

1e o = E [ - V ~ + (1 - v ) O ' o ] + a " AT [ 3b]w h e r e o t i s t h e t h e r m a l e x p a n s i o n c o e f f i c i e n t a n d A T i st h e d i ff e r e n c e b e t w e e n t h e t e m p e r a t u r e o f i n te r e s t ( r o o mt e m p e r a t u r e i n t h i s s t u d y ) a n d t h e s o l u t i o n i z i n g t e m p e r -a t u r e f o r t h e c o m p o s i t e s . W e n o t e t h a t w i th t h e a b o v ed e f i n i t i o n f o r t h e s t ra i n , t h e r e f e r e n c e s t a t e is th e o n e a tt h e s o l u ti o n i z in g t e m p e r a t u r e . T h e s t ra i n c o m p a t i b i l i t yr e l a t i o n s h i p i s

r deoe~ = e0 + - - [4]d r

S ubs t i t u t ion o f the s t r e s s - s t r a i n E q . [ 3 ] i n to E q . [ 41 y i e l ds1 d 22 d r 2 ( r 2o r ) - o r = 0 [ 5 ]

A f t e r i n t e g r a t i o n , t h e r a d i a l s t re s s i s g i v e n b yBor = A - - - [ 6 ]F 3

a n d f r o m E q . [ 1 ] t h e t a n g e n t i a l s t r e s s isBo0 = A + - - [ 712 r 3

T h e b o u n d a r y c o n d i t i o n s a p p r o p r i a t e fo r sm a l l c e r a m i cv o l u m e f r a c t i o n s a r e

O r ( r = a ) = - - P l [ 8 a ]o ' r ( r = b) = - P 2 [ 8b]u ( r = b ) = a , , . A T . b [ 8c ]

E q u a t i o n [ 8 c] r e p r e s e n t s t h e d i s p l a c e m e n t a t t h e m i d -p o i n t b e t w e e n t w o n e i g h b o r i n g p a r t ic l e s . S i n c e t h e s t r e s sm o d e i s c r i ti c a l ly i n f l u e n c e d b y t h e b o u n d a r y c o n d i t i o n s ,t h e i r v a l i d i t y ( e s p e c i a l l y E q s . [ 8 b ] a n d [ 8 c ] ) w i l l b e f u r -t h e r d i s c u s s e d i n S e c t i o n V .

S u b s t i t u t i o n o f E q s . [ 8 a ] a n d [ 8 b ] in t o E q s . [ 6 ] a n d[ 7 ] l e a d s t o t h e s t r e s s e s i n t e r m s o f t h e c o n t a c t p r e s s u r e s ,P 1 and P 2 , a s

Pl a 3 - P2 b 3 a 3 b 3 ( p 1 - Pc )O'r - - b3 _ a3 (b 3 _ a3 )r 3 [9a]

Px a 3 - P2 b 3 a 3 b 3 ( p 1 - P 2)+ [ 9b]o0 = b3 _ a3 2 ( b3 _ a3 ) r 3S u b s t i t u t i n g E q s . [ 9 a ] a n d [ 9 b ] i n t o E q . [ 3 b ] y i e l d sd i s p l a c e m e n t a s

(1 + v m ) a 3 b 3 ( P 1 - P 2 )u = 2 E m ( b 3 - a 3 ) r 2+ ( " ( 1 - 2 v ~ ) ( P l a 3 - P 2 b 3 ) )--~ m (~ ~ a T + a m ' A T r [ 10]

U t i li z i n g th e b o u n d a r y c o n d i t i o n o f E q . [ 8 c ], P 2 is g i v e n b y3a3( 1 - v m ) e 1

P 2 = a 3 + 2 b 3 + V m ( a 3 _ 4 b 3 ) [ 11]F u r t h e r s u b s t i t u t io n o f E q . [ 1 1 ] in t o E q s . [ 9 a ] a n d [ 9 b ]y i e l d s

- P l a 3 ( 1 + V , n )Or = a3 + 2b 3 + um( a 3 _ 4b3)

203b3(1- - 2Vm)__P2 ( 1 )- a 3 + 2 b 3 + l i r a ( a 3 _ 4b 3) ~-~ [12 a]

- p l a 3 ( 1 + v m)O 0 = a 3 + 2 b 3 + l t m ( a 3 - 4 b 3)

a 3 b 3 ( 1 - 2 v , , ) P , ( 1 )+ a 3 ; ~ ; ; m ~ a 3 ~ 4 b 3 ) ~ [ 1 2 b ]

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W i t h i n t h e p l a s t i c z o n e , t h e s t r a i n s a r e t h e s u m o f th ep l a s t i c a n d e l a s t i c s t r a i n s . S i n c e t h e e l a s t i c s t r a i n s a r er e l a t e d t o s t r e s s e s b y H o o k e ' s l a w , w e c a n w r i t e

d u 1. . . . ( 0 " r - - 2 t ' m tr O ) + a m " A T + e e r [25]e~ dr E m

u 1e0 . . . . [ - Vmt r ~ + (1 - Vm)tro]r Em+ a m 9 A T + ePo; a


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T h e e l a s t ic w o r k p e r u n i t v o l u m e o f a n e l e m e n t l o c a t e da t d i s t a n c e r i s

T a b l e I . S o m e V a l u e s o fI n t e r p a r t i c l e S p a c i n g ( E q u a t i o n [ 4 0 ] )1 ek b/awe( r ) = 2 [Orr(er - - ep - - O/m" A T) 0.001 4.497

0.002 3.591+ 2o '0(e0 - e p - t~m" A T) ] 0.005 2.6860 .010 2 .183_ 0 .020 1 .8026( 1 - 2vm ) t r2 [In ( r / a ) ] 2 0 .050 1 .450Em 0.100 1.275

2( 1 - 2Urn) ( 2 t r y - - 3P 0 t r y I n ( r / a ) 0.200 1.160+ 0.500 1.073Em1 2 ( 1 - 2Vm)O'yP~

3 ]- (1 - v,.)O'y - ~ (1 - 2 v . . ) P~ [381S i mi l a r l y , t he e l a s t i c s t r a i n ene r gy pe r un i t vo l ume s t o r edo u t s i d e t h e p l a s t i c z o n e i s

(1 + Yr.) Ory (1 +toOUt(r -- ( r p / b ) 6 + ( r p / r ) 66( 1 - 2 vm ) E~ 3 E , ,

[ 39]D . R e l a t i o n s h i p b e t w e e n 2 b a n d V o l u m e F r a c t i o n , 4 )

I t i s n o w n e c e s s a r y t o f i n d a r e l a t i o n s h i p b e t w e e n t h ec e r a m i c v o l u m e f r a c t i o n , ~ b, a n d i n t e r p a r t i c l e d i s t a n c e ,2 b . F o r a p o l y d i s p e r s e d s y s t e m o f p a r t ic l e s , a n a c c u r a t er e l a t io n s h i p i s d i ff i c u lt to o b t a i n . H o w e v e r , w h e n a l l o ft h e p a r t ic l e s a r e a s s u m e d t o h a v e t h e s a m e r a d i u s , a , i tc a n b e s h o w n f121 t h a t f o r r a n d o m l y d i s t r i b u t e d p a r t i c l e s ,t h e a v e r a g e h a l f d i s t a n c e , b , b e t w e e n t w o n e i g h b o r i n gp a r t i c l e s i s

a " e x p ( 8~ b) _ I X-2/3e-Xd xb = a + 6q~ l / 3 . .~ , [40]E q u a t i o n [ 4 0] i s k n o w n t o b e a c c u r a t e w h e n t h e v o l u m ef r a c ti o n , ~ b, a p p r o a c h e s z e r o . T h u s , t h e m a x i m u m ~b v a l u ei n th i s s t u d y i s li m i t e d t o 0 . 1 . T a b l e I l i s t s ' s o m e b / ar a t i o s o b t a i n e d w i t h E q . [ 4 0 ] .

I I I . E X P E R I M E N T A L P R O C E D U R E SC o m p o s i t e m a t e r i a ls o f t h e p r e a l l o y e d 2 0 2 4 A l p o w d e r s

a n d c e r a m i c p a r t i c l e s w e r e f a b r i c a t e d v ia h o t p r e s s i n ga n d h o t r o l li n g . T h e a v e r a g e s i z e s o f th e c e r a m i cp a r t ic l e s a r e l i s te d i n T a b l e I I . T h e c e r a m i c m a t e r i a lsw e r e b l e n d e d w i t h p r e a ll o y e d 2 0 2 4 A 1 p o w d e r s , v a c u u mh o t p r e s s e d a t 4 9 5 ~ f o r 2 h o u r s a s 5 0 - m m - d i a m e t e r ,1 3 - m m - l o n g d i s k s u n d e r a p r e s s u r e o f 3 0 0 0 p s i . T h eh o t - p r e s s e d s a m p l e s w e r e t h e n w r a p p e d w i t h t h i n A 1 f o i l si n o r d e r t o p r e v e n t a n y r e a c t i o n b e t w e e n t h e c o m p o s i t e sa n d t h e c o p p e r t u b e d u r i n g h e a t i n g a n d h o t r o l li n g , e n -c a p s u l a t e d w i th c o p p e r t u b e u n d e r v a c u u m , h e a t - t r e a t e dt o 5 1 5 ~ ( a b o v e t h e s o l id u s t e m p e r a t u r e ) , a n d h o t ro l l e dt o 2 - m m - t h i c k s h e e t s in f i v e p a s s e s . T h r e e d i f f e r e n t v o l -u m e p e r c e n t s o f c e r a m i c p a r t i c le s w e r e c o n s i d e r e d : 0 ,2 , a n d 1 0 p c t . T h e 0 v o l p c t s a m p l e s e r v e d a s a c o n t ro l .

T h e s a m p l e s w e r e s o l u t io n i z e d a t 4 9 0 ~ f o r 1 h o u r i na s a lt b a t h a n d s u b s e q u e n t l y w a t e r q u e n c h e d .

D i s l o c a t i o n s t r u c tu r e s w e r e a n a l y z e d f o r b o t h t h e c o m -p o s i t e s a n d t h e c o n t r o l a l l o y u s i n g a P H I L I P S * E M 3 0 1

*PHILIPS is a trademark of Phil ips Instruments Corporation,Mahwah, NJ.t r a n s m i s s i o n e l e c t r o n m i c r o s c o p e o p e r a t i n g a t 1 0 0 k V .T r a n s m i s s i o n e l e c t r o n m i c r o s c o p y s a m p l e s w e r e c h e m -i c a l ly t h in n e d w i t h a 1 0 p c t N a O H , 9 0 p c t H 2 0 s o l u ti o nt o a t h i c k n e s s o f le s s t h a n 1 5 0 / x m t o p r e v e n t i n tr o -d u c t i o n o f d i s l o c a ti o n s d u r i n g a t h i n n in g p r o c e s s . T h e n ,3 - m m - d i a m e t e r d i s k s w e r e p r e p a r e d a n d j e t p o l i s h e d w i t ha 3 0 p c t n i t r ic a c i d , 7 0 p c t m e t h a n o l s o l u t i o n u s i n g ap o t e n t i a l o f 3 5 V a t a t e m p e r a t u r e o f a b o u t - 3 0 ~ T h ed i s l o c a ti o n d e n s i t y , p , w a s d e t e r m i n e d f r o m a m i c r o -g r a p h u s i n g t h e f o r m u l a , p = 2 N / L t , w h e r e N i s t h en u m b e r o f d i s l o c a t i o n i n t e r se c t i o n s w i t h a g r i d l i n e o fl e n g t h , L , d i v i d e d b y t h e m a g n i f i c a t i o n , a n d t i s t h et h i c k n e s s o f t h e s a m p l e , c~31 I t s h o u l d b e n o t e d t h a t t h ed i s l o c a t io n d e n s i t y t h u s o b t a i n e d c a n b e l o w e r t h a n a na c t u a l v a l u e b e c a u s e o f a lo w o p e r a t i n g p o w e r o f th eE M 3 0 1 m i c r o s c o p e . T h e t h i n f o i l t h i c k n e s s w a s m e a -s u r ed u s i n g a J E O L - 1 0 0 C X t r a n sm i s s io n e l e c t r o n m i c r o -s c o p e o p e r a t i n g a t 1 2 0 k V . W i t h o u t l i q u i d N 2 c o l d s t a g e ,t h e f o il s u r f a c e w a s c o n t a m i n a t e d i n 5 to 1 0 m i n u t e s b yt h e f o c u s e d e le c t ro n b e a m . T h e t h i c k n e ss w a s m e a s u r e df r o m a m i c r o g r a p h s h o w i n g t h e p r o j e c t e d l e n g t h b e t w e e nt h e c o n t a m i n a t e d s p o t s o f u p p e r a n d l o w e r f o i l s u rf a c e su s i n g t = d / s i n 0 , w h e r e d i s t h e p r o j e c t e d l e n g t h a n d0 i s th e p r i m a r y t i lt in g a n g l e , e q u a l t o ~ 4 0 d e g i n t h i ss t u d y . T h e t h in f o i l t h i c k n e s s w a s m e a s u r e d a s a f u n c t io no f d i s ta n c e f r o m t h e A 1 / c e r a m i c i n t er f a c e . E a c h t h i c k -n e s s v a l u e a t e a c h p o i n t w a s i n c o r p o r a t e d f o r m e a s u r e -m e n t s . A c i r c u l ar t e s t li n e w a s u s e d t o e l i m i n a t e t h e e f f e c to f a n y p r e f e r r e d o r i e n t a t i o n o f a d i s l o c a t i o n s t r u c t u r e ,a n d t h e l e n g t h o f a g r i d l i n e w a s 7 6 c m .

T a b l e I I . A v e r a g e S i z e a n dV o l u m e F r a c t i o n o f M a t e r i a l sM at r i x Cer ami c Vol P c t

2024 AI S i C ( 22 / . t m) 0 , 2 , 102024 AI S i C ( 14 / zm ) 0 , 2 , 102 0 2 4 A l S i C ( 5 / t m ) 0 , 2 , 1 02024 AI T i C ( 30 / zm ) 0 , 2 , 10

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I V . R E S U L T SA . S t r e s s a n d S t r a i n E n e r g y D i s t ri b u t i o ni n C o m p o s i t e s

M a t e r i a l p a r a m e t e r s u s e d f o r t h e c a l c u l a t i o n a r e g i v e ni n T a b l e I I I . F i g u r e 2 p r e s e n t s th e n o r m a l i z e d s t r e s s w i t hr e s p e c t t o t h e y i e l d s t r e ss o f t h e m a t r i x a s a f u n c t i o n o ft h e n o r m a l i z e d d i s t a n c e f r o m t h e c e r a m i c c e n t e r i n t h eA I / S i C s y s t e m . T h e v a l u e o f u n i t y f o r r / a c o r r e s p o n d st o t h e m a t r i x / c e r a m i c i n t e r f a c e . T h e s t r e s s d i s t r i b u t i o n si n t h e p u r e l y e l a s t i c s t a t e a n d i n t h e e l a s t o p l a s t i c s t a t eare disp laye d for ~b = 0 (a s ingle par t ic le) and ~b = 0.00 3,r e spe c t ive ly . Ma t r ix s t r e s s com po nen ts , o -~ and t r0 , a ter e l a t i v e l y s h o r t - r a n g e d , d e c a y i n g e f f e c t i v e l y t o z e r o a ta p p r o x i m a t e l y r - - 3 a . I t is n o t e d t h a t t h e d i f f e r e n c e i nthe s t r e s s s ta te be tween the two cases i s s ign i f i can t wi th int h e c e r a m i c p a r t i c l e as w e l l a s i n t h e p la s t i c z o n e b u t n o tin the e la s t i c r eg ion of the m a t r ix . I ns ide the p la s t i c zone ,t h e t a n g e n t i a l s t re s s i s q u i te d i f f e r e n t , w i t h t h e s i g n b e i n gr e v e r s e d f r o m i ts c o u n t e r p a r t o f th e p u r e e l a s t i c c a s e .T h e p l a s t i c z o n e s i z e i s s e e n t o i n c r e a s e w i t h i n c r e a s i n gc e r a m i c v o l u m e f r a c t i o n .I n F i g u r e 3 , t h e n o r m a l i z e d s t r a i n e n e r g y d e n s i t y i sp l o t t e d a s a f u n c t i o n o f th e n o r m a l i z e d r a d i a l d i s t a n c ef o r t h e t h r e e d i f f e r e n t v o l u m e f r a c t i o n s . T h e s t r a i n e n e r -g i e s w e r e n o r m a l i z e d w i t h r e s p e c t t o t h e p l a s t i c s t r a i ne n e r g y p e r u n i t v o l u m e f o r ~b = 0 a t r = a . W h e n t h ep l a s t i c z o n e s o v e r l a p p e d e a c h o t h e r , t h e p l a s ti c s t r a in e n -e r gy dens i t i e s wer e ca lcu la ted th r ough supe r pos i t ion . Notet h a t t h e p l a s t i c s t r a i n e n e r g y d e n s i t y d e c r e a s e s i n v e r s e l yw i t h r 3 ( E q . [ 3 6 ] ) a n d i s s e e n t o i n c r e a s e w i t h v o l u m ef r a c t i o n o f c e r a m i c . I n t h e c a s e o f ~b = 0 , t h e t h e o r e t i c a lp l a s t i c z o n e s i z e , r p , i s f o u n d t o b e 2 . 4 1 a f o r A 1 / S i Cc o m p o s i t e s a n d 2 . 2 7 a f o r A 1 / T i C c o m p o s i t e s . T h e t h e -o r e t i c a l p l a s t i c z o n e s i z e , rp , i s f o u n d n o t t o i n c r e a s er ap id ly w i th an in c r ease in ~b, bu t p la s t i c zon es s ta r t too v e r l a p e a c h o t h e r a t ~b = 0 . 0 0 7 f o r A 1 / S i C c o m p o s i t e sas rp ~- b a t t h i s v o l u m e f r a c t i o n .B . D i s l o c a t i o n G e n e r a t i o n

D i s l o c a t i o n d e n s i t y g r a d ie n t s h o w i n g a h i g h e r d e n s i t ynea r the A1/S iC in te r f aces i s obse r ved in the a s - quenched~b = 0 . 0 2 a n d 0 . 1 A 1 / S i C c o m p o s i t e s ( F i g u r e s 4 a n d 5 ) .L i n e , h e l i c a l , a n d l o o p - t y p e d i s l o c a t i o n s c a n b e s e e n i nt h e m a t r i x . F o r A 1 / S i C c o m p o s i t e s w i t h ~b = 0 . 0 2 , t h ed i s l o c a t i o n s n e a r t h e A 1 / S i C i n t e r f a c e a r e g e n e r a l l y i nt a n g l e d a r r a n g e m e n t s , w h e r e a s d i s l o c a t i o n s a w a y f r o mt h e S i C p a r t i c l e a r e n o t , a s s h o w n i n F i g u r e 4 . A s t h eS iC vo lum e f r ac t ion inc r eased , d i s loca t ion dens i ty s eem edt o i n c r e a s e , a n d t h e d i s l o c a t i o n s a l s o o c c u p i e d a w i d e ra r e a ( F i g u r e 5 ) . A s e x p e c t e d , m o r e d i s l o c a t i o n s w e r eg e n e r a t e d a t a c o m e r o f a S i C p a r t i c l e , w h e r e a h i g h e rs t r e s s w a s c o n c e n t r a t e d d u r i n g c o o l i n g ( a r r o w m a r k e d ) .

A , ' 2 0 2 4 - - S i C

. . . . . . . . . ..++ ;,~/ /9 / / - + - + : 0 0 0 3 ) : 0 p u r e e l a s t i c 9/ /9 / / - - 0 ) e l a s t o - p l a s t i c 9/ / 0 . 0 0 3/ /

- 1 C / // /

-1,' I +r/ a

Fig. 2- -T he stress distribution in the purely elastic state and in th eelastoplastic state in A1/SiC composites. The matrix/ceramic inter-face locates at r/a = 1. The volume fraction effects are displayed for~b = 0.003. The stre sses are normaliz ed with respec t to try.

S i m i l a r to t h e f i n d in g s o f V o g e l s a n g e t a l . , m t h e i n -t e n s i t y o f d i s l o c a t i o n g e n e r a t i o n a t t h e A 1 / c e r a m i c i n t e r -f a c e i s r e l a t e d t o t h e s i z e a n d s h a p e o f S i C p a r t i c l e s .F i g u r e 6 re v e a l s a l o w i n t e n s i t y f o r d i s l o c a t i o n g e n e r a -t i o n i n t h e v i c i n i t y o f a s m a l l S i C p a r t i c l e w i t h d i a m e t e r

A

L .t -t -t~

6

A 1 2 0 2 4 / S I C. . . . e l a s t i c- - p l a s t i co1 : 0 . 02 : 0 .0 23 : 0 . 1

2 [ . . . . . 3 . . . . . .

L 2 2 r ? "r / a

Fig. 3--The strain energy distribution in the elastoplastic state in AI/SiC composites. The strain energies are normalized with respect tothe plastic strain energy per unit volume at r = a for the case of ~b = 0.The volume fraction effects are displayed for q~ = 0.02 and 0.1.

T a b l e I I I . M a t e r i a l P a r a m e t e r sYield Strength Elast ic Modu lus Thermal Expansion Therma lMaterial (MPa) (GPa) Poiss on's Ratio Co efficien t (K -a) M isfit Strain

2024 A1 76 73 1/3 24 X 1 0 - 6 - -S iC - - 510 0 .19 4 . 4 X 1 0 - 6 0.0091T i C - - 3 1 0 0 .1 9 7 . 4 X 10 -6 0 . 0077

678--V OLU ME 21A, MARCH 1990 METALLURGICAL TRANSACTIONS A


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Fi g . 4 - -D i s l o c a t i o n g e n e ra t io n n e a r a n A1 / S i C i n t e r fa c e fo r to = 0 .0 2 .Th e c e ra mi c p a r t i c l e d i a me t e r i s 7 .4 ~ m . No t e t h e sma l l i n t e rme t a ll i cc o mp o u n d s a s so c i a t e d wi t h a 2 0 2 4 A1 a l l o y .

0.8 /xm. No s igni f icant d is loca t ion dens i ty gradient i sobservable in this case . As the par tic le diameter in-c reases ( to -7 /x m in F igure 4 and - -1 7/x m in F igure 7) ,the intensity of dislocation generation increases signif i-cant ly . In compar ing the present r e su l t wi th those ofVoge lsang et al. tl] and Arsen ault and Fisher , t2] an inter-es t ing obse rva t ion i s that muc h m ore he l ica l d is loca t ionsseemed to be genera ted in the 2024 A1 matr ix than inthe 1100 AI or 6061 A1 matrices. No detailed studies wereper formed to f ind the cause of th is morphologica l d i f -fe rence , but one poss ib le r eason i s tha t for the present2024 A1 a l loy , the wa te r quenching r ight a f te r a so lu-t ioniz ing t r ea tment should have produced quenched- invacanc ies or vacancy loops , which migh t have he lped tonucleate helical dislocations. The 1100 and 6061 A1 alloysamples , however , were furnace - or a i r -cooled a f te rann eali ng . [1,2]

Figure 8 shows dislocatio n generation in a ~b = 0.0 2 A 1/TiC composi te , in which the pa r t ic le s ize i s 16 .2 /xm.As compared wi th an AI /S iC composi te in F igures 4 or

Fi g . 6 - -D i s l o c a t i o n g e n e ra t i o n i n t h e v ic i n i t y o f a sma l l S i C p a r ti c l efo r t o = 0 .0 2 . Th e c e ra mi c d i a me t e r i s 0 .8 / z m.

7, the dislocation density is found to be significantly less.This dec rease can be a t t r ibuted to smal le r d i f fe rences inboth the the rmal expans ion coe f f ic ien t and e la s t ic mod-ulus be tween A1 and TiC (Table I I I ) .Dis loca t ion dens i ty i s p lo t ted as a func t ion of a nor -mal ized d is tance f rom the A1/ce ramic in te r face inF igure 9 . A lso inc lud ed a re theore t ica l dens i ty curvesobta ined wi th the p las t ic s t r a in ene rgy te rm of Eq. [37].Theore t ica l d is loca t ion dens i ty i s approximated as Pt -~Po + fa r~ where/90 is the dislocation den sityin the control a l loy (-5 x 108 cm-2 ) , /z is the matr ixshear modu lus, b is the Burgers ve ctor , and f is a par-t i t ioning pa ramete r which re f lec ts the f rac t iona l amountof the plastic stra in energy stored as dislocations. Inplas t ic de format ion , pa r t of the p las t ic work i s d iss i -pa ted , and de fec ts o the r than d is loca tions a re produced.H ow e ve r , i n t he a bse nce o f ou r de t ai l e d know le dge onthe pa r t it ioning of a p las t ic w ork , the pa ramete r , f , wasarbitrar i ly taken as 0.5 in Figure 9. Each set of the ex-pe r imenta l da ta in F igure 9 represents dens i ty measure -ments f rom a TEM mic rograph.

V . D I S C U S S I O NResidual stress is an inherent character ist ic for com-posite materia ls. Wh en composites are cooled to the room

Fi g . 5 - -D i s l o c a t i o n g e n e ra ti o n n e ar a n M / S i C i n t er fa c e for tO = 0 .1 .Th e c e ra mi c p a r t i c l e d i a me t e r i s 7 .4 ~ m. F i g . 7 - -Di s l o c a t i o n g e n e ra t i o n i n t h e v i c i n i t y o f a l a rg e S i C p a r t i c l efo r t o = 0 .0 2 . Th e c e ra mi c d i a me t e r i s 1 7 .4 / z m.

METALLURGICAL TRANSACTIONS A VOLUME 21A, MARCH 1990-- 679


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F i g . 8 - - D i s l o c a t i o n g e n e r a t i o n i n a n A 1 / T i C c o m p o s i t e w i t h ~b = 0 . 0 2 .T h e p a r t i c l e s i z e i s 1 6 . 2 / ~ m .

tempera ture f rom the fabr ica t ion or so lu t ioniz ing tem-perature , residual stresses are introduced into thecomposi te s due to a mismatch in the rmal expans ioncoef f ic ien t be tween the ma tr ix and ce ramic phases .Led bett er an d Austin I~5~ meas ured the averag e residuals t re ss for 6061 AI composi te s conta in ing 30 vol pe t S iCpartic les v ia X-ray d i f f rac t ion ana lys is . A hydros ta t ictens i le s t r e ss was found in the a luminum matr ix , whi lea hydros ta t ic compress ive s t r e ss was found in the S iCpartic les. Arsen ault and T aya tt61 obs erved that the tensiley ie ld s t r e ss was grea te r than the compress ive y ie ld s t r e ssby - 1 3 MP a for 6061 A1 composites containing 20 vol petspherical SiC par tic les, suggesting a possibil i ty that a

2 0

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01 J I-" a a I i n J1.0 1.5 2.0Control Alloy " + + + to" 13 a

r / aF i g . 9 - - D i s l o c a t i o n d e n s i t y p r o f i l e a s a f u n c t i o n o f a n o r m a l i z e d d i s -t a n c e f r o m a n A l / c e r a m i c i n t e r f a c e . T h e t h e o r e t i c a l l y c a l c u l a t e d d e n -s i t i e s a r e d i s p l a y e d f o r ~ = 0 . 0 2 a n d 0 . 1 A I / S i C c o m p o s i t e s .

compressive residual stress existed in the matr ix. Thougha t f i rs t g lance the two measurements appear to cont rad ic teach o the r , we should note tha t r es idua l s t r e sses depe ndon seve ra l f ac tor s , such as the volume f rac t ion and d is -tr ibution of cera mic par tic les f~Tj and also possi bly on thede ta i ls o f hea t t r ansfe r dur ing coo l ing .When the ce ramic volume f rac t ion i s sma l l , we mayassume tha t each ce ramic pa r t ic le i s sur rounded by thematr ix phase , and the ave rage in te rpa r t ic le spac ing i s no tinf luenced by the presence of a ce ramic phase wi th alower the rmal expans ion coe f f ic ien t . In th is s i tua t ion ,spherical ceramic par tic les are in a hydrosta tic sta te andthe i r sur rounding matr ix ph ase suf fe r s a compress iv e ra -dial stress sta te . Thus, on the average, the compositesshould produce a compressive residual stress. On the otherhand, when the volume f rac t ion i s so h igh tha t the ce -ramic pa r t ic le s touch each o the r , the duc t i le ma tr ix phaseshould be cons ide red enc losed by the ce ramic phase . Inthis case , the ceramic par tic les are st i l l in a compressivestress sta te , as is the matr ix in the proximity of the ce-ramic par tic les. Ho weve r , the major ity of the matr ix phasewi th a h igher the rmal expans ion coe f f ic ien t shouldundergo shr inking on cool ing , r e su l t ing in a tens i le r e -s idua l s tr e ss. The ac tua l condi t ions of compos i te s ma yfa l l be tween the two ex t reme cases . In l igh t of th is d is -cuss ion , we note tha t the bound ary condi t ions (Eqs . [8a]through [8c ] ) employed in the present mode l a re basedon the approximation of a small volume fraction, ~b, andthus the mode l ' s p red ic t ion should y ie ld an over -es t ima t ion for the p las t ic s t r a in ene rgy dens i ty (and thedislocation density) a t a high q~. In this regard, otherboundary condi t ions re f lec ting the volum e f rac t ion of ce -ramic par tic les are cer ta inly desirable as a future study.According to the ea r ly s tudies on the inc lus ion s ize -dependent y ie ld str es s, I4'7'8! the e f fec t ive y ie ld s t r e ss ford is loca t ion genera t ion should approach the macroscopicyie ld s t r e ss of the ma tr ix phase i f the ce ramic pa r t ic les ize becomes of the orde r of a f ew mic rons and thece ramic -matr ix in te r face i s incoherent . There fore , thedislocation density gradient showing a higher density nearthe A1/ce ramic in te r face should be obse rved when thece ramic pa r t ic le s ize i s ove r a f ew mic rons (F igures 4 ,5 , and 7) . No s igni f icant d is loca t ion dens i ty gradien t ,however , i s obse rved when the pa r t ic le d iamete r i s le ssthan 1 /~m (Figure 6) . Sever al reasons can be offered forthis case . The stress due to the thermal misf i t could beless than the effective yie ld stress for dislocation gen-eration, t4,8] Altern atively , in ca se the p lastic zon e size issmal le r than the cha rac te r i s t ic d is loca t ion pu nching d is -tance, the dislocation generated fn 'st would possibly reachthe plastic zone front, thus l imiting the fur th er disloca -tion generation through the inf luence of the back stressf l 1I t should be a lso noted tha t a s tab le , po lygonized d is -loca t ion s t ruc ture can be deve loped a round a submic ronpar t ic le , a s n ice ly demons t ra ted in the h igh-vol tage e lec -t r o n m ic r o sc o p y w o r k o f V o g e l s a n g et al. t~]Although F igure 9 shows a fa i r agreemen t , a t leas tqua l i ta t ive ly , be tween the measured and ca lcu la ted d is -loca t ion dens i ty , we note tha t the ca lcu la ted d is loca t iondens i ty depends o n the pa r t i t ion ing pa ramete r , f , tha t ap-pea red in the formula p , ~ Po + f to P ( r ) / ( O . 5 tx b 2 ) . A sment ioned be fore , some of a p la s t ic work i s d iss ipa ted

6 8 0 - - V O L U M E 2 1 A , M A R C H 1 99 0 M E T A L L U R G I C A L T R A N S A C T IO N S A


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a s h e a t a n d s o m e i s s p e n t f o r p r o d u c i n g v a r i o u s d e f e c t s .O n e c a n e s t i m a t e t h e v a l u e o f f f r o m a m a c r o s c o p i c t e n -s i l e o r c o m p r e s s i o n t e s t . H o w e v e r , i t s h o u l d b e n o t e dt h a t t h e p r e s e n t m i c r o s c o p i c s i t u a t i o n i s q u i t e d i f f e r e n tf r o m a m a c r o s c o p i c c a s e ; t h u s , t h e u s e o f s u c h a n e x -p e r i m e n t a l v a l u e f o r f i s n o t n e c e s s a r i l y v a l i d . I n v i e wo f t h is u n c e r t a i n t y o f f , f u r t h e r st u d ie s a r e n e c e s s a r y f o ra u s e f u l a p p l ic a t i o n o f t h e p r e s e n t m i c r o m e c h a n i c s a n a l-y s i s . A s a n a l t e r n a t i v e , a n e f f e c t i v e p l a s t i c s t r a i n c a n b eu s e d f o r e s t i m a t i n g d i s l o c a t i o n d e n s i t i e s , b u t i t r e q u i r e sa m e a n f r e e p a t h f o r d i s l o c a t i o n s , w h i c h i s a g a i n a q u a n -t i ty a s a m b i g u o u s a s h e a t d is s i p a ti o n . ~ W h e n t w o c e -r am ic pa r t i c le s wi th a the r m a l m is f i t s t r a in appr oach eacho t h e r , t h e r e a r i s e s a n a d d i t i o n a l s t r e s s f i e l d d u e t o t h ed i f f e r e n c e in e l a s t ic c o n s t a n t s b e t w e e n t h e m a t r i x a n dc e r a m i c p h a s e . 04,~81 H o w e v e r , t h e s t r e n g t h o f t h i si n h o m o g e n e i t y - i n d u c e d s t r es s i s p r o p o r t i o n a l t o r - 6 ( asc o m p a r e d w i t h t h e r - s t e r m s o f E q s . [ 1 2 a ] a n d [ 1 2 b ] )a n d w a s n e g l e c t e d i n t h e p r e s e n t a n a l y s i s .N o i n s i t u T E M w o r k w a s p e r f o r m e d f o r t h e p re s e n t2 0 2 4 A I c o m p o s i t e s , b u t t h e w o r k o f V o g e l s a n g e t a l . t~o n 6 0 6 1 A 1 / S i C c o m p o s i t e s f o u n d t h a t m o s t o f t h e d is -l o c a t i o n s w e r e a n n e a l e d o u t a t 5 0 7 ~ T h u s , it a p p e a r sq u i t e r e a s o n a b l e t o u s e t h e s o l u t io n i z i n g t e m p e r a t u r e o f4 9 0 ~ a s th e r e f e r e n c e t e m p e r a t u r e a t w h i c h a l l o f th ed i s p l a c e m e n t s w e r e a s s u m e d t o d i s a p p e a r . T h e p l a s t i cr e l a x a t i o n o f th e m a t r i x s u r r o u n d i n g a t h e r m a l m i s f i t t i n gc e r a m i c p a r t i c l e c a n l e a d t o s u b s t a n t i a l c h a n g e s i n t h ea s s o c i a t e d s t r e s s f i e l d a n d s t r a i n e n e r g y . T h i s r e s u l t s i ne l a s t i c a n d p l a s t i c s t r a i n e n e r g y g r a d i e n t s a r o u n d t h e c e -r a m i c p a r t i c l e s . A l u m i n u m a l l o y m a t r i x / c e r a m i c c o m -p o s i te s a r e s o m e w h a t u n i q u e s i n c e t h e p r o p e rt i e s o f t h ea l u m i n u m m a t r i x i t s e lf c a n b e m o d i f i e d a n d v a r i e d s i g-n i f i c a n t l y t h r o u g h t h e p r e c i p i t a t i o n p r o c e s s . T h e d r a s t i cc h a n g e i n t h e s t r e s s f i e l d s u r r o u n d i n g t h e c e r a m i c p a r -t i c l e s , t h e r e f o r e , c a n a f f e c t d i f f u s i o n , k i n e t i c s o f p r e c i p -i t a te nuc lea t ion and gr owth , cohe r ency s ta te , e t c . I t w o u l db e o f g r e a t v a l u e t o u n d e r s t a n d t h e e f f e c t s o f t h e p l a s t i cr e l a x a t i o n o n t h e p r e c i p i t a t i o n k i n e t i c s .

ab(r r(r e(r e(r ye re oe~ePoEvolTA TurpP1P24 ,

N O M E N C L A T U R Es p h e r i c a l c e r a m i c r a d i u sh a l f o f i n t e r p a r t i c l e s p a c i n gr a d i a l s t r e s s c o m p o n e n tt a n g e n t ia l s t r es s c o m p o n e n te q u i v a l e n t s t r e s sy i e l d s t r e s s o f t h e m a t r i x p h a s er a d ia l s t r ai n c o m p o n e n tt a n g e n t i a l s t r a i n c o m p o n e n tr a d i a l p l a s t i c s t r a i n c o m p o n e n tt a n g e n t i a l p l a s t i c s t r a i n c o m p o n e n te l a s t i c m o d u l u ss h e a r m o d u l u sP o i s s o n ' s r a t i oc o e f f i c i e n t o f t h e r m a l e x p a n s i o nt e m p e r a t u r et e m p e r a t u r e o f m e a s u r e m e n t - s o l u t io n i z i n gt e m p e r a t u r er a d ia l d i s p l a c e m e n tp l a s t i c z o n e r a d i u si n t e r n a l p r e s s u r ee x t e r n a l p r e s s u r ev o l u m e f r a c t i o n o f t h e c e r a m i c

S u b s c r i p t sm m a t r i xc c e r a m i c

A C K N O W L E D G M E N T ST h e a u t h o r s a r e g r a t e f u l t o P r o f e s s o r R . J . A r s e n a u l tf o r h is c r i t ic a l r e ad i n g o f t h e m a n u s c r i p t a n d m a n y c o n -s t lu c t iv e s u g g e s ti o n s . T h e y a l so w i s h t o a c k n o w l e d g e t h ef i n a n c i a l s u p p o r t p r o v i d e d b y A L C O A ( c o n t r i b u t i o n s o fC T K a n d M R P ) a n d b y t h e D i v i s i o n o f M a t e r i a ls R e s e a r c h

o f th e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r G r a n t N o .D M R - 8 5 0 8 7 2 0 ( c on t ri b u ti o n s o f J K L a n d M R P ) .

V I . C O N C L U S I O N SF r o m t h e e x p e r i m e n t a l d a t a a n d t h e t h e o r e t ic a l m o d e l ,t h e f o l l o w i n g c o n c l u s i o n s w e r e m a d e :

1 . A t h e o r e t i c a l m o d e l f o r t h e p l a s ti c r e l a x a t i o n o f t h e r -m a l m i s f i t s t r e s s i n c o m p o s i t e s w a s d e v e l o p e d s u c ht h a t t h e m o d e l i n c o r p o r a t e s t h e v o l u m e f r a c t i o n o fs p h e r i c a l c e r a m i c p a r t i c l e s . I t s p r e d i c t i o n o f t h e p l a s -t i c st r a in e n e r g y g r a d i e n t a r o u n d t h e c e r a m i c p a r t i c l e sw a s f o u n d t o b e i n a f a i r a g r e e m e n t w i t h t h e o b s e r v e dd i s l o c a t i o n d e n s i t y g r a d i e n t .2 . A s t h e v o l u m e f r a c t i o n o f c e r a m i c p a r t i c l e s i n -c r e a s e d , t h e d i s l o c a t i o n d e n s i t y i n c r e a s e d , a n d t h ed i s l o c at i o n s t ru c t u r e b e c a m e m o r e t a n g l e d d u e t o t h ei n t e r a c t i o n b e t w e e n t h e r m a l m i s f i t s t r e s s f i e l d s .3 . T h e i n t e n s i t y o f d i s l o c a t i o n g e n e r a t i o n d u e t o t h e d i f-f e r e n t t h e r m a l c o n t r a c t i o n s s e e m e d t o b e s t r o n g l y i n -f l u e n c e d b y t h e c e r a m i c p a r t ic l e s i ze a n d a l s o b y t h ed i f f e re n c e in e l a s ti c m o d u l u s b e t w e e n t h e c e r a m i c a n dm a t r i x .

REF ER ENCE S1. M. Vogelsang, R.J . Arsenault, and R.M. Fisher:.Metal l . Trans. A,1986, vol. 17A, pp. 379-89.2. R.J. Arsenault and R.M. Fisher: S c r i p t a Me t a l l . , 1983, vol. 17,pp. 67-71.3. K.K. Chawla and M. Metzger: J . Ma t e r . S c i . , 1972, rot. 7,pp. 34-39.4. J.K. Lee, Y.Y. Earmme, H.I. Aaronson , and K.C. Russell: Me t a l l .Tra n s . A , 1980, vol. llA, pp. 1837-47.5. C.A. Hoffman: J . En g . Ma t e r . Te c h n o l . , 1973, vol. 95,

pp. 55-62.6. Y.Y. Earmme, W.C. Johnson, and J.K. Lee: Me t a l l . Tra n s . A ,1981, vol. 12A, pp. 1521-30.7. W.C. Johnson and J.K. Lee: A c t a M e t a l l . , 1983, vol. 31,pp. 1033-45.8. M.F. Ashby and L. Johnson: P h i l . M a g . , 1969, vol. 20,pp. 1009-22.9 . R . H i ll : Th e Ma t h e m a t i c a l Th e o ry o f P l a s t ic i t y , O x f o rd UniversityPress, London, 1950, p. 97.10. A. Mendelson: P l a s t i c i ~ - - T h eo r y a n d A p p li c a t io n s , MacMillanPublishing Company, New York, NY, 1968, p. 135.11. G.E. Dieter: M e c h a n i c a l M e t a l l u r g y, McGraw-Hill, New York,NY, 1961, p. 58.12. A.J. Ardell: A c t a M e t a l l . , 1972, vol. 20, pp. 61-71.

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13. P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Whelan, and D.W.Pashley: E l e c t r o n M i c r o s c o p y o f T h i n C r y s ta l s , 2nd ed., Krieger,New York, NY, 1977, p. 422.14. J.D. Eshelby: P r o c . R . S o c . , 1957, vol. 241A, pp. 376-96.15. H.M. Ledbetter and M.W. Austin: Ma t e r . S c i . En g . , 1987, vol. 89,pp. 53-61.

16. R.J. Arsenault and M. Taya: A c t a M e t a l l . , 1987, vol. 35,pp. 651-59.17. N.K. Asamoach and W.G. Wood: J . S t ra i n An a l . , 1970, vol. 5,pp. 88-97.18. W.C. Johnson and J.K. Lee: Me t a l l . Tra n s . A , 1979, vol. IOA,pp. 1141-49.

682--VOLUME 21A, MARCH 1990 METALLURGICAL TRANSACTIONS A

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