Fracture characteristics of a ductile-matrix/brittle-fiber composite
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K~- { E c f c X f , f 2 d V f + ~ m , m
d ( 1 - Vf)Z~t~/2 %
]'
[1]
Ef = m o d u l u s of the f i b e r s Ern ~ = a v e r a g e m o d u l u s of the m a t r i x o v e r the s t r a i n d Vf af am Ef Em
r a n g e w h e r e the s e c o n d a r y m o d u l u s is e n countered = d i a m e t e r of the f i b e r s = v o l u m e of the f i b e r s = the f r a c t u r e s t r e n g t h of the f i b e r s = the u l t i m a t e t e n s i l e s t r e n g t h of the m a t r i x = true fracture strain = the u n i f o r m s t r a i n in a t e n s i l e t e s t of the matrix.
The f i r s t t e r m in Eq. [1] r e p r e s e n t s the c o n t r i b u t i o n of the f i b e r s and is due to C o o p e r and K e l l y 8 and the s e c o n d t e r m , the c o n t r i b u t i o n of the m a t r i x . The f a c tor d (1 - ~'f)~
.j i s a t w o - d i m e n s i o n a l e s t i m a t e of the v o l u m e f r a c t i o n and i s v a l i d for l a m i n a t e s only. Eq. [1], in which the c o n t r i b u t i o n of the f i b e r s to the t o u g h n e s s is quite s u b s t a n t i a l , g i v e s s o m e w h a t l o w e r t o u g h n e s s v a l u e s than those determined experimentally. F o r b r i t t l e - f i b e r d u c t i l e - m a t r i x c o m p o s i t e s , on the o t h e r hand, C o o p e r and K e l l y s found the c o n t r i b u t i o n of the f i b e r s to the w o r k of f r a c t u r e , Gm, to be n e g l i g i b l e and d e s c r i b e d by Gm = 2 X ( 1 -
w h e r e E~ i s the s e c o n d a r y m o d u l u s of e l a s t i c i t y and i s given b y ~~
E'c = E f V f + Eme. (1 - Vf)
[2]
where YNGVE NAERHEIM is Researcher, Department of Materials Technology, Engineering Research Foundation at the Norwegian Institute of Technology (SINTEF), Trondheim, Norway. He is presently Euroscholar, Department of Industrial Metallurgy, University of Birmingham, Birmingham, England. This paper is based on M.Sc. Thesis in Department of Materials Science and Metallurgical Engineering, University of Cincinnati, Cincinnati, Ohio. Manuscript submitted December 9, 1974. METALLURGICAL TRANSACTIONS A
V/) V
[3]
w h e r e X is the u n i f o r m l o a d t r a n s f e r length given by
X:
(1-~- v d ) .... amd ~ 4~"
[4]
where U = the a r e a under the load vs e x t e n s i o n c u r v e for the m a t r i x d i v i d e d by the v o l u m e of the s p e c i men T = the s t r e n g t h of the m a t r i x in s h e a r . The o t h e r notation is a s above. The f r a c t u r e t o u g h n e s s can then b e c a l c u l a t e d f r o m n = [EcGm] ~/2
[5] VOLUME 7A, JANUARY 1976-63
F r o m Eqs. [3] and [4] it can be e x p e c t e d that a c o m p o s i t e with c o a r s e r f i b e r s w i l l be t o u g h e r than will one with f i n e r f i b e r s . Due to the n o n u n i f o r m i t y of the s t r a i n t h r o u g h o u t the s p e c i m e n , it i s d i f f i c u l t to o b t a i n an a c c u r a t e e s t i m a t e of U. Both m o d e l s d i s c u s s e d above a s s u m e that the f r a c t u r e t o u g h n e s s , K, is a m a t
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