KINETICS

OF

CRACK

PROPAGATION

V. N. Barabanov, and A. N. Sazhin

L.

L.

GRAPHITE

IN

UDC 666.764.4:620.178.74

Lyshov,

I n r e c e n t y e a r s considerable attention has been given to the investigation of the possibility of the o p e r a tion of p a r t s and components of s t r u c t u r e s under conditions of incipient failure. Since some m a t e r i a l s , e s p e cially graphite, have, right f r o m the beginning, a ramified network of m i c r o - and m a c r o p o r e s and c r a c k s (owing to the peculiarities of their raw m a t e r i a l and the production technology), the beginning of their o p e r a tion in actual s t r u c t u r e s may be viewed as the beginning of the stage of failure. The continuum theories of failure indicate that in the formulation of the law of c r a c k propagation two m o s t essential f a c t o r s of the p r o c e s s of failure m u s t be taken into account: the kinetics of c r a c k propagation and the energy dissipation during propagation of many initial m i c r o - and m a c r o c r a c k s and l a t e r during p r o p a g a tion of the a r t e r i a l c r a c k [1]. Here it is important to establish what kind of p r o c e s s c r a c k propagation is, whether continuous or d i s c r e t e , i.e., whether the rate of c r a c k propagation is a continuously i n c r e a s i n g f u n c tion of time or whether it is a discontinuous o r oscillating function with time-dependent mean value and a m p l i tude. The kinetic features of the p r o c e s s of failure have to be c o r r e l a t e d with the s t r u c t u r e of the m a t e r i a l and be determined by it. The great s t r u c t u r a l inhomogeneity, and consequently the inhomogeneity of the r e s i s t a n c e fields in any c r o s s section of the bulk of the tested graphite, as well as the r e s u l t s of f r a c t o g r a p h i c analysis of specimen f r a c t u r e under different kinds of loading, lead one to a s s u m e that the p r o c e s s of c r a c k propagation in graphites is of a clearly d i s c r e t e nature. This view is supported by the fact that the c r a c k s are sinuous and also by the noticeable change in rate of failure when one kind of loading is replaced by another. When specimens with a s m a l l m a r g i n of e l a s t i c e n e r g y are tested, the a f t e r - s t a r t p r o c e s s of c r a c k propagation is of an inhibiting i n t e r mittent nature, and often the a r t e r i a l c r a c k stops, not t r a v e r s i n g the entire "live" c r o s s section of the s p e c i men. In tests with a high loading rate o r large m a r g i n of e l a s t i c energy the rate of failure is noticeably f a s t e r and the f r a c t u r e b e a r s t r a c e s of ramification of the a r t e r i a l crack. The p r e s e n t communication p r e s e n t s the r e s u l t s of the investigation of the kinetics of c r a c k propagation in graphites of different c a t e g o r i e s . Some methodological questions connected with the investigation of the c r a c k propagation rate in fiat graphite specimens subjected to bending tests were examined in [2]. The method of r e c o r d i n g the rate of c r a c k propagation is based on utilizing the r e g u l a r i t i e s of the r e d i s tribution of the potentials of the e l e c t r i c field induced in the fiat specimen when it moves. The calibration

It1

" 1.5

,0 5

Fig. 1. A v e r a g e d c a l i bration curves for tensile testing of graphite s p e c i m e n s : 1) MPG-6; 2) VPP.

A 15 2.

55 651cr

Moscow. T r a n s l a t e d f r o m P r o b l e m y P r o c h n o s t i , No. 7, pp. 52-57, July, 1978. mitted March 21, 1977.

790

0039-2316/78/1007-0790507.50

Original article sub-

0 1 9 7 9 Plenum Publishing C o r p o r a t i o n

T A B L E 1. ite Brands Graphite

P h y s i c o m e c h a n i c a l and S t r u c t u r a l C h a r a c t e r i s t i c s of the I n v e s t i g a t e d G r a p h -

Density, g/cm3

Calculated

Max .grain

porosity, cma/g

size of filler, mm

o;, kgf/cm 2 in tension

o."b, kgf/ Ob.en' kgtf/ [ ]Raw materials cm z, in compression cruz

VPP

1,78--1,90 [ 0,075--0,104

1,2

150

68O

250

MG-1 MPG-6

1,61--1,70 1,74--1,83

0,15 0,15

120 320

450 1000

130 53O

0,161--0,219 0,083--0,110

Calcined coke and pitch The same Nonca lcined coke and pitch

* The table presents the mean values of o b, O_b, and Oben; the variation coefficient of these values for the Indicated brands of graphite attains 20%.

T A B L E 2. E x p e r i m e n t a l V a l u e s of M a x i m u m C r a c k P r o p a g a t i o n R a t e s Vma x and S t r e s s I n t e n s i t y F a c t o r s Kic f o r D i f f e r e n t R e l a t i v e N o t c h D e p t h s l/b in the I n v e s t i g a t e d G r a p h ites Graphite

t/b

VPP MG-a

0,125 0,125 0,,25

MPG-6. VPP

0,08 0,08

MPG-6

'~Ic, kgf/ mm3/2

4,8~0,5 4,2~0,4 1,6•

Vmax, m/sec

lib

Bending 100 1,5 20

0,25 025 0,25

Klc, kgf/ram3 Vmax,. m/sec

5,6• 5,3~0,5 2,1•

50 0,5 10

Tension

,i,6~0,9 4,1 +0,4

650 350

0,1513,92::0,71350 0,15 4,7::h0,5

250

curves r = f ( / c r ) w e r e not p l o t t e d by m o d e l i n g the f a i l u r e of the s p e c i m e n on e l e c t r o c o n d u c t i n g p a p e r , a s w a s done in [2], but by s l i t t i n g t h r e e to f i v e a c t u a l s p e c i m e n s of the i n v e s t i g a t e d b r a n d of g r a p h i t e and m e a s u r i n g this d e p e n d e n c e d i r e c t l y on t h e m . F i g u r e 1 c o n t a i n s the c a l i b r a t i o n c u r v e s f o r t e n s i l e t e s t s of g r a p h i t e s p e c i m e n s M P G - 6 and V P P . l a r c u r v e s w e r e p l o t t e d f o r the t e s t i n g of s p e c i m e n s of o t h e r b r a n d s of g r a p h i t e .

Simi-

T h e c r a c k p r o p a g a t i o n r a t e in s p e c i m e n s of g r a p h i t e b r a n d s M P G - 6 , V P P , and M G - 1 w e r e i n v e s t i g a t e d with two t y p e s of l o a d i n g : c o n c e n t r a t e d b e n d i n g of a n o t c h e d s p e c i m e n and t e n s i o n of a f l a t s p e c i m e n w i t h one lateral notch. S o m e p h y s i c o m e c h a n i c a l and s t r u c t u r a l c h a r a c t e r i s t i c s s e n t e d in T a b l e 1.

of the i n v e s t i g a t e d b r a n d s of g r a p h i t e a r e p r e -

T h e t e s t s w e r e c a r r i e d out on an I n s t r o n - 5 0 0 m a c h i n e . In b e n d i n g , the s p e e d of the m o b i l e g r i p w a s 20 mmflmin. Flat specimens 5 x 40 x 160 mm in size with an initial notch of variable length, were tested. The selection of the mentioned dimensions was governed by the possibilities provided by the method of crack r e cording. It was pointed out above that the method permits practically inertialness measurement of the crack propagation rate in flat specimens. The great width of the specimens was chosen to increase the crack scope for the sake of fuller recording of the supercritical propagation region. During the tests, oscillograms of failure were recorded. In addition to that, the critical stress intensity factor Klc [3] in concentrated bending was calculated from the fracture loads: K I c = - Y 6Mll/2 tb 2

,

in t e n s i o n of s p e c i m e n s with one l a t e r a l notch: plil 2 KIttY

tb

'

791

F i g . 2. M i c r o s t r u c t u r e of i n v e s t i g a t e d g r a p h i t e s : a , b, c) m i c r o s e c t i o n s u r f a c e s (tension) of g r a p h i t e s V P P , M G - 1 , M P G - 6 , r e s p e c t i v e l y ; g, h, i) s u r f a c e s c l e a v a g e s u r f a c e s of g r a p h i t e s V P P , M G - 1 , M P G - 6 , r e s p e c t i v e l y . w h e r e M i s the a p p l i e d b e n d i n g m o m e n t ; b, t , l a r e the w i d t h , t h i c k n e s s , and d e p t h , r e s p e c t i v e l y , of the notch in the s p e c i m e n ; Y is the p a r a m e t e r of K - c a l i b r a t i o n ; P is the f r a c t u r e l o a d . T h e t e n s i l e t e s t s w e r e c a r r i e d out w i t h g r a p h i t e s p e c i m e n s of b r a n d s M P G - 6 and V P P with s i z e s 5 • 65 x 165 and 7 x 80 • 180 r a m , r e s p e c t i v e l y . T h e s p e e d of the m o v i n g g r i p w a s 10 r a m / r a i n . T h e s p e c i m e n s w e r e h e l d in the g r i p s by b o l t s . T h e i n i t i a l n o t c h e s w e r e m a d e with a d i a m o n d d i s k 0.15 m m t h i c k ; the r a d i u s of the notch a t the a p e x did not e x c e e d 0.1 r a m . T h e i n v e s t i g a t i o n s s h o w e d t h a t in c o n s e q u e n c e of the m a c r o d e f e c t i v e n e s s of the s t r u c t u r e s u c h a s h a r p n e s s of the n o t c h is s u f f i c i e n t f o r t h e m . If the r a d i u s of the notch a p e x is m a d e s m a l l e r , i t d o e s not i n c r e a s e the s t r e s s c o n c e n t r a t i o n e f f e c t . ( E a c h e x p e r i m e n t a l p o i n t w a s o b t a i n e d f r o m 7-12 t e s t e d s p e c i m e n s . ) T h e v a l u e s of K i c in b e n d i n g and t e n s i l e t e s t s w e r e d e t e r m i n e d with a c o n f i d e n c e l e v e l of c~ = 0.95. r e s u l t s of t h e s e t e s t s a r e p r e s e n t e d in T a b l e 2.

The

I t can be s e e n t h a t the m a x i m u m c r a c k p r o p a g a t i o n r a t e in the s a m e m a t e r i a l d e p e n d s s t r o n g l y on the s p e c i m e n r i g i d i t y l / b , and a l s o on the m a r g i n of e l a s t i c ener~5, in the s p e c i m e n - m a c h i n e s y s t e m .

792

of g r a p h i t e s V P P , MG-1, M P G - 6 , r e s p e c t i v e l y ; d, e, f) r e p l i c a s of the c l e a v a g e of c l e a v a g e f a c e s of g r a p h i t e s V P P , MG-1, M P G - 6 , r e s p e c t i v e l y ; k, l, m)

In the f i r s t c a s e this is i l l u s t r a t e d by the r e d u c t i o n of the m a x i m u m c r a c k p r o p a g a t i o n r a t e Vma x with i n c r e a s i n g depth of the notch, in the second c a s e by an i n c r e a s e in Vma x in t e n s i l e t e s t s as c o m p a r e d with b e n d ing t e s t s . The l a r g e d i f f e r e n c e in the c r a c k p r o p a g a t i o n r a t e s in d i f f e r e n t b r a n d s of g r a p h i t e is due to the s u b s t a n t i a l d i f f e r e n c e s in t h e i r m a c r o - and m i c r o s t r u c t u r e s . The m a c r o s t r u c t u r e of the i n v e s t i g a t e d g r a p h i t e s includes the i n i t i a l h e t e r o d i s p e r s e d coke g r a i n s of the f i l l e r and the coked d i s p e r s e m e d i u m binding them. Of the two b a s i c e l e m e n t s of the m a c r o s t r u c t u r e m e n tioned a b o v e , the g r a i n s a r e s t r o n g e r [4, 5], and t h e r e f o r e the m e c h a n i s m of f a i l u r e in g r a p h i t e w a s viewed m a i n l y in the m a c r o s c o p i c a s p e c t [4-7]. It was e s t a b l i s h e d that the f a i l u r e o c c u r s along the g r a i n b o u n d a r i e s , along the coke f r o m the bonding a g e n t , and, in the m i c r o v o l u m e , p r e d o m i n a n t l y along the b o u n d a r i e s of c r y s t a l i i t e s and aIong t h e i r base p l a n e s , d i s r u p t i n g the w e a k bonds between them. In connection with that, both c h a r a c t e r i s t i c s - the fine g r a i n s and the d i s o r i e n t e d s t r u c t u r a l e l e m e n t s - of f i n e - g r a i n e d g r a p h i t e a r e bound to d e t e r m i n e its g r e a t e r s t r e n g t h c o m p a r e d with m e d i u m - and c o a r s e - g r a i n e d graphite and a l s o the d i f f e r e n c e s in f a i l u r e c h a r a c t e r i s t i c s and kinetic f e a t u r e s . The m i c r o s t r u c t u r e of the i n v e s t i g a t e d g r a p h i t e s is shown in Fig. 2, f r o m which it can be seen that the f i n e - g r a i n e d g r a p h i t e s MG-1 and MPG-6 have a much m o r e h o m o geneous s t r u c t u r e (cf. Fig. 2 a - c ) .

793

F i n e c o m m i n u t i o n of the f i l l e r e n s u r e s g r e a t e r g r a i n s t r e n g t h , d e n s e r and m o r e f a v o r a b l e p a c M n g , a m o r e h o m o g e n e o u s c h a r a c t e r of the p o r o u s s t r u c t u r e and p o r e d i s t r i b u t i o n in s p a c e , and a h o m o g e n e o u s g r a p h i t e m a c r o s t r u c t u r e w i t h o u t l a r g e c r a c k s and p o r e s w h i c h s u b s t a n t i a l l y r e d u c e the s t r e n g t h of the m a t e r i a l . A l l t h i s h e l p s i n c r e a s e the s t r e n g t h of the m a t e r i a l [5, 7]. E l e c t r o n m i e r o g r a p h s of the r e p l i c a s of the c l e a v a g e s u r f a c e s of the s p e c i m e n s a f t e r t e n s i l e t e s t s (ef. F i g . 2d-f) and p h o t o g r a p h s of the c l e a v a g e s u r f a c e o b t a i n e d with a s c a n n i n g e l e c t r o n m i c r o s c o p e (of. F i g . 2 g - m ) i n d i c a t e that the s t r u c t u r e of g r a p h i t e s MG-1 and V P P (on c a l c i n e d coke) e x h i b i t c l e a r l y e x p r e s s e d b o u n d a r i e s b e t w e e n g r a i n s and a r o u n d the g r a i n s , and that in the s t r u c t u r e of g r a p h i t e M P G - 6 (on u n c a l c i n e d coke) the open c o n t o u r s of the b o u n d a r i e s a r o u n d the f i l l e r p a r t i c l e s (which a r e l e s s c l e a r l y v i s i b l e ) p r e d o r n i n a t e on a c c o u n t of the s t r o n g c o h e s i o n and t h e i r m u t u a l i n t e r p e n e t r a t i o n on s e c t i o n s of c o n t a c t i n t e r a c t i o n in the p r o c e s s of the t e c h n o l o g i c a l a n n e a l i n g . A s a r e s u l t , i t s s t r u c t u r e is m o r e m o n o l i t h i c and h a s the a s p e c t of a c o n t i n u o u s f r a m e w o r k (of. F i g . 2m). T h u s , in c o n s e q u e n c e of the c a M n g , the g r a p h i t e M P G - 6 h a s s h o r t e r i n t e r g r a n u l a r b o u n d a r i e s a n d a m o r e u n i f o r m s t r e s s d i s t r i b u t i o n in the d e f o r m e d bulk; t h i s m e a n s that t h e r e a r e g r e a t e r o b s t a c l e s to f o r m a t i o n of an a r t e r i a l c r a c k . H o w e v e r , a f t e r the l a t t e r h a s f o r m e d due to the g r e a t e r c o n t i n u i t y and s o l i d i t y of the m a t e r i a l f o r m i n g the s t r u c t u r e of g r a p h i t e M P G - 6 , t h e r e h a v e to be f e w e r o b s t a c l e s i n h i b i t i n g i t s g r a d u a l p r o p a g a tion. T h e m e a n s p e e d of the c r a c k i n c r e a s e s and the c u r v e of change in the r a t e of c r a c k p r o p a g a t i o n a l o n g the p a t h of i t s m o v e m e n t is l e s s i r r e g u l a r . T h e s e a r e the a s s u m p t i o n s f o l l o w i n g f r o m the a n a l y s i s of the m a c r o and m i c r o s t r u c t u r e of the i n v e s t i g a t e d g r a p h i t e s . It w a s found a b o v e that in f o r m u l a t i n g the t h e o r y of c r a c k p r o p a g a t i o n the p r o b l e m of the e n e r g y d i s s i p a tion n m s t be e x a m i n e d . W i t h o u t c l a i m i n g that the p r o b l e m of energ~r d i s s i p a t i o n in the p r o c e s s of g r a p h i t e f a i l u r e h a s been c o m p l e t e l y s e t t l e d , we note (on the b a s i s of the M n e t i c f e a t u r e s of c r a c k f o r m a t i o n ) that when the c r a e k b e g i n s to m o v e at s u p r a e r i t i c a l s p e e d , t h e d i s s i p a t i o n o f e l a s t i c e n e r g y a c e u m u l a t e d in the s p e c i m e n - m a c h i n e s y s t e m is c o n s i d e r a b l y g r e a t e r in c o a r s e - g r a i n e d and m a c r o s t r u e t u r a l l y m o r e i n h o m o g e n e o u s m a t e r i a l of the V P P than in f i n e - g r a i n e d m a t e r i a l s M P G - 6 and M G - 1 . If we c o m p a r e the m e a n v a l u e s of Kie (ef. T a b l e 2) in d e p e n d e n c e on the v a l u e s of Vma x f o r the s a m e b r a n d s of g r a p h i t e , we find that they b e c o m e s m a l l e r when the s p e e d of the c r a c k s i n c r e a s e s . T h e s a m e t e n d e n c y w a s o b s e r v e d in [1] and [8] in r e g a r d to m e t a l s . F i g u r e 3 s h o w s the c u r v e s of c h a n g e in s p e e d of the a r t e r i a l c r a c k a t d i f f e r e n t s e c t i o n s of i t s m o v e m e n t u n d e r t e n s i o n , w h i c h b e a r s out the h y p o t h e s i s of i r r e g ~ a l a r j e r k y c r a c k p r o p a g a t i o n in g r a p h i t e s . It s h o u l d be b o r n e in m i n d t h a t the f r e q u e n c y of the p o i n t s of m e a s u r e m e n t of the s p e e d of the a r t e r i a l c r a c k a t d i f f e r e n t s e c t i o n s of i t s path is d e t e r m i n e d by the m o d u l a t i o n f r e q u e n c y of the s i g n a l f r o m the s p e c i m e n , w h i c h is p r o p o r t i o n a l to the c r a c k length [2]. T h e m o d u l a t i o n f r e q u e n c y in i t s t u r n i s l i m i t e d by the r e s o l u t i o n of the f i l m on w h i c h the f r a c t u r e o s c i l l o g r a m is p h o t o g r a p h e d . T h e r e f o r e in the t i m e i n t e r v a l d e t e r m i n e d by the m o d u l a t i o n f r e q u e n c y the t r u e s p e e d of the a r t e r i a l c r a c k i s a v e r a g e d , and t h i s c o r r e s p o n d s to the p o i n t s in F i g . 3. C o n n e c t i n g the p o i n t s of r e c o r d e d s p e e d s by s t r a i g h t l i n e s is s o m e w h a t a r b i t r a r y and is d e t e r m i n e d by the p o s s i b i l i t i e s of the a p p l i e d m e t h o d of m e a s u r i n g the s p e e d of the a r t e r i a l c r a c k . T h e a c t u a l j u m p s in s p e e d b e t w e e n the p o i n t s a r e p r o b a b l y e v e n m o r e and m a c r o p o r e s and i n i t i a l c r a c k s in g r a p h i t e s is c h i e f l y d e t e r m i n e d by the and the technologs; of t h e i r p r o c e s s i n g into g r a p h i t e . B e s i d e s , m a c r o p o r e s to the m e r g i n g of m i c r o p o r e s u n d e r the e f f e c t of s t r e s s c o n c e n t r a t i o n , both f i l l e r and a l o n g the g r a i n b o u n d a r i e s in the eoke of the b o n d i n g a g e n t .

a b r u p t . T h e p r e s e n c e of m i c r o p r o p e r t i e s of the r a w m a t e r i a l s f o r m in the p r o c e s s of l o a d i n g due i n s i d e the coke p a r t i c l e s of the

T h e a r t e r i a l c r a c k p r o p a g a t e s i r r e g u l a r l y f r o m one m a e r o p o r e o r i n i t i a l c r a c k to a n o t h e r , f r o m a l o w s t r e n g t h s e c t i o n to a h i g h - s t r e n g t h s e c t i o n and v i c e v e r s a . Since s u c h i n h o m o g e n e o u s s e c t i o n s in the i n v e s t i g a t e d b u l k of the m a t e r i a l a r e i r r e g ~ a l a r l y d i s t r i b u t e d , a c c e l e r a t i o n o r s l o w i n g down, and in s o m e e a s e s e v e n c o m p l e t e s t o p p i n g of the c r a c k , a r e p o s s i b l e a n y w h e r e along its entire path. B e s i d e s , the i r r e g u l a r i t y in the c r a c k p r o p a g a t i o n r a t e is due to the n o n u n i f o r m i t y of s t r e n g t h and d e f o r m a t i o n t y p i c a l of g r a p h i t e s , and a l s o the n o n u n i f o r m i t y in 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 of the l o c a l s e c t i o n s of the m a t e r i a l a h e a d of the c r a c k f r o n t .

794

Vcr , IT1 / ~;~c

800 500

Joo

~-~

1oo i 20


50

80

?0 l ~ , mm

Fig. 3. Change in the speed of the a r t e r i a l c r a c k in tension of s p e c i m e n s with d i f f e r e n t initial notches: 1, 2) graphite MPG-6 at l/b = 0.15 and 0.8, r e s p e c tively; 3, 4) g r a p h i t e V P P at l~ b -- 0.15 and 0.08, r e s p e c t i v e l y . T h u s , c r a c k p r o p a g a t i o n in g r a p h i t e is a d i s c r e t e p r o c e s s . The m i c r o - and m a c r o c r a c k s , and a l s o the a r t e r i a l c r a c k in g r a p h i t e , p r o p a g a t e in j u m p s whose magnitude is d e t e r m i n e d at once by the m a c r o - and m i c r o s t r u c t u r e o f t h e a c t u a l type of g r a p h i t e in question and its loading conditions. The kinetic d i f f e r e n c e of the p r o c e s s c o n s i s t s in the fact that the f a i l u r e of f i n e - g r a i n e d g r a p h i t e s MPG-6 and MG-1 o c c u r s wit]] r e l a t i v e l y s m a l l e r changes in the speed of the a r t e r i a l c r a c k f r o m section to s e c t i o n c o m p a r e d with the c o a r s e - g r a i n e d and m o r e d e f e c t i v e g r a p h i t e V P P .

An assessment of the deviation of the speed of the arterial crack on different sections from its mean value, measured according to the ratio Av/vav, showed that for fine-grained MPG-6 graphites it is within 0-4Cr for VPP graphite from 5 to 95%. LITERATURE 1.

2. 3. 4.

5.

6. 7. 8.

CITED

F. Erdogan, "Theory of crack propagation," in: Failure [Russian translation], A. Yu. Ishlinskii (editor), Vol. 2, Mir, Moscow (1975), pp. 521-615. V. M. Markochev, V. V. Zhitenev, and A. P. Bobrinskii, Zavod. Lab., 42, No. 2, 221-224 (1976). U. Brown and D. Srawley, Testing High-Strength Metallic Materials for Fracture Toughness in Plane Deformation [Russian translation], Mir, Moscow (1972). Yu. N. Rabotnov, V. N. Barabanov, G. M. Volkov, and L. I. Knoroz, "Local deformation of graphite materials in the process of high-temperature tension," Dokl. Akad. NaukSSSR, 224, No. 5, 1013-1065 (1975). Yu. S. Virgil'ev and V. N. Barabanov, "The mechanism of failure of graphite materials," Probl. Prochn., No. 7, 96-100 (1975). V. N. Barabanov, G. M. Volkov, and L. I. Knoroz, "The effect of high temperatures on the mechanism of d e f o r m a t i o n and f a i l u r e of g r a p h i t e m a t e r i a l s , " P r o b l . P r o e h n . , No. 6, 88-94 (1976). V. N. B a r a b a n o v and Yu. S. V i r g i l ' e v , R a d i a t i o n Strength of S t r u c t u r a l Graphite [in R u s s i a n ] , A t o m i z dat, Moscow (1976). G. M. K r a f f t and G. I r w i n , "The effect of the r a t e of c r a c k p r o p a g a t i o n , " in: A p p l i e d P r o b l e m s of F r a c t u r e T o u g h n e s s [Russian t r a n s l a t i o n ] , Ya. B. F r i d m a n and B. A. D r o z d o v s k i i ( e d i t o r s ) , M i r , Moscow

(1968).

795