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r e p r e s e n t s the m a t r i x . It is a s s u m e d that the c o m p o s i t e cylinder undergoes symmetrical plastic elongation. The p o w e r c o n s u m e d in this d e f o r m a t i o n , ~bT, is e x p r e s s e d in t e r m s of eight p a r a m e t e r s : 1) P a r t i c l e g e o m e t r y , T i / R i . 2) R e l a t i v e s p a c i n g in the r a d i a l d i r e c t i o n , R o / R i. 3) R e l a t i v e s p a c i n g in the a x i a l d i r e c t i o n , T o / T i . 4) R a t i o of the y i e l d s t r e n g t h of the p a r t i c l e to the y i e l d s t r e n g t h of the m a t r i x , a. 5) R a t i o of the s t r e n g t h of the m a t r i x - p a r t i c l e bond to the y i e l d s t r e n g t h of the m a t r i x , m. 6) R e l a t i v e e n v i r o n m e n t a l p r e s s u r e , i . e . , the r a t i o of the e n v i r o n m e n t a l p r e s s u r e to the y i e l d s t r e n g t h of the m a t r i x , P/(ao)matrix. 7) D e g r e e of p a r t i c l e d e f o r m a t i o n , i . e . , the a x i a l v e l o c i t y at the top of the p a r t i c l e c y l i n d e r r e l a t i v e to the a x i a l v e l o c i t y at the top of the m a t r i x c y l i n d e r , ~. If /3 = 0 the p a r t i c l e d o e s not d e f o r m . If ~ = T i / T o the p a r t i c l e d e f o r m s h o m o g e n e o u s l y with the m a t r i x . 8) D e g r e e of void opening, i . e . , the r e l a t i v e a x i a l v e l o c i t y of the m a t r i x m i n u s the r e l a t i v e a x i a l v e l o c i t y of the p a r t i c l e , both m e a s u r e d at the top of the p a r t i c l e c y l i n d e r , e.

1) G e o m e t r y of the p a r t i c l e s 2) Spacing b e t w e e n the p a r t i c l e s . 3) S t r e n g t h of the p a r t i c l e r e l a t i v e to the s t r e n g t h of the m a t r i x . 4) The s t r e n g t h of the p a r t i c l e - m a t r i x bond.

Thus

A n u m b e r of i n v e s t i g a t o r s 3'5's have t r e a t e d the p r o b l e m c o n s i d e r i n g one o r two of the f a c t o r s l i s t e d a b o v e , but a t r e a t m e n t of a l l the f a c t o r s s i m u l t a n e o u s l y was not a c h i e v e d until r e c e n t l y when the m o d e l of a p a r t i c l e in a m a t r i x was a n a l y z e d by A v i t z u r 9 using the u p p e r bound a p p r o a c h . T h i s m o d e l is e x p l a i n e d in d e t a i l in Ref. 9, thus only highlights a r e given h e r e . In this m o d e l , the m e t a l is r e g a r d e d a s a c o l l e c t i o n of contiguous r i g h t c i r c u l a r c y l i n d r i c a l e l e m e n t s . E a c h c o m p o s i t e e l e m e n t c o n s i s t s of a c y l i n d e r of r a d i u s R i and height Ti, r e p r e s e n t i n g the p a r t i c l e , c o a x i a l with a l a r g e r c y l i n d e r (of r a d i u s Ro and height To), which ELIEZER ROZOVSKY, formerly Graduate Assistant, Lehigh University, Bethlehem, Pa. 18015, is now with Gilbert Associates, Reading, Pa. 19603. W. C. HAHN, Jr. and BETZALEL AVITZUR are Professor and Professor and Director, respectively, Institute