Fracture Mechanical Approach to Metal-Matrix Composites

Application

of Fracture Mechanics

to Composite

Materials

edited by K. Friedrich © Elsevier Science Publishers B.V., 1989

Chapter

12

Fracture Mechanical Shojiro Department

Approach

to Metal-Matrix

Composites

OCHIAI of Metallurgy,

Faculty of Engineering,

Kyoto

University,

Sakyo-ku,

Kyoto 606, Japan

Contents Abstract 492 1. Introduction 492 2. Blunting of the notch tip due to matrix yielding and splitting 494 3. Influence of damage zone at the notch tip on l o a d - C O D curve 498 3.1. Matrix yielding 498 3.2. Splitting and transverse damage zone 499 4. Influence of splitting on fracture behaviour 502 5. Work done at notch tip and work of fracture 504 5.1. Pull-out of fibres from matrix 504 5.2. Debonding of fibres from matrix 505 5.3. Stress redistribution 505 5.4. Work along the fibre-matrix interface due to bridging of fibre 506 5.5. Work done to break the matrix 506 5.6. Work done to break the fibres 506 5.7. Work of fracture of some composite systems 507 6. Fracture criteria 508 6.1. Fracture criteria based on conventional and modified LEFM 509 6.1.1. Fracture criteria for metal-fibre-metal-matrix composites 509 6.1.2. Fracture criteria for brittle-fibre-metal-matrix composites 511 6.1.2.1. Critical strain energy release rate and critical stress intensity factor criterion 511 6.1.2.2. Intense-energy-region concept: Waddoups, Eisenman and Kaminski's fracture criterion 514 6.1.2.3. Mar-Lin criterion 515 6.1.2.4. Strain failure criterion (Poe and Sova's criterion) 516 6.1.2.5. R-cmve criterion 521 6.1.2.6. /-integral criterion 524 6.2. Fracture criteria particular to composites 525 6.2.1. Macroscopic approaches 525 6.2.1.1. Point- and average-stress criteria (Whitney and Nuismer's fracture model) 6.2.1.2. Pipes, Wetherhold and Gillespie's criterion 529 6.2.2. Microscopic approach 531 7. The influence of structural and environmental factors on fracture behaviour 535 7.1. Yield stress of the matrix 535 491

525

492

5. Ochiai

7.2. Interfacial bonding strength between fibre and matrix 536 7.3. Fibre diameter 537 7.4. Type of discontinuity 537 7.5. Specimen width 538 7.6. Specimen thickness 538 7.7. Laminate configuration in cross-plied composites 538 7.8. Sequence of lay-up in cross-plied composites 539 7.9. Test temperature 539 List of symbols 540 References 542

Abstract A d a m a g e z o n e c a n b e f o r m e d at t h e n o t c h - t i p b y v a r i o u s e v e n t s , s u c h as b r e a k a g e o f fibres, m a t r i x y i e l d i n g , s p l i t t i n g , d e b o n d i n g a t t h e i n t e r f a c e b e t w e e n fibres a n d m a t r i x , a n d , for u n i d i r e c t i o n a l fibre-reinforced c o m p o s i t e s , b r i d g i n g of fibres a n d p u l l - o u t o f fibres. F o r a n g l e - p l y c o m p o s i t e s , t h e d a m a g e z o n e c a n b e f o r m e d b y e v e n t s s u c h a s d e l a m i n a t i o n b e t w e e n p l i e s a n d f r a c t u r e of t h e w e a k e r p l i e s in addition to the a b o v e - m e n t i o n e d events. T h e formation of a d a m a g e zone results in a c o m p l e x i t y a n d m u l t i p l i c i t y o f t h e f r a c t u r e b e h a v i o u r o f c o m p o s i t e s . I n t h i s c h a p t e r w i t h t h e e m p h a s i s o n t h e i n f l u e n c e o f t h e d a m a g e z o n e , it is s h o w n h o w t h e d a m a g e z o n e affects t h e f r a c t u r e b e h a v i o u r a n d t h e w o r k at t h e n o t c h - t i p of m e t a l - m a t r i x c o m p o s i t e s . T h e n it is s h o w n h o w v a r i o u s f r a c t u r e m e c h a n i c a l c o n c e p t s p r o p o s e d u n t i l n o w c a n b e a p p l i e d . F i n a l l y , it is s h o w n h o w s t r u c t u r a l a n d e n v i r o n ­ m e n t a l f a c t o r s affect t h e d a m a g e z o n e a n d , t h e r e f o r e , t h e n o t c h e d s t r e n g t h .

1. Introduction T h e f r a c t u r e b e h a v i o u r o f n o t c h e d u n i d i r e c t i o n a l c o m p o s i t e s is affected b y m a n y f a c t o r s , s u c h as t h e m e c h a n i c a l p r o p e r t i e s o f t h e c o n s t i t u e n t s ( Y o u n g ' s a n d s h e a r m o d u l i o f fibre a n d m a t r i x , y i e l d s t r e s s a n d s t r a i n h a r d e n i n g coefficient o f t h e m a t r i x , s t r e n g t h a n d its s c a t t e r o f fibres), i n t e r f a c i a l c o n d i t i o n s b e t w e e n fibre a n d m a t r i x (interfacial b o n d i n g strength, existence of c h e m i c a l reaction layer a n d frictional shear stress w h i c h acts after interfacial d e b o n d i n g ) a n d geometrical conditions ( d i a m e t e r o f fibre, v o l u m e f r a c t i o n of fibre a n d size o f s p e c i m e n s ) . I n a n g l e - p l y c o m p o s i t e s , in a d d i t i o n t o t h e a b o v e - m e n t i o n e d f a c t o r s , s t a c k i n g s e q u e n c e a n d thickness of plies, the angle b e t w e e n plies a n d the interfacial b o n d i n g strength b e t w e e n p l i e s affect t h e b e h a v i o u r . T h e s e f a c t o r s affect t h e i n i t i a t i o n a n d g r o w t h o f crack-tip d a m a g e a n d , therefore, the fracture m o d e a n d n o t c h sensitivity, resulting in a m u l t i p l i c i t y a n d c o m p l e x i t y o f f r a c t u r e b e h a v i o u r o f c o m p o s i t e s . I n g e n e r a l , different c o m p o s i t e s y s t e m s s h o w different f r a c t u r e m o d e s a n d d a m a g e m e c h a n i s m s , r e q u i r i n g c o r r e s p o n d i n g different a n a l y t i c a l a n d e x p e r i m e n t a l t e c h n i q u e s . A s a r e s u l t , t h e r e is n o t y e t a c o n s e n s u s r e g a r d i n g t h e p r o p e r set o f c r i t e r i a for f r a c t u r e [ 1 ] . T h e difficulty t o d e s c r i b e t h e f r a c t u r e b e h a v i o u r o f c o m p o s i t e s w i t h a u n i q u e c r i t e r i o n is m a i n l y d u e t o t h e c o m p l e x i t y o f s t r e s s s t a t e at t h e n o t c h t i p . I n t h e c a s e

Fracture mechanical approach to metal-matrix

Matrix

ι

11

Ιι ι

composites

493

Fibre 11 11

ι/

I I I I

Brekage of Fibres (a)

Interfacial ing (d)

Debond-

Y i e l d i n g of (b)

Matrix

Bridging of Fibres (e)

Splitting (c)

Pull-out Fibres (f)

of

Fig. 1. Schematic representation of the events occurring at the notch tip in unidirectional composites.

of u n i d i r e c t i o n a l c o m p o s i t e s , t h e following events o c c u r at the n o t c h tip, s c h e m a t i ­ c a l l y s h o w n in fig. 1: ( a ) b r e a k a g e of fibres; ( b ) m a t r i x y i e l d i n g ; (c) s p l i t t i n g d u e t o d e b o n d i n g at fibre-matrix i n t e r f a c e o r d u e t o f a i l u r e o f m a t r i x ; ( d ) i n t e r f a c i a l d e b o n d i n g b e t w e e n fibre a n d m a t r i x ; (e) b r i d g i n g o f fibres w h e n t h e n o t c h e x t e n d s i n t o m a t r i x w i t h o u t b r e a k a g e o f fibres; a n d (f) p u l l - o u t o f fibres. O f c o u r s e , t w o o r m o r e of t h e s e events c a n also o c c u r s i m u l t a n e o u s l y , d e p e n d i n g o n m a t e r i a l system. I n a n g l e - p l y c o m p o s i t e s , in a d d i t i o n t o t h e s e e v e n t s , d e b o n d i n g a t t h e i n t e r f a c e b e t w e e n plies a n d t h e fracture of the plies occur, also d e p e n d i n g o n m a t e r i a l system. T h e d e l a m i n a t i o n in a n g l e - p l y c o m p o s i t e s is c a u s e d b y t r a n s v e r s e t e n s i l e s t r e s s a m o n g t h e p l i e s o r b y s h e a r s t r e s s , e s p e c i a l l y a t t h e free e d g e . T h i s p r o b l e m h a s b e e n i n v e s t i g a t e d first b y P i p e s a n d P a g a n o [ 2 ] , f o l l o w e d b y m a n y i n v e s t i g a t o r s using elastic theory [ 3 - 9 ] . I n u n i d i r e c t i o n a l m e t a l - m a t r i x c o m p o s i t e s ( M M C ) , t h e e v e n t s s h o w n in fig. 1 occur, constituting a d a m a g e z o n e . Especially yielding of a m a t r i x at t h e n o t c h tip is a f e a t u r e o f M M C . It is w e l l - k n o w n t h a t m a t r i x y i e l d i n g p l a y s a r o l e in t h e b l u n t i n g o f t h e n o t c h t i p . A n o t h e r e v e n t w h i c h c a u s e s b l u n t i n g is s p l i t t i n g t h a t results from l o n g i t u d i n a l cracking of the m a t r i x or d e b o n d i n g at t h e fibre-matrix i n t e r f a c e . T h e i n f l u e n c e o f m a t r i x y i e l d i n g a n d s p l i t t i n g at t h e n o t c h t i p o n b l u n t i n g will b e s h o w n in sect. 2. I n sect. 3 it will b e d i s c u s s e d h o w t h e l o a d - C O D ( c r a c k o p e n i n g d i s p l a c e m e n t ) c u r v e is affected b y t h e d a m a g e z o n e . I n sect. 4 , t h e i n f l u e n c e of s p l i t t i n g o n f r a c t u r e b e h a v i o u r will b e d e s c r i b e d . T h e v a r i o u s t y p e s o f w o r k w h i c h a r e d o n e a t t h e n o t c h t i p , will b e briefly s u m m a r i z e d in sect. 5. A s s t a t e d a b o v e , a n u m b e r o f f r a c t u r e c r i t e r i a h a v e b e e n p r o p o s e d u n t i l n o w , s o m e o f w h i c h will b e d i s c u s s e d in sect. 6. I n s e c t . 7, t h e i n f l u e n c e o f v a r i o u s s t r u c t u r a l a n d e n v i r o n m e n t a l p a r a m e t e r s , s u c h as y i e l d s t r e s s o f m a t r i x , i n t e r f a c i a l b o n d i n g s t r e n g t h , s t a c k i n g

494

5. Ochiai

c o n f i g u r a t i o n a n d s e q u e n c e o f l a y - u p in c r o s s - p l i e d c o m p o s i t e s , t e s t t e m p e r a t u r e , etc., will b e s u r v e y e d . 2. Blunting of the notch tip due to matrix yielding and splitting F o r m a t r i x y i e l d i n g a t t h e n o t c h t i p in u n i d i r e c t i o n a l M M C , s e v e r a l e x p e r i m e n t a l r e s u l t s h a v e b e e n r e p o r t e d . T i r o s h [ 1 0 ] o b s e r v e d t h a t t h e m a t r i x y i e l d e d a t t h e slit e n d s in l o n g , n a r r o w r e g i o n s p a r a l l e l t o t h e fibres a n d t h e l e n g t h o f t h i s y i e l d e d r e g i o n w a s p r o p o r t i o n a l t o t h e slit l e n g t h a n d t h e s q u a r e of t h e a p p l i e d s t r e s s . P o e a n d Sova [11] f o u n d from the r e a d i n g s of strain g a u g e s l o c a t e d r e m o t e from the slits t h a t t h e y i e l d e d r e g i o n e x t e n d e d far e n o u g h t o w a r d t h e s t r a i n g a u g e s t o d i s t u r b t h e far-field s t r a i n s , b u t f o r s h o r t e r slits, it d i d n o t e x t e n d far e n o u g h t o affect t h e m e a s u r e d s t r a i n s . P e t e r s [ 1 2 ] s h o w e d t h a t t h e n e t f r a c t u r e s t r e s s at t h e c r a c k e x t e n s i o n w a s s t r o n g l y r e l a t e d t o t h e s h e a r d e f o r m a t i o n a t t h e n o t c h t i p . A t s m a l l slits, t h e r e w a s m a i n l y a n e l a s t i c d e f o r m a t i o n w h i c h c a u s e s a s h a r p d r o p in t h e n e t f r a c t u r e s t r e n g t h . A t l a r g e r slit l e n g t h s , t h e s h e a r y i e l d i n g a n d t h e o n s e t o f s h e a r c r a c k s b l u n t e d t h e n o t c h e s , w h i c h r e s u l t e d in a less s h a r p d r o p of t h e n e t f r a c t u r e s t r e n g t h with increasing notch length. These results indicate that the extension of the region o f m a t r i x y i e l d i n g is d e p e n d e n t o n n o t c h l e n g t h a n d a p p l i e d s t r e s s , a n d t h e s h a p e o f t h i s r e g i o n is n a r r o w in w i d t h , b u t l o n g i n l o n g i t u d i n a l d i r e c t i o n ( p a r a l l e l t o t h e fibres). W h e n notched unidirectional M M C specimens are pulled u n d e r tension, the notch c a u s e s t e n s i l e stress c o n c e n t r a t i o n s at t h e t i p a n d s h e a r s t r e s s c o n c e n t r a t i o n s b e t w e e n c u t a n d i n t a c t fibres. T h e s h e a r stress c o n c e n t r a t i o n c a u s e s s h e a r y i e l d i n g o f t h e m a t r i x , d e b o n d i n g a t t h e fibre-matrix i n t e r f a c e if t h e b o n d i n g s t r e n g t h is l o w a n d f a i l u r e o f t h e m a t r i x if t h e e x e r t e d s h e a r s t r e s s e x c e e d s t h e s h e a r s t r e n g t h of m a t r i x , a s s c h e m a t i c a l l y s h o w n in fig. 2, in w h i c h t h e fibres b e t w e e n 0 t o η - 1 h a v e b e e n c u t t o i n t r o d u c e a n o t c h , a n d t h e c u t - e n d o f t h e fibres is t a k e n as y = 0. I n fig. 2 , it is a s s u m e d t h a t t h e m a t r i x y i e l d i n g ( M Y ) , i n t e r f a c i a l d e b o n d i n g ( I D ) a n d m a t r i x f a i l u r e ( M F ) o c c u r o n l y b e t w e e n c u t a n d i n t a c t fibres. If t h e e x e r t e d s h e a r stress τ b e t w e e n c u t a n d i n t a c t fibres is l o w e r t h a n b o t h t h e i n t e r f a c i a l b o n d i n g s t r e n g t h in s h e a r , r a n d t h e s h e a r y i e l d s t r e s s of t h e m a t r i x , r , as in ( a ) , t h e m a t r i x b e h a v e s e l a s t i c a l l y for a n y y. T h i s s i t u a t i o n will b e f o u n d w h e n t h e a p p l i e d stress o n a c o m p o s i t e is l o w . W i t h i n c r e a s i n g a p p l i e d s t r e s s a n d η < r ( b ) , d e b o n d i n g o c c u r s w h e n τ b e c o m e s e q u a l t o τ,· b e f o r e f r a c t u r e o f c o m p o s i t e s s i n c e τ i n c r e a s e s w i t h i n c r e a s i n g a p p l i e d s t r e s s . If r < r, ( c ) , y i e l d i n g o f t h e m a t r i x in s h e a r will o c c u r w h e n τ b e c o m e s equal to r before fracture of composites. Both d e b o n d i n g a n d y i e l d i n g o f m a t r i x o c c u r first a t y = 0 s i n c e t h e s h e a r stress c o n c e n t r a t i o n is m a x i m a l at y = 0. T h e r e g i o n s o f d e b o n d i n g a n d y i e l d i n g o f m a t r i x g r o w w i t h i n c r e a s i n g a p p l i e d stress. F o r τ < r , the matrix never yields since d e b o n d i n g occurs prior to y i e l d i n g a n d , t h e r e f o r e , t h e s h e a r stress c o n c e n t r a t i o n c a n n o t b e h i g h e n o u g h t o c a u s e yielding. F o r T > r , τ increases with increasing a p p l i e d stress after matrixy i e l d i n g , a n d d e b o n d i n g o c c u r s w h e n τ b e c o m e s e q u a l t o τ b e f o r e f r a c t u r e of c o m p o s i t e s if τ < r ( d ) , w h e r e r is t h e u l t i m a t e s h e a r s t r e n g t h of t h e m a t r i x . If r < r m a t r i x f a i l u r e o c c u r s u n d e r s h e a r w h e n τ ( o r s h e a r s t r a i n of m a t r i x ) b e c o m e s i ?

y

y

y

y

{

y

{

y

{

{

u

i 9

u

u

Fracture mechanical approach to metal-matrix

495

composites

Fig. 2. Schematic representation of occurrence of interfacial debonding ( I D ) , matrix yielding (MY) and matrix failure (MF) due to existence of cut fibres. In the region where neither I D , MY, nor M F occur, the matrix behaves elastically in shear ( M E region).

equal to r

u

(or s h e a r strain to failure of t h e m a t r i x ) before fracture of c o m p o s i t e s

( e ) . A n o t h e r p o s s i b l e m e c h a n i s m o f m a t r i x f a i l u r e for η > r

u

has been k n o w n to be

c a u s e d b y t h e t e n s i l e s t r e s s a h e a d of t h e n o t c h t i p i n t r a n s v e r s e d i r e c t i o n , a s will b e d i s c u s s e d i n sect. 4. T h e f a i l u r e o f t h e m a t r i x b y s h e a r o r t e n s i l e s t r e s s i n t r a n s v e r s e direction causes splitting. T h e m i c r o m e c h a n i c a l m o d e l t o c a l c u l a t e s t r e s s c o n c e n t r a t i o n s i n i n t a c t fibres w a s p r e s e n t e d first b y H e d g e p e t h [ 1 3 ] w h o u s e d t h e s h e a r - l a g a n a l y s i s m e t h o d , i n w h i c h composites are m o d e l l e d as being c o m p o s e d of tension-carrying

fibres

embedded

in p u r e l y s h e a r - c a r r y i n g m a t r i x . H e f o u n d t h a t t h e s t a t i c s t r e s s c o n c e n t r a t i o n f a c t o r i n t h e i n t a c t fibres for a n u m b e r o f η c u t fibres i n a t w o - d i m e n s i o n a l m o d e l is g i v e n by 4 χ 6 χ 8 χ · - ·Χ(2Η+2) " ~ 3 Χ 5 Χ 7 Χ · · · χ ( 2 η + 1) for s i t u a t i o n ( a ) i n fig. 2 w h e r e t h e m a t r i x b e h a v e s e l a s t i c a l l y . Z e n d e r a n d D e a t o n [14] d e m o n s t r a t e d the validity of this t y p e of analysis. V a n D y k e a n d

Hedgepeth

[ 1 5 ] , L o c k e t t [ 1 6 ] a n d Z w e b e n [ 1 7 ] a n a l y z e d t h e s t r e s s c o n c e n t r a t i o n s for s i t u a t i o n ( b ) i n fig. 2 ; V a n D y k e a n d H e d g e p e t h [ 1 5 ] , Z w e b e n [ 1 7 ] , H e d g e p e t h a n d V a n D y k e [ 1 8 ] , K o p y o v a n d O v c h i n s k y [ 1 9 ] , R e e d y [ 2 0 ] a n d O c h i a i et a l . [ 2 1 ] for s i t u a t i o n ( c ) , K o p y o v a n d O v c h i n s k y [ 1 9 ] a n d O c h i a i et a l . [ 2 2 ] f o r s i t u a t i o n ( d ) , a n d G o r e e a n d G r o s s [23] for situation (e).

496

S. Ochiai

In the matrix elastic ( M E ) region, t h e stress equilibrium e q u a t i o n u n d e r static l o a d i n g f o r t h e t w o - d i m e n s i o n a l m o d e l is g i v e n b y [ 1 3 ] E

r

A

^ + ^ ( U d / d f

N

+

-2U +

l

£ / „ _ , ) = 0,

N

(2)

m

where U is t h e d i s p l a c e m e n t of t h e N t h fibre, E t h e Y o u n g ' s m o d u l u s of t h e fibre, A t h e c r o s s - s e c t i o n a l a r e a o f t h e fibre, G t h e s h e a r m o d u l u s o f t h e m a t r i x , h t h e t h i c k n e s s of t h e m o d e l c o m p o s i t e a n d d t h e w i d t h o f m a t r i x . W h e n t h e fibres for Ν = 0 t o η - 1 a r e c u t , e q . (2) is m o d i f i e d f o r t h e M Y , I D a n d M F r e g i o n s s h o w n i n fig. 2, a s f o l l o w s , a s s u m i n g t h a t M Y , I D a n d M F o c c u r s o n l y b e t w e e n c u t a n d i n t a c t fibres. N

f

f

m

m

I n e q . ( 2 ) , as t h e m a t r i x is t r e a t e d t o b e h a v e e l a s t i c a l l y in t h e M E r e g i o n , t h e s h e a r s t r e s s b e t w e e n t h e fibres for Ν = n-1 a n d η is g i v e n b y

?{n-\)-n=-f-(Un-\-

U ).

(3)

n

T h e s h e a r s t r e s s b e t w e e n t h e fibres for Ν = - 1 a n d Ν = 0 is a l s o g i v e n b y e q . ( 3 ) , d u e t o s y m m e t r y of t h e m o d e l c o m p o s i t e . W h e n m a t r i x b e c o m e s p l a s t i c ( M Y r e g i o n ) , T( -»-n n

is g i v e n b y [ 1 7 , 1 8 , 2 3 ] (4)

' (n-l)-n '

if t h e m a t r i x b e h a v e s in a r i g i d e l a s t i c - p l a s t i c m a n n e r w i t h o u t s t r a i n - h a r d e n i n g , o r by [21,22]

ΓΓΛ-,-ί/,,

«->--'>

+

A

L

d

m

r] y

(5)

G j '

if t h e m a t r i x s t r a i n - h a r d e n s l i n e a r l y w i t h s t r a i n , w i t h t h e s t r a i n h a r d e n i n g coefficient β . W h e n d e b o n d i n g o c c u r s at the interface ( I D region) or w h e n matrix failure o c c u r s ( M F r e g i o n ) , a f r i c t i o n a l s h e a r s t r e s s r a c t s b e t w e e n t h e fibres f o r Ν = η - 1 a n d Ν = η. T h u s τ _ _ is g i v e n b y 0

f

( η

υ

η

(6)

'(n-l)-w "

The boundary conditions are: (i) S t r e s s o f c u t fibres is z e r o a t y = 0. (ii) D i s p l a c e m e n t o f fibres o t h e r t h a n t h e c u t o n e s is z e r o at y = 0. (iii) S t r e s s o f all fibres is e q u a l t o t h e r e m o t e s t r e s s σγ a t y = ±oo. (iv) D i s p l a c e m e n t o f all fibres is c o n t i n u o u s . (v) S t r e s s o f all fibres is c o n t i n u o u s . (vi) S h e a r stress b e t w e e n t h e fibres for Ν = n - \ a n d η a t t h e b o u n d a r y b e t w e e n M E a n d M Y r e g i o n s is r t h a t b e t w e e n M E a n d I D r e g i o n s is τ , t h a t b e t w e e n M Y a n d I D r e g i o n s is τ a n d t h a t b e t w e e n M Y a n d M F r e g i o n s is r . y9

χ

{

u

Fracture mechanical approach to metal-matrix

I

I

0

2

497

L

4

a = (a /Iy)(G A /E d h) f

composites

f

m

f

f

1/2

m

Fig. 3. Examples of variations of stress concentration factor ahead of the notch tip as a function of the non-dimensional stress level
f

{

y

m

(

f

m

Figure 3 [22] s h o w s a n e x a m p l e of the v a r i a t i o n of t h e tensile stress c o n c e n t r a t i o n f a c t o r Κ (0) γ

i n i n t a c t fibres a t y = 0 for η = 5 a s a f u n c t i o n o f n o n - d i m e n s i o n a l s t r e s s

level σγ, g i v e n b y

In this e x a m p l e , the ratio of frictional s h e a r stress at t h e interface to interfacial bonding strength, 7 γ / τ

Ϊ 5

is t a k e n t o b e 0.2. W h e n t h e r a t i o o f i n t e r f a c i a l b o n d i n g

s t r e n g t h t o s h e a r y i e l d s t r e s s o f m a t r i x , τ-J r , is t a k e n t o b e 2.0, a s i n fig. 3 a , y i e l d i n g y

of t h e m a t r i x begins at Β u p o n l o a d i n g a n d s u b s e q u e n t l y interfacial

debonding

o c c u r s a t C . N a m e l y , b e l o w t h e s t r e s s level c o r r e s p o n d i n g t o B , s i t u a t i o n ( a ) o f fig. 2 o c c u r s , b u t , b e y o n d B , s i t u a t i o n (c) o c c u r s a n d b e y o n d C , s i t u a t i o n ( d ) . It is c l e a r l y s h o w n t h a t t h e s t r e s s c o n c e n t r a t i o n f a c t o r is r e d u c e d d u e t o y i e l d i n g o f t h e m a t r i x a n d i n t e r f a c i a l d e b o n d i n g . I n fig. 3 b , t h e v a r i a t i o n s o f K^O) of (Jf a r e s h o w n f o r v a r i o u s v a l u e s o f r-J r . W h e n τ-J r y

y

as a function

is less t h a n u n i t y , i n t e r f a c i a l

d e b o n d i n g occurs p r i o r to yielding of the m a t r i x a n d t h e m a t r i x n e v e r s h o w s yielding u p o n l o a d i n g . F o r s u c h a c a s e , K (Q) X

l o w ( 7 . I n t h e c a s e o f rjr f

y

decreases rapidly with increasing a

f

even at

= 0.5, s i t u a t i o n ( a ) i n fig. 2 is f o u n d for E F in fig. 3 b

a n d s i t u a t i o n ( b ) for F K . W h e n τ-J r is g r e a t e r t h a n u n i t y , t h e y i e l d i n g o f t h e m a t r i x y

o c c u r s a t G . F o r s u c h a c a s e , w i t h i n c r e a s i n g rjr

y9

t h e s t r e s s level f o r i n t e r f a c i a l

498

S. Ochiai

d e b o n d i n g t o o c c u r i n c r e a s e s . If τ-J r is 1.2, d e b o n d i n g o c c u r s at H , a n d if τ-J r is 2.0, d e b o n d i n g o c c u r s at I. C o n s e q u e n t l y , K (Q) v a r i e s a l o n g E F K for rjr = 0.5, a l o n g E G H L for τ-J r = 1.2 a n d a l o n g E G I M for τ-Jr = 2.0. If τ-J r is h i g h e n o u g h t o s u p p r e s s d e b o n d i n g , Κ (0) v a r i e s a l o n g E G J . H e r e it s h o u l d b e n o t e d t h a t it d e p e n d s n o t o n l y o n τ-J r b u t a l s o o n t h e s t r e n g t h of c o m p o s i t e s w h e t h e r d e b o n d i n g o c c u r s o r n o t . F o r i n s t a n c e , if t h e s t r e n g t h of a c o m p o s i t e is l o w a n d it f r a c t u r e s at a s t r e s s level c o r r e s p o n d i n g t o H , d e b o n d i n g n e v e r o c c u r s a s l o n g a s r-J r is h i g h e r t h a n 1.2. O n t h e o t h e r h a n d , if t h e s t r e n g t h of a c o m p o s i t e is h i g h a n d it f r a c t u r e s at a stress level c o r r e s p o n d i n g t o I, for i n s t a n c e , d e b o n d i n g o c c u r s a s l o n g a s τ-J r is l o w e r t h a n 2.0. y

y

X

y

y

y

y

χ

y

y

y

I n t h e a b o v e e x a m p l e , t h e i n f l u e n c e of m a t r i x y i e l d i n g a n d d e b o n d i n g o n stress c o n c e n t r a t i o n in t h e fibres a h e a d o f t h e n o t c h t i p h a s b e e n d e m o n s t r a t e d . I n t h e c a s e w h e r e m a t r i x f a i l u r e o c c u r s , t h e stress c o n c e n t r a t i o n is r e d u c e d in a m a n n e r s i m i l a r t o t h a t for i n t e r f a c i a l d e b o n d i n g . T h u s it c a n b e c o n c l u d e d t h a t m a t r i x y i e l d i n g , i n t e r f a c i a l d e b o n d i n g a n d m a t r i x f a i l u r e c a u s e b l u n t i n g of t h e n o t c h t i p , l e a d i n g t o a r e d u c t i o n o f t h e stress c o n c e n t r a t i o n f a c t o r in fibres a h e a d of t h e n o t c h tip. 3 . Influence of d a m a g e zone at the notch-tip on l o a d - C O D curve 3.1.

Matrix

yielding

T h e g r o w t h of t h e p l a s t i c r e g i o n in u n i d i r e c t i o n a l c o m p o s i t e s c a n b e c a l c u l a t e d b y t h e s h e a r - l a g a n a l y s i s m e t h o d a n d a l s o b y t h e f o r m u l a t i o n of M c C l i n t o c k [ 2 4 ] for a t w o - d i m e n s i o n a l m o d e l in w h i c h it w a s a l s o a s s u m e d t h a t o n l y t h e m a t r i x b e t w e e n c u t a n d i n t a c t fibres s h o w s y i e l d i n g . A c c o r d i n g t o M c C l i n t o c k [ 2 4 ] , t h e l e n g t h o f p l a s t i c r e g i o n R is g i v e n b y p

(8) w h e r e σ is t h e a p p l i e d s t r e s s , c is h a l f o f t h e initial c r a c k l e n g t h , a n d t h e A a r e t h e o r t h o t r o p i c i n - p l a n e stiffness c o m p o n e n t s . T h e g r o w t h of t h e p l a s t i c r e g i o n c a u s e s d e p a r t u r e of t h e l o a d - C O D c u r v e f r o m t h a t p r e d i c t e d b y e l a s t i c t h e o r y . A w e r b u c h a n d H a h n [ 2 5 ] c a l c u l a t e d a l o a d - C O D r e l a t i o n for u n i d i r e c t i o n a l b o r o n a l u m i n i u m composites by taking the C O D as the s u m of a plastic C O D ( C O D ) a n d a n e l a s t i c C O D ( C O D ) . D e n o t i n g t h e finite-width c o r r e c t i o n f a c t o r b y Y, t h e w i d t h a n d t h i c k n e s s of t h e p l a t e s p e c i m e n s b y W a n d £ , t h e Y o u n g ' s a n d s h e a r m o d u l i of t h e c o m p o s i t e b y Ε a n d G, a n d t h e P o i s s o n ' s r a t i o of t h e c o m p o s i t e b y */, t h e C O D is [ 2 5 ] 0

tj

p

e

COD = C O D + COD, p

(9)

(10)

Fracture mechanical approach to metal-matrix

composites

499

50

0

80 120 160 C O D / pm

AO

200

2A0

Fig. 4. Comparison of experimental and analytical results of l o a d - C O D curves of unidirectional boronaluminium composites for various notch lengths. The solid curve is the theoretical curve predicted by eq. (9) and the dashed curve that predicted by eq. (11). (After Awerbuch and Hahn [25]: reproduced by permission from Technomic Publishing Co., Inc.)

COD

e

=

4a

Yc a 0

E BW L

(ID

'

(12) w h e r e a is t h e o r t h o t r o p i c c o r r e c t i o n f a c t o r a n d t h e s u b s c r i p t s L a n d Τ r e f e r t o longitudinal a n d transverse directions, respectively. A w e r b u c h a n d H a h n [25] com­ p a r e d e q . (9) w i t h e x p e r i m e n t a l l o a d - C O D c u r v e s a n d o b t a i n e d f a v o u r a b l e a g r e e ­ m e n t , a s s h o w n in fig. 4. T h e d a s h e d l i n e s i n fig. 4 r e p r e s e n t t h e i n i t i a l s l o p e of t h e l o a d - C O D c u r v e as p r e d i c t e d f r o m e q . ( 1 1 ) . It is e v i d e n t t h a t m a t r i x y i e l d i n g c a u s e s b l u n t i n g of t h e n o t c h t i p , l e a d i n g to larger C O D t h a n t h a t p r e d i c t e d o n the basis of e l a s t i c b e h a v i o u r . 3.2.

Splitting

and transverse

damage

zone

W h e n splitting occurs in c o m p o s i t e s , the l o a d - C O D b e h a v i o u r also departs from the linear o n e . F i g u r e 5 s h o w s a s c h e m a t i c r e p r e s e n t a t i o n of t h e l o a d - C O D curve of a u n i d i r e c t i o n a l b o r s i c - a l u m i n i u m s i n g l e - e d g e - n o t c h e d s a m p l e w i t h w e a k i n t e r facial b o n d i n g , a s p r e s e n t e d b y H o o v e r a n d A l l r e d [ 2 6 , 2 7 ] . A f t e r a n i n i t i a l l i n e a r region, t h e slope of t h e l o a d Ρ versus C O D curve b e g a n to d e c r e a s e at a l o a d d e n o t e d by P . C o i n c i d e n t with this deviation from linearity, audible cracking was h e a r d [ 2 6 , 2 7 ] . U p o n a d d i t i o n a l l o a d i n g , a c r a c k p a r a l l e l t o t h e fibres b e c a m e o b v i o u s d

S. Ochiai

500

COD Fig. 5. Typical l o a d - C O D curve and appearance of single edge-notched unidirectional borsic-aluminium composite whose fibre-matrix interface is weak. P shows the load at which debonding occurred and P the load at fracture of composite. (After Hoover and Allred [27]: reproduced by permission from Technomic Publishing, Co., Inc.) d

u

at t h e n o t c h t i p . T h i s c r a c k g r e w s t a b l y u n t i l it e x t e n d e d f r o m t h e n o t c h t i p t o t h e d o u b l e t a b s at b o t h e n d s o f t h e s a m p l e . A s t h i s c r a c k p r o g r e s s e d , t h e s l o p e of t h e curve was observed to increase gradually until the vertical crack e x t e n d e d over the entire gauge section, a n d the Ρ versus C O D record b e c a m e linear again. Further l o a d a p p l i c a t i o n t h e n c a u s e d final f r a c t u r e a c r o s s t h e r e m a i n i n g l i g a m e n t at a l o a d d e n o t e d b y P . C h a r a c t e r i s t i c a l l y , final f r a c t u r e o c c u r r e d in a c r o s s - s e c t i o n w h i c h did not contain the n o t c h a n d a p p e a r e d macroscopically very similar to the tensile f r a c t u r e in u n n o t c h e d s p e c i m e n s [ 2 6 , 2 7 ] . u

T h e g r o w t h of the splitting region c a n b e c a l c u l a t e d a p p r o x i m a t e l y by using the s h e a r - l a g a n a l y s i s [ 1 5 , 1 7 , 1 8 , 2 2 , 2 3 ] w h e r e t h e i n i t i a t i o n o f s p l i t t i n g is a s s u m e d t o o c c u r b y s h e a r s t r e s s . I n t h e c a s e w h e r e i n t e r f a c i a l b o n d i n g is w e a k , d e b o n d i n g at i n t e r f a c e l e a d s t o s p l i t t i n g , a s h a s b e e n s h o w n in fig. 2. I n t h e c a s e w h e r e i n t e r f a c i a l b o n d i n g is s t r o n g e n o u g h t o s u p p r e s s d e b o n d i n g , t h e f r a c t u r e o f m a t r i x a l s o l e a d s to splitting. G o r e e a n d G r o s s [23], employing the shear-lag analysis, determined the stress c o n c e n t r a t i o n in fibres a h e a d o f t h e n o t c h t i p a n d a l s o t h e C O D a s a f u n c t i o n of a p p l i e d s t r e s s , b y c o n s i d e r i n g t h e s i t u a t i o n s h o w n in fig. 6 a w h e r e s p l i t t i n g is a s s u m e d t o o c c u r w h e n t h e m a t r i x h a s f a i l e d in s h e a r a t its f a i l u r e s t r a i n , f o l l o w e d b y m a t r i x y i e l d i n g in s h e a r . I n t h e i r a n a l y s i s , it w a s a s s u m e d t h a t n o s h e a r stress e x i s t s in t h e r e g i o n w h e r e s p l i t t i n g h a s o c c u r r e d . T h i s a s s u m p t i o n c o r r e s p o n d s t o t h e c a s e o f r = 0 in fig. 2. T h e y c o m p u t e d t h e l o a d - C O D c u r v e u s i n g t h e i r t w o d i m e n s i o n a l m o d e l a n d c o m p a r e d it w i t h t h e e x p e r i m e n t a l d a t a o f A w e r b u c h a n d H a h n s h o w n in fig. 4. A s a r e s u l t , t h e y f o u n d t h a t t h e p r e d i c t e d l o a d - C O D c u r v e s c o u l d b e c l o s e l y fit t o t h e d a t a w h e n v a l u e s of y i e l d s t r e s s a n d stiffness o f m a t r i x were selected. This G o r e e - G r o s s a p p r o a c h was extended by D h a r a n i , Jones a n d G o r e e [ 2 8 ] , w h o i n c o r p o r a t e d t h e i n f l u e n c e o f t h e e x i s t e n c e of a t r a n s v e r s e d a m a g e z o n e in a d d i t i o n t o t h e s p l i t t i n g a n d y i e l d i n g o f t h e m a t r i x in t h e l o n g i t u d i n a l d i r e c t i o n a s s h o w n in fig. 6 b , b a s e d o n t h e o b s e r v a t i o n t h a t a c e r t a i n a m o u n t of s t a b l e t r a n s v e r s e e x t e n s i o n o f t h e i n i t i a l n o t c h in u n i d i r e c t i o n a l b o r o n - a l u m i n i u m c o m p o s i t e s o c c u r r e d in t h e f o r m o f b r e a k a g e o f a n a r b i t r a r y n u m b e r o f fibres a h e a d f

Fracture mechanical approach to metal-matrix

composites

501

-Transverse Damage

(a)

(b)

Fig. 6. (a) The two-dimensional model of damage zone at notch tip of Goree and Gross [23], and (b) that of Dharani, Jones and Goree [28]. In the former model, longitudinal damage due to splitting and matrix yielding is treated and in the latter model, longitudinal damage due to splitting, matrix yielding and transverse damage are treated.

of the initial n o t c h tip, s o m e t i m e s a c c o m p a n i e d by fracturing of t h e m a t r i x b u t m o r e often w i t h o u t any matrix fracture. Using this m o d e l , they c o m p u t e d l o a d - C O D curves of b o r o n - a l u m i n i u m c o m p o s i t e s a n d a g a i n c o m p a r e d these with t h e experi­ m e n t a l d a t a o f A w e r b u c h a n d H a h n , s h o w n in fig. 4. F o r a m o d e l l a m i n a t e h a v i n g s e v e n b r o k e n fibres, w h i c h c o r r e s p o n d s t o a n o t c h l e n g t h o f 1.27 m m f o r t h e s p e c i m e n o f A w e r b u c h a n d H a h n , t h e y i e l d s t r e s s a n d stiffness G / d w e r e c h o s e n t o give a b e s t fit a n d t h e t h u s d e t e r m i n e d v a l u e s w e r e u s e d for c a l c u l a t i o n o f t h e l o a d - C O D c u r v e s for 27 b r o k e n fibres, c o r r e s p o n d i n g t o a n o t c h l e n g t h o f 5 m m . T h e r e s u l t is s h o w n in fig. 7, s h o w i n g g o o d a g r e e m e n t . I n t h e i r w o r k , 88 M P a w a s c h o s e n a s t h e s h e a r y i e l d s t r e s s o f m a t r i x a n d 1 1 5 x l 0 G N / m a s G /d . T h e f o r m e r v a l u e is m

3

3

m

0

AO 80 COD / |jm

m

m

120

Fig. 7. The l o a d - C O D curves of centre-notched unidirectional boron-aluminium composites calculated by Dharani, Jones and Goree [28] compared with the experimental results of Awerbuch and Hahn [25] which are shown in fig. 4. (After Dharani, Jones and Goree [28]: reproduced with permission from Pergamon Journals Ltd.)

502

5. Ochiai

approximately equal to the value (60-130 M P a ) d e t e r m i n e d experimentally by Peters [ 2 9 ] for b o r o n - a l u m i n i u m c o m p o s i t e . T h e l a t t e r v a l u e g i v e s 20.4 G P a if d is t a k e n as t h e c e n t r e - l i n e d i s t a n c e b e t w e e n fibres. T h i s v a l u e is c l o s e t o t h e v a l u e o f 27 G P a for a l u m i n i u m . I n t h i s w a y , t h e i r a p p r o a c h d e s c r i b e s t h e t r e n d o f l o a d - C O D c u r v e s s u c c e s s f u l l y , as w e l l as t h e A w e r b u c h a n d H a h n ' s a p p r o a c h m e n t i o n e d a b o v e . m

4. Influence of splitting on fracture behaviour D u e t o t h e h i g h d e g r e e o f a n i s o t r o p y i n c o m p o s i t e s y s t e m s , e s p e c i a l l y in u n i d i r e c ­ t i o n a l s p e c i m e n s , s p l i t t i n g h a s b e e n o b s e r v e d in m a n y c o m p o s i t e s , a s s c h e m a t i c a l l y s h o w n i n fig. 8 a , w h i l e , i n i s o t r o p i c m a t e r i a l s , t h e p r o p a g a t i o n o f t h e n o t c h o c c u r s , in m o s t c a s e s , p e r p e n d i c u l a r t o t h e t e n s i l e a x i s , a s s h o w n in fig. 8 b . S p l i t t i n g in u n i d i r e c t i o n a l c o m p o s i t e s h a s b e e n r e p o r t e d to b e initiated by cracking of the matrix o r b y d e b o n d i n g at t h e i n t e r f a c e in l o n g i t u d i n a l d i r e c t i o n d u e t o " t e n s i l e stress in transverse d i r e c t i o n " [26,27,30,32], a n d to b e initiated by matrix failure or by interfacial d e b o n d i n g d u e to " s h e a r s t r e s s " at t h e n o t c h tip [33,34]. C o n c e r n i n g the "tensile stress" m e c h a n i s m , C o o k a n d G o r d o n [30] calculated the stress distribution a h e a d o f a c r a c k in a n a l l - b r i t t l e s y s t e m a n d f o u n d t h a t t h e t e n s i l e stress in t h e t r a n s v e r s e d i r e c t i o n r e a c h e s a m a x i m u m v a l u e w h i c h is one-fifth o f t h e t e n s i l e stress in t h e l o n g i t u d i n a l d i r e c t i o n . A c c o r d i n g t o t h i s m e c h a n i s m , s p l i t t i n g o c c u r s if ^ ,T/^C,L<0.2,

(13)

C

where a and a are t h e transverse a n d t h e l o n g i t u d i n a l strength of c o m p o s i t e s , r e s p e c t i v e l y . A l t h o u g h e q . (13) is v a l i d o n l y for a n a l l - b r i t t l e b o d y , b u t n o t for m e t a l - m a t r i x c o m p o s i t e s i n a r i g i d m a n n e r , G e r b e r i c h [ 3 1 ] f o u n d t h a t it c o u l d b e applied well to unidirectional stainless-steel-aluminium composites. According to G e r b e r i c h , s p l i t t i n g w a s o b s e r v e d for (r /a <0.24. In his s p e c i m e n s , a decreased but a i n c r e a s e d w i t h i n c r e a s i n g v o l u m e f r a c t i o n o f fibre V , l e a d i n g t o t h e o c c u r r e n c e of s p l i t t i n g a t h i g h V . M c G u i r e a n d H a r r i s [ 3 2 ] f o u n d in t h e i r unidirectional t u n g s t e n - ( a l u m i n i u m - 4 % copper) composite that splitting occurred at a b o u t JQ o f t h e r a t i o f o r a l o w w o r k - h a r d e n e d m a t r i x a n d a t a b o u t η f o r a h i g h work-hardened matrix. c T

c L

CjT

c L

CtL

c T

f

f

C o o p e r a n d K e l l y [ 3 3 ] f o u n d in t h e i r t u n g s t e n - c o p p e r c o m p o s i t e t h a t s p l i t t i n g o c c u r r e d w h e n o- /a < 1 / 1 3 . 2 . T h i s v a l u e is s m a l l e r t h a n t h e v a l u e o f 0.2 o f C o o k a n d G o r d o n [ 3 0 ] . T h e n t h e y c o n s i d e r e d t h a t t h e s p l i t t i n g o c c u r s b e c a u s e of s h e a r f a i l u r e o f t h e m a t r i x p a r a l l e l t o t h e fibres, a n d o b t a i n e d t h e c o n d i t i o n for s p l i t t i n g t o o c c u r for t h i s m e c h a n i s m b y u s i n g a t w o - d i m e n s i o n a l m o d e l . T h e c o n d i t i o n is expressed by cT

cL

(14) w h e n the matrix b e h a v e s elastically, a n d by (15)

Fracture mechanical approach to metal-matrix

(a)

composites

503

(b)

Fig. 8. Schematic representation of: (a) splitting parallel to tensile (fibre) axis, and (b) crack extension perpendicular to tensile axis in unidirectional composites.

w h e n m a t r i x b e h a v e s p l a s t i c a l l y , w h e r e V is t h e v o l u m e f r a c t i o n o f t h e m a t r i x , a n d y a n d y a r e y i e l d a n d f a i l u r e s t r a i n s o f m a t r i x in s h e a r , r e s p e c t i v e l y . E q u a t i o n s (14) a n d (15) g a v e V f > 0 . 0 1 a n d V > 0 . 7 3 a s t h e c o n d i t i o n s f o r s p l i t t i n g t o o c c u r for t h e i r t u n g s t e n - c o p p e r c o m p o s i t e . N a m e l y , if f a i l u r e o f m a t r i x o c c u r s in s h e a r w i t h n o p l a s t i c flow, s p l i t t i n g s h o u l d a l w a y s o c c u r for V > 0 . 0 1 , b u t if p l a s t i c flow o c c u r s in t h e m a t r i x b e f o r e f a i l u r e , t h e c o n d i t i o n o f V f > 0 . 7 3 is r e q u i r e d t o e n s u r e s p l i t t i n g . A s , in t h e i r e x p e r i m e n t , s p l i t t i n g o c c u r r e d f o r V f > 0 . 3 4 , t h e t w o v e r s i o n s of their analysis w e r e c o n s i d e r e d to i n c l u d e t h e e x p e r i m e n t a l result. m

y

u

f

f

F o r t h e i n i t i a t i o n o f s p l i t t i n g , t w o different m e c h a n i s m s h a v e b e e n p r o p o s e d , as m e n t i o n e d a b o v e . H o w e v e r , for t h e p r o p a g a t i o n o f s p l i t t i n g , it h a s b e e n s h o w n t h a t t h e " s h e a r s t r e s s " is t h e m a j o r c a u s e [ 3 0 , 3 5 ] . G a r g [ 3 5 ] o b s e r v e d t h e f r a c t u r e m o r p h o l o g y of a u n i d i r e c t i o n a l g r a p h i t e - e p o x y c o m p o s i t e a n d f o u n d t h a t initiation o f s p l i t t i n g is c a u s e d b y t h e c o m b i n a t i o n o f t r a n s v e r s e t e n s i l e s t r e s s a n d s h e a r s t r e s s , the former being m o r e i m p o r t a n t , b u t once the initiation occurs, the splitting p r o p a g a t e s primarily d u e to s h e a r stress. W h e n splitting occurs, the composite b e c o m e s notch-insensitive d u e to blunting o f t h e n o t c h t i p . T h e s t r e s s c o n c e n t r a t i o n a t t h e n o t c h t i p is r e d u c e d , a s h a s b e e n s h o w n in fig. 2. A c t u a l l y , w h e n s p l i t t i n g o c c u r s , t h e s t r e n g t h o f n o t c h e d s p e c i m e n s b e c o m e s high [33]. A n e x t r e m e e x a m p l e for w e a k l y b o n d e d u n i d i r e c t i o n a l t u n g s t e n c o p p e r c o m p o s i t e s , w h i c h w e r e all o b s e r v e d t o s p l i t p a r a l l e l t o t h e fibres, h a s b e e n r e p o r t e d b y C o o p e r a n d K e l l y [ 3 3 ] . I n t h e s e s p e c i m e n s , after s p l i t t i n g h a d o c c u r r e d , the r e m a i n d e r c o n s t i t u t e d a tensile test o n a n u n n o t c h e d s p e c i m e n of r e d u c e d cross-section. However, they also observed that, unless splitting was complete, n o t c h - i n s e n s i t i v i t y w a s n o t a c h i e v e d : i.e., t h e s t r e n g t h b a s e d o n t h e n e t c r o s s - s e c t i o n a l area was r e d u c e d . H o o v e r a n d Allred [26,27] also o b s e r v e d t h a t t h e net strength of

5. Ochiai

504

a n o t c h e d u n i d i r e c t i o n a l b o r s i c - a l u m i n i u m c o m p o s i t e , w h o s e l o a d - C O D c u r v e is p r e s e n t e d in fig. 5, w a s r e d u c e d in s p i t e of s p l i t t i n g . T h e s e r e s u l t s i n d i c a t e t h a t , a l t h o u g h s p l i t t i n g a c t s t o c a u s e b l u n t i n g of t h e n o t c h t i p , t h e r e still r e m a i n s s h a r p n e s s at t h e n o t c h t i p , g i v i n g r i s e t o s t r e s s c o n c e n t r a t i o n . T o characterize fracture of metals a n d o t h e r materials w h i c h , in general, s h o w n o splitting, fracture m e c h a n i c a l a p p r o a c h e s b a s e d on linear elastic fracture mechanics ( L E F M ) have widely been applied. In order to apply the L E F M approach, t h e c o n d i t i o n o f s e l f - s i m i l a r c r a c k e x t e n s i o n s h o u l d b e satisfied. T h e s p l i t t i n g in c o m p o s i t e m a t e r i a l s m a k e s it difficult t o a p p l y t h e L E F M a p p r o a c h t o c o m p o s i t e s in a r i g i d m a n n e r . 5. W o r k done at the notch tip and work of fracture W h e n t h e e v e n t s s h o w n in fig. 1 o c c u r , w o r k is d o n e in t h e r e g i o n s u r r o u n d i n g the notch tip, which contributes to fracture toughness. Concerning the work d o n e at t h e n o t c h t i p , a n u m b e r o f m e c h a n i s m s h a v e b e e n p r o p o s e d a n d a p p l i e d u n t i l n o w [31,33,36-53]. In general, two or m o r e m e c h a n i s m s act together to contribute t o t h e t o u g h n e s s a n d it d e p e n d s o n t h e c o m p o s i t e s y s t e m w h a t k i n d s o f m e c h a n i s m s act. As the f u n d a m e n t a l m e c h a n i s m s , o n e s h o u l d m e n t i o n t h e following m e c h a n i s m s for u n i d i r e c t i o n a l c o m p o s i t e s : (1) p u l l - o u t o f fibres; (2) d e b o n d i n g ; (3) s t r e s s r e d i s t r i b u t i o n ( o r r e l a x a t i o n o f s t r a i n e n e r g y f r o m fibre t o m a t r i x after a fibre b r e a k s ) ; (4) w o r k a l o n g t h e fibre-matrix i n t e r f a c e d u e t o b r i d g i n g o f fibre; (5) w o r k d o n e t o b r e a k m a t r i x ; a n d (6) w o r k d o n e t o b r e a k fibres. T h e s e m e c h a n i s m s will b e d i s c u s s e d in m o r e d e t a i l i n t h e r e m a i n d e r o f t h i s s e c t i o n . 5.1.

Pull-out

of fibres from

matrix

W o r k is d o n e d u r i n g p u l l - o u t o f fibres if a s h e a r s t r e s s is m a i n t a i n e d b e t w e e n fibre a n d m a t r i x d u r i n g t h e p u l l - o u t p r o c e s s . A c c o r d i n g t o C o t t r e l l [ 3 6 ] a n d K e l l y [ 3 7 , 3 8 ] , t h e w o r k d o n e in p u l l i n g o u t o f a s i n g l e fibre w i t h a l e n g t h s h o r t e r t h a n t h e c r i t i c a l l e n g t h / ( = l e n g t h n e c e s s a r y for a fibre t o s h o w its full s t r e n g t h ) is c

= j4nd rj

Wi

2

r

(/
(16)

c

w h e r e / is t h e l e n g t h o f t h e fibre, d t h e d i a m e t e r o f t h e fibre a n d r t h e s h e a r stress r e s i s t i n g p u l l - o u t , w h i c h w a s t a k e n t o b e c o n s t a n t in t h e i r m o d e l s . T h e s h e a r stress r is g i v e n b y t h e s h e a r s t r e s s o f t h e m a t r i x if t h e i n t e r f a c i a l b o n d i n g s t r e n g t h is h i g h o r b y a f r i c t i o n a l s h e a r s t r e s s b e t w e e n t h e fibre a n d t h e m a t r i x if t h e b o n d h a s f a i l e d . W h e n fibres a r e l o n g e r t h a n t h e c r i t i c a l l e n g t h , a f r a c t i o n o f IJl of fibres will b e p u l l e d o u t , w h i l e t h e r e s t will b e b r o k e n a t t h e p l a n e of m a t r i x f r a c t u r e , a s d e m o n s t r a t e d b y K e l l y a n d T y s o n [ 5 4 ] . F o r s u c h a c a s e , W is g i v e n b y f

s

s

s

po

Wl

0

= {ljl)(^d,rf)

(/>/ ). c

(17)

Fracture mechanical approach to metal-matrix

composites

505

W h e n fibres a r e c o n t i n u o u s , s o m e fibres f r a c t u r e i n a c r o s s - s e c t i o n o t h e r t h a n a t t h e f r a c t u r e s u r f a c e a n d p u l l - o u t o f fibres o c c u r s , t o o . T h e p r o b l e m o f w o r k o f p u l l - o u t o f fibres i n c o n t i n u o u s u n i d i r e c t i o n a l c o m p o s i t e s w a s c o n s i d e r e d b y C o o p e r [ 4 1 ] . H e a s s u m e d t h a t t h e s t r e n g t h o f fibres is e v e r y w h e r e c r , e x c e p t f o r w e a k points of strength σ a n d that the weak points are uniformly spaced a distance of / a p a r t a l o n g t h e l e n g t h o f fibres. T h e m e a n w o r k d o n e p e r fibre a g a i n s t p u l l - o u t is g i v e n b y f o

ΐ w

w

W

s po

W

s po

= ^d rJl

for

f

/ <[(ow

n o

-^ )/a w

= [(a -a^)l /a^ ](nd r /24l ) Uo

c

0

f

s

n o

]/ ,

for

w

(18)

c

/ >[K -^J/^, ]/ . w

0

0

c

(19)

T h e total w o r k of pull-out c a n b e calculated b y multiplying this b y t h e n u m b e r of fibres p e r u n i t a r e a [ = Vf/kird -]. 2

5.2. Debonding

of fibres from

matrix

I n c o n d i t i o n s w h e r e fibre p u l l - o u t is p o s s i b l e , t h e w o r k d o n e i n b r e a k i n g t h e b o n d contributes t o the fracture energy. O u t w a t e r a n d M u r p h y [39] estimated this w o r k based o n t h e consideration that the stored elastic energy in the d e b o n d e d length of a fibre i m m e d i a t e l y b e f o r e i t s f r a c t u r e c a n n o t b e r e d i s t r i b u t e d i n t h e c o m p o s i t e w h e n t h e fibre s n a p s . W h e n t h e r e l e a s e d e n e r g y c o m i n g f r o m t h e fibre c a u s e s d e b o n d i n g , t h e w o r k f o r d e b o n d i n g p e r fibre is g i v e n b y

debond = [ ^ f V ?

u

/8E ]/ , f

(20)

d

w h e r e / is t h e d e b o n d e d l e n g t h a n d a is t h e s t r e n g t h o f fibre. M a r s t o n e t a l . [ 4 5 ] c o n s i d e r e d t h a t t h e a v e r a g e l e n g t h o f e x p o s e d fibre will b e o n e - q u a r t e r t h e c r i t i c a l l e n g t h s i n c e t h e l o n g e s t e x p o s e d fibre will b e o n e - h a l f t h e critical l e n g t h a n d t h e s h o r t e s t fibre will h a v e z e r o l e n g t h . U s i n g t h i s c o n c e p t , t h e w o r k p e r fibre t o c a u s e d e b o n d i n g is g i v e n b y d

W

fu

= nd a l /32E .

s

2

dehond

(21)

2

f

fu c

f

T h e w o r k d u e t o d e b o n d i n g can also b e expressed from the viewpoint of creation of n e w c y l i n d r i c a l s u r f a c e s . W h e n t h e c r e a t e d i n t e r f a c e b e t w e e n fibre a n d m a t r i x h a s a s u r f a c e e n e r g y % , t h e w o r k p e r fibre c a n b e e s t i m a t e d b y t h e p r o d u c t o f t h e area of t h e interface a n d y , f

i f

^debond

=

277

"^Λ%Γ,

w h e n l = \l a s a p p r o x i m a t e d a b o v e , t h i s w o r k is e x p r e s s e d a s d

c

5.3. Stress

(22) {7rd y l . r

if c

redistribution

A c c o r d i n g t o F i t z - R a n d o l p h [ 4 3 ] , e n e r g y is d i s s i p a t e d o n fibre f r a c t u r e , s i n c e w h e n a fibre s n a p s , t h e s t r e s s a t t h e b r o k e n e n d s falls t o z e r o a n d b u i l d s u p o v e r a distance | / from t h e e n d . A s s u m i n g that t h e stress b u i l d s u p linearly from t h e b r o k e n c

S. Ochiai

506

e n d , t h e s t o r e d e n e r g y l o s t f r o m a fibre d u r i n g its f r a c t u r e is g i v e n b y W =wd a J /24E . s

2

5.4.

(23)

2

redis

c

r

Work along

the

fibre-matrix

interface

due to bridging

of fibre

W h e n a b r i d g i n g o f fibre o c c u r s , t h e fibre will d o w o r k o n t h e m a t r i x . T h i s w o r k is g i v e n b y t h e p r o d u c t o f t h e i n t e r f a c i a l f o r c e a n d t h e d i s p l a c e m e n t o f t h e fibre w i t h r e s p e c t t o t h e m a t r i x [ 4 2 ] . T h e w o r k d o n e b y t h e fibre o n t h e e l a s t i c m a t r i x p e r fibre, W , is g i v e n b y P i g g o t [ 4 2 ] a s s

mtC

W^=irdWj96r E , s

(24)

Îr

w h e r e E is t h e Y o u n g ' s m o d u l u s o f t h e fibre w h i c h is r e s t r a i n e d f r o m d e c r e a s i n g diameter. Taya a n d D a i m a r u [52] showed t h e work for a plastically deformable matrix, W to be fr

s

mp

W , s

= Wd r(o-f -a' )8 ,

(25)

2

m p

U

f

0

w h e r e a' is t h e stress o f t h e fibre i n a p l a s t i c a l l y d e f o r m i n g m a t r i x a n d <5 is t h e C O D a t t h e fibre l o c a t i o n . f

0

5 . 5 . Work done to break

the

matrix

T o c r e a t e a n e w s u r f a c e o f t h e m a t r i x , w o r k m u s t b e d o n e . W h e n t h e m a t r i x is b r i t t l e a n d fails b y s e l f - s i m i l a r c r a c k e x t e n s i o n w i t h o u t b e i n g affected b y t h e e x i s t e n c e o f fibres, t h e w o r k o f f r a c t u r e o f t h e m a t r i x will b e t w o t i m e s t h e Griffith's s u r f a c e energy of t h e matrix material. I n metal-matrix composites, t h e matrix deforms p l a s t i c a l l y u n d e r t h e i n f l u e n c e o f fibres, w h i c h m a k e s it difficult t o e s t i m a t e t h e w o r k of fracture o f a matrix i n a rigid m a n n e r . C o o p e r a n d Kelly [ 3 3 ] , in their study of u n i d i r e c t i o n a l t u n g s t e n - c o p p e r c o m p o s i t e s , o b s e r v e d m a s s i v e p l a s t i c flow a n d s h o w e d t h a t t h e w o r k o f f r a c t u r e o f t h i s c o m p o s i t e is d u e t o p l a s t i c d e f o r m a t i o n o f t h e b r i d g e s o f t h e c o p p e r m a t r i x left after t h e fibres h a d f r a c t u r e d . I n t h i s c o m p o s i t e s y s t e m , t h e f r a c t u r e s t r a i n o f fibres w a s m u c h less t h a n t h a t o f t h e m a t r i x . T h e y obtained a n expression for t h e extent of the zone of plastic deformation of the m a t r i x p a r a l l e l t o t h e a x i s o f t h e fibres b y c o n s i d e r i n g t h e t r a n s f e r o f t h e t e n s i l e load carried b y the bridges of matrix. T h e n they obtained a n expression for work of f r a c t u r e o f t h i s s y s t e m , W , a s m

W

= a s Ml-V ) /V ,

(26)

2

m

mu

m

f

r

where a and s a r e t h e u l t i m a t e t e n s i l e stress a n d t h e f a i l u r e t o s t r a i n i n t e n s i o n of t h e m a t r i x , r e s p e c t i v e l y . mu

m

u

5.6. Work done to break

the fibres

W h e n fibres a r e b r i t t l e , t h e w o r k n e c e s s a r y t o b r e a k t h e fibres is s m a l l , c o r r e s p o n d ­ i n g t o a p p r o x i m a t e l y t w o t i m e s t h e f r a c t u r e s u r f a c e e n e r g y . W h e n fibres a r e d u c t i l e , t h e p l a s t i c d e f o r m a t i o n o f t h e fibres c o n t r i b u t e s t o t h e w o r k o f f r a c t u r e . G e r b e r i c h

Fracture mechanical approach to metal-matrix

composites

507

[ 3 1 ] s h o w e d t h a t t h e h e i g h t o f a s e v e r e l y d e f o r m e d fibre s t r i p is a b o u t 2d

f

work per unit area, W

f9

W =2ar e drV f

u

where a

fu

fu

(27)

f9

and e

and the

is g i v e n b y

f u

are t h e tensile strength a n d t h e strain to failure of

fibres,

respectively. 5.7.

Work

of fracture

of some

composite

systems

A m o n g the m e c h a n i s m s described above, t w o or m o r e of t h e m m a y act together to contribute to the toughness, d e p e n d i n g o n composite systems. F o r instance, a c o m b i n a t i o n of t h e w o r k s i n s e c t s . 5 . 1 , 5.2 a n d 5.3 w a s a p p l i e d t o d e s c r i b e t h e w o r k o f f r a c t u r e o f b o r o n - e p o x y a n d c a r b o n - p o l y e s t e r c o m p o s i t e s b y M a r s t o n et al. [ 4 5 ] , t h a t o f s e c t s . 5 . 1 , 5.2, 5.5 a n d 5.6 t o g l a s s - p o l y e s t e r c o m p o s i t e s b y H a r r i s et al. [ 4 6 ] . F o r m e t a l - m a t r i x c o m p o s i t e s , e.g., f o r t u n g s t e n - c o p p e r c o m p o s i t e s [ 3 3 , 3 6 - 3 8 , 4 1 ] , t h a t o f s e c t . 5.1 w a s c o n s i d e r e d t o b e a m a j o r f a c t o r w h e n t h e fibres a r e s h o r t , b u t t h a t o f s e c t . 5.5 w a s c o n s i d e r e d t o b e a m a j o r o n e w h e n t h e fibres a r e c o n t i n u o u s . H o w e v e r , w h e n t h e c o m p o s i t e s y s t e m is different, different m e c h a n i s m s h a v e b e e n applied. F o r ductile fibre-aluminium-matrix c o m p o s i t e s , a c o m b i n a t i o n of t h e works d e s c r i b e d i n s e c t s . 5.5 a n d 5.7 g a v e g o o d a g r e e m e n t w i t h e x p e r i m e n t a l d a t a , a s reported by Gerberich [31]. C o n c e r n i n g the m a c r o s c o p i c w o r k of fracture of c o m p o s i t e s as a w h o l e , T a y a a n d D a i m a r u [52] f o u n d in their c a r b o n - a l u m i n i u m c o m p o s i t e s t h a t a c o m b i n a t i o n o f t h e w o r k s d e s c r i b e d i n s e c t i o n s 5.1 a n d 5.4 a n d e l a s t i c s t r a i n e n e r g y r e l e a s e r a t e describes well the total w o r k . F o r b o r o n - a l u m i n i u m c o m p o s i t e s , K y o n o , Hall a n d T a y a [53] m e a s u r e d t h e w o r k of fracture of t h e c o m p o s i t e as a w h o l e a n d analyzed t h e i r d a t a w i t h e m p h a s i s o n t h e i n f l u e n c e o f d e g r a d a t i o n o f fibres d u e t o e x p o s u r e at h i g h t e m p e r a t u r e s . I n t h e i r a n a l y s i s , t h e w o r k o f f r a c t u r e w a s g i v e n b y a c o m b i n a ­ t i o n o f t h e w o r k s d e s c r i b e d i n s e c t i o n s 5.1 a n d 5.4 a n d t h e e l a s t i c s t r a i n e n e r g y release rate similarly t o t h a t for c a r b o n fibre-aluminium composite. In borona l u m i n i u m c o m p o s i t e , t h e strength was r e d u c e d d u e to chemical reactions at the i n t e r f a c e . A c c o r d i n g l y , t h e w o r k of f r a c t u r e v a r i e d w i t h i n c r e a s i n g e x p o s u r e t i m e . Figure 9 s h o w s t h e variation of t h e fracture surface energy (half of w o r k of fracture) as a function of e x p o s u r e t i m e , t o g e t h e r with t h e p r e d i c t i o n s b a s e d o n their m o d e l . I n t h e i r a n a l y s i s , t h e w o r k a l o n g t h e fibre-matrix i n t e r f a c e w a s c a l c u l a t e d u s i n g e q . ( 2 5 ) a n d t h e w o r k o f p u l l - o u t w a s c a l c u l a t e d u s i n g e q . ( 1 9 ) i n w h i c h t h e fibre s t r e n g t h at w e a k p o i n t s a w a s s e t t o b e e q u a l t o fibre b u n d l e s t r e n g t h u s i n g a W e i b u l l d i s t r i b u t i o n f u n c t i o n [ 5 5 ] . T h e s o l i d c u r v e i n fig. 9 s h o w s t h e r e s u l t o f a c a l c u l a t i o n in w h i c h t h e b u n d l e s t r e n g t h w a s t a k e n f r o m t h e i r e x p e r i m e n t a l d a t a o f a t e n s i l e t e s t o n e x t r a c t e d fibres, a n d t h e d a s h e d c u r v e s h o w s t h e r e s u l t i n w h i c h t h e b u n d l e strength w a s t a k e n from t h e empirical e q u a t i o n of Dicarlo [ 5 6 ] , describing the a v e r a g e s t r e n g t h o f d e g r a d e d fibres as a f u n c t i o n of e x p o s u r e t e m p e r a t u r e a n d t i m e , a n d a n a s s u m e d v a l u e o f t h e W e i b u l l m o d u l u s w = 5. A g o o d a g r e e m e n t b e t w e e n the e x p e r i m e n t a l a n d analytical results w a s o b t a i n e d for t h e range of e x p o s u r e time i n v e s t i g a t e d . It w a s clarified i n t h e i r w o r k t h a t t h e t o u g h n e s s o f t h e r m a l l y e x p o s e d f w

5. Ochiai

508

50h

0

1

5 10

50 100

t / ks Fig. 9. Fracture surface energy of unidirectional boron-aluminium composite y as a function of exposure time t for: (a) V = 0.3, and (b) V = 0.5. Ο shows measured values by three-point bending test. and show the predicted curves based on Kyono, Hall and Taya's calculation using experimentally obtained values of the Weibull modulus w and strengths of fibres, and using Dicarlo's empirical expression describing strength of fibres and an assumed value of w = 5, respectively. (After Kyno, Hall and Taya [53]: reproduced with permission of Chapman and Hall Ltd.) F

f

f

b o r o n - a l u m i n i u m composites decreased with increasing exposure time similar to t h e strength for u n n o t c h e d s p e c i m e n s . 6. Fracture criteria T h e e x i s t e n c e o f a d a m a g e z o n e i n c l u d i n g t h e e v e n t s s h o w n in fig. 1 affects t h e f r a c t u r e b e h a v i o u r . A s s h o w n in fig. 3 , t h e l o a d - C O D c u r v e s d e v i a t e f r o m t h e e l a s t i c curve d u e to the d a m a g e z o n e at n o t c h tip. W h e n splitting o c c u r s , t h e crack d o e s n o t e x t e n d p e r p e n d i c u l a r t o t h e t e n s i l e a x i s a n d n o s e l f - s i m i l a r c r a c k g r o w t h is o b s e r v e d , a s s h o w n in fig. 4 . A l s o , w o r k is d o n e a t t h e n o t c h t i p , a s s h o w n in fig. 5, w h i l e in b r i t t l e h o m o g e n e o u s m a t e r i a l s , o n l y w o r k t o c r e a t e t h e f r a c t u r e s u r f a c e p e r p e n d i c u l a r t o t h e t e n s i l e a x i s is n e e d e d . T h e s e f e a t u r e s o b s e r v e d in m a n y c o m ­ posite systems seem to indicate that the linear elastic fracture m e c h a n i c s ( L E F M ) c a n n o t b e a p p l i e d t o c o m p o s i t e s in a r i g i d m a n n e r . H o w e v e r , in o r d i n a l m e t a l s , a p l a s t i c z o n e a p p e a r s at t h e n o t c h t i p , t o w h i c h v a r i o u s k i n d s of f r a c t u r e - m e c h a n i c a l a p p r o a c h e s , e s s e n t i a l l y b a s e d o n L E F M , h a v e s u c c e s s f u l l y b e e n a p p l i e d for c h a r a c ­ t e r i z a t i o n . If t h e d a m a g e z o n e in c o m p o s i t e s c o u l d b e t r e a t e d s i m i l a r t o t h e p l a s t i c z o n e in m e t a l s , t h e c o n v e n t i o n a l L E F M a p p r o a c h e s c o u l d b e a l s o a p p l i e d t o composites.

Fracture mechanical approach to metal-matrix

509

composites

C o n c e r n i n g t h e a p p l i c a b i l i t y o f L E F M t o c o m p o s i t e s , m u c h effort h a s

been

i n v e s t e d u n t i l n o w . H o w e v e r , t h e r e a r e still different o p i n i o n s o n t h e a p p l i c a b i l i t y of L E F M . F u r t h e r m o r e , d u e t o t h e c o m p l e x i t y o f f r a c t u r e b e h a v i o u r , v a r i o u s t y p e s of a p p r o a c h e s h a v e b e e n p r o p o s e d as f r a c t u r e c r i t e r i a . T h e r e s u l t s o f t h e a p p l i c a t i o n o f f r a c t u r e c r i t e r i a b a s e d o n c o n v e n t i o n a l a n d m o d i f i e d L E F M will b e d i s c u s s e d in sect. 6 . 1 . T h e n , a p p r o a c h e s p a r t i c u l a r t o c o m p o s i t e m a t e r i a l s w h i c h a r e o u t o f t h e f r a m e o f s o - c a l l e d L E F M will b e d i s c u s s e d in sect. 6.2. 6.1.

Fracture

criteria

based

on conventional

and modified

LEFM

I n m e t a l - m a t r i x c o m p o s i t e s , d u c t i l e m e t a l l i c fibres, s u c h as s t a i n l e s s s t e e l , m o l y b ­ d e n u m a n d t u n g s t e n o r b r i t t l e n o n - m e t a l l i c fibres s u c h a s g r a p h i t e ( c a r b o n ) , b o r o n a n d s i l i c o n c a r b i d e , a r e e m b e d d e d in a d u c t i l e m e t a l - m a t r i x . W h e n d u c t i l e

fibres

a r e u s e d , t h e d u c t i l i t y o f fibres c o n t r i b u t e s t o f r a c t u r e t o u g h n e s s , b u t w h e n b r i t t l e fibres

a r e u s e d , s u c h a n effect c a n n o t b e e x p e c t e d .

6.1.1.

Fracture

criteria for metal-fibre-metal-matrix

composites

For metal-fibre-metal-matrix composites, Gerberich [31], McGuire and Harris [32] a n d A r c h a n g e l s k a a n d Mileiko [57] have s h o w n that a modified L E F M a p p r o a c h c a n b e a p p l i e d b y t a k i n g t h e e n e r g y a b s o r p t i o n c a p a c i t y o f fibres a n d m a t r i x i n t o consideration, unless splitting occurs. G e r b e r i c h [ 3 0 ] e s t i m a t e d t h e w o r k of m a t r i x a n d fibres p e r u n i t a r e a o f u n i d i r e c ­ t i o n a l s t a i n l e s s - s t e e l - a l u m i n i u m c o m p o s i t e s b y u s i n g e q s . (26) a n d ( 2 7 ) , r e s p e c t i v e l y , for l o n g i t u d i n a l f r a c t u r e t e s t w h e r e t h e n o t c h w a s i n t r o d u c e d p e r p e n d i c u l a r t o t h e fibres ( t e n s i l e ) a x i s . T a k i n g t h e e n e r g y d i s s i p a t i o n G t o b e t h e s u m o f t h e fibres a n d m a t r i x c o n t r i b u t i o n s (G = W + W ) , t h e r e l a t i o n b e t w e e n G a n d t h e s t r e s s i n t e n s i t y f a c t o r K is g i v e n b y c

c

m

f

c

c

K

2 c

= EiG ,

(28)

c

w h e r e E' is t h e s e c o n d a r y Y o u n g ' s m o d u l u s [E V + (da /ds) V ]. I n e q . ( 2 8 ) , t h e effect of a n i s o t r o p y is n e g l e c t e d . H o w e v e r , t h i s effect is s m a l l in t h i s c o m p o s i t e system [31]. As a result, G e r b e r i c h o b t a i n e d t h e following e x p r e s s i o n for K, L

f

r

m

m

c

1/2

(29)

^mu^mu

F i g u r e 10a s h o w s t h e c o m p a r i s o n o f e x p e r i m e n t a l d a t a w i t h t h e p r e d i c t i o n g i v e n b y e q . ( 2 9 ) . T h e a g r e e m e n t is r e a s o n a b l e w i t h b o t h i n c r e a s i n g t r e n d s w i t h V a n d the quantitative values b e i n g predicted within a b o u t 2 0 % . In the case of a transverse f r a c t u r e t e s t w h e r e t h e n o t c h w a s i n t r o d u c e d p a r a l l e l t o fibres, K is a p p r o x i m a t e l y given by f

c

K ^{E a e [\d ^/V,y -d ]} \ /2

c

T

mu

mu

f

l/

f

(30)

w h e r e E is t h e p r i m a r y t r a n s v e r s e Y o u n g ' s m o d u l u s . I n e q . ( 3 0 ) , o n l y t h e v o l u m e o f m a t r i x m a t e r i a l i n v o l v e d in p l a s t i c e n e r g y d i s s i p a t i o n is c o n s i d e r e d . F i g u r e 10b T

510

S. Ochiai

0

0.1

0.2

OA

0.3

Vf

0

0.5

0.1

0.2

OA

0.3

v

0.5

f

Fig. 10. Measured values of the critical stress intensity factor K for (a) longitudinal and (b) transverse fracture tests for unidirectional stainless steel-aluminium composites, compared with predictions based on eq. (29) for longitudinal fracture and on eq. (30) for transverse fracture shown with solid curves. (After Gerberich [31]: reproduced with permission from Pergamon Journals Ltd.) c

shows the c o m p a r i s o n of the e x p e r i m e n t a l d a t a with the prediction by eq. (30). T h e p r e d i c t i o n represents t h e t r e n d in t h e d a t a very well. I n a l o n g i t u d i n a l f r a c t u r e t e s t G e r b e r i c h a l s o o b s e r v e d t h a t t h e g a p s 2v b e t w e e n b r o k e n fibres b e c a m e i n c r e a s i n g l y l a r g e r w i t h g r e a t e r d i s t a n c e s b e h i n d t h e n o t c h tip, a n d c o m p a r e d t h e g a p s with t h e theoretical d i s p l a c e m e n t in t h e c o n t i n u u m model given by G o o d i e r a n d Field [58], 2v = [ 2 c s e c ( 0 ) a / 7 r E ] { c o s ( 0 ) l n [ s i n ~ ( 0 - 0 ) / s h T ( 0 + 0 ) ] 2

0

2

y s

2

c

2

2

+ cos(0 )ln[[sin(0 ) + sin(0)] /[sin(0 )-sin(0)] ]}, 2

2

2

w h e r e 0 = πσ/2σ 2

2

a n d c o s θ = x/c

γ5

0

the centre of the crack, a

ys

(31)

2

s e c 0 , a n d t h e χ c o o r d i n a t e h a s its o r i g i n at 2

is t h e y i e l d s t r e s s a n d σ is t h e a p p l i e d s t r e s s . T h e g a p s

w e r e m e a s u r e d after u n l o a d i n g . T h e m e a s u r e d e s t i m a t e a t e a c h f r a c t u r e d fibre w a s given by 2^ = gap + 2 d e , f

(32)

f u

b a s e d o n t h e o b s e r v a t i o n t h a t t h e fibres w e r e s e v e r e l y d e f o r m e d t o a d i s t a n c e a b o u t e q u a l t o o r s l i g h t l y g r e a t e r t h a n t h e fibre d i a m e t e r o n e i t h e r s i d e o f t h e f r a c t u r e . F i g u r e 11 s h o w s t h e c o m p a r i s o n o f t h e e x p e r i m e n t a l d a t a w i t h t h e p r e d i c t i o n b a s e d o n eq. (31), indicating t h a t the fracture p r o c e s s can b e d e s c r i b e d well by the conventional continuum model. M c G u i r e a n d H a r r i s [ 3 2 ] m e a s u r e d t h e p l a s t i c s t r i p h e i g h t of t h e fibres, h , n

t h a t of t h e m a t r i x , estimated G G

c

c

in u n i d i r e c t i o n a l

p

tungsten-aluminium

composites

and and

by

= h V a p

h,

m

m u

e

m u

+ h V a n

f

r u

s

f u

.

T h e y f o u n d t h a t t h i s v a l u e a g r e e s fairly w e l l w i t h t h e G d e t e r m i n e d by fracture m e c h a n i c a l tests.

(33) c

value

experimentally

Fracture mechanical

I

approach

to metal-matrix

composites

511

o 4

>

CN

φ 0.4 ω ο

ο

Q.

ω Q

0.2 ο 0

2 -3 - 2 - 1 0 1 Distance from Crack T i p / m m

Fig. 11. Comparison of experimentally determined displacement 2v by longitudinal fracture test for unidirectional stainless steel-aluminium composites with the theoretical continuum model given by eq. (31). (After Gerberich [31]: reproduced with permission of Pergamon Journals Ltd.)

Archangelska a n d Mileiko [57] p r o p o s e d a m o d e l b a s e d o n the conventional L E F M concept c o m b i n e d with the plastic instability a p p r o a c h p r o p o s e d by Mileiko [ 5 9 ] , w h i c h m a k e s it p o s s i b l e t o p r e d i c t t e n s i l e s t r e n g t h a n d f a i l u r e s t r a i n o f u n n o t c h e d ductile-fibre-ductile-matrix composites. According to their model, G and K are given by

c

c

Oc =

[(^c,f) /^f](e 2

K =

c u

/e

)V +[(X , ) /^ ](e 2

f u

f

c

m

m

[(E V +E Vm ) G ]

c

f

f

m

v

c u

/e

m u

)V

r

(34) (35)

C

where K and K a r e t h e c r i t i c a l stress i n t e n s i t y f a c t o r o f t h e fibre a n d t h e m a t r i x , r e s p e c t i v e l y , a n d e is t h e f a i l u r e s t r a i n o f u n n o t c h e d c o m p o s i t e a s a w h o l e , w h i c h can be calculated by Mileiko's model [59]. They carried out experiments using unidirectional i r o n - a l u m i n i u m composites a n d f o u n d that the experimental results supported their model concept. cf

cm

cu

6.1.2. Fracture criteria for brittle-fibre-metal-matrix composites F o r brittle-fibre-metal-matrix c o m p o s i t e s , v a r i o u s types of a p p r o a c h e s h a v e b e e n p r o p o s e d b a s e d o n c o n v e n t i o n a l a n d m o d i f i e d L E F M c o n c e p t s , s o m e o f w h i c h will be discussed below. 6.1.2.1. Critical strain energy release rate and critical stress intensity factor criterion. T h e c r i t i c a l s t r a i n e n e r g y r e l e a s e r a t e G is r e l a t e d t o c r i t i c a l s t r e s s i n t e n s i t y factor K by c

c

K

2 C

=

EG

C

(36)

S. Ochiai

512

for i s o t r o p i c m a t e r i a l s , a n d C'Kl=G

C9

C'= (èA

(37) 1 1

A

)

2 2

1 / 2

[(A

2 2

/A

1 1

)

1 / 2

for a n i s o t r o p i c m a t e r i a l s . W h e n G

+ (2A c

1 2

or K

c

+ A

6 6

)/2A ] u

i s a materia l constan t independen t o f

crack lengt h a n d s p e c i m e n size , i t ca n b e u s e d a s a fractur e criterion . K to th e n o t c h e d strengt h cr K=

Ya (nc )

c

N

N

c

i s relate d

by

1/2

(38)

0

F o r c o m p o s i t e s i n w h i c h b r i t t l e fibres a r e e m b e d d e d i n a m e t a l m a t r i x , i t h a s b e e n e x a m i n e d i f thi s fractur e criterio n ca n b e a p p l i e d t o predic t th e n o t c h e d strength . C o o p e r a n d Kell y [33 ] s h o w e d that , i f splittin g doe s no t occur , th e fractur e o f t u n g s t e n - c o p p e r composite s i s governe d b y thi s criterion . C o o p e r a n d Kell y [33 ] a n d Tetelma n [60 ] considere d tha t th e n o t c h e d strengt h o f M M C ca n b e predicte d b y t a k i n g t h e w o r k o f p l a s t i c d e f o r m a t i o n o f t h e m a t r i x [ e q . ( 2 6 ) ] a s t h e critica l strain energ y releas e rate . H a n c o c k a n d S w a n s o n [61 ] a n d Waszcza k [62 ] indicate d tha t conventiona l concepts o f fractur e t o u g h n e s s ca n b e usefu l i n characterizin g th e toughnes s o f b o r o n - a l u m i n i u m composites . Wrigh t a n d Iannuzz i [63 ] foun d i n thei r b o r o n aluminium composite s tha t stabl e crac k growt h occurred . The y estimate d th e crac k extension A c fro m th e matchin g o f compliance , a n d a d d e d th e thu s estimate d value s o f àc t o t h e i n i t i a l c r a c k l e n g t h . T h e c r i t i c a l f a i l u r e s t r e s s w a s p l o t t e d a g a i n s t t h e t h u s d e t e r m i n e d l e n g t h o f t h e c r a c k , a s s h o w n i n fig. 12 . T h e d a t a o b t a i n e d f r o m specimens o f variou s thicknesse s exhibi t a slop e o f - 0 . 5 . Thi s indicate s tha t th e analytical expressio n fo r t h e critica l stres s intensit y facto r describe s a d e q u a t e l y th e fracture characteristic s [63] . H o o v e r [64 ] s h o w e d tha t th e valu e o f K i s constan t c

3.0,

Ο

ι .cw

•

1.52

Δ

2.18

•

l.09(Kreider and Dardi 11972) )

0.5|

5

10

15

2 0 25

50

100

Y c / mm 2

Fig. 12. Variation of notched strength of unidirectional boron-aluminium composites plotted in log-scale against Y c, where Y is the finite-width correction factor and c is the crack length at fracture. (After Wright and Iannuzzi [63]: reproduced with permission from the Metallurgical Society of AIME.) 2

Fracture mechanical approach to metal-matrix

composites

513

o v e r s i g n i f i c a n t r a n g e s o f s p e c i m e n g e o m e t r y in h i s b o r o n - a l u m i n i u m c o m p o s i t e , indicating that K

values c a n b e u s e d as a fracture criterion.

c

H o o v e r a n d Allred [64,65] m e a s u r e d a p a r a m e t e r L

c

defined by

__dV\ de

(39) δ= δ

ε

w h e r e V is t h e p o t e n t i a l e n e r g y a n d δ is t h e d e f l e c t i o n in t h r e e - p o i n t b e n d i n g test. L c a n b e viewed as t h e p o t e n t i a l energy release rate regardless of crack length or critical d e f l e c t i o n v a l u e s if it is a m a t e r i a l c o n s t a n t . T h i s L is e q u a l t o G for e l a s t i c m a t e r i a l s . T h i s L c a n a l s o b e r e g a r d e d a s t h e r a t e o f d e c r e a s e of w o r k r e q u i r e d t o initiate fracture with increasing crack length. T h e energy to crack p o p - i n per unit thickness was determined by measuring the area u n d e r load Ρ versus δ curves. T h e n , t h e t h u s d e t e r m i n e d energy w a s p l o t t e d against c . T h e slope of t h e energy v e r s u s c c u r v e c o r r e s p o n d s t o L if t h i s c u r v e is l i n e a r . F i g u r e 13 s h o w s o n e o f t h e r e s u l t s [ 6 4 ] . T h e e n e r g y v e r s u s c p l o t is l i n e a r o v e r a s u b s t a n t i a l r a n g e o f c r a c k l e n g t h . L b e i n g i n d e p e n d e n t o f c r a c k l e n g t h s u g g e s t s t h a t t h e r e is a c r i t i c a l p o t e n t i a l e n e r g y r e l e a s e r a t e w h i c h c o n t r o l s c r a c k i n i t i a t i o n in t h e s p e c i m e n s . H o o v e r [ 6 4 ] m e a s u r e d L f o r v a r i o u s u n i d i r e c t i o n a l b o r o n - a l u m i n i u m c o m p o s i t e s w i t h different m a t r i x y i e l d s t r e s s a n d fibre d i a m e t e r . H e f o u n d t h a t L w a s i n d e p e n d e n t of m a t r i x c

c

c

c

0

0

c

0

c

c

c

Fig. 13. Normalized energy to crack initiation versus crack length for a unidirectional boron-aluminium (6061) composite. (After Hoover [64]: reproduced with permission from Technomic Publishing Co. Inc.)

514

S. Ochiai

y i e l d s t r e n g t h a n d fibre d i a m e t e r . H e p r o p o s e d t h a t L i s a r e f l e c t i o n o f t h e s t r a i n e n e r g y s t o r e d i n fibres a t c r a c k i n i t i a t i o n a n d , t h u s , i s a m e a s u r e o f t h e p a r a m e t e r w h i c h c o n t r o l s c o m p o s i t e c r a c k - i n i t i a t i o n s t r a i n i n t h e fibres a t t h e n o t c h r o o t . 6.1.2.2. Intense-energy-region concept: Waddoups, Eisenman and Kaminskïs fracture criterion. W a d d o u p s , E i s e n m a n a n d K a m i n s k i [ 6 6 ] h a v e p r o p o s e d a m o d e l based o n L E F M concep t whic h assume s th e existenc e o f a n intens e energ y regio n w i t h a siz e a * , e x t e n d i n g p e r p e n d i c u l a r t o t h e l o a d d i r e c t i o n , a s s c h e m a t i c a l l y s h o w n i n fig. 1 4 f o r a c i r c u l a r h o l e . T h e c o n t r i b u t i o n o f s u c h a n i n t e n s e e n e r g y r e g i o n o n fracture strengt h ca n b e solve d b y applyin g L E F M a n d th e inherent-fla w concep t of B o w i e [ 6 7 ] . T h e s t r e s s i n t e n s i t y f a c t o r K fo r m o d e I a t f r a c t u r e fo r a g i v e n circular hol e i s give n b y c

Q

K = a%(va*) f(a*/R)

(40 )

1/2

Q

9

w h e r e cr ^ i s t h e i n f i n i t e - w i d t h p l a t e f r a c t u r e s t r e s s , R t h e h o l e r a d i u s a n d the

finite-width

f(a*/R)

c o r r e c t i o n f a c t o r fo r a h o l e . D e n o t i n g t h e u n n o t c h e d s t r e n g t h b y

cr , th e rati o o f th e n o t c h e d t o u n n o t c h e d strength s fo r a h o l e i s give n b y 0

^/a =l/f{a*/R).

(41 )

0

W h e n a sli t i s i n t r o d u c e d i n t h e c o m p o s i t e i n s t e a d o f a h o l e , K given by [1],

K = a~[ir(c +a*)V , /2

Q

0

Q

a n d σ^/°~o

are

(42)

σ

Intense Energy Region

σ Fig. 14. Schematic representation of Waddoups, Eisenman and Kaminski's model [66] for a hole, assuming the existence of an intense energy region with a size a*.

Fracture

I

I

0

0.1

mechanical

I

I

0.2 R

approach

to metal-matrix

I

I

0.3 0À / mm

0.5

composites

515

I

0,6

Fig. 15. Comparison of experimental results with the prediction based on Waddoups, Eisenman

and

Kaminski's criterion for boron-aluminium

and

[±45/0 ] 2

s

laminates containing a circular hole. The af

a* corresponding to the broken curves refer to the a* values which fit best to the experimental data for smallest and largest hole radii, respectively, and a* corresponding to solid curve, refers to the average value of a*. (After Awerbuch and Hahn [1]: reproduced with "permission from Technomic

Publishing

Co. Inc.)

^ / c r = [a7(co+fl*)r ·

(43)

0

I n t h i s m o d e l , t h e c h a r a c t e r i s t i c l e n g t h a * is r e g a r d e d a s a m a t e r i a l c o n s t a n t . A w e r b u c h a n d M a d h u k a r [1] collected experimental data o n the n o t c h e d strength of u n i d i r e c t i o n a l a n d cross-plied b o r o n - a l u m i n i u m c o m p o s i t e s , a n d a n a l y z e d the collected d a t a by applying this a n d other a p p r o a c h e s . A n e x a m p l e of the application of t h i s m o d e l is s h o w n in fig. 15, w h e r e t h e t w o d a s h e d c u r v e s c o r r e s p o n d t o t h e v a l u e s o f a * w h i c h b e s t fit t h e e x p e r i m e n t a l d a t a o f t h e s m a l l e s t ( a f ) a n d l a r g e s t ( a * ) hole radii, respectively, while the solid curve c o r r e s p o n d s to the average value o f a * . A g o o d a g r e e m e n t is f o u n d b e t w e e n 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 p r e d i c t i o n . 6.1.2.3. Mar-Lin criterion. Linear elastic fracture mechanics predicts the m o d e - I f r a c t u r e s t r e s s o f a w i d e s h e e t of a h i g h - s t r e n g t h m e t a l c o n t a i n i n g a c r a c k t o b e given by ^=K,c(7rc )-

1

/

2

0

,

(44)

where K is t h e c r i t i c a l s t r e s s i n t e n s i t y f a c t o r f o r m o d e I. M a r a n d L i n [ 6 8 , 6 9 ] p r o p o s e d t h a t t h e f r a c t u r e o f c o m p o s i t e s is g o v e r n e d b y t h e e q u a t i o n lc

d-S = H ( 2 c ) - S

(45)

m

c

0

w h e r e H is t h e c o m p o s i t e t o u g h n e s s w h i c h h a s t h e d i m e n s i o n o f s t r e s s x ( l e n g t h ) » a n d is a p r o p e r t y of t h e l a m i n a t e m a t e r i a l a n d l a y - u p , a n d t h e e x p o n e n t m is t h e p a r a m e t e r r e l a t e d t o t h e s t r e n g t h o f t h e s i n g u l a r i t y at t h e t i p o f c r a c k . I n t h e M a r - L i n m o d e l , t h e d i s c o n t i n u i t y a t its t i p w a s m o d e l l e d as if it w e r e a c r a c k w i t h its t i p a t w

c

x

5. Ochiai

516

t h e i n t e r f a c e of a b i - m a t e r i a l , i.e., t h e m a t r i x a n d t h e fibre. T h e m o d e l p i c t u r e s t h e c r a c k as b e i n g in t h e m a t r i x - t y p e m a t e r i a l p o i s e d t o p r o p a g a t e i n t o t h e material. T h e singularity at t h e crack tip, m , x

fibre-type

is 0.5 f o r h o m o g e n e o u s e l a s t i c m a t e r i a l s .

O n t h e o t h e r h a n d , t h e s i n g u l a r i t y at t h e c r a c k t i p w h i c h t o u c h e s t h e b i - m a t e r i a l interface d e p e n d s o n t h e ratio of the s h e a r m o d u l i of the t w o materials a n d the values of Poisson's ratio [ 7 0 - 7 2 ] . F o r the present c o m b i n a t i o n

of b o r o n

and

a l u m i n i u m , t h e s i n g u l a r i t y w a s c a l c u l a t e d t o b e 0.34 [ 6 9 ] . When the two constants, m

l

a n d H , w e r e c h o s e n t o fit b e s t w i t h t h e e x p e r i m e n t a l c

d a t a , t h e e x p e r i m e n t a l r e s u l t s w e r e d e s c r i b e d w e l l b y e q . ( 4 5 ) , a s s h o w n i n fig. 16a for a h o l e . T a k i n g 2 c

as t h e length of the discontinuity, the results were also

0

d e s c r i b e d w e l l b y e q . ( 4 5 ) , a s s h o w n in fig. 16b w h e r e C C D , S C D , H E D a n d D E D r e f e r t o c i r c u l a r h o l e s , c e n t r e - s l i t s , h o l e s w i t h slits a n d s y m m e t r i c e d g e slits, r e s p e c ­ tively. T h e r e s u l t s s h o w n i n fig. 16b i n d i c a t e t h a t t h e l e n g t h of d i s c o n t i n u i t y a n d n o t t h e s h a p e (i.e., h o l e , slit o r h o l e w i t h slits) is t h e c o n t r o l l i n g p a r a m e t e r . T h e and H

m

l

w e r e e s t i m a t e d t o b e 0.30 a n d 7 5 0 M P a ( m m ) ' , r e s p e c t i v e l y , for b o r o n 0

c

3

a l u m i n i u m [ ± 4 5 / 0 ] l a m i n a t e s . T h e e x p e r i m e n t a l l y e s t i m a t e d v a l u e o f 0.3 for 2

s

m

l

w a s c l o s e t o t h e p r e d i c t e d v a l u e o f 0.34. T h u s M a r a n d L i n [ 6 8 , 6 9 ] c o n c l u d e d t h a t the results are described well by c o n s i d e r i n g t h e crack p r o p a g a t i o n at the interface in a bi-material. 6.1.2.4.

Strain

failure

criterion

(Poe and Sova's

criterion

[11,73]). Poe and Sova

[ 1 1 ] c o n s i d e r e d t h a t a l a m i n a t e fails w h e n e v e r t h e fibre s t r a i n s r e a c h a c r i t i c a l level

1.0 0.8 0.6 Best Fit

V C L = 0.14(2R)

0.2 I—L 1 L_ 0.6 0.8 1

2 U 2R / mm

6

8 10

1.0 0.8 b° 8

Z

(b)

0.6

O.Ah Average of 3 Data

0.2l—L Q6 Q8 1

2 U 2C / mm

6

8 10

0

Fig. 16. The notched- to unnotched strength ratio of a boron-aluminium [ ± 4 5 / 0 ] laminate plotted against: (a) 2R, and (b) 2 c on log-log scale. In (b), the length of discontinuity for holes ( C C D ) , centre-slit ( S C D ) , holes with slits ( H E D ) and symmetric edge slits ( D E D ) is taken as 2 c . (After Mar and Lin [69]: reproduced with permission from Technomic Publishing Co. Inc.) 2

s

0

0

Fracture mechanical approach to metal-matrix

517

composites

in t h e p r i n c i p a l l o a d - c a r r y i n g l a m i n a e a n d p r o p o s e d a g e n e r a l fracture t o u g h n e s s parameter Q

a s a m a t e r i a l c o n s t a n t . T h e p r o c e d u r e t o c a l c u l a t e Q a n d its p h y s i c a l

c

c

m e a n i n g are as follows. First, t h e fracture t o u g h n e s s was e x p r e s s e d by b o t h a stress-intensity a n d s t r a i n - i n t e n s i t y f a c t o r . T h e s t r e s s - i n t e n s i t y f a c t o r for t h e a centre-slit, K , = a [7r(c

Q

N

where cr

+ p) sec(7TCo/W)]

0

c

1 / 2

,

(46)

is t h e n o t c h e d s t r e n g t h of t h e

N

d a m a g e z o n e at f a i l u r e a n d [sec(nc / 0

finite-width

W)]

l/2

0

strength σ . Thus, p 0

and K

c

specimen, p

is t h e

W h e n t h e r e is n o slit ( 2 c = 0 ) , t h e s t r e n g t h c r

finite-width

c

is t h e size of

correction

factor.

in e q . ( 4 6 ) e q u a l s t h e u n n o t c h e d

N

are related by

Q

p = (K /a ) /7T.

(47)

2

c

Q

0

By c o m b i n i n g e q . ( 4 6 ) w i t h e q . ( 4 7 ) , K

Q

K

= K J[l

Q

- K J(7TC a )] 2

Q

K

2

Q

0

is e x p r e s s e d a s (48)

l/2

0

9

= < r [ * r c s e c ( 7 T C / W)]

(49)

1/2

Qe

N

where K

0

0

9

is t h e e l a s t i c c r i t i c a l s t r e s s - i n t e n s i t y f a c t o r . F o r l o n g slits, K ~

Qe

a

specimen with

was defined by

Q

K

finite-width

Q

K . Qe

The

s t r e s s - s t r a i n b e h a v i o u r o f b o r o n - a l u m i n i u m l a m i n a t e s w a s v e r y n o n - l i n e a r . It w a s t h o u g h t t h a t t h e effect o f a n o n - l i n e a r c o m p l i a n c e c o u l d l a r g e l y b e a v o i d e d

by

analyzing t h e d a t a in t e r m s of strain r a t h e r t h a n stress. T h e r e f o r e , t h e d a t a w e r e also a n a l y z e d with a strain-intensity factor defined as K

= s [7r(c

£

0

where ε

+ p)

0

s e c ( 7 T C / W)] , 0

is t h e r e m o t e s t r a i n a n d ρ

0

(50)

l/2

e

ε

is t h e d a m a g e z o n e size f o r s t r a i n . By f o l l o w i n g

the s a m e p r o c e d u r e w h i c h was used to derive eq. (48), we obtain K

= K J[l~

Q

eQ

where K

K J(vC J] , 2

eQ

(51)

l/2

o£

is t h e c r i t i c a l s t r a i n - i n t e n s i t y f a c t o r , s

eQ

tu

is t h e u n n o t c h e d u l t i m a t e t e n s i l e

strain, a n d ^

Q

e

= e o c O c o sec(7TC / 0

W)]

(52)

l/2 9

is t h e e l a s t i c c r i t i c a l s t r a i n - i n t e n s i t y f a c t o r , w h e r e s

0c

is t h e r e m o t e s t r a i n a t f a i l u r e .

S i m i l a r t o e q . ( 4 7 ) , t h e size o f t h e d a m a g e z o n e at f a i l u r e is Psc = (K /e ) /m

(53)

2

eQ

tu

T h e strains in t h e p r i n c i p a l l o a d - c a r r y i n g l a m i n a e w e r e d e r i v e d from t h e solution for s i n g u l a r s t r e s s e s i n a n a n i s o t r o p i c , h o m o g e n e o u s s h e e t c o n t a i n i n g a t h r o u g h - t h e thickness crack. F o r a n o r t h o t r o p i c material with m o d e - I loading u n d e r p l a n e stress, t h e singular stresses j u s t a h e a d of the crack tip in t h e p l a n e of t h e crack can b e simplified to ,'(£ £ )' 1 / 2

X

V

S. Ochiai

518

w h e r e the c o o r d i n a t e s χ a n d y refer to t h e directions p e r p e n d i c u l a r a n d parallel to t e n s i l e a x i s , r e s p e c t i v e l y . r is z e r o b e c a u s e o f s y m m e t r y . T h e s t r a i n s a r e g i v e n b y xy

K m Εγ(2πτ)

y

(55)

/2

λ

0

w h e r e β is t h e m a t r i x o f c o n s t i t u t i v e p r o p e r t i e s , g i v e n b y l/E

-Vy lE

[β] =

0

-Vyx/Ey

x

X

0

l/Ey

y

0

0

(56)

l/G

xy

T h e s i n g u l a r s t r a i n s in t h e p r i n c i p a l d i r e c t i o n s of t h e i t h l a m i n a a r e g i v e n b y

f

"

ε

χ

Sy

(57)

7xy^

i

w h e r e [ T ] is t h e t e n s o r t r a n s f o r m a t i o n m a t r i x , a n d t h e s u b s c r i p t s 1 a n d 2 refer t o d i r e c t i o n s p a r a l l e l a n d t r a n s v e r s e t o t h e fibres, r e s p e c t i v e l y . D e n o t i n g t h e a n g l e of fibre o r i e n t a t i o n for t h e i t h l a m i n a t e b y a, [ T] is g i v e n b y sin

2

a

cos

sin 2a

a

cos" a

sin a

-sin 2a

— \sin 2α

| sin 2 a

—cos 2 a

[T] =

2

(58)

Substituting eq. (55) into eq. (57), w e h a v e «1

£»(2wr)

(59)

6

1/2

?12

where

'(E E y

/2

x

y

= [Τ],[βΊ<

(60) 0

ξΐ2

A s s u m i n g t h a t t h e l a m i n a t e fails w h e n t h e fibre s t r a i n s r e a c h a critical level in t h e principle load-carrying laminae, e ( 2 7 r r ) i n e q . (59) will b e a c o n s t a n t a t f a i l u r e . C o n s e q u e n t l y , s e t t i n g Κ = K , t h e Kq^^i/Ey i n e q . ( 5 9 ) [ w h e r e ( £ , ) , refers t o t h e p r i n c i p a l l o a d - c a r r y i n g l a m i n a e ] will b e c o n s t a n t a t f a i l u r e , i n d e p e n d e n t o f l a m i n a t e o r i e n t a t i o n . T h u s t h e g e n e r a l f r a c t u r e t o u g h n e s s p a r a m e t e r Q is d e f i n e d b y 1 / 2

1

Q

c

(61) Q

c

is g i v e n i n t e r m s o f K

eQ


l

by (62)

Fracture mechanical approach to metal-matrix

composites

519

F o r t h e 0° p r i n c i p a l l o a d - c a r r y i n g l a m i n a e , e q . ( 6 0 ) r e s u l t s in (ti)

= \-v (E /E ) \

(63)

l/

i

yx

x

y

a n d for b o t h + 4 5 ° a n d —45° p r i n c i p a l l o a d - c a r r y i n g l a m i n a e , (fi),- = \ [1 " PyxiEJEy) ' ] 1 2

[1 + (E /E ) ' ].

(64)

1 2

y

x

T h e g e n e r a l fracture t o u g h n e s s p a r a m e t e r Q defines t h e critical level of strains in t h e p r i n c i p a l l o a d - c a r r y i n g p l i e s . T h e v a l u e s o f Q w e r e c a l c u l a t e d u s i n g t h r e e different a p p r o a c h e s t o s h o w t h e effects o f t h e n o n - l i n e a r s t r e s s - s t r a i n b e h a v i o u r : (1) Q = ξιΚ } with eq. (62) a n d average K v a l u e s f o r all s p e c i m e n s t e s t e d ; (2) Q = ( K /E w i t h e q . (61) a n d a v e r a g e K v a l u e s for all s p e c i m e n s t e s t e d ; and c

c

c

ε(

c

1

Q

eQ

y

Q

(3)
Q

uy

uy

where E

y9

uy

is t h e

F i g u r e 17 s h o w s t h e c a l c u l a t e d v a l u e s o f Q f o r s p e c i m e n s t e s t e d . T h e Q = ξ\Κ v a l u e s a r e i n t e r m e d i a t e b e t w e e n t h e Q = ξ Κ /E a n d Q = £iK /E values, as expected, a n d generally closer to the latter. F o r a linear elastic material these values s h o u l d h a v e b e e n e q u a l . B u t for a n o n - l i n e a r m a t e r i a l like b o r o n - a l u m i n i u m c o m ­ p o s i t e s , t h e Q = ξχΚςι/E a n d t h e Q = ξ Κ /E values give u p p e r a n d l o w e r b o u n d s , r e s p e c t i v e l y . E x c e p t for [ ± 4 5 ] l a m i n a t e s , t h e ξ\Κ^ values are approxi­ m a t e l y e q u a l . F o r [ ± 4 5 ] l a m i n a t e s , t h e v a l u e is m u c h t o o h i g h d u e t o y i e l d i n g o f s p e c i m e n s w i t h s m a l l slits. F o r [ 0 ] l a m i n a t e s , t h e v a l u e is c l o s e t o t h e o t h e r s b u t c

c

uy

c

c

χ

0

y

c

λ

0

y

c

Q

ε0

uy

2 s

2 s

6 T

Fig. 17. Values of the general fracture toughness parameter Q for the boron-aluminium laminates. (After Poe and Sova [11]: reproduced with permission from the National Aeronautics and Space Administration.) c

S. Ochiai

520

slightly e l e v a t e d d u e t o t h e e x t e n s i v e d a m a g e a t t h e e n d s o f l o n g slits. T h e n for t h e s u b s e q u e n t calculations of the n o t c h e d strength, the Q values of [ 0 / ± 4 5 ] , [ ± 4 5 / 0 ] a n d [ 0 / ± 4 5 ] l a m i n a t e s w e r e a v e r a g e d , w h i c h is s h o w n in fig. 17 b y t h e d a s h e d line. T h e n this v a l u e of Q w a s u s e d to calculate K and K for e a c h l a m i n a t e o r i e n t a t i o n , a n d t h e n o t c h e d s t r e n g t h w a s p r e d i c t e d . F i g u r e 18 s h o w s t h e result of t h e calculation, w h e r e the strengths are multiplied by t h e finite-width c o r r e c t i o n f a c t o r . D i f f e r e n t s y m b o l s a r e u s e d for different s p e c i m e n w i d t h s . T h e strengths predicted with the K values agree well with t h e test values. c

2

s

2

s

s

c

eQ

Q

eQ

T h e r e s u l t s h o w n in fig. 18 i n d i c a t e s t h a t Q is r e a s o n a b l y i n d e p e n d e n t of t h e p r o p o r t i o n o f 0° a n d ± 4 5 ° p l i e s . T h e r e l a t i o n b e t w e e n K a n d Q is d e s c r i b e d o n l y o n the basis of the elastic c o n s t a n t s of t h e l a m i n a t e a n d the o r i e n t a t i o n angle of p r i n c i p a l l o a d - c a r r y i n g fibres. S i n c e e l a s t i c c o n s t a n t s c a n b e p r e d i c t e d , Q c a n b e c

Q

c

c

tf

I—ι—ι—ι—ι—ι—ι

I—ι—ι—ι—ι—ι—ι

Fig. 18. Comparison of measured notched strengths multiplied by the finite-width correction factor [sec(7rc / W ) ] with the prediction based on Poe and Sova's general fracture toughness (Q ) criterion for unidirectional and cross-plied boron-aluminium laminates. (After Poe and Sova [11]: reproduced with permission from the National Aeronautics and Space Administration.) 1 / 2

0

c

Fracture mechanical approach to metal-matrix

d e t e r m i n e d from tests of o n e lay-up, a n d K

composites

521

can then be predicted without additional

Q

t e s t i n g for o t h e r l a y - u p s o f t h e s a m e m a t e r i a l . If Q

is p r o p o r t i o n a l t o o n l y t h e u l t i m a t e t e n s i l e s t r a i n o f t h e

c

fibres,

e , f u

as

m e a s u r e d o n a u n i d i r e c t i o n a l l a m i n a t e , as i n d i c a t e d for several c o m p o s i t e systems [11], K

f o r all l a m i n a t e s c a n b e p r e d i c t e d f r o m o n l y t h e u n i d i r e c t i o n a l t e n s i l e

Q

p r o p e r t i e s a n d t h e c o n s t a n t QJ ε .

T h e prediction b a s e d on this idea was d e m o n ­

ΐχί

s t r a t e d t o a g r e e w i t h t h e e x p e r i m e n t a l d a t a , e x c e p t for l a m i n a t e s t h a t d e l a m i n a t e d o r split e x t e n s i v e l y [ 7 3 ] . P o e [ 7 3 ] f o u n d t h a t QJ e

is 1.5 ( m m )

fu

o n a v e r a g e for

1 / 2

v a r i o u s l a y - u p s a n d for v a r i o u s c o m p o s i t e s y s t e m s . T h e f a i l i n g s t r a i n r a t i o ( t h e r a t i o of n o t c h e d strain ε

tu

2

0

tu

0

of l a m i n a t e s ) w a s p r e d i c t e d by

= [1 + « ( 6 * „ / 1 . 5 ε J ] "

W)] /e l/2

e [sec(7TC / N

to u n n o t c h e d strain s

Ν

1 / 2

Γ

,

(65)

a n d t h i s s t r a i n r a t i o w a s c o n v e r t e d t o a s t r e s s r a t i o . T h e r e s u l t is s h o w n in fig. 19, showing again good agreement between prediction and experimental data. 6.1.2.5. R-curve criterion. If t h e d a m a g e z o n e p l a y s a r o l e a n a l o g o u s t o t h e p l a s t i c z o n e in m e t a l s , t h e p o s s i b i l i t y a r i s e s t h a t t h e r e s i s t a n c e c u r v e (K -curve) method, w h i c h h a s b e e n u s e d extensively to characterize fracture b e h a v i o u r of metals [ 7 4 ] , might also b e a p p l i c a b l e to c o m p o s i t e s . A c c o r d i n g to this m e t h o d , t h e d e v i a t i o n of C O D f r o m l i n e a r i t y in l o a d - C O D c u r v e s is a s s u m e d t o b e e n t i r e l y t h e r e s u l t o f s e l f - s i m i l a r c r a c k e x t e n s i o n . T h e p r o c e d u r e is briefly d i s c u s s e d h e r e . F r o m s t r a i g h t l i n e p o r t i o n s in t h e l o a d - C O D c u r v e s at v a r i o u s c r a c k l e n g t h s , t h e c o m p l i a n c e - c r a c k l e n g t h r e l a t i o n ( c a l i b r a t i o n c u r v e ) is m e a s u r e d in a d v a n c e . T h e n , in t h e l o a d - C O D d i a g r a m o f t h e s p e c i m e n , a s t r a i g h t l i n e is d r a w n f r o m t h e o r i g i n t o a s e l e c t e d p o i n t o n t h e l o a d - C O D c u r v e a n d t h e c o m p l i a n c e is m e a s u r e d . T h i s c o m p l i a n c e is m a t c h e d R

I 0

I

I

!

I

I

10

20

30

U0

50

2c

0

I 60

/ mm

Fig. 19. Measured notched strengths multiplied by the finite-width correction factor compared with the predictions based on the general fracture toughness criterion for unidirectional and cross-plied boronaluminium laminates. In the calculation, eq. (65) was employed, in which the QJe was assumed to be 1.5 ( m m ) . (After Poe [73]: reproduced with permission from Pergamon Journals Ltd.) fu

1 / 2

S. Ochiai

522

t o t h e c o m p l i a n c e c u r v e a n d t h e effective c r a c k l e n g t h c, g i v e n b y c = c + Ac,

(66)

0

is d e t e r m i n e d , w h e r e A c is t h e q u a s i - c r a c k e x t e n s i o n . R e p e a t i n g t h i s p r o c e d u r e , a different v a l u e o f effective c r a c k l e n g t h a s a f u n c t i o n o f t h e l o a d is o b t a i n e d . T h e crack growth resistance K

is g i v e n b y

R

K =

Y Œ(TTC)

R

w h e r e Y is t h e

1/2

(67)

finite-width

correction factor. At low a p p l i e d stress, stable crack

g r o w t h o c c u r s b u t , b e y o n d a c r i t i c a l s t r e s s , u n s t a b l e c r a c k g r o w t h o c c u r s . T h e critical stress c a n b e d e t e r m i n e d b y t h e t a n g e n t p o i n t b e t w e e n K

R

c u r v e d e f i n e d b y Κ = Ya(7rc) , c o n c e p t , t h e K -c R

in eq. (67) a n d the Κ

with σ as a p a r a m e t e r . A c c o r d i n g to t h e K ^ - c u r v e

i/2

curve, the K

R

value at u n s t a b l e crack g r o w t h a n d the crack

e x t e n s i o n Ac a t u n s t a b l e f a i l u r e a r e m a t e r i a l c o n s t a n t s , i n d e p e n d e n t o f n o t c h size a n d s p e c i m e n size. G a g g a r a n d B r o u t m a n [ 7 5 ] s h o w e d t h a t t h e K -cuvve

concept can be applied to

R

randomly oriented discontinuous H a h n [76] s h o w e d that the K

fibre-epoxy

or polyester composites. Morris and

c u r v e p r o v i d e s full i n f o r m a t i o n o n t h e

R

fracture

r e s i s t a n c e u p t o final f r a c t u r e of a n g l e - p l y g r a p h i t e - e p o x y c o m p o s i t e s . H o w e v e r , the results presented by Morris a n d H a h n [76] indicate that the tangent b e t w e e n K -curve R

a n d IC-eurve d o e s n o t exist until fracture a n d Ac at

point

fracture

d e p e n d s o n the initial crack length. O c h i a i a n d Peters [77,78] o b s e r v e d

similar

t r e n d s for c r o s s - p l i e d g r a p h i t e - e p o x y c o m p o s i t e s . W e l l s a n d B e a u m o n t [ 7 9 ] s h o w e d that w h e n cross-plied l a m i n a t e s , such as c a r b o n - e p o x y a n d K e v l a r - e p o x y , n o t c h - s e n s i t i v e , t h e R-curve

are

m e t h o d can predict the n o t c h e d strength well, while

w h e n t h e y a r e n o t c h - i n s e n s i t i v e , t h i s m e t h o d is i n v a l i d . O n t h e a p p l i c a b i l i t y o f t h e K -curve R

concept to b o r o n - a l u m i n i u m

composites,

there are o p p o s i t e o p i n i o n s . A w e r b u c h a n d H a h n [25] f o u n d that the iC^-curve m e t h o d c a n n o t b e a p p l i e d in t h e i r b o r o n - a l u m i n i u m c o m p o s i t e s s i n c e g r o w t h , w h i c h a p p e a r s in t h e form of m a t r i x cracking a l o n g t h e

fibre

w i t h i n i n d i v i d u a l p l i e s , d e l a m i n a t i o n , fibre b r e a k s at w e a k l o c a t i o n s ,

damage direction fibre-matrix

d e b o n d i n g , e t c . , c a n n o t b e c h a r a c t e r i z e d as s e l f - s i m i l a r c r a c k e x t e n s i o n . I n t h e i r e x p e r i m e n t a l s t u d y for u n i d i r e c t i o n a l c o m p o s i t e s p e c i m e n s [ 2 5 ] w h e r e a n i n t e r f e r o m e t r i c d i s p l a c e m e n t g a u g e , w h i c h is v e r y s e n s i t i v e t o t h e a p p e a r a n c e o f c r a c k - t i p d a m a g e , w a s u s e d t o m e a s u r e C O D , t h e K -curve R

b e c a m e different from

that

measured with a compliance gauge. This means that attention should be paid to t h e m e t h o d b y w h i c h C O D is d e t e c t e d . W h e n t h e s e n s i t i v e i n t e r f e r o m e t r i c d i s p l a c e ­ m e n t g a u g e w a s u s e d , t h e l o n g i t u d i n a l p l a s t i c d e f o r m a t i o n at t h e c r a c k t i p c a u s e d s i g n i f i c a n t i n c r e a s e i n C O D , r e s u l t i n g i n a l a r g e p l a t e a u in t h e l o a d - C O D c u r v e . B e c a u s e o f s u c h a flat l o a d - C O D c u r v e , e m p l o y m e n t of t h e c o m p l i a n c e m a t c h i n g m e t h o d for n o t c h e s l o n g e r t h a n 5.1 m m r e s u l t e d i n effective c r a c k l e n g t h s l o n g e r t h a n t h e w i d t h of t h e s p e c i m e n . W i t h a c o m p l i a n c e g a u g e , w h i c h is n o t so s e n s i t i v e t o d a m a g e g r o w t h , t h e K -curve R

c o u l d b e o b t a i n e d , a s s h o w n in fig. 2 0 . F i g u r e 20

indicates that the tangent point between the K

R

curve a n d Κ curve d i d n o t exist

at a n y p o i n t o t h e r t h a n t h e fracture p o i n t , a n d t h e d a m a g e g r o w t h Ac at fracture

Fracture mechanical approach to metal-matrix

ο

^ /

^

0

/

523

2Co=5,23

2Ç El35"/ o°

composites

2Cp=7,59

/

>2co--l0.18

·' / / / -

' /

i

J

.•2co=12.85

Ο

ο

0

~ ~ Α

8

Effective

12 Crack

16

20

Length / mm

Fig. 20. Fracture resistance of unidirectional boron-aluminium composites for various crack lengths obtained with a compliance gauge. (After Awerbuch and Hahn [25]; reproduced with permission from Technomic Publishing Co. Inc.)

w a s very large ( c o m p a r a b l e with c ) a n d w a s d e p e n d e n t o n c . If t h e K 0

0

R

value at

fracture w a s t a k e n as a fracture criterion, this value w a s also d e p e n d e n t o n the i n i t i a l c r a c k l e n g t h , a s k n o w n f r o m fig. 2 0 . T h e s e r e s u l t s i n d i c a t e t h a t t h e A c a n d K

R

value at fracture are n o t material constants. Wright a n d Iannuzzi [63] o b t a i n e d

Ac at fracture also for u n i d i r e c t i o n a l b o r o n - a l u m i n i u m c o m p o s i t e s . I n their results, the values of Ac were also d e p e n d e n t o n initial crack length. O n the other h a n d , M a h i s h i a n d A d a m s [80] s h o w e d the possibility of applying this m e t h o d to the s a m e c o m p o s i t e system. T h e y calculated t h e crack initiation a n d p r o p a g a t i o n i n t h e m a t r i x f r o m b r o k e n fibre e n d s i n a u n i t c o m p o s i t e c o n s i s t i n g o f a s i n g l e b r o k e n fibre e m b e d d e d i n a n a l u m i n i u m s h e a t h b y u s i n g a

finite-element

m e t h o d . As a result, a substantial a m o u n t of slow a n d stable crack growth w a s found prior to an unstable catastrophic failure. T h e result that a continually increas­ ing load h a d to b e a p p l i e d to m a i n t a i n the stable crack g r o w t h indicates a n increasing r e s i s t a n c e t o c r a c k g r o w t h . T h i s s i t u a t i o n is s i m i l a r t o t h a t o b s e r v e d i n m e t a l s . Mahishi a n d A d a m s [80] calculated the energy release rate G by G = (7r -7r )/kc, B

where π

Β

(68)

A

and π

Α

are total potential energy before the incremental crack growth

a n d after t h e crack i n c r e m e n t , respectively. T h e total p o t e n t i a l e n e r g y

π

was

calculated as t h e s u m of t h e strain energy of t h e system a n d t h e potential energy of t h e externally a p p l i e d l o a d . F r o m t h e relation G = K /E ,

(69)

2

R

the K

R

L

v a l u e w a s c o m p u t e d . F i g u r e 21 s h o w s t h e c o m p u t e d K

R

v a l u e s f o r different

524

S. Ochiai

1200, 1000 800 E 60 0

O CL ς:

Û 200

0

0.25

0.50

0.75

1.00

Crack Length(Normalized) (r +Ac)/r m i

f

r

Fig. 21 . Crack-growt h resistanc e curve s o f unidirectiona l boron-aluminiu m composite s fo r differen t r / r ratios , calculate d wit h th e single-fibr e mode l i n whic h crac k propagatio n fro m fibre int o matri x was considered . (Afte r Mahish i an d Adam s [80] : reproduce d b y permissio n fro m Technomi c Publishin g Co. Inc. ) f

m

r / r r a t i o ( r a n d r a r e m a t r i x a n d fibre r a d i i , r e s p e c t i v e l y , i n t h e i r s i n g l e fibre model) plotte d agains t crac k growt h normalize d wit h respec t t o r . Th e slop e o f e a c h K -curve a b r u p t l y c h a n g e d afte r a c e r t a i n a m o u n t o f c r a c k p r o p a g a t i o n , indicating th e point s o f crac k instability . W h e n th e K valu e c o r r e s p o n d i n g t o th e point o f instabilit y wa s take n a s a m e a s u r e o f fractur e toughness , th e fractur e t o u g h n e s s v a l u e o f 12 0 M P a V m , c o r r e s p o n d i n g t o r / r = 0.7 1 w a s i n c l o s e a g r e e m e n t w i t h t h e e x p e r i m e n t a l l y d e t e r m i n e d v a l u e o f 112. 7 M P a Vm fo r t h e u n i d i r e c tional b o r o n - a l u m i n i u m composite s o f A w e r b u c h a n d H a h n [25] . I n thi s model , only th e crac k p r o p a g a t i o n int o th e matri x wa s considered , bu t n o othe r event s observed i n actua l composite s wer e incorporated . O n thi s point , th e applicabilit y o f t h e K -cmve t o a c t u a l c o m p o s i t e s i s n o t n e c e s s a r i l y d e m o n s t r a t e d b y t h e i r m o d e l . H o w e v e r , fo r t h e c a s e w h e r e c r a c k p r o p a g a t i o n f r o m fibre e n d s i n t o t h e m a t r i x i s the controllin g process , thei r concep t coul d b e applied . f

m

m

f

m

R

R

f

m

R

6.1.2.6. J-integral criterion. I n t h e L E F M a p p r o a c h , i t i s i m p l i c i t l y a s s u m e d t h a t the crack-ti p plasticit y i s negligibl y small . I n metals , however , w h e n th e plasti c zon e at t h e c r a c k t i p i s n o t s m a l l , t h e c r i t i c a l e n e r g y r e l e a s e r a t e G i s i n f l u e n c e d b y t h e plastic z o n e . F o r thi s p r o b l e m , t h e / - i n t e g r a l a p p r o a c h [81 ] ca n b e applied . I n thi s a p p r o a c h , th e energ y releas e rat e J i s give n b y c

J=

(70)

-dV/dc,

w h e r e V i s t h e p o t e n t i a l e n e r g y . T h e r e l a t i o n fo r / i s t h e g e n e r a l i z e d r e l a t i o n fo r the energ y releas e rat e d u e t o crac k propagation , allowin g existenc e o f a plasti c z o n e . A t instability , / i s e x p r e s s e d a s J w h i c h i s e q u a l t o G fo r t h e elasti c cas e b u t n o t fo r t h e p l a s t i c c a s e . c

c

Fracture mechanical approach to metal-matrix

525

composites

H o o v e r a n d A l l r e d [ 6 5 ] e s t i m a t e d t h e / - i n t e g r a l for a u n i d i r e c t i o n a l a l u m i n i u m c o m p o s i t e , u s i n g t w o définitions of

borsic-

J, c

\-dP/dc) d8,

(71)

s

0 P.

(dô/dc)

dP.

P

(72)

JO

w h e r e Ρ is t h e l o a d p e r u n i t l e n g t h o f c r a c k f r o n t a n d δ is t h e l o a d - p o i n t d i s p l a c e m e n t . A s / is a p a t h - i n d e p e n d e n t i n t e g r a l [ 8 1 ] , J o f e q . (71) s h o u l d b e e q u a l t o t h a t of e q . ( 7 2 ) . T h e e x p e r i m e n t a l r e s u l t s of H o o v e r a n d A l l r e d [ 6 5 ] , h o w e v e r , s h o w e d t h a t t h e s e v a l u e s w e r e different f r o m e a c h o t h e r . A l s o , t h e y f o u n d t h a t t h e J v a l u e s d e f i n e d b y e q s . (71) a n d (72) d e p e n d e d o n c r a c k l e n g t h . F r o m t h e s e r e s u l t s , t h e y c o n c l u d e d t h a t t h e J - i n t e g r a l a p p r o a c h is n o t a n effective f r a c t u r e c r i t e r i o n for t h e i r s p e c i m e n s . A c c o r d i n g t o t h e m , a p o s s i b l e r e a s o n f o r t h i s is t h a t t h e c o m p o s i t e is n o t a h o m o g e n e o u s , i s o t r o p i c m a t e r i a l , a s is t h e o r e t i c a l l y r e q u i r e d . c

c

6.2. Fracture

criteria particular

to

composites

I n sect. 6 . 1 , v a r i o u s a p p r o a c h e s b a s e d o n c o n v e n t i o n a l a n d m o d i f i e d L E F M w e r e surveyed. Together with these a p p r o a c h e s , various a p p r o a c h e s out of the frame of conventional L E F M have b e e n proposed. T h e latter a p p r o a c h e s can be divided into two categories; m a c r o m e c h a n i c a l a n d micromechanical a p p r o a c h e s . In the m a c r o m e c h a n i c a l a p p r o a c h e s , t h e c o m p o s i t e is i d e a l l y a h o m o g e n e o u s a n i s o t r o p i c m a t e r i a l , w h i c h m a k e s it p o s s i b l e t o e s t i m a t e t h e s t r e s s - s t a t e m a t h e m a t i c a l l y b y applying conventional or modified conventional mechanics. In the micromechanical a p p r o a c h e s , a h e t e r o g e n e i t y is i n c o r p o r a t e d w h o s e effect is, h o w e v e r , difficult t o e s t i m a t e u n l e s s a s i m p l i f i c a t i o n is m a d e . T h e l a t t e r a p p r o a c h e s h a v e b e e n a p p l i e d m a i n l y t o u n i d i r e c t i o n a l c o m p o s i t e s s i n c e , a t p r e s e n t , it is difficult t o f o r m u l a t e m a t h e m a t i c a l l y t h e c o m p l e x i t y for c r o s s - p l i e d c o m p o s i t e s . I n sect. 6 . 2 . 1 , m a c r o s c o p i c a p p r o a c h e s a n d t h e i r a p p l i c a t i o n s will b e d i s c u s s e d a n d in sect. 6.2.2, m i c r o m ­ echanical approaches. 6.2.1. Macroscopic approaches 6.2.1.1. Point- and average-stress criteria (Whitney and Nuismefs fracture model [ 8 2 , 8 3 ] ) . W h i t n e y a n d N u i s m e r [ 8 2 , 8 3 ] c o n s i d e r e d t h a t t h e a p p l i c a t i o n of L E F M is q u e s t i o n a b l e s i n c e c r a c k s o f t h e t y p e s o b s e r v e d in m e t a l s d o n o t f o r m in r e s i n matrix composites u n d e r repeated load, and, unlike metals, a positive correlation b e t w e e n t h e u n n o t c h e d t e n s i l e s t r e n g t h for a c o m p o s i t e a n d its f r a c t u r e t o u g h n e s s seems to exist ( n a m e l y t h e higher t h e tensile strength, t h e h i g h e r the fracture t o u g h n e s s ) . T h e n t h e y p r o p o s e d a f a i l u r e m o d e l b a s e d o n t h e t h e o r e t i c a l stress d i s t r i b u t i o n n e a r t h e n o t c h t i p . T h e y a s s u m e d t h a t f r a c t u r e o c c u r s w h e n t h e stress o v e r s o m e d i s t a n c e d in f r o n t o f t h e n o t c h t i p is e q u a l t o o r g r e a t e r t h a n t h e s t r e n g t h o f t h e u n n o t c h e d m a t e r i a l σ ( p o i n t - s t r e s s c r i t e r i o n ) o r w h e n t h e a v e r a g e stress o v e r s o m e d i s t a n c e a e q u a l s t h e u n n o t c h e d strength (average-stress criterion), as 0

0

0

5. Ochiai

526

b

2c

*

0

*

Fig. 22. Schematic representation of: (a) the point-stress criterion and (b) average-stress criterion, proposed by Whitney and Nuismer [82,83].

s c h e m a t i c a l l y s h o w n in fig. 22 for a c e n t r e - n o t c h , w h e r e a is t h e s t r e s s in t e n s i l e d i r e c t i o n a s a f u n c t i o n o f χ a t y = 0. d a n d a a r e r e g a r d e d a s m a t e r i a l c o n s t a n t s . W h e n a circular hole with a radius R exists, the a can be expressed a p p r o x i m a t e l y by y

0

0

y

w h e r e σ°° is t h e a p p l i e d s t r e s s at infinity (y = ± o o ) , a n d Κγ

is t h e o r t h o t r o p i c stress

c o n c e n t r a t i o n f a c t o r f o r a n infinitely w i d e p l a t e g i v e n b y

F o r t h e c o o r d i n a t e s y s t e m g i v e n in fig. 2 2 , Κ γ is w r i t t e n a s X ? = 1 + [2[(E /E ) -

v ] + E /G ] .

l/2

y

(75)

1/2

x

yx

In the point-stress criterion, d

0

y

yx

is d e f i n e d b y

cr (at x = R + d ) = a , v

0

(76)

0

f r o m w h i c h t h e r a t i o o f t h e n o t c h e d t o u n n o t c h e d s t r e n g t h s , σ^/σ , b y c o m b i n i n g e q . (73) w i t h e q . ( 7 6 ) . T h i s r a t i o is g i v e n b y 0

σ~/σ

is c a l c u l a t e d

= 2/[2 + ζ + 3ζΪ-(Κ~-3)(5ζΪ-7ζΪ)1

(77)

2

0

w h e r e ζ = R/(R

+ d ).

ι

F o r t h e average stress criterion, a

0

0

a dx = a . y

0

is d e f i n e d b y

(78)

Fracture mechanical approach to metal-matrix

C o m b i n i n g e q . ( 7 8 ) w i t h e q . ( 7 3 ) , ο~κ/σ

is g i v e n b y

0

σ£/σ

= 2 ( 1 - ζ )/[2

0

2

w h e r e ζ = R/(R

+

2

- ζ\-ζ\+

(Κψ-3)(ζ -

ζ\)],

6

2

(79)

α ). 0

W h e n a slit w i t h a l e n g t h 2 c e x i s t s , t h e σ 0

a

527

composites

is g i v e n b y

γ

= (a x)/(x -c ) . x

y

2

2

(80)

1/2

0

C o m b i n i n g e q . ( 8 0 ) w i t h e q s . ( 7 6 ) a n d ( 7 8 ) , σ^/σ

is g i v e n b y

0

σ^/σ

= (1-ζ ) , 2

0

(81)

ι/2

3

f o r t h e p o i n t - s t r e s s c r i t e r i o n , w h e r e £ = c /(c + 3

ο-Ζ/σ

0

= [{\-ζ )/{\

0

d ),

0

and

0

+ ζ )ν ,

(82)

/2

Λ

4

for t h e a v e r a g e stress criterion, w h e r e ζ = c / ( c + a ) . T h e K 4

0

0

0

Q

is g i v e n b y

K =a [TTC (l-C )y , 2

Q

0

0

(83)

/2

3

for t h e p o i n t - s t r e s s c r i t e r i o n a n d K

= ^ [ 7 r c ( l - £ ) ( 1 + ζ )] \

(84)

ι/

Q

0

0

4

4

for t h e a v e r a g e - s t r e s s c r i t e r i o n . F o r n o t c h e d s a m p l e s , t h e a * in t h e W a d d o u p s , E i s e n m a n a n d K a m i n s k i m o d e l is r e l a t e d t o t h e c h a r a c t e r i s t i c l e n g t h a for a v e r a g e s t r e s s c r i t e r i o n b y a * = \a 0

Q

from

t h e c o m p a r i s o n of e q . ( 4 3 ) w i t h e q . ( 8 2 ) . A l t h o u g h t h e s e m o d e l s a r e b a s e d

on

different c o n c e p t s , t h e y a r e o p e r a t i v e l y e q u i v a l e n t f o r a c e n t r e n o t c h [ 1 ] . F i g u r e 2 3 s h o w s t h e v a r i a t i o n o f σ^/σ

of a b o r o n - a l u m i n i u m [ 0 / 9 0 ]

0

2 s

laminate

as a function of crack length, a n a l y z e d b y A w e r b u c h a n d M a d h u k a r [1] for their

Average Stress C r i t . : a

ι 0

0.1

Q

= 1.81 mm

ι

ι

ι

0.2

0.3

0.4

2C

Q

/

ι 0.5

0.6

W

Fig. 23. Comparison of experimental results with the predictions based on point- and average-stress criteria for boron-aluminium [ 0 / 9 0 ] laminates containing a centre notch. (After Awerbuch and Madhukar [1]: reproduced with permission from Technomic Publishing Co. Inc.) 2 s

528

S. Ochiai

c o l l e c t e d d a t a . T h e u s e o f c o n s t a n t v a l u e s for d a n d a , w h i c h w e r e t a k e n t o fit best the experimental results, yields a very g o o d agreement between experiments and predictions. 0

0

K is d e p e n d e n t o n n o t c h l e n g t h for b o t h c r i t e r i a , as k n o w n f r o m e q s . (83) a n d ( 8 4 ) . K is l o w a t s m a l l 2cJ W b u t it i n c r e a s e s w i t h i n c r e a s i n g 2c /W and a p p r o a c h e s asymptotically a constant value given by Q

Q

K

9

0

= a (2nd ) ,

(85)

1/2

Q

0

0

for t h e p o i n t - s t r e s s c r i t e r i o n , a n d K

= a (^a ) ,

(86)

i/2

Q

0

0

for t h e a v e r a g e - s t r e s s c r i t e r i o n . F i g u r e 2 4 [ 1 ] s h o w s t h e v a r i a t i o n o f K

of t h e

Q

s a m p l e s w h o s e d a t a h a v e b e e n s h o w n in fig. 2 3 , w h e r e t h e s o l i d a n d d a s h e d c u r v e d l i n e s r e p r e s e n t t h e p r e d i c t i o n s a c c o r d i n g t o e q s . (83) a n d ( 8 4 ) , r e s p e c t i v e l y , a n d t h e c o r r e s p o n d i n g s t r a i g h t l i n e s r e p r e s e n t t h e a s y m p t o t i c v a l u e s a c c o r d i n g t o e q s . (85) a n d (86). T h e results i n d i c a t e that using c o n s t a n t values of d a n d a 0

0

for K

Q

yields

a v e r y g o o d a g r e e m e n t b e t w e e n e x p e r i m e n t s a n d p r e d i c t i o n s , a s w e l l as t h a t for CO /

σ /σ . Ν

0

T h e point-stress criterion was modified by K a r l a k [84] w h o studied

fracture

strength of various types of g r a p h i t e - e p o x y l a m i n a t e s with a h o l e , a n d c o n c l u d e d that d

0

is n o t a m a t e r i a l c o n s t a n t b u t is r e l a t e d t o t h e h o l e r a d i u s . H e s h o w e d t h a t

t h e p o i n t - s t r e s s c r i t e r i o n c a n well p r e d i c t f r a c t u r e s t r e n g t h w h e n d

0

d ocR

1/2

is g i v e n b y (87)

0

Pipes, Wetherhold a n d Gillespie [85-87] c o m b i n e d the point-stress criterion with t h e o b s e r v a t i o n of K a r l a k [ 8 4 ] m e n t i o n e d a b o v e , a n d p r o p o s e d a n e w m o d e l t o d e s c r i b e a n d t o p r e d i c t f r a c t u r e s t r e n g t h , w h i c h is d i s c u s s e d b e l o w . 50

Point Stress C r i t .

0.1

0.2

0.3

2C /

OA

K - 3 5 . 2 MPa-m Q

0.5

0.6

W

0

Fig. 24. Critical stress intensity factor K based on point- and average-stress criteria for boron-aluminium [ 0 / 9 0 ] laminates containing a centre notch. Individual data are those shown in fig. 23. (After Awerbuch and Madhukar [1]: reproduced with permission from Technomic Publishing Co. Inc.) Q

2 s

Fracture mechanical

approach to metal-matrix

composites

529

6.2.1.2. The Pipes, Wetherhold and Gillespie's criterion [ 8 5 - 8 7 ] . P i p e s , W e t h e r h o l d a n d Gillespie [ 8 5 - 8 7 ] p r o p o s e d a m o d e l to d e s c r i b e t h e strength of n o t c h e d c o m ­ posites by i n t r o d u c i n g the following n e w p a r a m e t e r s : the n o t c h sensitivity factor C a n d t h e e x p o n e n t i a l p a r a m e t e r m, f o l l o w e d b y a n o t c h e d s t r e n g t h - r a d i u s ( o r slit length) s u p e r p o s i t i o n m e t h o d . This s u p e r p o s i t i o n m e t h o d allows s u p e r p o s i t i o n of n o t c h e d s t r e n g t h t o a s i n g l e m a s t e r c u r v e f o r all l a m i n a t e s h a v i n g a c o m m o n s t r e s s concentration factor Κγ. In their model, the point-stress criterion was used, but the characteristic length d w a s t a k e n t o b e a f u n c t i o n o f h o l e r a d i u s (slit l e n g t h ) b a s e d u p o n K a r l a k ' s o b s e r v a t i o n t h a t d is n o t a m a t e r i a l c o n s t a n t b u t is r e l a t e d t o t h e h o l e r a d i u s [ 8 4 ] . In their m o d e l , d was given by 0

0

0

d ocR ,

(88)

m

0

for a h o l e . σ^/σ

for a h o l e u n d e r t h e c o n d i t i o n s m e n t i o n e d a b o v e is g i v e n b y

0

c7^/c7 = 2[2 + [ / ( ^ ) ] - + 3 | y ( ^ ) ] 2

4

0

-(K~-3)(5[f(R)r -mR)r )r\ 6

w h e r e f(R) f(R)

m

s

is g i v e n b y = 1+ J R

(

W

-

1

)

^ -

M

)

C (

|

)

,

(90)

w h e r e R is t h e r e f e r e n c e r a d i u s , a n d C a n d m a r e t h e n o t c h s e n s i t i v i t y f a c t o r a n d the e x p o n e n t i a l p a r a m e t e r , respectively, as m e n t i o n e d a b o v e . C a n d m are the v a r i a b l e s for a g i v e n in e q . ( 8 9 ) , i n f l u e n c i n g n o t c h e d s t r e n g t h . T h e l a r g e r C, t h e m o r e n o t c h - s e n s i t i v e b e c o m e s t h e l a m i n a t e for a g i v e n h o l e r a d i u s , m a c t s t o c h a n g e t h e s l o p e o f a^/a -R curve, m ranges from zero, at w h i c h the p r o p o s e d criterion b e c o m e s identical to the point-stress criterion of W h i t n e y a n d N u i s m e r [82,83], to u n i t y , a t w h i c h t h e f r a c t u r e s t r e n g t h b e c o m e s i n d e p e n d e n t of h o l e r a d i u s . S i m i l a r l y , 0 " N / ο for slit n o t c h is g i v e n b y 0

0

σ

a

^/^o=[l-(l + 4 " m

where d

0

1 )

/C'- )- ] 1

2

l / 2

,

(91)

for a slit n o t c h is g i v e n b y

d =cZ/C.

(92)

0

I n e q . ( 9 2 ) , C is t h e s l i t - n o t c h s e n s i t i v i t y f a c t o r , c o r r e s p o n d i n g t o C for a h o l e - n o t c h sensitivity factor [87]. F i g u r e 2 5 s h o w s t h e v a r i a t i o n o f σ^/σ a s a f u n c t i o n o f log(R) o f u n i d i r e c t i o n a l b o r o n - a l u m i n i u m composites where the materials I to III have identical volume f r a c t i o n o f fibre a n d i d e n t i c a l u n n o t c h e d s t r e n g t h , b u t differ in y i e l d s t r e s s of m a t r i x b e t w e e n I a n d ( I I a n d I I I ) (42.8 M P a for I a n d 29 M P a for I I a n d I I I ) a n d in p r o c e s s i n g d a t a b e t w e e n I I a n d I I I . It is e v i d e n t t h a t , w h e n a set o f v a l u e s of C a n d m is d e t e r m i n e d t o fit b e s t t h e e x p e r i m e n t a l r e s u l t s , t h e O~n/°O v e r s u s log(R) r e l a t i o n is w e l l d e s c r i b e d . T h e r e s u l t s in fig. 25 s h o w t h a t m a t e r i a l I e x h i b i t s a m u c h g r e a t e r n o t c h s e n s i t i v i t y ( C ] = 1.16 m m ) t h a n t h e m a t e r i a l s I I a n d I I I ( C = 0.39 m m and C = 0.34 m m ) , i n d i c a t i n g t h a t a h i g h y i e l d s t r e s s of t h e matrix has a deleterious influence. 0

- 1

- 1

n

- 1

H I

530

S. Ochiai

Log(R)

(R/mm)

Fig. 25. Comparison of experimental results with the predictions by eq. (90) for unidirectional boronaluminium composites with a hole. (After Pipes, Wetherhold and Gillespie [85]: reproduced with permission from Technomic Publishing Co. Inc.)

W h e n Κγ is c o n s t a n t f o r a g i v e n c o m p o s i t e m a t e r i a l s y s t e m , o r a s y s t e m of l a m i n a t e c o n f i g u r a t i o n s , it is p o s s i b l e t o s u p e r i m p o s e all t h e n o t c h e d s t r e n g t h d a t a representing various values of t h e n o t c h sensitivity C a n d e x p o n e n t i a l p a r a m e t e r m to a single master curve. In the case of a l a m i n a t e with a hole, letting R = 1 inch (25.4 m m ) , a r a d i u s shift p a r a m e t e r a is d e f i n e d a s f o l l o w s 0

c

a= c

(C*/C)

1 / ( m

-

1 }

( 0 < m < 1).

(93)

T h e n o t c h e d s t r e n g t h d a t a for a n o t c h s e n s i t i v i t y C a r e t h e n s u p e r i m p o s e d u p o n t h a t of t h e n o t c h s e n s i t i v i t y C * b y s h i f t i n g t h e s t r e n g t h a t r a d i u s R t o as i l l u s t r a t e d in fig. 2 6 a , w h e r e R* is g i v e n b y R* = a R. c

(94)

Fig. 26. (a) Radius-notch sensitivity factor superposition, and (b) exponential shift parameter. (After Pipes, Wetherhold and Gillespie [85]: reproduced with permission from Technomic Publishing Co. Inc.)

Fracture mechanical approach to metal-matrix

composites

531

W h e n t h e s t r e n g t h d a t a for m a t e r i a l s o r l a m i n a t e s o f different e x p o n e n t i a l p a r a m e t e r s a r e t o b e s u p e r i m p o s e d , a s e c o n d shift p a r a m e t e r is d e f i n e d : a

m

= (

m

- l ) / (

m

* - l )

(0
(95)

T h e n o t c h e d s t r e n g t h d a t a f o r a g i v e n e x p o n e n t i a l p a r a m e t e r , m, is s u p e r i m p o s e d u p o n t h a t for m * b y s h i f t i n g t h e s t r e n g t h a t r a d i u s R t o R*,

a s i l l u s t r a t e d in fig.

2 6 b , w h e r e l o g ( # * ) is g i v e n b y \og(R*)

= a

m

(96)

log(#).

I n t h i s w a y , w h e n t w o m a t e r i a l s a r e c h a r a c t e r i z e d b y t w o sets o f c o n s t a n t s , ( C , m) a n d ( C * , ra*), t h e σ^/σ v e r s u s R c u r v e f o r a set o f ( C , m ) c a n b e s u p e r i m p o s e d o n t h e c u r v e f o r a set o f ( C * , m * ) , a n d v i c e v e r s a . 0

W i t h t h i s p r o c e d u r e , t h e d a t a for m a t e r i a l s I t o I I I , s h o w n in fig. 2 5 , c a n b e s u p e r i m p o s e d o n a m a s t e r c u r v e w i t h C = 0.39 m m a n d m = 0.5, a s s h o w n in fig. 27. It is p o s s i b l e t o c h o o s e o t h e r m a s t e r c u r v e s b y t a k i n g different sets o f ( C , m). According to Pipes, Wetherhold and Gillespie [85-87], the superposition prin­ ciple allows quantitative c h a r a c t e r i z a t i o n of t h e influence of t h e l a m i n a t e stacking sequence u p o n n o t c h e d strength, since does not vary with laminate sequence. I n a d d i t i o n , it is p o s s i b l e t o c o n s t r u c t r e l a t i o n s f o r n o t c h e d s t r e n g t h v e r s u s h o l e ( o r slit) size f o r a n y s t a c k i n g s e q u e n c e , g i v e n t h a t for a t l e a s t o n e o t h e r s t a c k i n g s e q u e n c e a n d t h e s t r e n g t h f o r t w o h o l e (slit) sizes in t h e d e s i r e d s t a c k i n g s e q u e n c e . - 1

6.2.2. Microscopic approach A s d i s c u s s e d in s e c t s . 2 a n d 3 , m a t r i x y i e l d i n g a n d s p l i t t i n g r e s u l t i n b l u n t i n g o f t h e n o t c h t i p . A s a r e s u l t , t h e u n n o t c h e d s t r e n g t h c h a n g e d d u e t o o c c u r r e n c e of 1.0 0

0.75h

t?

0.50

0.25

0

-1

ο

II

•

III

0

2 L o g R (R/mm)

Fig. 27. Master curve for unidirectional boron-aluminium composites, obtained by the application of the superposition method to the data shown in fig. 25. (After Pipes, Wetherhold and Gillespie [85]: reproduced with permission from Technomic Publishing Co. Inc.)

532

5. Ochiai

t h e s e e v e n t s . F o r c r o s s - p l i e d c o m p o s i t e s , it is, h o w e v e r , difficult t o s o l v e t h e g r o w t h of d a m a g e z o n e analytically d u e to the complexities m e n t i o n e d already. O n the o t h e r h a n d , for u n i d i r e c t i o n a l c o m p o s i t e s , t h e g r o w t h o f d a m a g e z o n e c a n b e s o l v e d a p p r o x i m a t e l y b y t h e s h e a r - l a g a n a l y s i s m e t h o d d i s c u s s e d in sect. 2. A s s h o w n in sect. 2, t h e s t r e s s c o n c e n t r a t i o n in t h e i n t a c t fibres in f r o n t o f t h e n o t c h t i p is r e d u c e d m u c h b y m a t r i x y i e l d i n g a n d s p l i t t i n g . S u c h a r e d u c e d s t r e s s c o n c e n t r a t i o n will l e a d t o a h i g h e r n o t c h e d s t r e n g t h t h a n t h a t p r e d i c t e d b y e l a s t i c m e c h a n i c s . T h i s effect can b e m o d e l l e d as follows. A s s u m i n g t h a t t h e f r a c t u r e o f a u n i d i r e c t i o n a l c o m p o s i t e o c c u r s w h e n t h e fibres a h e a d o f t h e d a m a g e z o n e a r e b r o k e n , t h e s t r e s s level o f t h e fibre a t c o m p o s i t e f r a c t u r e σγ is g i v e n b y N

^ N = *fu,o/Ki(<>),

(97)

f f

where
Γ

f u 0

f

Γ ? Ν

This i d e a has b e e n a p p l i e d by Z w e b e n [17], G o r e e a n d G r o s s [23], R e e d y [20] a n d D h a r a n i , J o n e s a n d G o r e e [28]. Z w e b e n [17] applied this idea to g r a p h i t e - e p o x y c o m p o s i t e s a n d o b t a i n e d a fairly g o o d c o r r e l a t i o n b e t w e e n p r e d i c t i o n a n d e x p e r i ­ mental results. For unidirectional b o r o n - a l u m i n i u m composite, G o r e e a n d Gross [23] e m p l o y e d a m o d e l in w h i c h the longitudinal d a m a g e (splitting a n d matrix y i e l d i n g ) , s h o w n in fig. 6 a , w a s c o n s i d e r e d . D h a r a n i , J o n e s a n d G o r e e [ 2 8 ] e m p l o y e d a m o d e l in w h i c h l o n g i t u d i n a l a n d t r a n s v e r s e d a m a g e s , s h o w n in fig. 6 b , w e r e c o n s i d e r e d . T h e y c a l c u l a t e d t h e relation of c r / σ to crack length using the a b o v e i d e a , b a s e d o n t h e m o d e l s s h o w n in fig. 6. I n b o t h m o d e l s , t h e v a l u e s o f r a n d G /d w e r e s e l e c t e d t o fit b e s t t h e l o a d - C O D c u r v e s a s s h o w n in sect. 3 , a n d t h e t h u s s e l e c t e d v a l u e s w e r e u s e d t o c a l c u l a t e c r / c r . T h e r e s u l t s a r e s h o w n in fig. 28 [ 2 8 ] . B o t h m o d e l s p r e d i c t t h e s a m e t r e n d in s t r e n g t h c u r v e s a s t h a t g i v e n b y t h e e x p e r i m e n t a l r e s u l t s of A w e r b u c h a n d H a h n [ 8 8 ] . T h e m o d e l o f l o n g i t u d i n a l d a m a g e gives a m u c h l a r g e r d e c r e a s e in s t r e n g t h , w h i l e t h e m o d e l o f l o n g i t u d i n a l -f- t r a n s v e r s e d a m a g e predicts strength values very close to the experimental results. A c c o r d i n g t o D h a r a n i , J o n e s a n d G o r e e [ 2 8 ] , t h e r e s u l t s s h o w n in fig. 28 i n d i c a t e a n a p p r o x i ­ m a t e l y c o n s t a n t t r a n s v e r s e d a m a g e - z o n e s i z e , i n d e p e n d e n t of i n i t i a l n o t c h l e n g t h , w h i c h is in a g r e e m e n t w i t h t h e p r e d i c t i o n b a s e d o n W h i t n e y a n d N u i s m e r ' s m o d e l a l r e a d y d i s c u s s e d in t h i s s e c t i o n . H o w e v e r , in t h i s m o d e l , fibre s t r e s s in f r o n t of t h e n o t c h is less s e v e r e t h a n t h e s q u a r e - r o o t b e h a v i o u r t h a t is a s s u m e d in W h i t n e y a n d Nuismer's model. N

0

y

m

m

N

0

R e e d y [ 2 0 ] c a l c u l a t e d t h e s t r e s s c o n c e n t r a t i o n o f t h e i n t a c t fibres a h e a d of t h e n o t c h tip for the case w h e r e matrix at the n o t c h tip d e f o r m s plastically with strain-hardening b e y o n d the yield stress, b u t w i t h o u t splitting. In his calculation, h e u s e d t h e p r i n c i p l e o f s t a t i o n a r y c o m p l e m e n t a r y e n e r g y for a d e f o r m a t i o n t h e o r y o f p l a s t i c i t y . H e u s e d t h e t h u s c a l c u l a t e d s t r e s s in t h e i n t a c t fibres a h e a d o f t h e n o t c h t i p a n d p r e d i c t e d t h e s t r e n g t h o f c o m p o s i t e s in t h e f o l l o w i n g m a n n e r . W h e n

Fracture mechanical approach to metal-matrix

composites

533

1.0 Experimental

Longitudinal Damage Model

0.2 ^

0

20

10

30

50

60

70

Number of Broken Fibres Fig. 28. Comparison of experimental notched strengths of unidirectional boron-aluminium composites reported by Awerbuch and Hahn [88] with the predictions based on the longitudinal damage model of Goree and Gross [23], shown in fig. 6a, and those based on the longitudinal + transverse damage model of Dharani, Jones and Goree [28], shown in fig. 6b. As both models deal with a two-dimensional composite, the crack length in actual specimens is given by the product of the number of cut fibres and fibre spacing. In the calculation, the number of fibres in the transverse damage zone N C was taken to be 7 in the Dharani, Jones and Goree model. (After Dharani, Jones and Goree [28]: reproduced with permission from Pergamon Journals Ltd.)

t h e n o t c h - t i p fibre s t r e s s r e a c h e s σ γ , t h e a p p l i e d b o u n d a r y l o a d s w e r e h e l d fixed a n d t h e n o t c h w a s e x t e n d e d t o t h e n e x t fibre b y s l o w l y r e d u c i n g t h e fibre s t r e s s a t the notch from a t o z e r o . T h e n o t c h - t i p fibre s t r e s s w a s t h e n r e d e f i n e d as t h e fibre s t r e s s a t t h e t i p o f t h e e x t e n d e d n o t c h . If t h e n e w s t r e s s w a s less t h a n σ , t h e n t h e n o t c h w a s c o n s i d e r e d t o h a v e g r o w n s t a b l y . I n t h a t e v e n t , t h e p r o c e s s of l o a d i n g to t h e fracture criterion given by eq. (97) a n d s u b s e q u e n t n o t c h extension t o t h e n e x t fibre w a s r e p e a t e d . W h e n t h e n e w s t r e s s a t t h e t i p e x c e e d s o - , t h e notch was considered to have grown unstably since the fracture criterion was e x c e e d e d a t t h e e x t e n d e d n o t c h l e n g t h w i t h o u t f u r t h e r i n c r e a s e in t h e a p p l i e d l o a d . F i g u r e 29 s h o w s t h e a v e r a g e a p p l i e d s t r e s s o n t h e c o m p o s i t e , σ , i n u n i d i r e c t i o n a l b o r o n - a l u m i n i u m c o m p o s i t e s at the initiation of n o t c h g r o w t h a n d also t h e m a x i m u m v a l u e r e a c h e d d u r i n g s t a b l e n o t c h g r o w t h , p l o t t e d a s a f u n c t i o n o f 2 c / W for o- ο = 3.45 a n d 5.52 G P a a s l o w e r a n d u p p e r b o u n d s , r e s p e c t i v e l y . T h e l o w e r b o u n d of or o = 3.45 G P a w a s c h o s e n s o t h a t t h e a n a l y s i s w o u l d c o r r e c t l y p r e d i c t t h e u n n o t c h e d strength of c o m p o s i t e , a n d t h e u p p e r b o u n d of a = 5.52 G P a w a s t a k e n f r o m a n u p p e r - b o u n d e s t i m a t e o f n o t c h - t i p fibre s t r e n g t h . T h e r e s u l t s s h o w n in fig. 29 i n d i c a t e s t h a t t h e n o t c h e d s t r e n g t h d e c r e a s e s m o s t r a p i d l y a s t h e c r a c k l e n g t h i n c r e a s e s for s m a l l v a l u e s o f 2 c / W. A n i n c r e a s e in t h e a p p l i e d s t r e s s d u r i n g n o t c h g r o w t h o c c u r r e d o n l y f o r i n t e r m e d i a t e i n i t i a l n o t c h l e n g t h s a n d is m o s t p r o n o u n c e d for t h e h i g h v a l u e o f c r = 5.52 G P a . T h e i n c r e a s e in l o a d d u r i n g s t a b l e n o t c h g r o w t h w a s r e l a t i v e l y s m a l l . All d a t a s h o w n in fig. 29 a r e b o u n d e d b y t h e t w o c u r v e s υ > 0

f u 0

ί χ ι 0

f u 0

Ν

0

fu

fu

f u 0

0

f u 0

S. Ochiai

534

2λ V

f

0.476

M=5 T = 3 9 . 5 MPa y

L--102 mm W=25.4 mm

Ο CL ID

Experimental F a i l u r e

Stress

lannuzzi(1974) Waszczak(l976) Wright,Welch(l978) Present

1.2

fu,o - 5 . 5 2 GPa

0.6

r

0.25

0.50

2c

0

0.75

/ W

Fig. 29. Comparison of the experimental notched strengths with the predictions based on Reedy's calculation for unidirectional boron-aluminium composites. The values employed in the calculation are shown, together with the results, where M is the stress exponent in shear strain hardening deformation of matrix. (After Reedy [20]: reproduced with permission from Pergamon Journals Ltd.)

for a = 3A5 a n d 5.52 G P a . T h e u p p e r b o u n d in fig. 29 is n o t p a r t i c u l a r l y t i g h t a n d t h a t is p r o b a b l y t h e c o n s e q u e n c e of t h e h i g h e r v a l u e o f a chosen. The curve for c r o = 3.45 G P a p r o v i d e s a r e a s o n a b l e e s t i m a t e o f t h e n o t c h e d s t r e n g t h . f u 0

f u 0

fu

In this section, only the fracture criteria that have actually b e e n applied to metal-matrix composites, have been discussed. These criteria have b e e n s h o w n to d e s c r i b e well t h e e x p e r i m e n t a l r e s u l t s for s p e c i m e n s a n a l y z e d . A w e r b u c h a n d M a d h u k a r [ 1 ] c o l l e c t e d a n u m b e r of e x p e r i m e n t a l d a t a o n b o r o n - a l u m i n i u m c o m ­ posites, together with g r a p h i t e - e p o x y a n d graphite-polyimide composites, and a n a l y z e d t h e m by a p p l y i n g t h e criteria of W a d d o u p s , E i s e n m a n a n d K a m i n s k i [66], Whitney and Nuismer [82,83], Pipes, Wetherhold and Gillespie [85-87], M a r and Lin [68,69], K a r l a k [84] a n d Poe a n d Sova [11]. As a result, they f o u n d that these criteria c a n describe well the collected d a t a [ 1 ] . O n e r e a s o n for this might b e attributed to the semi-empirical n a t u r e of m o s t criteria, b e c a u s e t h e characteristic p a r a m e t e r s , s u c h a s m a t e r i a l c o n s t a n t s , a r e d e t e r m i n e d b y fitting t o e x p e r i m e n t a l d a t a . O n c e t h e s e p a r a m e t e r s h a v e b e e n d e t e r m i n e d e x p e r i m e n t a l l y for a g i v e n c o m p o s i t e , t h e s e c a n b e u s e d t o p r e d i c t t h e n o t c h e d s t r e n g t h for t h i s g i v e n c o m p o s i t e . O n this p o i n t , these criteria are of practical use. A t p r e s e n t , it is difficult t o p r e d i c t t h e n o t c h e d s t r e n g t h in a t h e o r e t i c a l m a n n e r d u e t o t h e c o m p l e x i t y o f t h e d a m a g e z o n e a t t h e n o t c h t i p . If t h e m o r p h o l o g y of t h e d a m a g e z o n e ( i n c l u d i n g t h e e v e n t s s h o w n in fig. 1) a n d t e n s i l e s t r e s s d i s t r i b u t i o n s in fibre a n d m a t r i x in e a c h p l y , a n d s h e a r s t r e s s d i s t r i b u t i o n s a t i n t e r f a c e s b e t w e e n fibre a n d m a t r i x in e a c h p l y a n d b e t w e e n p l i e s , c o u l d b e e x p r e s s e d as a f u n c t i o n

Fracture mechanical approach to metal-matrix

composites

535

o f a p p l i e d s t r e s s , i n t e r f a c i a l b o n d i n g s t r e n g t h s b e t w e e n fibre a n d m a t r i x a n d b e t w e e n p l i e s , s t a c k i n g c o n f i g u r a t i o n a n d s e q u e n c e , size a n d s h a p e o f d i s c o n t i n u i t y , size o f specimens,

etc., the

fracture

behaviour

of metal-matrix

composites

could

be

described in m o r e detail a n d n o t c h e d strength could b e p r e d i c t e d theoretically. F o r t h i s a i m , f u r t h e r s t u d y is n e e d e d , e s p e c i a l l y t o clarify t h e b e h a v i o u r o f t h e d a m a g e z o n e a t t h e n o t c h t i p in v a r i o u s t y p e s o f c r o s s - p l i e d a n d

unidirectional

composites. 7. Influence of structural and environmental factors on fracture behaviour T h e f r a c t u r e b e h a v i o u r o f m e t a l - m a t r i x c o m p o s i t e s is i n f l u e n c e d b y m a n y s t r u c ­ tural a n d e n v i r o n m e n t a l p a r a m e t e r s , s u c h as yield stress of the m a t r i x , interfacial b o n d i n g s t r e n g t h b e t w e e n fibre a n d m a t r i x , fibre d i a m e t e r , t y p e o f d i s c o n t i n u i t y ( h o l e , slit o r r e c t a n g u l a r c u t - o u t , e t c . ) , s p e c i m e n w i d t h , s p e c i m e n t h i c k n e s s , l a m i n a t e c o n f i g u r a t i o n in c r o s s - p l i e d c o m p o s i t e s , s e q u e n c e o f l a y - u p in c r o s s - p l i e d l a m i n a t e s a n d t e s t t e m p e r a t u r e . I n t h i s s e c t i o n , as t h e i n f l u e n c e s o f t h e s e f a c t o r s f o r g r a p h i t e epoxy, boron-aluminium and graphite-polyimide composites have been summarized briefly b y A w e r b u c h a n d M a d h u k a r [ 1 ] , t h o s e f o r m e t a l - m a t r i x c o m p o s i t e s will b e s h o w n in s o m e detail, by discussing t h e review of A w e r b u c h a n d M a d h u k a r a n d other experimental d a t a reported until n o w . 7.1.

Yield stress

of the

matrix

P i p e s , W e t h e r h o l d a n d G i l l e s p i e [ 8 5 ] f o u n d in t h e i r u n i d i r e c t i o n a l b o r o n a l u m i n i u m c o m p o s i t e s t h a t a m a t e r i a l w i t h a h i g h m a t r i x y i e l d s t r e s s h a d less f r a c t u r e t o u g h n e s s t h a n a m a t e r i a l w i t h a l o w e r m a t r i x y i e l d s t r e s s , a s s h o w n in fig. 2 5 , w h i l e their u n n o t c h e d strengths were identical. G o r e e a n d Jones [89] carried out fracture t e s t s a n d c a l c u l a t i o n o n d a m a g e g r o w t h , u s i n g t h e m o d e l s h o w n in fig. 6 b . T h e y f o u n d t h a t t h e t r a n s v e r s e d a m a g e g r o w t h ( s t a b l e n o t c h e x t e n s i o n ) w a s l a r g e r in t h e softer m a t r i x ; t h e s t r o n g e r T6 m a t r i x substantially r e d u c e d t h e yielding of t h e matrix in t h e c o m p o s i t e , a n d , t h u s , r e d u c e d t h e a m o u n t o f d a m a g e t o l e r a n c e . T h e r e s u l t of increasing yield strength of the matrix m a t e r i a l w a s , t h u s , to m a k e t h e c o m p o s i t e more notch-sensitive and, actually, weaker t h a n a composite having a more ductile, energy-absorbing matrix. H o w e v e r , they stated that o n e s h o u l d b e careful in extend­ i n g t h i s r e a s o n i n g t o a n e v e n s t r o n g e r m a t r i x m a t e r i a l in w h i c h t h e t r a n s i t i o n is m a d e from matrix yielding to longitudinal splitting d u e to matrix failure. F o r e x a m p l e , a n e p o x y m a t r i x is a m a t r i x in w h i c h little y i e l d i n g is e x h i b i t e d b u t s p l i t t i n g o c c u r s at t h e n o t c h t i p u n d e r l o w l o a d . O n c e a s m a l l s p l i t t i n g f o r m s , it will q u i c k l y e x t e n d t o t h e m a c h i n e g r i p s . T h e s t r e s s c o n c e n t r a t i o n s in fibres a h e a d o f t h e n o t c h t i p s a r e t h e n r e m o v e d . T h u s , t h e c o n c l u s i o n t h a t a stiffer o r s t r o n g e r m a t r i x will w e a k e n t h e n o t c h e d c o m p o s i t e is v a l i d o n l y f o r m a t r i c e s t h a t a r e n o t b r i t t l e e n o u g h for s p l i t t i n g . M c D a n i e l s a n d S i g n o r e l l i [ 9 0 ] c o n d u c t e d i m p a c t t e s t s o n u n i d i r e c t i o n a l b o r o n - 1 1 0 0 , 2024, 5052 a n d 6061 a l u m i n i u m c o m p o s i t e s . T h e y f o u n d t h a t the use of a d u c t i l e m a t r i x ( 1 1 0 0 a l u m i n i u m ) g a v e a h i g h i m p a c t s t r e n g t h d u e t o a l a r g e matrix shear deformation.

S. Ochiai

536

O n t h e o t h e r h a n d , H o o v e r [ 6 4 ] f o u n d in t h e i r u n i d i r e c t i o n a l b o r o n - a l u m i n i u m composites that the fracture toughness was raised due to heat-treatment. According t o H o o v e r , t h e d i f f e r e n c e in f r a c t u r e t o u g h n e s s b e t w e e n c o m p o s i t e s w i t h a l o w a n d c o m p o s i t e s w i t h a h i g h y i e l d s t r e n g t h m a t r i x is t h e r e s u l t o f t h e c h a n g e in e n e r g y p a r t i t i o n i n g b e t w e e n t h e fibres a n d t h e m a t r i x . H e n c e , t h e m a x i m u m f r a c t u r e t o u g h n e s s is o b t a i n e d w h e n t h e s t r a i n e n e r g y s t o r e d in t h e m a t r i x p r i o r t o c r a c k i n i t i a t i o n is m a x i m i z e d ; i.e., m a t r i x y i e l d s t r e n g t h is m a x i m i z e d . R e e d y [ 9 1 ] s h o w e d t h a t t h e i n f l u e n c e of t h e y i e l d s t r e s s o f t h e m a t r i x o n t h e n o t c h e d s t r e n g t h is different w h e n t h e l o a d i n g c o n d i t i o n is v a r i e d . H e c a l c u l a t e d t h e l o a d - C O D c u r v e s a n d n o t c h e d s t r e n g t h b y t h e m e t h o d s h o w n in sect. 6.2.2. I n h i s c a l c u l a t i o n , in o r d e r t o s h o w t h e d e p e n d e n c e o f stress c o n c e n t r a t i o n in fibres at t h e n o t c h tip o n l o a d i n g c o n d i t i o n , h e e m p l o y e d a u n i f o r m t r a c t i o n b o u n d a r y condition a n d displacement condition. As a result, he found that the traction b o u n d a r y c o n d i t i o n c a n c a u s e extremely h i g h stress c o n c e n t r a t i o n s , even w h e n the y i e l d s t r e n g t h o f t h e m a t r i x is l o w . U n d e r t h i s c o n d i t i o n , w h e n y i e l d s t r e n g t h is l o w , the greater the h a r d e n i n g e x p o n e n t , the greater b e c o m e s the stress c o n c e n t r a t i o n . U n d e r t h e d i s p l a c e m e n t b o u n d a r y c o n d i t i o n , t h e fibre s t r e s s c o n c e n t r a t i o n i n c r e a s e s m o n o t o n o u s l y w i t h i n c r e a s i n g y i e l d s t r e s s , b u t it is s m a l l in c o m p a r i s o n w i t h t h a t for t r a c t i o n b o u n d a r y c o n d i t i o n , e s p e c i a l l y w h e n t h e y i e l d s t r e s s of t h e m a t r i x is low. In m o s t l a b o r a t o r y testing, the grips are e x p e c t e d to b e rigid e n o u g h to prevent t h e m o r e s e v e r e t r a c t i o n - l i k e c o n d i t i o n . H o w e v e r , if t h e y i e l d s t r e n g t h o f t h e m a t r i x is l o w , a t r a c t i o n - l i k e c o n d i t i o n is p o s s i b l e in a n a c t u a l s t r u c t u r e . R e e d y [ 9 1 ] c o n d u c t e d fracture tests using wedge-action type grips a n d confirmed that the e m p l o y m e n t of t h i s t y p e o f g r i p p r o d u c e d a c o n d i t i o n w h i c h , o n a v e r a g e , fell b e t w e e n t h e extremes of d i s p l a c e m e n t a n d traction c o n d i t i o n s . H e stated t h a t his calculated results are a strong w a r n i n g against the use of test results o b t a i n e d with rigid grips t o d e s i g n s t r u c t u r e s s u b j e c t t o t r a c t i o n - l i k e l o a d s w h e n t h e m a t r i x is r e l a t i v e l y soft.

7.2. Interfacial

bonding

strength

between

fibre

and

matrix

I n u n i d i r e c t i o n a l c o m p o s i t e s , w h e n t h e i n t e r f a c i a l b o n d i n g s t r e n g t h is l o w , s p l i t t i n g o c c u r s a n d c a u s e s b l u n t i n g o f t h e n o t c h - t i p , w h i c h will l e a d t o a n i n c r e a s e in n o t c h e d s t r e n g t h , a s s h o w n in sect. 2. H o o v e r a n d A l l r e d [ 2 7 ] e x a m i n e d t h e i n f l u e n c e o f t h e fibre-matrix b o n d i n g strength on the splitting process, using unidirectional borsica l u m i n i u m c o m p o s i t e s . I n t h e i r a n a l y s i s , t h e t r a n s v e r s e t e n s i l e s t r e n g t h w a s u s e d as a m e a s u r e o f fibre-matrix b o n d i n g s t r e n g t h , w h i c h w a s a l t e r e d t h r o u g h v a r i a t i o n s in h o t - p r e s s u r i n g d u r i n g c o m p o s i t e f a b r i c a t i o n . I n c o m p o s i t e s w i t h w e a k b o n d i n g , t h e n e t - s e c t i o n stress a t d e b o n d i n g w a s o b s e r v e d t o d e c r e a s e w i t h i n c r e a s i n g c r a c k l e n g t h . V a r i a t i o n s in c r a c k l e n g t h d i d n o t s i g n i f i c a n t l y a l t e r t h e b a s i c f r a c t u r e m e c h a n i s m in w h i c h t h e d e b o n d i n g p r o c e s s l e d t o n o t c h i n s e n s i t i v i t y . I n c r e a s e s in t h e fibre-matrix b o n d i n g s t r e n g t h c a u s e d a m a r k e d i n c r e a s e in t h e s t r e s s a t d e b o n d i n g , b u t d i d n o t c h a n g e t h e g e n e r a l d e b o n d i n g p r o c e s s . T h r o u g h t h e u s e of c r o s s - p l i e d composites, the transverse strength was increased further a n d d e b o n d i n g was sup­ pressed. This led to a notch-sensitive fracture b e h a v i o u r a n d a substantial decrease

Fracture mechanical approach to metal-matrix

composites

537

in f r a c t u r e t o u g h n e s s c o m p a r e d t o c o m p o s i t e s w h i c h s h o w e d d e b o n d i n g p r i o r t o failure. 7.3. Fibre

diameter

A c c o r d i n g t o r e p o r t e d d a t a a n d c o n c e p t s [ 3 1 , 3 3 , 6 3 , 9 0 ] , in b o t h d u c t i l e fibre- a n d b r i t t l e - f i b r e r e i n f o r c e d m e t a l s , t h e t o u g h n e s s b e c o m e s h i g h w h e n t h i c k fibres a r e e m p l o y e d . F o r d u c t i l e - f i b r e c o m p o s i t e s , t h e i n c r e a s e in f r a c t u r e t o u g h n e s s h a s b e e n a t t r i b u t e d t o t h e l a r g e a m o u n t of p l a s t i c d e f o r m a t i o n s o f t h e m a t r i x a n d fibres [eqs. (26), (27) a n d (29)]. A c c o r d i n g to M c D a n e l s a n d Signorelli [90] w h o m e a s u r e d i m p a c t t o u g h n e s s of b o r o n - f i b r e ( w h i c h is b r i t t l e ) - a l u m i n i u m c o m p o s i t e s , t h e i n c r e a s e in t o u g h n e s s d u e t o t h e i n c r e a s e in fibre d i a m e t e r is a t t r i b u t e d t o a l a r g e s h e a r p l a s t i c d e f o r m a t i o n b e t w e e n fibres, w h i c h is l a r g e w h e n t h e i n t e r f i b r e s p a c i n g is l a r g e u n d e r a fixed v o l u m e f r a c t i o n o f fibre. A n o t h e r explanation was given by Wright a n d Iannuzzi [63] w h o obtained the e x p e r i m e n t a l r e s u l t t h a t t h e t o u g h n e s s of c r o s s - p l i e d b o r o n - a l u m i n i u m w i t h fibres h a v i n g a d i a m e t e r o f 142 μ ΐ η w a s a p p r e c i a b l y g r e a t e r t h a n t h a t m e a s u r e d f r o m t h e m a t e r i a l c o n t a i n i n g t h e s m a l l e r 102 μ η ι fibres. A c c o r d i n g t o t h e m , t h e d i f f e r e n c e s reflect t h e s t r e n g t h o f u n n o t c h e d s p e c i m e n s . I n t h e i r s p e c i m e n s , t h e y f o u n d a t e n d e n c y f o r t h e s m a l l e r fibres t o fail b y s p l i t t i n g . T h e d i f f e r e n c e in t o u g h n e s s w a s c o n s i d e r e d t o r e s u l t f r o m t h e i n h e r e n t l y w e a k e r fibres p r e s e n t in t h e m a t e r i a l s c o n t a i n i n g t h e s m a l l - d i a m e t e r fibres. 7.4. Type of

discontinuity

A l t h o u g h only a limited n u m b e r of r e p o r t s h a v e b e e n p u b l i s h e d c o n c e r n i n g t h e effect of t y p e of d i s c o n t i n u i t y ( h o l e , slit, h o l e w i t h slits, o r r e c t a n g u l a r c u t - o u t ) , a c o m m o n c o n c l u s i o n h a s b e e n d r a w n for b o r o n - a l u m i n i u m c o m p o s i t e . M a r a n d L i n [69] c o n d u c t e d fracture tests o n [ ± 4 5 / 0 ] l a m i n a t e s with various discontinuities ( h o l e , slit, o r h o l e w i t h slits) a n d f o u n d t h a t t h e l e n g t h o f d i s c o n t i n u i t y , a n d n o t t h e s h a p e , is t h e c o n t r o l l i n g p a r a m e t e r f o r f r a c t u r e , a s h a s b e e n s h o w n in fig. 16b. J o h n s o n , Bigelow a n d B a h e i - E l - D i n [92] c o n d u c t e d fracture tests o n b o r o n a l u m i n i u m [ ± 4 5 ] , [ 9 / ± 4 5 ] a n d [ 0 / ± 4 5 ] l a m i n a t e s w i t h a h o l e o r a slit a n d f o u n d t h a t w h e n t h e l e n g t h o f t h e d i s c o n t i n u i t y ( l e n g t h o f t h e slit o r d i a m e t e r o f t h e h o l e ) was the same, the strength reduction was approximately the same. Dharani, Jones a n d G o r e e [ 2 8 ] a n a l y z e d t h e s t r e s s c o n c e n t r a t i o n in t h e fibres a h e a d o f t h e c r a c k a n d the extent of the transverse d a m a g e by a p p l y i n g shear-lag analysis to t h e m o d e l s h o w n in fig. 6 b . T h e c a l c u l a t i o n w a s c a r r i e d o u t n o t o n l y for t h e slit b u t a l s o for a h o l e a n d for a r e c t a n g u l a r c u t - o u t in u n i d i r e c t i o n a l b o r o n - a l u m i n i u m c o m p o s i t e s . V e r y little d i f f e r e n c e w a s f o u n d for t h e t h r e e t y p e s o f i n i t i a l d i s c o n t i n u i t y ; t h e e x t e n t of the transverse d a m a g e z o n e at failure w a s a p p r o x i m a t e l y c o n s t a n t , i n d e p e n d e n t o f t h e i n i t i a l s h a p e a n d l e n g t h o f t h e d i s c o n t i n u i t y , a n d v e r y little d i f f e r e n c e w a s f o u n d f o r t h e m a x i m u m fibre stress a n d s t r e s s d i s t r i b u t i o n in f r o n t o f t h e d a m a g e zone. 2

2 s

s

2

s

s

538

S. Ochiai

7.5. Specimen

width

In most cases, notched strength data are plotted against the non-dimensional n o t c h l e n g t h 2c / W ( o r 2R/ W for a h o l e ) , w h e r e e x p e r i m e n t a l n o t c h e d s t r e n g t h is c o n v e r t e d t o a n i n f i n i t e - w i d t h s t r e n g t h t h r o u g h t h e finite-width c o r r e c t i o n f a c t o r . W h e n d a t a a r e a n a l y z e d in t h i s w a y , t h e r e is a t e n d e n c y t h a t t h e w i d e r t h e s p e c i m e n s , the m o r e notch-sensitive b e c o m e the subject laminates. Poe a n d Sova [11] a n d J o h n s o n , Bieglow a n d B a h e i - E l - D i n [92] o b s e r v e d t h a t t h e w i d e r s p e c i m e n s failed at l o w e r s t r e s s e s for t h e s a m e 2cJ W in u n i d i r e c t i o n a l a n d c r o s s - p l i e d b o r o n a l u m i n i u m c o m p o s i t e . A w e r b u c h a n d M a d h u k a r [1] a n a l y z e d the d a t a of P o e a n d Sova [11] following the p r o c e d u r e described above a n d found that the characteristic l e n g t h a o f t h e a v e r a g e - s t r e s s c r i t e r i o n differs s l i g h t l y a m o n g t h e s p e c i m e n s w i t h different w i d t h s ( w i d e s p e c i m e n s s h o w a l a r g e a in c o m p a r i s o n v/ith n a r r o w s p e c i m e n s ) . A c c o r d i n g t o t h e m , a w i d t h effect s h o u l d b e e x p e c t e d c o n s i d e r i n g t h e e x t e n s i v e n o t c h - t i p d a m a g e z o n e size d e v e l o p e d p r i o r t o c a t a s t r o p i c f r a c t u r e . 0

0

0

7.6. Specimen

thickness

H a n c o c k a n d S w a n s o n [ 6 1 ] f o u n d in t h e i r b o r o n - a l u m i n i u m c o m p o s i t e s t h a t t h e i n c r e a s e in t h i c k n e s s c o u l d r e d u c e t h e c r i t i c a l s t r e s s i n t e n s i t y f a c t o r . T h e y a s c r i b e d t h i s effect t o c h a n g e s o f t h e m e t a l l u r g i c a l s t r u c t u r e o r o f t h e i n t e r n a l s t r e s s , o r a c o m b i n a t i o n of b o t h , resulting from h e a t - t r e a t m e n t . Wright a n d I a n n u z z i [63] o b s e r v e d n o t h i c k n e s s d e p e n d e n c e in t h e i r r e s u l t s o b t a i n e d o n c r o s s - p l i e d b o r o n a l u m i n i u m c o m p o s i t e s . G o r e e a n d J o n e s [ 8 9 ] o b s e r v e d t h a t t h e d a m a g e z o n e size of u n i d i r e c t i o n a l b o r o n - a l u m i n i u m c o m p o s i t e s with o n e to eight plies was very s i m i l a r , i n d e p e n d e n t o f t h i c k n e s s . T h e y a l s o f o u n d t h a t t h e m a x i m u m C O D at f r a c t u r e w a s a l m o s t i d e n t i c a l for t h e t h i c k n e s s e s t e s t e d . 7.7. Laminate

configuration

in cross-plied

composites

T h e n o t c h s e n s i t i v i t y is affected b y t h e l a m i n a t e c o n f i g u r a t i o n . A c c o r d i n g t o A w e r b u c h a n d M a d h u k a r [ 1 ] , σ^/σ of the experimental results of M a d h u k a r [93] for b o r o n - a l u m i n i u m c o m p o s i t e s d e c r e a s e s i n t h e o r d e r [ 9 0 ] -» [ 0 / ± 4 5 / 9 0 ] -» [ 0 ] = [ ± 4 5 ] ^ [ 0 / 4 5 / 0 ] - > [ 0 / 9 0 ] in t h e r a n g e o f 0 < 2 c / w < 0 . 6 , w h i l e t h e u n n o t c h e d s t r e n g t h σ d e c r e a s e s in t h e o r d e r [ 0 ] ^ [ 0 / ± 4 5 / 0 ] ^ [ 0 / ± 4 5 / 9 0 ] ^ [ ± 4 5 ] - > [ 9 0 ] . A c c o r d i n g t o t h e s e a u t h o r s , t h e p l a s t i c d e f o r m a t i o n o f t h e m a t r i x in u n i d i r e c t i o n a l [ 0 ] s a m p l e s served as a crack-arresting m e d i u m a n d a d d i t i o n a l crack extension a l t e r n a t e d w i t h t h e c o m p l e t i o n o f s u c c e s s i v e l o n g i t u d i n a l p l a s t i c z o n e s , r e s u l t i n g in a n i r r e g u l a r final f r a c t u r e s u r f a c e . I n o t h e r c r o s s - p l i e d s a m p l e s , h o w e v e r , n o l o n g i ­ t u d i n a l p l a s t i c z o n e s w e r e o b s e r v e d , t h e f r a c t u r e s u r f a c e w a s fairly c o - p l a n a r , a n d t h e c r a c k - t i p d a m a g e w a s fairly l o c a l i z e d a n d it p r o p a g a t e d a s a n e a r l y s e l f - s i m i l a r c r a c k e x t e n s i o n . T h e d i f f e r e n c e in n o t c h s e n s i t i v i t y a m o n g v a r i o u s c o n f i g u r a t i o n s m i g h t b e a t t r i b u t e d t o a d i f f e r e n c e in d a m a g e z o n e . 0

8

2 s

s

0

2 s

s

8

0

8

s

s

2 s

8

8

A c c o r d i n g t o P o e [ 7 3 ] , t h e n o t c h e d s t r e n g t h is d e t e r m i n e d b y t h e f a i l u r e o f t h e major load-carrying ply, a n d the n o t c h e d strength of u n i d i r e c t i o n a l a n d cross-plied

Fracture mechanical approach to metal-matrix

539

composites

b o r o n - a l u m i n i u m composite can be analyzed by using the general fracture toughness p a r a m e t e r Q a n d f r a c t u r e s t r a i n o f fibres, a s s h o w n in fig. 19. A c c o r d i n g t o P o e a n d S o v a ' s c o n c e p t [ 1 1 ] , t h e d i f f e r e n c e in c o n f i g u r a t i o n r e s u l t s in a different s t r a i n s t a t e in t h e m a j o r l o a d - c a r r y i n g p l y , w h i c h l e a d s t o a d i f f e r e n c e i n n o t c h e d s t r e n g t h . c

7.8. Sequence

of lay-up

in cross-plied

composites

P r e w o [ 9 4 ] u s e d b o r o n - a l u m i n i u m l a m i n a t e s w i t h l a y - u p s e q u e n c e s of: ( A ) [ ( 9 0 / 0 ) ] ; (B) [ ( 9 0 / 0 ) ] ; (C) [ ( 9 0 / 0 ) ] ; a n d ( D ) [ 9 0 0 ] . F o r these lay-ups, he f o u n d from t h e c o m p a r i s o n of fracture t o u g h n e s s d a t a a n d from t h e c o m p a r i s o n of fracture m o r p h o l o g y that t h e s e q u e n c e of lay-up has a strong influence. C o m p a r i s o n of the fracture t o u g h n e s s b e t w e e n (A) a n d ( D ) t y p e s s h o w e d t h a t t h e segregation o f 0° a n d 90° p l i e s c a u s e s a 2 0 % i n c r e a s e i n f r a c t u r e t o u g h n e s s o v e r t h e a l t e r n a t i n g p l y l a y - u p s e q u e n c e . H e e x p l a i n e d h i s d a t a in t e r m s o f a r e s t r i c t i o n o n m a t r i x d e f o r m a t i o n a n d f r a c t u r e i n r e g i o n s p a r a l l e l t o 0° fibre d i r e c t i o n ; n a m e l y , t h e a l t e r n a t i n g - p l y l a y - u p s e q u e n c e p r e v e n t s m o r e effectively t h e c r a c k - t i p b l u n t i n g , while t h e segregated-ply l a y - u p p e r m i t s a greater a m o u n t of m a t r i x d e f o r m a t i o n to t a k e p l a c e in r e s p o n s e t o t h e transverse a n d s h e a r stresses o c c u r r i n g at t h e crack tip. 2

4

8

3

3

3

6

6

s

P o e a n d S o v a [ 1 1 ] f o u n d little if a n y d i f f e r e n c e in t h e n o t c h e d f r a c t u r e t o u g h n e s s values K and K , g i v e n b y e q s . (48) a n d ( 5 1 ) , r e s p e c t i v e l y , for [ 0 / ± 4 5 ] a n d [ ± 4 5 / 0 ] laminates, w h e r e a s the ultimate tensile strength a n d strains of u n n o t c h e d l a m i n a t e s w e r e n o t i c e a b l y different. T h u s , t h e s e s t a c k i n g s e q u e n c e s h a d t h e f e a t u r e that they changed the u n n o t c h e d strength but not the notched strength, while the s t a c k i n g s e q u e n c e c o n t a i n i n g 0° a n d 90° p l i e s g a v e a c h a n g e i n f r a c t u r e t o u g h n e s s b u t n o t in t h e u n n o t c h e d tensile p e r f o r m a n c e . Q

2

eQ

2

s

s

7.9. Test

temperature

W r i g h t a n d I a n n u z z i [ 6 3 ] c o n d u c t e d f r a c t u r e t e s t s a t t e m p e r a t u r e s u p t o 589 Κ for u n i d i r e c t i o n a l a n d c r o s s - p l i e d [ ± 4 5 ] t y p e b o r o n - a l u m i n i u m c o m p o s i t e s . T h e y f o u n d t h a t , in t h e c a s e o f u n i d i r e c t i o n a l c o m p o s i t e s , t h e f r a c t u r e s t r e n g t h i n c r e a s e d w i t h i n c r e a s i n g t e m p e r a t u r e , b u t , o n t h e o t h e r h a n d , in t h e c r o s s - p l i e d c o m p o s i t e , the fracture strength decreased with increasing temperature. They attributed the increase in n o t c h e d strength of t h e former c o m p o s i t e s to t h e l o w e r i n g of t h e stress c o n c e n t r a t i n g effect o f t h e n o t c h d u e t o t h e d e c r e a s e in y i e l d s t r e n g t h o f t h e m a t r i x a n d w o r k - h a r d e n i n g c a p a c i t y of t h e m a t r i x at h i g h e r t e m p e r a t u r e s , a n d t h e d e c r e a s e in n o t c h e d s t r e n g t h in t h e l a t t e r c o m p o s i t e s t o t h e r e d u c e d l o a d - b e a r i n g c a p a c i t y of t h e m a t r i x a t h i g h t e m p e r a t u r e s . According to Awerbuch and M a d h u k a r [1,93], b o r o n - a l u m i n i u m [ 0 ] and [ 0 / ± 4 5 / 9 0 ] c o m p o s i t e s b e c a m e m o r e n o t c h sensitive at elevated t e m p e r a t u r e s , while little if a n y effect o f t e s t t e m p e r a t u r e o n t h e n o t c h s e n s i t i v i t y o f [ 0 / 9 0 ] and [ 0 / ± 4 5 / 0 ] c o m p o s i t e s w a s r e c o r d e d . A c c o r d i n g t o t h e s e a u t h o r s , it h a s b e e n e x p e c t e d for [ 0 ] c o m p o s i t e s t h a t t h e i n c r e a s e d m a t r i x ductility s h o u l d e n h a n c e the crack-arresting m e c h a n i s m resulting from matrix shear deformation, but the reverse was recorded. In order to explain this result, they considered that strength 8

s

2 s

s

8

S. Ochiai

540

d e g r a d a t i o n o f fibres d u e t o c h e m i c a l r e a c t i o n s b e t w e e n fibres a n d m a t r i x a t h i g h temperatures should be taken into account. Actually, w h e n interfacial reaction takes p l a c e , t h e s t r e n g t h o f b o r o n fibres is r e d u c e d , a s h a s b e e n o b s e r v e d b y m a n y r e s e a r c h e r s [ 5 6 , 9 5 - 1 0 1 ] . W h e n s u c h a d e g r a d a t i o n o c c u r s , t h e w o r k o f f r a c t u r e is r e d u c e d , a s h a s b e e n s h o w n i n fig. 9. If d e g r a d a t i o n o f fibres is t h e m a j o r c a u s e of a c h a n g e in n o t c h s e n s i t i v i t y , n o definite e x p l a n a t i o n c a n b e m a d e c o n c e r n i n g t h e effect of t e s t t e m p e r a t u r e o n f r a c t u r e s t r e n g t h , a n d a d d i t i o n a l i n v e s t i g a t i o n s a r e needed [1]. List of s y m b o l s Stress,

strain

and

strength

U y δ

d i s p l a c e m e n t o f fibre in s h e a r - l a g a n a l y s i s shear strain deflection

ε

tensile strain

ε e

Γ υ

m u

E

t u

e

c u

σ cr

f N

cr ο σ fu

Γ w

σ cr

ί 0

fu

σγ σγ σ cr crjvj τ T 0

N

F

TJ r r r s

u

y

Stress Ki(0) Κγ

t e n s i l e s t r a i n t o f a i l u r e o f fibres tensile strain to failure of m a t r i x t e n s i l e s t r a i n t o f a i l u r e o f u n n o t c h e d c o m p o s i t e r e i n f o r c e d w i t h b r i t t l e fibres tensile strain to failure of u n n o t c h e d c o m p o s i t e reinforced with ductile fibres tensile stress s t r e s s level o f fibre a t f r a c t u r e o f n o t c h e d c o m p o s i t e fibre strength in u n n o t c h e d c o m p o s i t e s t r e n g t h o f fibre a t w e a k p o i n t s i n C o o p e r ' s m o d e l s t r e n g t h of fibre e v e r y w h e r e e x c e p t f o r w e a k p o i n t s in C o o p e r ' s m o d e l u l t i m a t e t e n s i l e s t r e n g t h o f fibre s t r e s s in fibre a t infinity n o r m a l i z e d fibre s t r e s s , (= (a / r )(G A / E d h) ) unnotched strength notched strength n o t c h e d s t r e n g t h c o r r e c t e d b y finite-width c o r r e c t i o n f a c t o r ( = Υ σ ) s h e a r stress f r i c t i o n a l s h e a r s t r e s s b e t w e e n fibre a n d m a t r i x , w h i c h a c t s a f t e r i n t e r f a c i a l debonding i n t e r f a c i a l s h e a r s t r e n g t h b e t w e e n fibre a n d m a t r i x s h e a r stress resisting pull-out f a i l u r e s t r e n g t h o f m a t r i x in s h e a r y i e l d stress o f m a t r i x i n s h e a r 1/2

f

y

m

f

f

m

Ν

concentration

factors

and stress

intensity

factors

s t r e s s c o n c e n t r a t i o n f a c t o r in fibres i n t a c t t o c u t fibres o r t h o t r o p i c s t r e s s c o n c e n t r a t i o n f a c t o r f o r a n infinitely w i d e p l a t e

Fracture mechanical approach to metal-matrix

K f c

c r i t i c a l s t r e s s i n t e n s i t y f a c t o r o f fibre

K

cm

critical stress intensity factor o f m a t r i x

K

541

critical stress intensity factor o f c o m p o s i t e

Q

K

c r a c k g r o w t h r e s i s t a n c e i n R-curve

R

^

composites

Q

concept

elastic critical stress intensity factor i n t h e P o e - S o v a fracture criterion

e

Κ

strain intensity factor in the P o e - S o v a fracture criterion

K

critical strain intensity factor i n t h e P o e - S o v a fracture criterion

ε

eQ

K

elastic critical strain intensity factor in t h e P o e - S o v a fracture criterion

eQe

Work

of

fracture

G

c

critical strain energy release rate

/ L

c

/ integral critical potential energy release rate ( = G for elastic materials) c

Wdebond

W

s po

Wredis

W Wm, W s

me

P

m

w o r k d u e t o d e b o n d i n g p e r fibre w o r k d o n e i n p u l l i n g - o u t o f s i n g l e fibre w o r k d u e t o r e d i s t r i b u t i o n o f s t r e s s p e r fibre e l a s t i c w o r k p e r fibre d u e t o b r i d g i n g o f fibre p l a s t i c w o r k p e r fibre d u e t o b r i d g i n g o f fibre work needed t o break matrix

W 7T

w o r k n e e d e d t o b r e a k fibre t o t a l p o t e n t i a l e n e r g y after c r a c k i n c r e m e n t

7τ

total potential energy before incremental crack growth

f

A

Β

Damage

zone

2c 2Ac

effective c r a c k l e n g t h quasi-crack extension

h h

plastic strip height of ductile plastic strip height o f matrix

n

p

R p ρ p 2v

fibre

length of plastic region size o f d a m a g e z o n e a t f a i l u r e f o r s t r e s s i n P o e - S o v a f r a c t u r e c r i t e r i o n size o f d a m a g e z o n e f o r s t r a i n i n P o e - S o v a f r a c t u r e c r i t e r i o n size o f d a m a g e z o n e a t f a i l u r e f o r s t r a i n i n P o e - S o v a f r a c t u r e c r i t e r i o n g a p b e t w e e n b r o k e n fibres a t f r a c t u r e s u r f a c e

p

c

ε

ec

Geometry A Β 2c d d h f

r

m

0

c r o s s - s e c t i o n a l a r e a o f fibre thickness of plate specimen initial crack length d i a m e t e r o f fibre inter-fibre s p a c i n g thickness of two-dimensional model composite employed in shear lag analysis

542

S. Ochiai

l

l e n g t h of

h

critical length

/„

fibre

'd

d i s t a n c e b e t w e e n w e a k p o i n t s in C o o p e r ' s m o d e l debonded length

η

n u m b e r of c u t

Ύ

r a d i u s of fibre m a t r i x r a d i u s i n single-fibre m o d e l

r R W m

fibres

r a d i u s of h o l e w i d t h of p l a t e s p e c i m e n finite-width correction factor

Y Mechanical

property

Ε Ε'

primary Young's modulus secondary Young's modulus

G

shear modulus

βο

s h e a r s t r a i n h a r d e n i n g coefficient o f m a t r i x after y i e l d i n g Poisson's ratio of c o m p o s i t e

Parameters a*

in various fracture

criteria

size of i n t e n s e e n e r g y r e g i o n in W a d d o u p s - E i s e n m a n - K a m i n s k i criterion

fracture

c h a r a c t e r i s t i c l e n g t h in a v e r a g e s t r e s s c r i t e r i o n shift p a r a m e t e r s in P i p e s - W e t h e r h o l d - G i l l e s p i e f r a c t u r e c r i t e r i o n

c

n o t c h s e n s i t i v i t y f a c t o r for a h o l e in P i p e s - W e t h e r h o l d - G i l l e s p i e f r a c t u r e criterion

c

n o t c h s e n s i t i v i t y f a c t o r for a slit n o t c h in fracture criterion

Pipes-Wetherhold-Gillespie

c h a r a c t e r i s t i c l e n g t h in p o i n t s t r e s s c r i t e r i o n

d H m mi

e x p o n e n t i a l p a r a m e t e r in P i p e s - W e t h e r h o l d - G i l l e s p i e f r a c t u r e c r i t e r i o n p a r a m e t e r r e l a t e d t o s t r e n g t h o f s i n g u l a r i t y a t n o t c h t i p in M a r - L i n c r i t e r i o n

Qc

g e n e r a l f r a c t u r e t o u g h n e s s p a r a m e t e r in P o e - S o v a f r a c t u r e c r i t e r i o n

0

c

c o m p o s i t e t o u g h n e s s in M a r - L i n f r a c t u r e c r i t e r i o n

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