Fracture Mechanics of Anisotropic Materials

Application

of Fracture Mechanics

to Composite

Materials

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

Chapter

1

Fracture Mechanics J.G.

of Anisotropic

Materials

WILLIAMS

Mechanical

Engineering Department,

Imperial College, London,

UK

Contents Abstract 3 1. Introduction 4 2. Basic considerations 5 3. G determinations 9 3.1. Method of analysis 10 3.2. Double cantilever beam ( D C B ) tests 12 3.3. A variable-ratio mixed-mode test 15 3.4. Large displacements in D C B tests 15 3.5. Transverse splitting from notches 18 3.6. Buckling under compression 20 4. Stability 22 5. Cracks in anisotropic sheets 25 5.1. Basic method 25 5.2. The crack problem 29 5.3. The calculation of G 32 5.4. The calculation of Κ 33 6. Damage zones 35 7. Conclusions 36 List of symbols 37 References 38

Abstract A r e v i e w o f t h e b a s i c e n e r g y r e l e a s e r a t e , G, a n a l y s i s for l i n e a r b u t a n i s o t r o p i c e l a s t i c m a t e r i a l s is g i v e n i n c l u d i n g t h e d e f i n i t i o n in t e r m s o f c o n t o u r s . F o r m a n y c o m p o s i t e s it is n o t e d t h a t c r a c k g r o w t h is s e l f - s i m i l a r b e c a u s e o f d e l a m i n a t i o n a n d t h i s l e a d s t o a r a t h e r s i m p l e g e n e r a l s c h e m e o f d e t e r m i n i n g G via l o c a l m o m e n t s a n d f o r c e s . I n a d d i t i o n , it is p o s s i b l e t o p a r t i t i o n t h i s G v a l u e i n t o m o d e s I a n d II in a s i m p l e b u t r i g o r o u s m a n n e r . S e v e r a l s o l u t i o n s a r e g i v e n for t e s t s p e c i m e n s which give v a r i o u s forms of m i x e d - m o d e l o a d i n g , i n c l u d i n g c o n s t a n t a n d variable r a t i o s . S o m e d i s c u s s i o n o f t h e r a t h e r c o m p l i c a t e d c a s e of G for b u c k l e d l a m i n a t e s 3

4

J.G.

Williams

u n d e r c o m p r e s s i o n is a l s o g i v e n . F o r c r a c k e d p l a t e s it is n e c e s s a r y t o r e s o r t t o t h e stress i n t e n s i t y f a c t o r , K, s o l u t i o n s v i a c o m p l e x - f u n c t i o n a n a l y s i s a n d t h i s is d e v e l o p e d in s o m e d e t a i l t o r e n d e r it a c c e s s i b l e t o t h e g e n e r a l r e a d e r a n d t h e relationships b e t w e e n G a n d Κ for t h e a n i s o t r o p i c case are derived. T h e very i m p o r t a n t r e s u l t t h a t Κ is a l m o s t t h e s a m e for a n i s o t r o p i c a n d i s o t r o p i c m a t e r i a l s is d e r i v e d a n d d e m o n s t r a t e d v i a n u m e r i c a l r e s u l t s . S o m e d i s c u s s i o n o f d a m a g e z o n e sizes is a l s o g i v e n .

1. Introduction Conventional fracture mechanics deals with h o m o g e n e o u s , isotropic materials a n d has b e e n highly successful b e c a u s e so m a n y practically useful materials are r e a s o n a b l e a p p r o x i m a t i o n s t o t h e s e a s s u m p t i o n s . T h e s a m e is t r u e , o f c o u r s e , for stress a n a l y s i s in g e n e r a l , a n d m o s t e l a s t i c i t y t e x t s c o n t a i n o n l y a p a s s i n g r e f e r e n c e t o a n i s o t r o p y . T h e n o t a b l e e x c e p t i o n s t o t h i s h a v e b e e n w o r k g e n e r a t e d as a r e s u l t of efforts t o d e s i g n l o a d b e a r i n g s t r u c t u r e s in w o o d [ 1 ] . I n p a r t i c u l a r e a r l y a i r c r a f t s t r u c t u r e s e m p l o y e d w o o d a n d t h e r e w a s m u c h i n t e r e s t in c a l c u l a t i n g 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 s a r o u n d h o l e s [ 2 , 3 ] . T h e u s e o f fibre c o m p o s i t e s t o m a k e l a m i n a t e s a n d a l s o t h e d e s i g n of p l y w o o d h a s p r o d u c e d a c o n s i d e r a b l e l i t e r a t u r e a n d in p a r t i c u l a r t h e t e x t b y L e k h n i t s k i i [ 4 , 5 ] w h i c h e m p l o y e d a d e v e l o p m e n t of t h e w e l l - k n o w n M u s k h e l i s h v i l i [ 6 ] c o m p l e x - n u m b e r f o r m of s t r e s s f u n c t i o n a n a l y s i s t o p r o d u c e a w i d e r a n g e of s o l u t i o n s t o p r o b l e m s o f p r a c t i c a l i m p o r t a n c e . In m o r e r e c e n t t i m e s , m a n y c o m p u t e r c o d e s h a v e b e e n d e v e l o p e d e m p l o y i n g finite e l e m e n t s a n d b o u n d a r y i n t e g r a l s w h i c h will give s o l u t i o n s for a n i s o t r o p i c m a t e r i a l s . F r a c t u r e m e c h a n i c s h a s b e e n i n v e s t i g a t e d in s o m e d e t a i l for w o o d [ 7 - 9 ] a n d f o u n d t o b e a v e r y u s e f u l t o o l for d e s i g n p u r p o s e s . It is i n t e r e s t i n g t o n o t e t h a t m a t e r i a l v a r i a b i l i t y is l a r g e for w o o d , e v e n b y c o m p o s i t e s s t a n d a r d s , a n d yet t h e m e t h o d p r o v e d u s e f u l . I n s p i t e o f t h i s , t h e u s e o f f r a c t u r e m e c h a n i c a n a l y s i s for composites has been rather limited. This w o u l d a p p e a r to stem from a suspicion that conventional analysis could not c o p e with the anisotropy a n d i n h o m o g e n e i t y of c o m p o s i t e s a n d t h a t s o m e o t h e r s c h e m e w a s n e c e s s a r y . A n i s o t r o p y c a n b e i n c l u d e d in t h e a n a l y s i s b y e m p l o y i n g t h e a p p r o p r i a t e m e t h o d s a n d , a l t h o u g h i n h o m o g e n e i t y is a l w a y s a p r o b l e m , it c a n b e d e a l t w i t h via size p a r a m e t e r s . F r a c t u r e m e c h a n i c s is a macro t h e o r y a n d d e f i n e s p a r a m e t e r s w h i c h c h a r a c t e r i z e f a i l u r e o v e r r e g i m e s of k n o w n size. T h e p a r t i c u l a r p r o b l e m o f c o m p o s i t e s is t h a t t h e sizes m a y b e l a r g e a n d r a t h e r c a r e f u l d e v e l o p m e n t o f t h e m e t h o d s is n e c e s s a r y . T h u s w h a t is r e q u i r e d is t h e development of t h e b a s i c m e t h o d s a n d n o t t h e i r a b a n d o n m e n t . T h i s h a s b e e n r e c o g n i z e d in a g r o w i n g i n t e r e s t a n d l i t e r a t u r e in r e c e n t y e a r s , a n d is reflected in this v o l u m e . W h i l e m u c h r e m a i n s t o b e d o n e in t h i s d e v e l o p m e n t t h e r e is a l a r g e useful analysis available via a n i s o t r o p i c elasticity a n d fracture m e c h a n i c s M a n y rather daunting p r o b l e m s lead to surprisingly simple solutions, a n d a w i d e b a s e f r o m w h i c h t o t a c k l e t h o s e e s p e c i a l l y difficult p r o b l e m s , i n h o m o g e n e i t y , i n h e r e n t in c o m p o s i t e s . T h i s c h a p t e r will give a r e v i e w

b o d y of analysis. provide such as of t h e s e

Fracture mechanics of anisotropic

5

materials

a v a i l a b l e r e s u l t s a n d m e t h o d s . M o s t of t h e a n a l y s i s will a s s u m e l i n e a r , e l a s t i c a n d h o m o g e n e o u s b e h a v i o u r s o t h a t t h e r e s u l t s m a y b e d e s c r i b e d as l i n e a r e l a s t i c f r a c t u r e m e c h a n i c s ( L E F M ) b u t for a n i s o t r o p i c m a t e r i a l s . T h e f r a c t u r e s a n a l y z e d a r e t h u s b r i t t l e in c h a r a c t e r h a v i n g s m a l l d a m a g e z o n e s in c o m p a r i s o n w i t h o t h e r d i m e n s i o n s in a c c o r d a n c e w i t h L E F M . A s in L E F M , s o m e c o n s i d e r a t i o n will b e g i v e n t o c r a c k tip z o n e sizes.

2. Basic considerations W e s h a l l e m p l o y t h e s c h e m e for d e v e l o p i n g f r a c t u r e m e c h a n i c s d e s c r i b e d in ref. [ 1 0 ] in w h i c h t w o a s s u m p t i o n s a r e m a d e : (1) All b o d i e s c o n t a i n c r a c k s o r flaws, a n d f r a c t u r e m e c h a n i c s is c o n c e r n e d w i t h a n a l y s i n g t h e g r o w t h of s u c h c r a c k s ; a n d (2) T h e c r a c k g r o w t h m a y b e c h a r a c t e r i z e d in t e r m s of t h e e n e r g y p e r u n i t a r e a n e c e s s a r y t o c r e a t e n e w s u r f a c e a r e a , t h e c r a c k r e s i s t a n c e R. T h e first a s s u m p t i o n p r e c l u d e s all d i s c u s s i o n of c r e a t i n g flaws in o t h e r w i s e p e r f e c t b o d i e s w h i c h is n o t a s e r i o u s r e s t r i c t i o n in c o m p o s i t e s a n d is a r g u a b l y n o r e s t r i c t i o n at all for r e a l m a t e r i a l s . T h e s e c o n d a s s u m p t i o n d o e s n o t i m p l y t h a t R is a c o n s t a n t b u t m a y v a r y w i t h a n y n u m b e r of v a r i a b l e s . F o r s i m p l i c i t y of p r e s e n t a t i o n w e s h a l l c o n s i d e r a c r a c k of l e n g t h α in a s h e e t of u n i f o r m t h i c k n e s s Β u n d e r g o i n g s e l f - s i m i l a r p r o p a g a t i o n s o t h a t t h e c h a n g e of c r a c k a r e a is g i v e n b y dA =

Bda.

T h e a n a l y s i s is b a s e d u p o n t h e e n e r g y b a l a n c e d u r i n g a t i m e i n t e r v a l at for a c r a c k m o v i n g at v e l o c i t y à fo r w h i c h w e c a n w r i t e U =U +U +U +BRa c

d

s

k

(1)

9

where U i s th e externa l wor k performed , U th e energ y dissipation , U th e store d e l a s t i c e n e r g y a n d U t h e k i n e t i c e n e r g y . S u c h a r e l a t i o n s h i p e n a b l e s al l s i t u a t i o n s to b e analyze d includin g thos e involvin g visco-elasti c dissipatio n ( U ) a n d high-rat e p r o c e s s e s (U ). H e r e w e s h a l l a d o p t t h e u s u a l s t a t i c L E F M a s s u m p t i o n s t h a t al l d i s s i p a t i o n i s e m b o d i e d i n R, a n d a l s o i g n o r e k i n e t i c e n e r g y . W e s h a l l t h e n defin e the paramete r "energ y releas e rate " G whic h i s writte n a s e

d

s

k

d

k

(2) a n d m a y b e regarde d a s th e crac k drivin g forc e sinc e w e m a y write , a t fracture , f r o m e q . (1 )

BGa= Ù -Ù = BRa, c

s

i.e. , G = R.

It i s u s u a l t o d e r i v e G s e p a r a t e l y a n d t h e n d e f i n e G = R a s f r a c t u r e b u t t h e n t o n o t e t h a t i f G> R t h e s y s t e m i s u n s t a b l e s i n c e U wil l i n c r e a s e . E n e r g y b a l a n c e is , of c o u r s e , a l w a y s m a i n t a i n e d vi a eq . ( 1 ) . G m a y b e d e r i v e d fo r a g e n e r a l b o d y c o n t a i n i n g a c r a c k o f l e n g t h a a s s h o w n i n fig. 1 w h i c h h a s a l o a d Ρ a p p l i e d g i v i n g k

a d e f l e c t i o n u. T h e l o a d - d e f l e c t i o n d i a g r a m , s h o w n in fig. 2, for t h e c r a c k l e n g t h a is O A a n d n e e d n o t b e l i n e a r , e v e n for l i n e a r e l a s t i c m a t e r i a l s , s i n c e l a r g e d i s p l a c e ­ m e n t s m a y o c c u r . If w e n o w s t a t e t h a t at A t h e c r a c k i n c r e a s e s b y da s o t h a t b o t h t h e l o a d a n d t h e d i s p l a c e m e n t c h a n g e g i v i n g p o i n t A'. S i n c e t h e b o d y is e l a s t i c , t h e u n l o a d i n g l i n e for a + da is O A ' . N o w f r o m e q . ( 2 ) , BGda

=

dU -dU , e

s

a n d f r o m fig. 2, d(7 = O A ' B ' - O A B s

and

dU

e

= BAA'B',

i.e.,

BG da = ( O A B + Β Α Α ' Β ' ) - O A ' B ' , w h i c h is t h e s h a d e d a r e a in fig. 2. T h i s is a n i m p o r t a n t r e s u l t p r a c t i c a l l y s i n c e G c a n b e f o u n d g r a p h i c a l l y if s u c h a d i a g r a m is m e a s u r e d . T h e r e s u l t m a y b e w r i t t e n

A

Fig. 2. The load-deflection curve.

Fracture mechanics of anisotropic

materials

7

as d f — Pau, da J

iu

BG = P

\a

and

U = s

Γ J

Ρ du.

(3)

T w o o t h e r f o r m s of t h i s e q u a t i o n a r e a l s o o f v a l u e , dU^ BG = — da

(4)

a n d s i n c e U = \u

d P , the c o m p l e m e n t a r y energy, we m a y write

c

dU +— c

U =Pu-U , c

i.e.,

s

BG =

1

da a n d n o t e t h a t for a linear system U = U , c

BG = +-

s

giving (5)

dda l/ P const.

T h e loads Ρ on the b o u n d a r y , m a y be t r a n s p o s e d to a n y c o n t o u r Γ a r o u n d the c r a c k t i p , a s s h o w n in fig. 3 , s i n c e t h e i n t e r v e n i n g m a t e r i a l is a s i m p l e e l a s t i c , n o n - s i n g u l a r , s t r e s s s y s t e m . T h u s , for t h e c o n t o u r l e n g t h Γ w e m a y w r i t e 1

dU

Β

da

l

e

du

dw \ s

w h e r e σ , a, u , w are, respectively, the n o r m a l a n d shear stresses a n d d i s p l a c e m e n t s η

s

n

s

o n t h e b o u n d a r y . U m a y b e w r i t t e n in t e r m s of t h e s t r a i n e n e r g y d e n s i t y f u n c t i o n s

W, s

and 1 dl/ Β

da

s

d da J

WLdA r

N o w if w e l o c a t e a c o o r d i n a t e s y s t e m at t h e c r a c k t i p as s h o w n in fig. 3 , t h e n at

Fig. 3. General crack tip contour.

J.G.

8

Williams

y

Fig. 4. Circular contour.

c r a c k g r o w t h for a n y p o i n t ; dx = -da, f o l l o w i n g e x p r e s s i o n for

a n d dA = dx dy = -da

dy, a n d w e h a v e t h e

G,

(6)

T h i s is t h e w e l l - k n o w n c o n t o u r i n t e g r a l e x p r e s s i o n for

[ 1 1 ] a n d is t r u e for a n y

elastic system. T w o i m p o r t a n t results arise directly from this expression w h e n we a n a l y z e t h e l o c a l c r a c k t i p s t r e s s e s . If w e u s e t h e p o l a r c o o r d i n a t e s y s t e m s h o w n in fig. 4, t h e n t h e s i n g u l a r s t r e s s field at t h e c r a c k t i p is σ*Γ-"/(0). N o w for a l i n e a r e l a s t i c m a t e r i a l b o t h W a n d σ d w / d r t a r e p r o p o r t i o n a l t o σ

and

1

s

t h u s t o r~ ". 2

If w e n o w c o n s i d e r a c i r c u l a r c o n t o u r as s h o w n in fig. 4 , t h e n

dy = r c o s θ dd a n d ds = r dd, a n d e q . (6) t a k e s t h e f o r m

N o w G c a n n o t d e p e n d o n t h e v a l u e o f r c h o s e n , i.e., b e p a t h d e p e n d e n t , s i n c e G m u s t b e s i n g l e v a l u e d , a n d h e n c e (1 —2n) = 0 a n d

Η(θ)άθ

m u s t b e finite. T h u s η = \ for l i n e a r l y e l a s t i c m a t e r i a l s , i n c l u d i n g a n i s o t r o p i c o n e s . T h e c o m p u t a t i o n o f G f r o m a l o c a l field is g r e a t l y s i m p l i f i e d b y c o l l a p s i n g t h e c o n t o u r o n t o t h e c r a c k m o v e m e n t as s h o w n in fig. 5, dy = 0, a n d s o t h e r e is n o s t o r e d - e n e r g y c o n t r i b u t i o n , b u t t h e r e a r e l a r g e c h a n g e s i n , for e x a m p l e , cr a n d n

I n i t i a l l y , for s o m e p o i n t a d i s t a n c e S f r o m

the e n d of the g r o w t h , the

o- (r,e)

cr = 0 a n d u = u (S,

n

= a (ka-S,0) n

w i t h u = 0, b u t n

m a t e r i a l is l i n e a r , t h e w o r k d o n e is \σ (Δα η

finally

n

n

- S, 0) u (S, n

n

π).

u. n

stress

Since the

π) a n d w e m a y w r i t e e q . (6)

Fracture mechanics of anisotropic

materials

9

as \[a (Aa n

Δα

- S, 0) w ( S , ττ) + σ ( Δ α - 5 , 0) w ( S , ττ)] d & n

5

s

T h i s is e q u i v a l e n t t o c o n s i d e r i n g t h e m o v e d c r a c k profile a n d a p p l y i n g t h e initial s t r e s s e s t o r e s t o r e t h e g r o w t h ; o f t e n r e f e r r e d t o a s c r a c k c l o s u r e f o r c e s . N o w for a linear system we have

so we m a y write 1/2

G =

. Δid α Jo Jo

\ Δ α\Aa-S

a n d u s i n g ξ = S/Δα "I /

r

dS

[/n(0) g n ( ^ ) + / ( 0 ) s

g (7T)], s

t h e first t e r m b e c o m e s

\ 1/2

T h e s e c o n d t e r m c a n b e c o n s i d e r e d as m a d e u p of t h e o p e n i n g , or m o d e - I , p a r t a n d s l i d i n g , o r m o d e - I I , p a r t a n d w e h a v e finally,

G = G, + G „ , G, = k / n ( 0 )

g (ir), n

(7)

It s h o u l d b e n o t e d t h a t e q s . (6) a n d (7) a s s u m e c o l l i n e a r c r a c k e x t e n s i o n u n d e r general l o a d i n g . F o r h o m o g e n e o u s materials this d o e s n o t often occur, b u t for highly a n i s o t r o p i c f r a c t u r e b e h a v i o u r , a s in l a m i n a t e s , it u s u a l l y d o e s , l e a d i n g t o c o n s i d e r ­ able simplification of t h e analysis. 3. G determinations S i n c e a b a s i c a s s u m p t i o n o f f r a c t u r e m e c h a n i c s is t h a t t o u g h n e s s is d e f i n e d b y R a n d t h a t f r a c t u r e o c c u r s w h e n G = R, it is i m p o r t a n t t o h a v e c o n v e n i e n t s c h e m e s

10

J.G.

Williams

Fig. 6. Delamination.

for t h e d e t e r m i n a t i o n o f G I n s o m e c a s e s it is p o s s i b l e t o d e t e r m i n e G w i t h o u t r e c o u r s e t o t h e l o c a l s t r e s s field a n d o n l y r e m o t e f o r c e s n e e d b e c o n s i d e r e d . M o s t l o a d i n g s o n d e l a m i n a t i o n s in t h i n s h e e t s a r e o f t h i s f o r m a n d it is p o s s i b l e t o d e r i v e a g e n e r a l r e s u l t for all f o r m s o f l o a d i n g , w h i c h is u s e f u l in b o t h t e s t i n g a n d p r o d u c t design [12]. 3.1.

Method

of

analysis

F i g u r e 6 s h o w s s u c h a d e l a m i n a t i o n in a s h e e t o f t h i c k n e s s 2h w h i c h is l o c a t e d a d i s t a n c e h f r o m o n e s u r f a c e a n d h f r o m t h e o t h e r . It will b e a s s u m e d t h a t t h e p r o p a g a t i o n of t h e d e l a m i n a t i o n will b e s e l f - s i m i l a r a n d t h u s c o n s i d e r a b l e s i m ­ p l i f i c a t i o n is p o s s i b l e . W e will a l s o l i m i t o u r a t t e n t i o n t o a u n i f o r m d e l a m i n a t i o n of w i d t h Β s o t h a t all p o s s i b l e l o a d i n g s a r e s h o w n in fig. 7 w h e r e t h e u n c r a c k e d p o r t i o n h a s a b e n d i n g m o m e n t M + M , a n a x i a l f o r c e P +P * a n d a s h e a r f o r c e Q + Q . In the cracked portion, these b e c o m e M P a n d Q o n t h e u p p e r (h ) s e c t i o n a n d M , P a n d Q o n t h e l o w e r (h ). C o n s i d e r i n g first t h e m o m e n t s a l o n e w e c o n s i d e r t h e c o n t o u r A B C D at t h e t i p o f t h e c r a c k Ο w h i c h m o v e s da t o C V F o r a n y b e a m w i t h a n a x i a l m o d u l u s E a n d a s e c o n d m o m e n t of a r e a I , t h e e n e r g y p e r u n i t l e n g t h of b e a m is x

2

x

x

2

x

2

1 ?

2

2

2

M

2

EI

2

x

x

2

x

1

x

x

2

X

X

F o r t h e m a t e r i a l w i t h i n t h e c o n t o u r w e m a y t h u s c a l c u l a t e t h e c h a n g e in s t o r e d e n e r g y w h e n t h e c r a c k m o v e s , as dU

=

M

M\

_2E l

2E I

xx x

XX

(M

+

x

M)

2

2

da.

2E I

2

XX

0

W e m a y n o w u s e e q . (5) d i r e c t l y s i n c e t h i s is a l i n e a r s y s t e m , a n d w r i t e ~

1

dU

Β

da

a

\M

M\

2

xx

X

(8)

I M const.

* P, + P is applied at h ( = h + [PJ(P, + P )]hi) general loads, there are interactions of Ρ and M. 2

l

2

t 0

gi

y e

z e r o

moments, arising from axial loads. For

Fracture mechanics of anisotropic

C

materials

11

Β

Fig. 7. Loadings on a delamination.

where I = hB(2hf

= $1,

0

1 = ±Bh\

I = &Bh\ = f x

/,

I =hBh\ 2

= {\-ξγΐ

N o t e t h a t a l t h o u g h t h i s is a n a n i s o t r o p i c m a t e r i a l , t h e s i m p l e stress s t a t e o n l y r e q u i r e s the modulus along the crack direction E = α ( s e e sect. 5.1). A s i m i l a r a n a l y s i s for t h e f o r c e s gives λ

n

η

•(ΡΛΡ2)

2

(9)

w h e r e A = Bh. T h e s h e a r f o r c e s r e q u i r e t h e c o n s i d e r a t i o n of t h e s h e a r s t r e s s d i s t r i b u t i o n a n d s i n c e t h e s e u s u a l l y a r i s e f r o m t h e m o m e n t g r a d i e n t s , i.e., dM da we can use a p a r a b o l i c distribution, giving G =

3

q

6 6

ri/dMA

105 Α [ ξ \ ά α )

2

1 (1-ξ)\άα)

/dM \ 2

2

/dM,

dM \ l

\âa

d a ) ]

2

2

9

1

j

w h e r e a is t h e s h e a r c o m p l i a n c e ( s e e sect. 5.1). A l t h o u g h R is t a k e n a s t h e b a s i c c r i t e r i o n o f f r a c t u r e , h e r e t h e r e is e v i d e n c e t h a t it is different u n d e r m o d e - I a n d m o d e - I I l o a d i n g . It is t h u s i m p o r t a n t t o s e p a r a t e o r p a r t i t i o n G i n t o t h e t w o c o m p o n e n t s . I n i s o t r o p i c m a t e r i a l s t h i s is m u c h less i m p o r t a n t s i n c e a n y t h i n g o t h e r t h a n m o d e - I is r a r e l y c o l l i n e a r . I n b e n d i n g , p u r e m o d e II o n l y o c c u r s w h e n t h e t w o a r m s h a v e t h e s a m e r a d i u s o f c u r v a t u r e in t h e s a m e d i r e c t i o n . N o w t h e r a d i u s of c u r v a t u r e of a b e a m ρ is g i v e n b y ( M / £ J ) so if w e h a v e a m o m e n t M „ o n t h e u p p e r a r m , w e h a v e 66

- 1

_ Ε , Iχ _ λ

EI

U 2

12

J.G.

w h e r e ψΜ

is t h e m o m e n t o n t h e l o w e r a r m for t h i s e q u a l i t y , i.e.,

η

3

(V)

ί

ψ-

Williams

M o d e I is m a d e u p of e q u a l m o m e n t s in o p p o s i t e s e n s e s o n t h e t w o a r m s , s o w e h a v e M = M]

Mj

u

and

Μ -ψΜ 2

Mi =

χ

M

= ψΜ

2

and

λΧ

+M \ M +M 2

M„ =

x

If w e s u b s t i t u t e t h e s e e x p r e s s i o n s for M

.

(11)

and M

x

in e q s . (8) w e h a v e

2

(l-ξ)

ξ

3

165/L

i.e.,

x

J

On expanding, the cross-product term becomes M,M„

1

2

= 0,

ψ

1(1- •ξΥ ξ

as r e q u i r e d b y p a r t i t i o n i n g , a n d _a M\

l +ψ

u

'

β/

_a„

16(l-£)

o X 3

1-f

° " ™ Κ 7

(

{Μζ-ψΜ,)

2

β / 16(1-^) (1 + ^ ) '

3

3

a„ 3 ( l - g ) ( M + M , )

2

2

1

+

*

)

=

Λ?

16^(1 + ^)

·

( 1 2 )

with G = G + G . F o r a x i a l l o a d s , p a r t i t i o n i n g is effected b y x

u

P = P +P X

U

X

and

p = p 2

n

+

f r o m w h i c h G = 0, a n d G is f o u n d d i r e c t l y f r o m e q . ( 9 ) . F o r s h e a r f o r c e s , G „ = 0 a n d Gi is f o u n d d i r e c t l y f r o m e q . ( 1 0 ) * . x

n

These results are of particular i m p o r t a n c e since they enable G to be d e t e r m i n e d s o l e l y f r o m t h e l o c a l m o m e n t s a n d l o a d s at t h e c r a c k e d s e c t i o n . T h u s o n l y a c o n v e n t i o n a l - b e a m t y p e o f a n a l y s i s is r e q u i r e d t o find t h e s e q u a n t i t i e s a n d n o r e c o u r s e t o e n e r g i e s is n e c e s s a r y . T h e a p p l i c a t i o n of t h e m e t h o d will n o w b e i l l u s t r a t e d b y a n u m b e r of e x a m p l e s . 3.2. Double

cantilever

beam

(DCB)

tests

T h e d o u b l e c a n t i l e v e r b e a m m e t h o d t e s t c o n f i g u r a t i o n s h o w n in fig. 8 is t h e m o s t c o m m o n l y u s e d for m o d e - I t e s t s w h e n s y m m e t r i c a l l o a d s a r e a p p l i e d , as in fig. 8a. We thus have M

2

= -M

x

= Pa,

* Global partitioning can give non-symmetric deformations which are not compatible with local solutions.

Fracture mechanics of anisotropic

a) mode I

materials

13

b) mode II

a

c) mixed mode

^

Fig. 8. The double cantilever beam (DCB) test.

a n d h e n c e f r o m e q s . (12) G„ = 0

and

1 + tft

^P'a BI

G,

2

16(1

~ξ)

3

i.e., p u r e m o d e I for all v a l u e s of ξ. F o r t h e u s u a l c a s e of ξ = \, ψ = 1, t h i s b e c o m e s the well-known result

«

p 2 c

2

(13)

ΒΕ Γ

1

η

T h e shear correction m a y be found, since dM

dM,

2

£

=

=

da

ρ

da

a n d f r o m e q . (10) w e h a v e ~

l3O α^ f ( l -Pf ) ' 2

6 6

a n d for ξ = \ w e h a v e a t o t a l G v a l u e g i v e n b y T

12PV BEhl 2

3

u

14-

10 loUii/U/

(14)

-

It s h o u l d b e n o t e d t h a t t h i s r e s u l t is t r u e for a n y profile of b e a m , a n d n o t j u s t t h e p a r a l l e l v e r s i o n s h o w n , s i n c e it is t h e local v a l u e s of B, E a n d h w h i c h a r e r e q u i r e d . u

J.G. Williams

14

If parallel beams are used and the total deflection at the load point, 8, is measured then from simple beam theory, Pa 3

18--2

-3E 11 / '

and Ell/can be found from the slope of the loading diagram prior to fracture, P / 8. This may be used in G calculations, so no separate modulus measurements are needed. Alternatively, eq. (13) may be written as 3 P8 G I =="2 Ba·

(15)

Corrections are needed for end rotations by adding J a 66 / 11 all h to the crack lengths [13]. Both this and the shear correction are sometimes approximated by using an empirical compliance oc an, n < 3. Even for slender beams the effects of these corrections can be significant, particularly on apparent modulus values. Figure 8b shows the same specimen in pure mode-II loading since

for

g ==!

and from eqs. (12),

G I == 0

and

p 2a2 h3 ° ll

9

Gil

=4

(16)

B2E

Note that this test is always in pure mode II, whatever the value of 1 M ==Pa-1 1+l/J

and

M 2 -l/JM1 == 0

giving

M

2

==Pa~·

1 + l/J'

g,

since

and

G I == o.

This is self-evident since the common boundary conditions for each arm lead to equal curvatures and GIl is given by the total moment M 1 + M 2 • Care must be taken in the test to ensure free sliding at the load-point interface by using a lubricant or small roller. A mixed-mode version of the DCB test is shown in fig. 8c in which an off-centre crack is used but with an unloaded upper arm so that, M 1 == 0

and

M 2 == Pa,

and from eqs. (12) 2

2

a 11_ p _ a - - -1- - G __ 1B/ 16(I-g)3(1+l/J)'

(17)

giving a fixed-ratio test in which, GIl == 3(1- g)4 GI g2

(18)

Fracture mechanics of anisotropic

15

materials

P(l-1/L)

Pl/L Fig. 9. Variable-ratio mixed-mode test.

A similar test m a y be d e s i g n e d using e n d - n o t c h e d t h r e e - p o i n t b e n d s p e c i m e n s to which a similar analysis can be applied. A s a f o o t n o t e it is w o r t h n o t i n g t h a t c o r r e c t i o n s for r e s i d u a l s t r e s s e s c a n b e m a d e via t h e s e m e t h o d s b y o b s e r v i n g a n y i n i t i a l d i s p l a c e m e n t s o r c u r v a t u r e i n c r a c k e d s e c t i o n s , c a l c u l a t i n g t h e m o m e n t s c a u s i n g t h e m , a n d t h e n i n c l u d i n g t h e m in t h e G calculations. 3.3.

A variable-ratio

mixed-mode

test

Figure 9 s h o w s a testing configuration w h i c h gives a c o n t i n u o u s l y varying ratio of G / u

G , f o r / < a < L. F o r 0
t h e t e s t is p u r e m o d e I I , b u t t h e r e a f t e r t h e r e

a r e different m o m e n t s o n t h e u p p e r a n d l o w e r a r m s . If w e c a l c u l a t e t h e f o r c e

P

x

b e t w e e n t h e t w o b e a m s at t h e r i g h t h a n d , w e h a v e

_2 a

2\a)

LJ '

giving the b e n d i n g m o m e n t s M, =

M =P,a,

P/(l-f)-P,a,

2

a n d , from eqs. (12), Pl

2 2

3Γ

(l\ V 2

^

Pl

2 2

9Γ

/a\1

2

T h u s , t h e t e s t c o n d i t i o n s c h a n g e f r o m p u r e m o d e II w h e n a = I t o p u r e m o d e I w h e n a = L. 3.4.

Large

displacements

in DCB

tests

In t e s t i n g t h i n l a m i n a t e s w i t h t o u g h m a t r i c e s , t h e d i s p l a c e m e n t s s o m e t i m e s b e c o m e l a r g e in c o m p a r i s o n w i t h a, as s h o w n in fig. 10, w h i c h c a n g i v e rise t o n o n - l i n e a r

J.G.

16

Williams

Fig. 10. Large displacements in the mode-I DCB test.

l o a d - d i s p l a c e m e n t d i a g r a m s o f t h e f o r m s h o w n in fig. 2. T h e a n a l y s i s m a y b e c o n d u c t e d in t e r m s o f local m o m e n t s a g a i n , a n d G , is g i v e n e x a c t l y b y e q . (13) b u t i n s t e a d of a w e h a v e x, t h e d i s t a n c e f r o m t h e c r a c k t i p t o t h e l i n e of a c t i o n of P, i.e., ϋ

λ 1

= - ^ - . BE I

(20)

U

A s δ i n c r e a s e s , χ < a, a n d s o Ρ m u s t i n c r e a s e t o give t h e s a m e G a n d h e n c e t h e u p w a r d c u r v a t u r e of t h e l i n e , χ c a n b e m e a s u r e d d i r e c t l y d u r i n g t h e t e s t , o r t h e i n c r e m e n t a l a r e a m e t h o d m a y b e u s e d t o find G b u t n e i t h e r m e t h o d is p a r t i c u l a r l y e a s y . I n p r a c t i c e , it is e a s i e r t o m e a s u r e 8 a n d a a n d h e n c e c a l c u l a t e x. T h i s m a y be done using a finite-displacement a n a l y s i s for b e a m s u s i n g t h e a n g l e s o f r o t a t i o n [ 1 4 ] in t e r m s of t h a t a t t h e l o a d p o i n t a , as s h o w n in fig. 10, s u c h t h a t , {

V

2a

3E I

= -J

χ =——-sin

u

a,

U

where Ιι=

f

άφ ι . . = Jo V s i n a - sin φ a

F o r s m a l l a , l =2a x

δ 2fl

? J

=

Pa 3£„7

1

and 7 and

[

a

and = 2

3

I= 2

a

χ = a,

Jo

sind - = = = = = . vsina-sin

» giving

( Ί Λ λ

(21)

Fracture mechanics of anisotropic

materials

17

TABLE 1 a

(rad)

hi

h

0.066 0.133 0.198 0.263 0.326 0.387 0.446 0.504 0.558 0.610 0.660 0.707 0.752

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

*Ί

cos a

0.994 0.977 0.953 0.914 0.869 0.815 0.755 0.688 0.617 0.542 0.464 0.385 0.304

0.990 0.961 0.913 0.848 0.770 0.681 0.585 0.485 0.386 0.292 0.206 0.131 0.072

2

the usual result. F o r large a, I a n d I m u s t b e evaluated numerically (or via elliptic x

2

integral tables) a n d we m a y express the result as a correction factor o n the smalld i s p l a c e m e n t v a l u e , i.e., 4 suna in a

Gi

?2 2

/

P V \

8

I

2

T a b l e 1 g i v e s t h e s e f u n c t i o n s for a r a n g e o f a v a l u e s s o t h a t if 8/2a in a t e s t , F

x

may be found and hence G

x

is m e a s u r e d

d e t e r m i n e d . E n d b l o c k s c a n affect t h e s e

c o r r e c t i o n s a n d c a n b e i n c l u d e d [ 1 4 ] . A u s e f u l a p p r o x i m a t i o n for m o d e r a t e (8/a

8/a

< 1) v a l u e s is, F =

\ - U 8 / a ) \

x

I n m o d e I I , t h e a n a l y s i s is s o m e w h a t m o r e c o m p l i c a t e d b e c a u s e o f t h e c h a n g e in J at t h e c r a c k e d s e c t i o n , a s s h o w n in fig. 1 1 . H e r e w e m u s t i n c l u d e t h e a n g l e a

x

at

this point, a n d the results are [16]

αφ {

Vsin a - sin φ

^ - Ι ^ Γ ' . ^

h Jo

8_]_[ a

C

a

4

ά

.

φ

,

ν sin a Η- 3 s i n a —4 s i n φ

and

x

sin(/>d

(

{

J L J «, V s i n a — s i n φ

J

3

4 sin φ

a

άφ

Vsin a + 3 sin a — 4 sin φ x

0

a n d t h e c o r r e c t i o n f a c t o r is G„

gT

4(sina-sina )

^

ï

71

-

F l

^

"

3

P a 2

2

°·"Ϊ6Ϊ3Ε^7·

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

x

( 2 3 )

must be found; however, a good

J.G.

18

Williams

γ /

V / V

•L-

δ

_t

Fig. 11. Large displacements in mode II.

a p p r o x i m a t i o n m a y b e f o u n d if a

x

is s m a l l , w h i c h is t r u e a s a/L^>

1 [ 1 5 ] , in w h i c h

case, (24) As a / 0 ,

F

computed F

u

3.5.

-» c o s a ( s h o w n in t a b l e 1) a n d t h i s s h o u l d b e u s e d if t h e a p p r o x i m a t e 2

u

< cos

Transverse

2

a.

splitting

from

notches

The general m e t h o d developed here may be applied to the situation illustrated in fig. 12 in w h i c h a s p e c i m e n w i t h a n o t c h l e n g t h α in a w i d t h w fails s u c h t h a t a c r a c k C r u n s n o r m a l t o t h e n o t c h . S u c h f a i l u r e s o c c u r in c o m p o s i t e s w h e n t h e r e a r e n o t c h e s n o r m a l t o t h e fibre d i r e c t i o n . If w e a s s u m e t h a t t h e s p l i t C is o u t s i d e t h e l o c a l s t r e s s field o f t h e n o t c h t h e n w e m a y u s e t h e l a m i n a t e a n a l y s i s a n d w r i t e a = h,

w = 2/ï ,

}

w-a

=

h . 2

F o r t h e t e n s i o n c a s e , s h o w n i n fig. 12a , w e h a v e P = 0 a n d P = Ρ a n d if w e a s s u m e x

2

parallel grips with n o i n d u c e d m o m e n t s then, from eq. (9),

w i t h ξ= a/w.

T h i s m a y b e w r i t t e n in t e r m s o f t h e n o r m a l f r a c t u r e m e c h a n i c s f o r m ,

E G= XX

U

w h e r e cr = P/2hB

Υ σ α, 2

2

is t h e g r o s s s t r e s s a n d Y (a/w) 2

is a c a l i b r a t i o n f a c t o r . ( N o t e t h a t

Fracture mechanics of anisotropic

2h

materials

19

ι

—w

κ hi h

Ρ/2

Ρ/2 b) Three point bending

a) Tension Fig. 12. Transverse splitting from notches.

E

is t h e m o d u l u s in t h e c r a c k d i r e c t i o n a n d n o t in t h e d i r e c t i o n o f t h e n o t c h . )

u

Thus we have EG n

=

u

1

Υ σ α, 2

2

η

(25)

2(1-a/w)'

R e f e r e n c e [ 1 6 ] gives a s o l u t i o n for t h i s c a s e d e r i v e d f r o m t h e s h e a r l a g m o d e l w h i c h i n c l u d e s a c o r r e c t i o n f o r t h e l o c a l s t r e s s field effects, a n d t e n d s t o e q . (25) for l a r g e C. For pin loading on the central axis, m o m e n t s are i n d u c e d o n h giving Μ , = 0 a n d M = Ρηξ a n d w e n o w h a v e a m o d e - I c o m p o n e n t a s w e l l a s a m o d e - I I c o m p o n e n t . W e can n o w write 2

2

EG U

=

U

Υ σ α, 2

2

ητ

where

_(

2

3

Y

I T

2 I I T

a

/

w

y

g

2\\-a/w) 1 2 ( 1 - a/w)

/

w d

(a/w) + ( l - a / w ) ' 3

3

Γ L

™

3(l-a/w)(a/w) (a/w)

3

+

(26)

(l-a/w)

F o r t h r e e - p o i n t b e n d i n g , s h o w n in fig. 1 2 b , w e h a v e n o a x i a l l o a d s a n d M M = P/4(LC). If w e n o w u s e t h e l o c a l b e n d i n g s t r e s s ,

x

2

3 σ = 8

P(L-C) BW{\-a/wY

= 0 with

J.G.

20

TABLE 2 Calibration factors for transverse splitting, EG Y

Y

2

0.500 0.555 0.625 0.714 0.833 1.000 1.250 1.667 2.500 5.000

σ α. 2

η

Y

2

2

1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

= Υ

2

X

a/w

Williams

0.0 0.0 0.009 0.096 0.635 3.0 10.84 36.05 147.7 1348.0

II

^11 Β

τ

0.500

0.0

0.500

0.761 1.202 1.931 2.976 4.000 4.464 4.505 4.808 6.849

0.003 0.025 0.118 0.441 1.333 3.348 8.175 25.64 184.9

0.616 0.769 0.946 1.071 1.000 0.714 0.405 0.192 0.069

then

1

(a/w)

2

6 (\ -a/w)\(a/wf

Y 1

II Β

-a/wfY (27)

l — a/w

1

2

+ (\

"2 (a/w)

+

3

(l-a/w) ' 3

T h e s e c a l i b r a t i o n f a c t o r s a r e g i v e n in t a b l e 2. 3.6.

Buckling

under

compression

F i g u r e 13 s h o w s a c r a c k e d s e c t i o n u n d e r c o m p r e s s i o n a n d s e r v e s as a useful v e h i c l e for i n d i c a t i n g h o w t h e G a n a l y s i s m a y b e useful in d e s i g n . If w e c o n s i d e r t h e c a s e o f a d e l a m i n a t i o n v e r y c l o s e t o t h e s u r f a c e s o t h a t ξ = h /2h < 1, t h e n w e m a y c o n s i d e r t w o p o s s i b l e f o r m s of b u c k l i n g b e h a v i o u r : g l o b a l a n d l o c a l (as s h o w n ) . T h e r e is a l s o a t h i r d p o s s i b i l i t y of a l o c a l b u c k l i n g s u p e r i m p o s e d o n t h e c o m p r e s s i o n s i d e . T h i s is s i m p l y t h e s t r u c t u r a l b e h a v i o u r w h i c h m a y b e a n a l y z e d v i a t h e u s u a l x

Local

S

^

ItI

.r-'

$ S

S

Buckling

v

-χ

v

L

L...

•

•

...

X Global

Fig. 13. Laminate under compression.

Buckling

Fracture mechanics of anisotropic

materials

21

E u l e r s t r u t t h e o r y . S i n c e / is a p p r o x i m a t e l y c o n s t a n t a l o n g t h e s e c t i o n (ξ < 1), t h e g l o b a l b u c k l i n g l o a d for t h i s b u i l t - i n e n d c a s e is .

2TT

E Bh

2

3

n

3

c

L

(28)

2

w h i l s t for l o c a l b u c k l i n g ^

p

E

^

6

__

( 2 9 )

1

a

1

T h u s , for l o c a l b u c k l i n g t o p r e c e d e g l o b a l b u c k l i n g w e h a v e P
and

C

£= ^ < f 2h L

(30)

P r i o r t o b u c k l i n g t h e r e is n o e n e r g y r e l e a s e s i n c e t h e r e is u n i f o r m s t r e s s i n g b u t in the b u c k l e d states G exists. F o r simple global buckling we m u s t c o m p u t e the m o m e n t at t h e c r a c k e n d s . If M

0

is t h e m o m e n t at t h e b e a m e n d , t h e n t h e b e a m e q u a t i o n

for t h i s c a s e is ά ν

M -P v

2

0

dx "

c

E SI

2

'

U

w h e r e ν is t h e d i s p l a c e m e n t of t h e b e a m ( s e e fig. 13) a n d t h e b o u n d a r y c o n d i t i o n s are df ν -= — = 0 dx

at

χ = 0

du ^ — = 0 dx

at

χ = L,

and

giving a m o m e n t d i s t r i b u t i o n of M = M

0

c o s ax.

Eu%I

A l s o , w e h a v e sin aL = 0 g i v i n g a l o w e s t b u c k l i n g m o d e o f aL=n and thus the s o l u t i o n for P in e q . ( 2 8 ) . T h e m o m e n t at t h e c r a c k e n d is g i v e n w h e n x = ( L - a ) , i.e., M = M COS π ( 1 - a/L) = - M c o s πα/L. M is d e t e r m i n e d b y t h e d i s p l a c e ­ m e n t after b u c k l i n g , δ , a n d m a y b e f o u n d via energy, since c

a

0

0

0

Β

, . 8 P=2 = 2 [ Jo 2 L

n

F o r ξ<1,

M — Efilt 2

LM

2

dx = -

u

e q s . (12) g i v e s ^

0 0

E 16I U

= 0, a n d

J.G.

22

Williams

This may be rewritten as, G„ = G „ ( | - l ) ,

(31)

where 2ττ _

(h h \

4

n

A

x

8

c

~

3

L

F o r l o c a l b u c k l i n g t h e r e is a m o d e - I c o n t r i b u t i o n f r o m b e n d i n g t h a t m a y b e c o m p u t e d as in t h e p r e v i o u s c a s e a n d a l s o a m o d e - I I c o n t r i b u t i o n f r o m t h e m i s m a t c h of a x i a l l o a d s s i n c e t h a t in t h e b u c k l e d s e c t i o n r e m a i n s c o n s t a n t w h i l e t h a t in t h e t h i c k e r section rises. T h e results are

G^G l

and

~ y ( \ - a / L )

0

Or

J

(32)

4-

nil

G =

where

u

G ^\j-y(l-a/L) 0

Va 2 f t / T h u s for γ < 1 , l o c a l b u c k l i n g will p r e c e d e g l o b a l b u c k l i n g , a n d G a n d G will rise u n t i l 8/8 = 1. At t h i s p o i n t g l o b a l b u c k l i n g will t a k e o v e r a n d e q . (31) p e r t a i n s , g i v i n g a l i n e a r i n c r e a s e in G w i t h δ, b u t r e s t a r t i n g a t z e r o for 8/8 = 1. If t h e l o c a l b u c k l i n g p e r s i s t s o n t h e c o m p r e s s i o n s i d e o f t h e g l o b a l d e f o r m a t i o n t h e n its c o n t r i b u ­ t i o n t o G r e m a i n s at t h e 8/8 =l value. T h e n a t u r e of b o t h l o a d a n d total G b e h a v i o u r for t h e s e t h r e e c a s e s is i l l u s t r a t e d in fig. 14 a n d s e r v e s t o e m p h a s i z e t h a t t h e b e h a v i o u r o f s u c h a s t r u c t u r e is q u i t e c o m p l i c a t e d , e v e n for t h i s v e r y s i m p l e s y s t e m . T h e b u c k l i n g c o n t r o l s t h e c h a n g e s in G a n d t h i s in t u r n is d e t e r m i n e d b y t h e g e o m e t r y of t h e s y s t e m . A x i a l s p l i t t i n g o c c u r s w h e n G r e a c h e s s o m e critical c o n d i t i o n a n d t h i s c a n b e in a n y of t h r e e m o d e s of b u c k l i n g . x

u

C

u

c

c

4.

Stability

T h e s t a b i l i t y o f a c r a c k is i m p o r t a n t in b o t h t e s t i n g a n d d e s i g n , s i n c e c r a c k s w h i c h " j u m p " a r e u n d e s i r a b l e in b o t h c a s e s . A s m e n t i o n e d in sect. 2, w e s h a l l d e f i n e u n s t a b l e h e r e as t h e c o n d i t i o n w h e n G > R s o t h a t t h e k i n e t i c e n e r g y o f t h e s y s t e m i n c r e a s e s v i a a n i n c r e a s e in c r a c k s p e e d . T h e i n s t a b i l i t y c o n d i t i o n m a y b e w r i t t e n as [ 1 2 ] dG Tda"

dR >

T "da-

If R is c o n s t a n t t h e n dR/da

(

3

3

)

= 0, b u t w h e n R i n c r e a s e s w i t h c r a c k g r o w t h , t h e

" i n c u r v e effect", t h e n t h e s y s t e m is m o r e likely t o b e s t a b l e . I n a n y r e a l s i t u a t i o n ,

Fracture mechanics of anisotropic

materials

23

p r e d i c t i o n c a n o n l y b e m a d e p r e c i s e l y if R(a) a n d G(a) a r e k n o w n ; h o w e v e r , a useful p r a c t i c a l g u i d e t o c r a c k b e h a v i o u r c a n b e o b t a i n e d q u i t e s i m p l y b y c o n s i d e r i n g t h e l i m i t i n g c a s e of dR/da =0 a n d d e f i n i n g G u n d e r c o n s t a n t d i s p l a c e m e n t c o n d i ­ t i o n s . I n t h i s c a s e , t h e r e is n o e x t e r n a l w o r k p e r f o r m e d s o it is a b e s t c a s e in t h a t o t h e r l o a d i n g s y s t e m s a r e l i k e l y t o b e m o r e u n s t a b l e . I n t e s t i n g w i t h stiff m a c h i n e s , c r a c k g r o w t h is effectively a t fixed d i s p l a c e m e n t s o t h e c o n d i t i o n is h e l p f u l in d e s c r i b i n g s u c h t e s t s . W e s h a l l t h u s d e f i n e a s y s t e m a s stable w h e n , dG da

^0,

(34)

u const.

b u t r e c o g n i z e t h a t c h a n g e s o f l o a d i n g o r R(a)

m a y c h a n g e this condition.

24

J.G.

Williams

T h e a n a l y s i s is b e s t c o n d u c t e d in t e r m s o f t h e c o m p l i a n c e C o f t h e b o d y C(a)

= u/P,

(35)

w h e r e w e a s s u m e l i n e a r l o a d - d e f l e c t i o n b e h a v i o u r (i.e., i g n o r i n g l a r g e - d i s p l a c e m e n t effects). If w e r e t u r n t o e q . ( 3 ) w e h a v e BG = P-

d Γ — P du, da J

du da

a n d on substituting for Ρ w e have _ u du d u BG = —-—--— C da da J C

u

dC

2C

da

T h e stability c o n d i t i o n , eq. (3), m a y n o w b e written as dG

u(

C"

2

da ~ 2

2C'

\BC

B'C

2

2

BC

BC

3

2

w h e r e t h e p r i m e s d e n o t e d i f f e r e n t i a t i o n w i t h r e s p e c t t o a. T h i s c o n d i t i o n m a y b e written as ^

C(a)

C (C"

B'\

,

x

m a y b e f o u n d f r o m t h e u s u a l a n a l y s i s o f t h e g e o m e t r y u s e d , b u t it is o f t e n

m o r e c o n v e n i e n t t o p r o c e e d f r o m t h e G s o l u t i o n . If w e r e t u r n t o e q . (36) a n d s u b s t i t u t e f o r u in t e r m s o f Ρ w e h a v e P

dC , da'

2

BG =

2

and, hence, [ 2BG a

C=

—T-da + C

where C

0

0

is t h e c o m p l i a n c e f o r α = 0.

If w e c o n s i d e r t h e D C B i n m o d e - I l o a d i n g , t h e n f r o m e q . ( 1 3 ) w e h a v e Pa 2

G\ —

2

ΒΕ Ι' η

a n d , h e n c e , for c o n s t a n t Β a n d h values, 2a ~3E I~ U

3

8a

3

E Bh

39

u

s i n c e C = 0 for t h i s c a s e . T h u s , o

Fracture mechanics of anisotropic

materials

i.e., s t a b l e b e h a v i o u r . F o r t h e c a s e of t h e p r o f i l e d s e c t i o n w i t h Β c o n s t a n t hoca ,

C

2/3

25

and

is c o n s t a n t a n d B' = C " = 0, g i v i n g Γ = 0 a n d s t a b l e b e h a v i o u r ( d G / d a

is z e r o for fixed l o a d in t h i s c a s e ) . If h is c o n s t a n t a n d B oc α a s u s e d in s o m e l a m i n a t e t e s t s , t h e n B'/ Β = C"/ C = 1/ a a n d Γ = 0 a g a i n , a n d d G / d a = 0 for c o n s t a n t load. F o r t h e m o d e - I I t e s t , e q . (16) gives 2BG

9a

2

U

C

'

P

~2E Bh '

2

3

u

and, hence, C

_ 3 a

3

+ L

3

24EI for a u n i f o r m s e c t i o n , g i v i n g

a n d h e n c e for 0
unstable,

Γ>1;

0.55
stable,

Γ<1.

F o r t h e t a p e r e d c a s e , Boca,

G „ is c o n s t a n t at c o n s t a n t l o a d a s in m o d e I, a n d a g a i n

Γ = 0, i n d i c a t i n g s t a b l e b e h a v i o u r .

5. Cracks in anisotropic sheets C e r t a i n configurations of cracks a n d l o a d i n g s d o n o t l e n d t h e m s e l v e s to analysis via t h e m e t h o d s d i s c u s s e d p r e v i o u s l y a n d it is m o r e c o n v e n i e n t t o p r o c e e d via l o c a l stress fields at t h e c r a c k t i p . T h i s is, of c o u r s e , t r u e for i s o t r o p i c m a t e r i a l s a l s o a n d l e a d s t o t h e stress i n t e n s i t y f a c t o r a n a l y s i s . I n o r d e r t o p r o c e e d w i t h t h i s a p p r o a c h h e r e w e m u s t b e a b l e t o s o l v e stress-field p r o b l e m s in a n i s o t r o p i c m e d i a a n d t h e s e c a n b e v e r y difficult. T h e o n l y v i a b l e m e t h o d is u s i n g c o m p l e x a n a l y t i c f u n c t i o n t h e o r y , w h i c h is d i s c u s s e d i n g r e a t d e t a i l b y L e k h n i t s k i i [ 4 ] . H e r e , w e will c o n f i n e o u r s e l v e s t o j u s t t h o s e p a r t s n e e d e d for t h e c r a c k p r o b l e m a n d d e r i v e t h e r e s u l t s g i v e n b y S i h et al. [ 1 7 , 1 8 ] . 5.1.

Basic

method

C o n s i d e r first a s t a t e o f p l a n e stress d e s c r i b e d w i t h c a r t e s i a n c o o r d i n a t e s c o m ­ p o n e n t s σ , σ a n d τ . In the a b s e n c e of b o d y forces the e q u i l i b r i u m r e l a t i o n s h i p s are χ

ν

χν

J.G.

26

Williams

T h e s e m a y b e satisfied in t h e u s u a l w a y b y u s i n g t h e A i r y s t r e s s f u n c t i o n ψ(χ, defined such that

dx

dx

y),

dy

The strains e , e a n d y a r e d e f i n e d in t e r m s o f t h e d i s p l a c e m e n t s in t h e χ a n d y o f d i r e c t i o n s , u a n d v, r e s p e c t i v e l y , x

y

xy

du

dv

du

dV

Θλ:

3_y

dy

dx

E l i m i n a t i o n of the d i s p l a c e m e n t s leads to the compatibility c o n d i t i o n ,

8 ^

3 ^

+

dy

dx

=

a ^ dx

H o o k e ' s l a w for compliances, e = a a x

u

e = ασ y

2ι

χ

x

(41)

dy

a general

+ a (r

+

+ α σ,

+ ατ,

l2

22

y

}

isotropic material

may

be written

in t e r m s

of

ar, l6

26

xy

(42)

χν

Jxy = «61 ^ + «62^ + α Τ . 66

χν

S y m m e t r y r e q u i r e s t h a t a = « 2 1 a n d α = α , s o t h e r e a r e six i n d e p e n d e n t c o n s t a n t s . A n i m p o r t a n t p r a c t i c a l c a s e is t h a t of o r t h o t r o p y in w h i c h t h e p r i n c i p a l e l a s t i c d i r e c t i o n s a r e o r t h o g o n a l , a n d if t h e y c o i n c i d e w i t h t h e c o o r d i n a t e d i r e c t o r s , t h e n « 1 6 « 2 6 0, a n d t h e r e is n o i n t e r a c t i o n o f s h e a r a n d t e n s i o n w i t h o n l y four c o n s t a n t s . T h e six c o n s t a n t s for a n o r t h o t r o p i c m a t e r i a l in a c o o r d i n a t e s y s t e m r o t a t e d t h r o u g h a n a n g l e 0 m a y b e e x p r e s s e d as 1 2

=

1 6

6 2

=

C

1 1

= a

+ Zl(l-cos20) +

r(l-cos40),

C

2 2

= «22-4(l-cos20)+

r(l-cos40),

1 1

C =a

-4r(l-cos40),

C

-

66

66

(43) = a

l2

C

l2

1 6

=

C= 26

r(l-cos40),

4 sin 2 0

+2rsin40,

4 sin 2 0

+2rsin40,

where Δ =\{a -a ) 22

u

and

Γ = |(2α

1 2

+ a

6

6

-a

2

2

-a ). u

T h e o r t h o t r o p i c coefficients a r e o f t e n e x p r e s s e d a s m o d u l i a n d P o i s s o n ' s r a t i o s ,

Fracture mechanics of anisotropic

materials

27

F o r t h e i s o t r o p i c c a s e , t h e n u m b e r of c o n s t a n t s r e d u c e s t o t w o , s i n c e a

22 = —

a

=

n

,

= 2—^r-

a

66

and

a

l 2

= - — .

(45)

I n t h e s u b s e q u e n t a n a l y s i s w e u s e t h e p a r a m e t e r s for a n o r t h o t r o p i c m a t e r i a l ,2

λ

22

,

= —

and

a

2012+066

χ =—-

,

(46)

w h i c h m a y b e c a l c u l a t e d d i r e c t l y f r o m t h e v a l u e s o f a, o r v i a a 2 a n d a , a n d t h e 2

c o m p l i a n c e a t ±\ττ Cu(W)

u

is C (\n),

=

22

since, from eq. (43),

C (W)

= C (W)

n

4

=\(a

22

—

=

l

l

+α

λ

+ 2α

22

1 2

+ α ), 6 6

i.e.,

+ À + 2^. 2

«11 T h e s o l u t i o n for t h e g e n e r a l c a s e m a y b e d e d u c e d b y s u b s t i t u t i n g e q s . ( 4 2 ) i n t o e q . (41) a n d t h e n r e p l a c i n g t h e s t r e s s e s b y t h e e x p r e s s i o n s in t e r m s o f φ

from

eqs. (39), giving

aV

0ii—z~2a 3 /

1 6

aV a/ax

—-,

,

h(2a

1 2

+ a

x 6 6

aV ay ax

) — 52 — 2 - 2 a 2

2 6

aV aV ^+0 2—z ay ax ax 3

2

4

= 0.

,

x

(47)

For the isotropic case, this reduces to —

+2

Α

a/

+ - ^ = 0,

Ψ 2

2

ay ax 2

ax

2

V V = 0,

i.e.,

4

the b i h a r m o n i c e q u a t i o n of c o n v e n t i o n a l elasticity t h e o r y . S o l u t i o n s of eq. (47) c a n b e c o n v e n i e n t l y e x p r e s s e d in t e r m s of f u n c t i o n s o f t h e c o m p l e x v a r i a b l e ζ = χ +

μγ,

w h e r e μ = a + i/3, a a n d β a r e r e a l c o n s t a n t s a n d i =

>/ T. =

T h i s c a n b e s e e n b y r e c o u c h i n g t h e v a r i o u s d e r i v a t i v e s , i.e.,

aψ ay 4

dV az

τ—4 = μ τ 4 4

-

_

4.1V

— m
a n d eq. (47) b e c o m e s ψ

1 ν

[α μ -2α 4

η

1 6

μ

3

+ (2α

+ α )μ -2α μ

+ α]

2

12

66

26

22

T h u s t h e s o l u t i o n m a y b e e x p r e s s e d a s any

= 0.

(48)

f u n c t i o n o f ζ for w h i c h μ t a k e s t h e

f o u r r o o t s o f t h e c h a r a c t e r i s t i c e q u a t i o n in b r a c k e t s in e q . ( 4 8 ) . It h a s b e e n s h o w n in ref. [ 4 ] t h a t t h e s e r o o t s a r e e i t h e r c o m p l e x o r p u r e l y i m a g i n a r y a n d o c c u r as t w o conjugate pairs, ζ = α + \β Χ

λ

ΐ9

ζ = α -\β Χ

Χ

λ >

ζ = α + \β , 2

2

2

ζ = 2

α -[β, 2

28

J.G.

Williams

a n d t h e s o l u t i o n is ψ = ψ (ζ ) λ

+ ψ (ζ )

γ

χ

+ φ (ζ )

χ

2

+

2

ψ (ζ ). 2

2

N o w it c a n b e s h o w n b y c o n s i d e r i n g a n y p o l y n o m i a l f u n c t i o n of ζ t h a t for a n a n a l y t i c f u n c t i o n (i.e., a f u n c t i o n of w h i c h all d e r i v a t i v e s exist) w e h a v e t h a t ^(z ) + ^(z ) = 2 R e ^ ( z ) , I

1

1

so the solution r e d u c e s to ψ = 2Κε[ψ (ζ ) χ

+ ψ (ζ )1

χ

2

(49)

2

w h e r e R e m e a n s t h e r e a l p a r t of. For the special case of isotropy there are equal roots, μ specific f o r m of eq. ( 4 9 ) ,

χ

ψ = 2Κ^[ζψ\(ζ)

+

= μ = i, a n d w e h a v e a 2

φ (ζ)], 2

w h i c h is t h e b a s i c r e s u l t u s e d b y M u s k h e l i s h v i l i [ 6 ] . T h e e x p r e s s i o n s for s t r e s s e s in e q s . (39) m a y n o w b e r e w r i t t e n , e.g., d \b 2

ο- =-

1

χ

2

dy

= 2Κε[μ ψ';(ζ )

+

2

χ

χ

μ\φ' \ζ )\ 2

2

a n d t h e d i s p l a c e m e n t s f r o m e q . (40) b e c o m e , e.g., du

du dz

du

dx

dz dx

dz

•a a xx

+aa

x

x2

+

y

ar, X6

xy

and on substituting and integrating we have w-2Re[(^ a

+ a -^ a )iA (z ) + (^^a

2

,

n

1 2

1

1 6

1

1

ignoring rigid-body rotations. It is c o n v e n t i o n a l t o r e p l a c e ψ'(ζ) σ=

2Κε[μ φ[(ζ )

a =

2Κϊ[φ' (ζ )

χ

χ

y

r

xy

+

2

χ

χ

+

χ

= -2 Κ^[μ φί(ζ ) ι

w i t h φ(ζ),

2

2

2

1 6

)^ (z )], 2

2

s o w e h a v e t h e final e x p r e s s i o n s

φ' {ζ )1 2

2

2

2

2

(50)

9

ρ φ (ζ )]

ν=

2 Re[g,0 (z ) + g 0 (z )],

χ

1

-/x a

2

2 Κε[ρ φ (ζ )+ χ

1 2

μ φ' (ζ )1

u =

χ

+ a

2

+ μ φ (ζ )]

ι

u

2

1

2

2

2

2

9

2

where Pl,2 = Μ 1,2^11 + ^ 1 2 - ^ 1 , 2 ^ 1 6 , «1,2 = Μΐ,2«12 + « 2 2 / Μ 1,2 ~ «26 ·

T h e f o r m s of φ c a n b e d e d u c e d b y v a r i o u s m e t h o d s , a n d a w i d e v a r i e t y of s o l u t i o n s a r e g i v e n in ref. [ 4 ] ( a l t h o u g h n o t for t h e c r a c k p r o b l e m ) . E v a l u a t i n g t h e c o m p l e x e x p r e s s i o n c a n b e q u i t e l e n g t h y s i n c e , in g e n e r a l , t h e c o m p l e x v a l u e s of μ m u s t b e

Fracture mechanics of anisotropic

materials

29

f o u n d . T h i s p r o c e s s is c o n s i d e r a b l y e a s e d b y t h e u s e o f c o m p l e x - n u m b e r facilities in s o m e c o m p u t e r c o d e s . T h e i m p o r t a n t o r t h o t r o p i c c a s e is m o r e t r a c t a b l e s i n c e eq. (48) b e c o m e s + 2 * μ + λ = 0.

4

2

μ

(51)

2

Thus μι,2 = ^ = [ ( ^ + λ )

±( τ-λ)

1 / 2

] = ω , ί,

1 / 2

Λ

1

χ>λ,

2

and (52)

Mi,2 = ^ U - ^ )

1

/

+ i(^+ A)

2

1 / 2

],

*<λ.

For both conditions, Mi + / * 2 5.2.

i>/2(* + λ )

=

The crack

and

1 / 2

μμ ι

=

2

-λ.

problem

C o n s i d e r a t i o n s of p a t h i n d e p e n d e n c e of G for linear elastic materials, including the general a n i s o t r o p i c case, r e q u i r e (see sect. 2) t h a t in t h e region of t h e crack t i p t h e s t r e s s field will h a v e t h e f o r m r

- 1 / 2

w h e r e r is t h e r a d i a l d i s t a n c e f r o m t h e c r a c k

tip. Since t h e c o m p l e x variable h a s t h e form ζ = x + μγ, w e c a n w r i t e t h i s i n t e r m s o f p o l a r c o o r d i n a t e s , s e e fig. 4 , a s ζ = r ( c o s θ + μ s i n Θ), so that w e w o u l d expect a stress function of t h e form φ'(ζ)

= Αζ'

ι /

\

w h e r e Λ is a c o m p l e x c o n s t a n t . If w e w r i t e F

= (cos θ + μ

l i 2

ι

sin 0 ) ~

α

1 / 2

,

then t h e stresses b e c o m e 2 v r

a =-^Re[A F v

i

l

+

AF] 2

2

9

2 r

x v

= - - = Re[/X! A

Vr

x

F, + /x i4 F ]. 2

2

2

T h e constants a r e defined b y t h e stress-free crack face c o n d i t i o n s , 0 =

±7f,

σ> = τ

χ > ;

= 0,

F

1

2

= -i,

30

J.G.

Williams

a n d t h e stress i n t e n s i t y f a c t o r s a r e d e f i n e d a s in t h e i s o t r o p i c c a s e , i.e., 0 = 0,

K

= a yj27rr

{

and

y

Κ

= r

χι

\ 2πτ,

F

/

xy

x2

= 1,

giving the conditions

0 = Im04, + A ), 2

0 = Im( μΑ χ

K /2V2^

+

χ

μ Α ), 2

= Re(A

X

+ A) = A +

x

-K /2\Î2tt

2

2

A,

x

2

= R e ( μ A + μ A ) = μΑ

u

1

1

2

2

χ

χ

+

μΑ, 2

2

w h e r e I m m e a n s i m a g i n a r y p a r t of. T h u s ,

Α = λ 1

-^2

2V27r(^ -/x ) ( K , + W / * ) , 2

l

A

2

=

2

2V2TT(^ -/X ) 1

(Κ, + Χ , , / μ ι ) ,

2

(note that Α are complex). T h e s t r e s s e s m a y t h e n b e d i v i d e d i n t o s y m m e t r i c (K ) parts, 1 > 2

x

and skew symmetric

(K ) u

(μ Ρ -μ Ρ ) χ

2

2

χ

.μι - μ 2 1

(μ Ρ -μ Ρ ) χ

2

2

χ

.μι - μ 2 μιμ2

(F,-F )

and

2

.μι - μ 2

(53)

1 (

μι - μ 2 1

(

μ ι -μι 1 y/Znr

(μχΡχ-μιΡι)

Ιμ\~μι

T h e d i s p l a c e m e n t s a r e in t e r m s of φ(ζ) u=:4V~rRe[p (A /F ) x

x

x

+

= 2Az

p (A /F )l 2

2

2

a n d a g a i n w e m a y w r i t e t h e s e in t w o p a r t s 1 />ι/>2 μ Ρι\ Re ^ λ / ^ τ γ γ „ J . μ ι l- μ \/ μ ιPi /? μF2/>Γ /, 2

2

2

x

1 / 2

s o t h a t , e.g.,

Fracture mechanics of anisotropic

,

Γ

77

materials

31

and (54)

\ίϊτη

λ

^1 7Γ

T h e b o u n d a r y c o n d i t i o n s a r e , of c o u r s e , m e t b y t h e s e r e l a t i o n s h i p s . I n m o d e I, e.g., at 0 = 0, F = F = 1, a n d {

2

Κε(-μ μ ), ι

r

2

= 0,

xy

a n d at 0 = π, F = F = - i , x

^

2

Κ, σ =~ϊ=Κφμ μ ),

= - ^ = R e ( - i ) = 0,

χ

ι

r

2

xy

= 0,

i.e., s t r e s s - f r e e c r a c k f a c e s . T h e f o r m o f t h e s t r e s s d i s t r i b u t i o n m a y b e w r i t t e n in r e a l f u n c t i o n s for t h e o r t h o t o p i c c a s e (χ> λ ) , e q . ( 5 2 ) , s i n c e ^

/

.

F = (cos

.

Λ

ν _

0 + ιω sin 0)

a n d if ( c o s 0 - ί ω sin 0 )

1

/

7

2

= -

(cos

θ-ϊω

52

sin 9

.

0)

(cos 0 4 - ω sin

2

1 / 2

—-rrr, 1 / 2

7

0)

'

= a + i6, t h e n

1 / 2

a = - ^ [ c o s 0 + Vcos 0 + w s i n 2

2

1 6=-^[-cos 0Wcos

2

2

0]

2

0 + w sin 2

1 / 2

,

0],

2

a n d h e n c e w e h a v e , for e x a m p l e , Κ

1

λ

~V2^?

[^(cos 0+ // )

c u ( c o s 0 + H ) /2'

1 / 2

1

2

ν 2(ω -ω )ί

ay

H

/

1

2

2

2

~

where //

= (cos 0 + ω , sin 2

1 2

2

2

2

0)

1 / 2

.

F o r a slightly a n i s o t r o p i c material we c a n write ω ~\ λ

— δ,

ω = 1 + δ, 2

and Η -> 1 - δ s i n 0, 2

λ

Η ^ 1 + δ s i n 0, 2

2

1

//,

J.G.

32

Williams

and we have K,

/ l + c o s îy 0\

ΓίττΛ κ, "

1 / 2

Γ

2

sin 0 2

2 1 + co s 0 cos 50[ 1+ sin(§0 ) - s i n ( £ 0 ) ] ,

c o s ( | 0 ) [ l + s i n 0 - s i n Q 0 ) ]= 2

V 2OT

1

1+ s i i r 0

2

the isotropi c solution . 5.3. 77i e calculation

of G

G m a y b e f o u n d u s i n g e q . (7 ) a n d t h e r e l a t i o n s h i p s fo r / a n d g f r o m e q s . (53 ) a n d ( 5 4 ) . F o r m o d e I , fo r e x a m p l e ,
y

n

/ (0) = Κ,/ν^τ", n

and /2τγ

J g . I m ^ - ^

\

μ>\-μ2

)

\

μ\~μ2

I

+ J g n l m f - ^ - ) '

\μι-μ2/.

a n d , s i m i l a r l y , for m o d e II / (0) s

= K„/V2T7,

and /2TT|

π

L

\μ\-μ2/

O n s u b s t i t u t i n g for ρ a n d g w e h a v e 2TT

gn(^)=

gsW

= +

1

2

2

77

fïïr

^ ι ( μ + μ ) + Κι ^ιι" Ι" Κι(/*ι α 2 1 niι L I —J, μι^2

a

Im [ Κ„(μ,, + μ ) + Ki/Xi/x ]-

u

2

2

77

F r o m e q . (7) w e h a v e

Ο = -α ΚΑΐΑ

Κλ{μΧ

λ

+

μ 2 )

+

Κ

\

η

22

L

MlM2

J

and G „ = + a X n \\m[K ( n

μ + μ ) + Κ μ

u

χ

2

λ

λ

μ ]. 2

F o r t h e o r t h o t r o p i c c a s e w e m a y u s e e q s . (52) A t i + A t = iV2 ( * + λ )

1 / 2

2

"(^ + A)' _j /2λ /2

and and

μμ ι

2

=

-λ,

G „ = /£„
1/2'

(55)

Fracture mechanics of anisotropic

5.4.

The calculation

Reference

materials

33

of Κ

[4] gives t h e stress function

for a n e l l i p t i c a l h o l e in a n

infinite

a n i s o t r o p i c plate l o a d e d with a uniform p r e s s u r e ρ a n d a s h e a r stress t within the h o l e b u t n o t l o a d e d at t h e b o u n d a r y . If w e t a k e t h e l i m i t i n g c a s e o f a c r a c k ( m i n o r axis zero) t h e n these b e c o m e

2(μ, -M2)' Φ2(Ζ )

- ( ' - •μι

=

2

ρ)

2(μ·ι

(56) fi,

where

zVz - a 2

/=-! +

2

z - a 2

ζ = χ

'

2

+ μγ,

w h e r e χ a n d y a r e m e a s u r e d f r o m t h e c e n t r e o f t h e c r a c k ( t h e c r a c k t i p is at χ = a). N o t e t h a t o n t h e b o u n d a r i e s ζ > a a n d / - » 0, g i v i n g z e r o s t r e s s e s a n d , a l o n g t h e c r a c k l i n e , y = 0 a n d ζ = χ. W i t h i n t h e c r a c k χ < a, s o R e [ / ] = - 1 a n d , e.g., a

= 2Re[cj>' (z )

y

l

ί-μ Ρ + μιΡ\^ Λ μι-μι I . 2

+ ' (x )] = Re

l

2

cj-, = /? R e [ / ] =

2

i.e.,

-p.

T h u s a u n i f o r m t e n s i o n σ at t h e b o u n d a r y a n d z e r o p r e s s u r e in t h e c r a c k c a n b e modelled by a d d i n g this uniform tension σ to the σ

ν

c o m p o n e n t a n d p u t t i n g ρ = σ,

g i v i n g t h e f o l l o w i n g e x p r e s s i o n s for t h e s t r e s s e s ,

( - μ ι Α + μιίτ)

Ιμ\-

Γ σ

= σ Re

a

K

1 - ( - Μ 2 / 1 + Μ1/2)

+ cr,

(57)

Ιμ\S j ^

_

{

f

+

x

f

i

)

ίμχ-μι

J

A l o n g t h e c r a c k l i n e , y = 0, w e h a v e ,

xVx 2

f = Â = f

2

= -i

+

χ

cr = σ- R e [ - / X ! / x / ] , x

2

—a

2

»

and

σ\, - ( J R e [ / ] + σ,

F o r χ < a, w i t h i n t h e c r a c k ; σ = τ ν

σ

χ

= -λσ.

χν

= 0, a s r e q u i r e d a n d for t h e o r t h o t r o p i c c a s e ,

C l o s e t o a n d o u t s i d e t h e c r a c k t i p w e c a n w r i t e χ = a + r, r < a, g i v i n g

f=-l+Vâ/2r

34

J.G.

TABLE 3 C N plate, Y/yf^ a/W

for L/W

χ = 0.55

λ = 0.1 2

0.2 0.4 0.6 0.8

1.05 1.19 1.41 1.85

Williams

= 2 [19].

0.6

0.65

0.7

0.75

1.0 (iso)

small (shear lag [15])

0.2

0.3

0.4

0.5

1.0

0.0

1.03 1.14 1.34 1.80

1.03 1.12 1.31 1.80

1.03 1.12 1.31 1.80

1.03 1.11 1.31 1.80

1.025 1.10 1.30

1.04 1.08 1.21 1.57

-

and σ = a^/a/2r,

σ

ν

T h u s , K = σ \/2πΓ l

i.e., σ /σ χ

γ

= λσ( — 1 + V a / 2 r ) ,

χ

= σ\ίπα,

γ

r

xy

= 0.

e x a c t l y a s i n t h e i s o t r o p i c c a s e , b u t σ -> Α σ ν ' a / 2 r =

λσ ,

χ

= A a n d λ = 1 for isotropy. A similar derivation gives the s a m e K

u

γ

a s in

the isotropic case. T h i s r e s u l t ( a s p o i n t e d o u t b y Sih et a l . [ 1 7 ] ) is r a t h e r s u r p r i s i n g , b u t is a n i m p o r t a n t s i m p l i f i c a t i o n s i n c e for s m a l l c r a c k l e n g t h s in p l a t e s (a/w<

1), w e w o u l d

e x p e c t t h e i s o t r o p i c r e s u l t . F o r a finite p l a t e w e w r i t e K= 2

Y (a/w)

σ α,

2

2

a n d m o s t o f t h e i n c r e a s e in Y

2

a b o v e π is d u e t o t h e e l e v a t i o n of t h e n e t s e c t i o n

stress a s a c o n s e q u e n c e o f e q u i l i b r i u m a n d s t r e s s - c o n t r o l l e d b o u n d a r y c o n d i t i o n s [10]. Thus one would expect Y

2

for

finite

n o t t o differ g r e a t l y f r o m t h e i s o t r o p i c v a l u e s , e v e n

a/w.

O n e a d d i t i o n a l effect in Y , 2

h o w e v e r , w o u l d b e e x p e c t e d t o give s o m e d i f f e r e n c e

in t h e a n i s o t r o p i c c a s e . T h i s is t h e c h a n g e in t h e infinite p l a t e Κ w h i c h r e s u l t s f r o m the stress-free edges. T h e s e arise from t h e relaxation of σ

along the edges, and

σ

is a f u n c t i o n o f t h e e l a s t i c c o n s t a n t s . A t t h e c r a c k c e n t r e a = - A C T , a n d for y>

a,

χ

y

d e c a y s in t h e f o r m a/2(a/y) , 2

χ

i.e., i n d e p e n d e n t o f t h e e l a s t i c c o n s t a n t s .

T a b l e 3 gives n u m e r i c a l results b y B o w i e a n d Freese [19] u s i n g c o m p l e x stress f u n c t i o n s a n d b o u n d a r y c o l l o c a t i o n for t h e c e n t r e - n o t c h e d p l a t e . T h e y p o i n t o u t t h a t t h i s is a v e r y efficient n u m e r i c a l s c h e m e , e v e n for t h e i s o t r o p i c c a s e . T h e r a t h e r limited r a n g e of λ

2

v a l u e s u s e d is f r o m 0.1 t o 0.5 a n d o f χ f r o m 0.5 t o 0.75, a n d

t h e r e is little v a r i a t i o n f r o m t h e i s o t r o p i c c a s e . H e r e t h e free e d g e s a r e r e m o t e f r o m t h e c r a c k s o t h e s m a l l effect is n o t s u r p r i s i n g . A l s o g i v e n is t h e r e s u l t f r o m a s h e a r l a g m o d e l [ 1 6 ] w h i c h is effectively λ = 0 , a n d χ s m a l l . A g a i n t h e i s o t r o p i c r e s u l t is r e p r o d u c e d a p p r o x i m a t e l y . T a b l e s 4 a n d 5 a r e d a t a t a k e n f r o m a m e t h o d g i v e n in ref. [ 2 0 ] w h i c h a r e c a l c u l a t e d u s i n g a

finite-element

solution a n d a c o n t o u r integral

t o give G f r o m w h i c h Κ is d e d u c e d u s i n g e q . ( 5 5 ) . F o r t h e d o u b l e - e d g e n o t c h e d p l a t e t h e r e s u l t s d o n o t differ g r e a t l y f r o m t h e i s o t r o p i c v a l u e s b u t t h e r e is s o m e decrease

for t h e h i g h e r λ v a l u e s . F o r t h e s i n g l e - e d g e n o t c h e d p l a t e ( t a b l e 5) t h e r e

a r e d a t a for a m u c h l a r g e r r a n g e o f χ a n d λ

2

values. For high λ

2

v a l u e s , t h e r e is

v e r y little d e p a r t u r e f r o m t h e i s o t r o p i c c a s e , e v e n for χ u p t o 4 0 0 , b u t for l o w λ , 2

Fracture mechanics of anisotropic

35

materials

TABLE 4

D E N plate, Y/sJ^r [20]. a/W

* = 1.5

25.0

1.0 (iso)

λ = 0.2

4.0

1.0

1.04 1.07 1.15

1.13 1.16 1.22

2

1.14 1.20 1.28

0.4 0.5 0.6

TABLE 5

SEN plate, Y/y/lf [20]. a/W

0.0 0.05 0.2 0.3 0.4 0.5 0.6

* = o.o

10.0

1.17 1.11

1.08 1.10 1.34 1.63 2.07 2.77 3.94

1.62 2.18 3.16 4.86

2

2

2

-

λ =1

λ = 0.05

λ = 20 100.0

400.0

χ = 0.0

5.0

20.0

* = 1.0 (iso) 1.12 1.13 1.37 1.66 2.11 2.89 4.03

1.02

1.01

1.18

1.02

1.01

-

-

-

-

-

1.87 2.38 3.15 4.40

1.78 2.24 2.89 3.90

1.89 2.34 3.04 4.27

3.10 3.87 5.05 6.28

2.38 3.00 3.79 4.89

i.e., w i t h t h e stiff d i r e c t i o n n o r m a l t o t h e c r a c k , t h e r e a r e h i g h e r v a l u e s for t h e l a r g e r χ v a l u e s . T h e s e e x t r e m e c a s e s d o p r e s e n t c o m p u t a t i o n a l difficulties, a n d t h e r e is s o m e d o u b t a b o u t t h e a c c u r a c y . ( T h e v a l u e s g i v e n in t a b l e s 4 a n d 5 a r e c o r r e c t e d v e r s i o n s o f t h o s e in ref. [ 2 0 ] k i n d l y s u p p l i e d b y t h e a u t h o r . ) O v e r a l l , it w o u l d a p p e a r t h a t t h e i s o t r o p i c finite-width c o r r e c t i o n f a c t o r s a r e a d e q u a t e e x c e p t w h e r e e x t r e m e a n i s o t r o p y is i n v o l v e d w h e n it m a y b e u s e f u l t o c o m p u t e Y f a c t o r s . 2

6. D a m a g e z o n e s T h e n o t i o n of a c r a c k t i p z o n e is of p a r t i c u l a r i m p o r t a n c e in c o n v e n t i o n a l f r a c t u r e m e c h a n i c s b e c a u s e its s i z e c o m p a r e d w i t h t h e o t h e r d i m e n s i o n s o f t h e c r a c k e d b o d y d e t e r m i n e s t h e stress s t a t e w i t h i n t h e z o n e a n d t h u s t h e v a l u e of e n e r g y a b s o r b e d in f r a c t u r e . H o w far s u c h i d e a s c a n b e t r a n s f e r r e d t o c o m p o s i t e s w h e r e t h e u s u a l plastic deformation processes are replaced by m o r e general d a m a g e , including m i c r o c r a c k i n g , h a s n o t b e e n e s t a b l i s h e d . F i r s t , it is n e c e s s a r y t o h a v e a d a m a g e criterion a n d while this m a y b e e x p e c t e d to b e s o m e form of critical stress criteria for a n a n i s o t r o p i c c o m p o s i t e , it c o u l d b e a f u n c t i o n o f d i r e c t i o n . I n a d d i t i o n , t h e f o r m of s t r e s s d e p e n d e n c e o f t h e d a m a g e p r o c e s s w o u l d a l s o b e n e e d e d if t h e size a n d s h a p e o f t h e d a m a g e z o n e a t t h e c r a c k t i p is t o b e d e t e r m i n e d . T h e l o c a l s t r e s s field a r o u n d t h e c r a c k t i p is d e f i n e d in e q . ( 5 3 ) s o if s u c h a c r i t e r i o n is k n o w n t h e n , in p r i n c i p l e , t h e d a m a g e z o n e c a n b e d e f i n e d in t e r m s of t h e Κ v a l u e s .

36

J. G.

Williams

S i n c e s u c h c r i t e r i a a r e n o t e s t a b l i s h e d t h e y will n o t b e p u r s u e d h e r e , b u t r e c o u r s e will b e m a d e t o t h e v e r y s i m p l e n o t i o n t h a t a test m a y b e p e r f o r m e d in s i m p l e t e n s i o n in a g i v e n d i r e c t i o n a n d a d a m a g e s t r e s s σ m e a s u r e d . If a n o t c h is t h e n m a d e n o r m a l t o t h a t d i r e c t i o n a n d t h e test r e p e a t e d t h e n t h e e x t e n t o f t h e d a m a g e z o n e m a y b e e s t i m a t e d f r o m t h e e x p r e s s i o n for σ v i a t h e d e f i n i t i o n of K i.e., ά

ν

l9

(58)

-(-V.

a n d a s i m i l a r e x p r e s s i o n for s h e a r u s i n g K a n d a s h e a r d a m a g e s t r e s s . T h e s t r e s s s t a t e is n o t s i m p l e t e n s i o n , o f c o u r s e , b u t t h i s is i g n o r e d h e r e . I n t h e i s o t r o p i c c a s e , t h e w h o l e f r a c t u r e a n a l y s i s m a y b e c o n d u c t e d in e i t h e r G ox Κ t e r m s , b u t for c o m p o s i t e s G seems to b e m o r e physically m e a n i n g f u l as a criterion of failure so t h a t t h e c r i t i c a l z o n e size at f r a c t u r e s h o u l d b e w r i t t e n in t e r m s of G in t h i s c a s e via e q s . ( 5 5 ) , i.e., u

l c

G

-

>

K

(*

a

+

*)

1/2

giving, 1

G

2π

σα ά

22

l c

V2 λ {χ + λ)

(59)

S i m i l a r l y , o n e m a y v i s u a l i z e a c r a c k - t i p o p e n i n g d i s p l a c e m e n t in t h e d a m a g e z o n e , given by £ic=

O /a . lc

d

(60)

7. Conclusions T h e m e t h o d s outlined here s h o w that the use of L E F M to define the energy release r a t e for a n i s o t r o p i c m a t e r i a l s a r e sufficiently d e v e l o p e d t o p r o v i d e t h e b a s i s of a w o r k i n g analysis. Simplifications, such as self-similar crack g r o w t h , r e n d e r m a n y c a s e s s u r p r i s i n g l y s i m p l e a n d t h e n o t i o n s o f s o l u t i o n s f r o m b e a m t h e o r y in t e r m s of local m o m e n t s l e a d t o u s e f u l g e n e r a l m e t h o d s o f a n a l y s i s . T h e p a r t i t i o n i n g o f m o d e s o f l o a d i n g s e e m s t o b e n e c e s s a r y a n d a g a i n c a n b e c a r r i e d o u t in s i m p l e t e r m s . It is c o n s i d e r e d b a s i c t h a t t h e r e a l c r i t e r i o n o f f r a c t u r e is t h e e n e r g y r e l e a s e r a t e . F o r c r a c k e d p l a t e s it is n e c e s s a r y t o c o n s i d e r l o c a l fields, b u t t h e r a t h e r c o m p l i c a t e d analysis leads to the result t h a t t h e stress intensity factors are a l m o s t the s a m e as the isotropic case a n d can b e easily c o n v e r t e d to the energy release r a t e . C l e a r l y , t h e r e is m u c h still t o b e d o n e , b u t t h e r e is a s o u n d b a s i s h e r e o f t r a c t a b l e a n a l y s i s o n w h i c h t o b a s e t h i s effort.

Fracture mechanics of anisotropic

materials

37

List of s y m b o l s a

crack length; compliance, when subscripted; complex constant

Λ

crack area

Β

plate thickness

C

c r a c k l e n g t h in t r a n s v e r s e s p l i t t i n g ; c o m p l i a n c e

e

direct strain

Ε

modulus

/

f u n c t i o n of θ

F

f u n c t i o n o f 0;

g

f u n c t i o n of θ

F c a l i b r a t i o n f a c t o r for l a r g e d i s p l a c e m e n t s x

G

e n e r g y r e l e a s e r a t e , s e p a r a t e d in G , for m o d e I, G

h

thickness of l a m i n a t e

n

for m o d e I I

H

f u n c t i o n of θ

i J Κ / L M ρ Ρ q Q r R S t u U

V-i s e c o n d m o m e n t o f a r e a ; i n t e g r a l s for l a r g e d i s p l a c e m e n t s stress intensity factor d i s t a n c e in b e a m t e s t ; v a r i a b l e r a t i o d i s t a n c e in b e a m t e s t ; v a r i a b l e r a t i o bending moment f u n c t i o n o f μ a n d a; p r e s s u r e load f u n c t i o n of μ a n d a shear force r a d i u s ; d i s t a n c e f r o m c r a c k t i p ; w h e n s u b s c r i p t e d , z o n e size crack resistance d i s t a n c e in l i n e z o n e s h e a r stress displacement energy

υ W x, y Y ζ

displacement energy density; width cartesian coordinates finite-width correction factor c o m p l e x v a r i a b l e , z = x + μγ

a y Γ δ Δ θ λ μ ν

s l o p e o f b e a m at l o a d p o i n t d i m e n s i o n l e s s l o a d in s t r u t s ; s h e a r s t r a i n stability p a r a m e t e r ; c o m p l i a n c e function d i s p l a c e m e n t at l o a d p o i n t ; d i s p l a c e m e n t a t c r a c k t i p c o m p l i a n c e f u n c t i o n ; l e n g t h of c r a c k e x t e n s i o n angle from crack line compliance ratio >/α /α c o m p l e x n u m b e r , μ = a + \b Poisson's ratio 22

η

38

J.G.

Williams

ξ σ r φ χ

t h i c k n e s s r a t i o in l a m i n a t e s ; d i m e n s i o n l e s s l e n g t h n o r m a l stress s h e a r stress slope of b e a m ; c o m p l e x stress function c o m p l i a n c e r a t i o , (2a + a )/2a

Φ ω

[ ( 1 ζ)Ιζ] , s t r e s s f u n c t i o n function of θ

l2

-

66

u

3

References [1] R.F.S. Hearmon, An Introduction to Applied Anisotropic Elasticity (Oxford University Press, Oxford, UK, 1969). [2] G.I. Taylor and A.E. Green, Proc. R. Soc. London A 173 (1939) 162. [3] A.E. Green, Proc. R. Soc. London A 184 (1945) 231, 289, 301. [4] S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Trans. P. Fern (Holden Day, San Francisco, CA, 1963). [5] S.G. Lekhnitskii, English Translation (2nd Ed.) of Revised 1977 Russian Edition (MIR Publishers, Moscow, 1981). [6] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity (Noordhoff, Groningen, The Netherlands, 1953). [7] A.P. Schniewind and R.A. Pozniak, Eng. Fract. Mech. 2 (1971) 273. [8] J.A. Johnson, Wood Science 6(2) (1972) 151. [9] J.D. Barrett, Eng. Fract. Mech. 9 (1976) 711. [10] J.G. Williams, Fracture Mechanics of Polymers (Horwood-Wiley, New York, 1984). [11] J.R. Rice, in: Fracture, Vol. 2, ed. H. Liebowitz (Academic Press, New York, 1968) ch. 3. [12] J.G. Williams, Int. J. Fract. 36 (1988) 101-119. [13] J.G. Williams, Comp. Sci. & Tech (1989). [14] J.G. Williams, J. Compos. Mater. 4(21) (1987) 330. [15] J.G. Williams, Proc. ICCM IV London 3 (1987) 33. [16] J. Nairn, J. Compos. Mater. 22 (1988) 561. [17] G.C. Sih, P.C. Paris and G.R. Irwin, Int. J. Fract. Mech. 1 (1965) 189-203. [18] G.C. Sih and H. Liebowitz, in: Fracture, Vol. 2, ed. H. Liebowitz (Academic Press, N e w York, 1968) ch. 2. [19] D.L. Bowie and C E . Freese, Int. J. Fract. Mech. 8 (1972) 49-58. [20] J. Sweeney, J. Strain. Anal. 21(2) (1986) 99-107.