1063 A SIMPLE MODEL FOR PREDICTING CRACK ARREST

P. H. Melville Central Electricity Research Laboratories Kelvin Avenue, Leatherhead, Surrey KT22 7SE UK tel: Leatherhead ?4488 R e c e n t l y Hahn et al. [1-3] have shown t h a t under c e r t a i n c o n d i t i o n s t h e k i n e t i c energy a s s o c i a t e d with a moving c r a c k can be r e a b sorbed by the c r a c k , so t h a t t h e c o n d i t i o n f o r c r a c k a r r e s t i s t h a t t h e t o t a l s t r a i n energy r e l e a s e d i s equal t o the i r r e v e r s i b l e work used in the c r e a t i o n o f the new s u r f a c e s and in the p l a s t i c deformation. Since the s t r a i n energy a s s o c i a t e d with a s t a t i o n a r y c r a c k i s a f u n c t i o n o f t h e crack l e n g t h and t h e l o a d i n g system, t h e t o t a l s t r a i n energy r e l e a s e d d u r i n g the time o f c r a c k p r o p a g a t i o n i s a f u n c t i o n o n l y o f t h e i n i t i a l and f i n a l s t a t e s , and i s i n d e p e n d e n t o f t h e time taken b e f o r e t h e c r a c k a r r e s t s . The c o n d i t i o n f o r crack a r r e s t is thus

faaf

Gdda =

o

f

a

af

Gsda =

o

faaf

Rda

(1)

o

where Gd i s t h e dynamic s t r a i n energy r e l e a s e r a t e , Gs i s t h e e q u i v a l e n t s t a t i c v a l u e f o r a s t a t i o n a r y c r a c k o f t h e same l e n g t h , R i s t h e r a t e a t which e n e r g y i s absorbed i r r e v e r s i b l y , and a and a¢ a r e t h e i n i t i a l and f i n a l c r a c k l e n g t h s . This i s a t v a r i a n c ~ with ~he more commonly used c r i t e r i o n f o r c r a c k a r r e s t , t h a t the s t r a i n energy r e l e a s e r a t e should j u s t f a l l t o a c r i t i c a l a r r e s t v a l u e Gd = G a

C2)

which would apply i f a l l t h e k i n e t i c e n e r g y i s absorbed by a n e l a s t i c v i b r a t i o n s in the m a t e r i a l . I t i s a l s o a p p a r e n t l y a t v a r i a n c e with t h e t h e o r e t i c a l work o f Eshelby [ 4 ] , Freund [ 5 ] , and Glennie and Willis [6] who give for an arbitrarily moving crack with instantaneous velocity v G d = (i - v / c r) G s

{_3)

where c r i s t h e Rayleigh v e l o c i t y , and where t h e r e is no a c c e l e r a t i o n dependence o f Gd. In t h i s case t h e i n t e g r a l s o f Gd and Gs in el) cannot be e q u a l . However, t h e s e s o l u t i o n s a r e v a l i d f o r f i n i t e c r a c k s o n l y f o r s h o r t time; b e f o r e s t r e s s waves a r e r e f l e c t e d back from t h e s i d e s o f the specimen or from the o t h e r end o f t h e c r a c k . In p r i n c i p l e a t l e a s t a r e f l e c t e d s t r e s s wave p a s s i n g t h e t i p o f t h e c r a c k could g i v e an a d d i t i o n a l t e n s i l e component h e r e a l l o w i n g t h e c r a c k t o propagate further. F u r t h e r d i s c u s s i o n o f the motion o f c r a c k s in f i n i t e samples i s given by B e r g k v i s t [ 7 ] . For an i d e a l b r i t t l e m a t e r i a l R r e p r e s e n t s j u s t a v e l o c i t y i n d e pendent s u r f a c e e n e r g y , and thus t h e l e n g t h a t which t h e c r a c k a r r e s t s i s given d i r e c t l y from (1). However, where t h e r e i s p l a s t i c deforma-

Int Journ of Fracture 11 (1975)

1064 t i o n , and more e s p e c i a l l y where t h e r e i s s t r a i n r a t e d e p e n d e n c e i t i s n e c e s s a r y t o know t h e v e l o c i t y o f t h e c r a c k b e f o r e (1) may be u s e d . Kanninen [3] has u s e d a c o m p l i c a t e d n u m e r i c a l p r o c e d u r e t o o b t a i n t h e m o t i o n o f a c r a c k i n a d o u b l e c a n t i l e v e r beam. E x t e n s i o n t o more complicated systems presents a sizeable problem, but for engineering applications all that is required is a quick approximation giving a reasonable figure for the velocity. I t i s known t h a t when t h e c r a c k f i r s t s t a r t s t o move (3) s h o u l d b e s a t i s f i e d , and t h a t a t c r a c k a r r e s t t h e c o n d i t i o n o f (1) s h o u l d be f u l f i l l e d . Further analytic solutions a r e t h o s e o f Broberg [ 8 , 9 ] who shows t h a t f o r a c r a c k moving a t c o n stant velocity Gd = fCv/cr) Gs

(4)

-- [1 - (V/Cm)2] Gs where the approximation is valid for low velocities, 0.38 c I = 0.69 c_ is the limiting velocity in Mott's ution [10]. In ~act the functions given in (3) and different [ii], and can be considered here to he the now required is a simple interpolation between (i), obvious choice is

Gd

d

~-~ [(1 - (V/Cm)2]

/a

and where c = • ,m quasl-statlc sol(4) are not very same. What is (3), (4). An

Gs(a' ) da'

(5)

0

since this gives the same function as obtained by Mott, except that in Mott ! s theory the lower limit of integration is 0 not a_. Justification for the use of a o (in addition to it allowing (3) ~o be satisfied) is that if this were set equal to zero for a wedge loaded system the integral would be infinite, and experiments by Kuppers [12] have also shown that the velocity of a moving crack correlates much better with a/a O than simply with the crack length. Real justification for the use of this equation comes from comparison with the numerical solutions of Kanninen (see Figure i). Eqn. (5) does not quite give the constant velocity propagation as found by Kanninen, and of course smooths out any effect due to reflection of stress waves from the end of the sample. However, the time for propagation, which is the most important parameter, agrees to within %5%, which is well sufficient for any engineering application, for situations where kinetic energy can he absorbed at the crack tip.

Acknowledgment: T h i s work was c a r r i e d o u t a t t h e C e n t r a l E l e c t r i c i t y R e s e a r c h L a b o r a t o r f e s , and i s p u b l i s h e d by p e r m i s s i o n o f Cent r a l E l e c t r i c i t y G e n e r a t i n g Board. REFERENCES

[1]

G. T. Hahn, R. G. Hoagland, M. F. Kanninen, and A. R. Rosenf i e l d , Dynamic Crack Propagation, N o r d h o f f , Leyden (1973) 649-662.

Int Journ of Fracture Ii (1975)

1065

[2]

G. T. Hahn, A. G. Hoagland, M. F. Kanninen and A. R. R o s e n f i e l d ,

Proceedings of the Third International Conference on Fracture, Munich 1973. [3]

M. F. Kanninen, International Journal of Fracture 10 (1974) 415-4301.

[4]

J. D. Eshelby, Journal of the Mechanics and Physics of Solids 17 (1969) 177-199.

[5]

L. B. Freund, Journal of the Mechanics and Physics of Solids 20 (1972) 141-152.

[6]

E. B. Glennie and J . R. W i l l i s , Journal of the Mechanics and Physics of Solids 19 (1971) 11-30.

[7]

H. B e r g k v i s t , Journal of the Mechanics and Physics of Solids 22 (1974) 491-502.

[8]

K. B. Broberg, Arkiv Fizik 28 (1960) 159-192.

[9]

K. B. Broberg, in Recent Progress in Applied Mechanics, Almquist and Wiksell, Stockholm (1967) 125-151.

[I0]

N. F. Mott, Engineering 165 (1948) 16-18.

[11]

L. B. Freund, Journal of the Mechanics and Physics of Solids 20 (1972) 129-140.

[12]

H. Kuppers, International Journal of Fracture Mechanics 3 (1967) 13-17.

2 July 1975 (rev 4 August 1975)

17S / t, ,t

150

/I I

12S

~ IO0

~ 7s $0

lS

/

/,

I

I

I

50

I00

150

I 200

TIME. ps FIG. I

COMpARI$0NOF RESULTSOF EOUATION(5) (DASHEDL|NE) WITH NUMERICALSOLUTIONOF KANNINEN( leJ/4 FIG.3) (FULL LINE) FOR CRACKLEliGTH AGAINSTTIME

Int Journ of Fracture II (1975)