INTERACTION

BETWEEN

AND TWO-MEDIA V.

E.

FAST

CRACK

INTERFACE

Sereda

and

V.

M.

Finkel'

UDC 620.192.45

An analysis of the s t r e s s system in the vicinity of the tip of a c r a c k running orthogonal to the interface of m a t e r i a l s with different elastic constents was c a r r i e d out for the f i r s t time by Zak and Williams [1]. L a t e r this problem attracted the attention of a l a r g e number of r e s e a r c h e r s [2-5]. Using their r e s u l t s , the conclusion may be drawn that the effect of two media on a c r a c k is not always the s a m e . It is determined by the mutual relation of the elastic c h a r a c t e r i s t i c s of l a y e r s and the orientation of the c r a c k . It should be noted that despite a large number of investigations concerned with the analysis of static c r a c k propagation conditions in brittle m a t e r i a l s , the fast c r a c k s have not been sufficiently investigated. In the experimental work being described the interaction of dynamic c r a c k s with the media interface was studied. Experimental

Procedure

Methods of dynamic photoelasticity using polarization-optical apparatus were used in an investigation of the kinetics of fracture and the stress system at the fast crack tip during its interaction with the interface of materials with different elastic characteristics [6]. The process was recorded by a n S F R - i M h i g h - s p e e d a m e r a in monochromatic light l =546 p. Three-layer specimens measuring 150 x 300 • 5 m m were used as model materials; they were castfrom ED-6 epoxy resin. The different mechanical properties of layerswere obtained by varying the amount of the plasticizer dlbutylphthalate or using a rigid plate layer cut from a glass sheet. The ratio of elastic moduli was EI/E 2 = 4. The fracture was initiated by producing an edge stress concentratloninthe form of acuttotake a knife bl ade with a detonator. The explosion of the charge was effected by discharging a bank of condensors (C =0.3 pF) at the m o m e n t of puncture of the discharge gap by a high-voltage impulse from the control panel of the S F R camera. The rate of fracture was controlled by changing the elastic potential accumulated in the specimen stretched on a press. The stretching load was monitored by a strain-gauge circuit incorporating a T D - 3 strain-gauge device. In the case under consideration the average crack growth rate varied within the range 250-350 m/sec.

Fig. 1.

F r a m e s showing the interaction between dynamic c r a c k and i n t e r face.

T a m b o v Institute of Chemical Engineering, T r a n s l a t e d from P r o b l e m y P r o c h n o s t i , No. 12, pp. 24-27, D e c e m b e r , 1977. Original article submitted July 1, 1976.

1426

0039-2316/77/0912-1426 $07.50 9 1978 Plenum Publishing C o r p o r a t i o n

EI/Ez I=lO0

0.1

,!

0'920

I0

i/

0

/F

0.6

o.6 9 "ZO I0

10 Z.mm

0

10 Z,mm

O.4

'

8

1__---

16 l.mm

C b R e l a t i o n K / K 0 a s a f u n c t i o n of the d i s t a n c e of c r a c k t i p f r o m inte r f a c e . a

F i g . 2.

Experimental

Results

T h e i n t e r a c t i o n b e t w e e n a f a s t c r a c k a n d the i n t e r f a c e of m a t e r i a l s with d i f f e r e n t e l a s t i c p r o p e r t i e s . r e s u l t s in a c h a n g e of the s t r e s s s y s t e m a t the c r a c k t i p and in the f r a c t u r e p r o p a g a t i o n r a t e . F i g u r e 1 s h o w s a f r a m e of a f r a c t u r e of a c o m p o s i t e m a t e r i a l with the r a t i o of e l a s t i c i t y m o d u l i of i n d i v i d u a l l a y e r s E I / E 2 = 4 . T h e c r a c k a p p r o a c h e d the i n t e r f a c e f r o m the s i d e of the m a t e r i a l with a h i g h e r m o d u lus of e l a s t i c i t y with an i n i t i a l p r o p a g a t i o n r a t e of 330 m / s e c at an a n g l e c l o s e to 90 ~ T h e p h o t o e l a s t i c p a t t e r n s c a n be u s e d to follow the a p p r o a c h of the c r a c k to the i n t e r f a c e which a t the s t a r t is a c c o m p a n i e d b y a c o m p l e x c h a n g e of the s t r e s s s y s t e m a t the c r a c k t i p . T h e " s t r e s s r o s e t t e , " r e p r e s e n t i n g a c o m b i n a t i o n of i s o c h r o m a t i c f r i n g e s w h i c h j o i n s a t the c r a c k t i p , a t f i r s t i n c r e a s e s (the d i s t a n c e f r o m c r a c k t o l n t e r f a c e b o u n d a r y l = 15 m m ) , a n d s u b s e q u e n t l y d e c r e a s e s (10< l< 15). H o w e v e r , f r o m the m o m e n t when the d i s t a n c e b e t w e e n c r a c k a n d i n t e r f a c e d e c r e a s e s to 10 m m , a d i s t i n c t t e n d e n c y is o b s e r v e d in the c r a c k f o r an i n c r e a s e in the n u m b e r of f r i n g e s . T h e s t r e s s i n t e n s i t y m o n o t o n i c a l l y i n c r e a s e s u n t i l the c r a c k c r o s s e s the i n t e r f a c e . A f t e r the c r o s s i n g the s t r e s s i n t e n s i t y m a r k e d l y d e c r e a s e s a n d s u b s e q u e n t l y , with the i n c r e a s i n g d i s t a n c e f r o m the c r a c k tip to the i n t e r f a c e , s t a r t s to g r o w a g a i n . T h i s c o n t i n u e s until the d i s t a n c e f r o m the c r a c k t i p to the i n t e r f a c e e x c e e d s 5 r a m . S u b s e q u e n t l y , a t l > 5 r a m , the s t r e s s c o n c e n t r a t i o n d e c r e a s e s a n d a t l> 10 m m r e m a i n s c o n s t a n t a s long a s f i l m i n g c o n t i n u e s . L e t us m a k e a q u a n t i t a t i v e a s s e s s m e n t I n t e r f a c e . I r w i n [7], B r a d l y a n d K o b o y a s h i factor K from photoelastic patterns using as A c c o r d i n g to t h i s m o d e l the m a x i m u m s h e a r the c r a c k t i p a s f o l l o w s

of the s i t u a t i o n a r i s i n g f r o m the i n t e r a c t i o n b e t w e e n c r a c k a n d in [8] d e v e l o p e d a m e t h o d f o r d e t e r m i n i n g the d y n a m i c s t r e s s i n t e n s i t y a m o d e l the s t r e s s d i s t r i b u t i o n in the v i c i n i t y of a s t a t i c c r a c k . s t r e s s e s in the p o l a r c o o r d i n a t e s y s t e m r , O c a n be d e s c r i b e d f o r

w h e r e a0x is the s t r e s s a t the p o i n t x =a (a is the c r a c k l e n g t h ) . up to e l i m i n a t e s t r e s s g0x u s i n g the c o n d i t i o n ~8e

I r w i n s h o w e d that a s e c o n d e q u a t i o n c a n be s e t

0.

F r o m t h e s e two e q u a t i o n s it is p o s s i b l e to f i n d the unknown q u a n t i t i e s (r0x and K. A n a l y z i n g t h i s m e t h o d , B r a d l y a n d K o b o y a s h i found t h a t the a c c u r a c y in d e t e r m i n i n g the s t r e s s i n t e n s i t y f a c t o r d e p e n d s to a c o n s i d e r a b l e e x t e n t on a n g l e e m a t w h i c h the Tm v a l u e r e a c h e s its m a x i m u m . F o r e x a m p l e , r e d u c i n g a n g l e e m f r o m

F i g . 3.

Frames

s h o w i n g the f r a c t u r e of a l a m i n a r s p e c i m e n .

1427

Fig. 4.

F r a m e s showing the propagation of a c r a c k in a composite m a t e rial.

90 to 70~ e r r o r made in determining this factor i n c r e a s e s f r o m 3 to 70%. the authors suggested that s t r e s s a0x be determined from equation GOx ~

~

K

In o r d e r to reduce the e r r o r ,

(2)

"

In this case the substitution of relation (2) into (1) produces K 0 a), 9 ,. = 2---~7~t ( ,~,

(3)

where [(e, r, a)----(sin~e + 2 V ~ s i n

8 sin ~--~- _~)0~.

Selecting points for m e a s u r i n g s t r e s s e s on two i s o c h r o m a t i c f r i n g e s rl, 0t and r2, 02, the following exp r e s s i o n is obtained ~( 9 ~ . , - ~,m = ~ "

r, V ; 7 - r, VT, V~, ,

thus

f2~

-- fl V~

(4)

Equation (4) will be used in processing experimental data. To clarify the effect of the interface on the crack consider the relative stress intensity coefficient K/K0' where K0 is the stress intensity factor at the crack tip 20 mm from the interface, a value which is determined for each layer separately. The calculation results of the relative s t r e s s intensity coefficient K/K 0 are given in Fig. 2a. The nonmonotonic type of change of this coefficient on the initial section of crack propagation is due to the effect of discharge waves generated at the moment when the crack starts and by the action of the variable cleavage force produced by the knife edge during the explosion. At a distance of about 10 mm from the interface the s t r e s s intensity coefficient begins to increase continuously and near the interface of layers exceeds the initial value 1.47 times. After the crack has crossed the interface the s t r e s s intensity coefficient K/K0 decreases to 0.92K/K 0 for l = 20 ram. A further crack development facilitates the growth of the s t r e s s intensity coefficient (K/K 0= I . 2 for /= 5.5 mm) with its subsequent reduction to a constant value at /> 10 ram.

Thus, for E 1/E 2 =4 the e l a s t i c interaction of c r a c k and interface b e c o m e s apparent at a distance of less than 10 m m . The p r e s e n c e of an interface on the dynamic c r a c k path i n c r e a s e s the s t r e s s intensity atthe c r a c k tip and, consequently, facilitates the failure p r o c e s s . However, after the c r a c k has c r o s s e d the interface the

l' mm ~'lOO ,5

4~j~ E,t l J t~,

5 0

1428

JZ

~i4~,psec

Fig. 5. Kinetics of i n t e r action between c r a c k and interface.

e f f e c t of this i n t e r f a c e b e c o m e s r e v e r s e d . In the i n t e r f a c e r e g i o n ( / < 5 mm) the c o e f f i c i e n t of s t r e s s I n t e n s i t y is s m a l l e r than away f r o m i t . Consequently, in this c a s e the i n t e r a c t i o n of the i n t e r f a c e with a d y n a m i c c r a c k r e t a r d s the l a t t e r . F i g u r e 3 shows f r a m e s of a f r a c t u r e of a c o m p o s i t e s p e c i m e n f o r the c a s e when the c r a c k s a p p r o a c h the i n t e r f a c e f r o m the side with a l o w e r e l a s t i c i t y m o d u l u s . The i n i t i a l r a t e of f r a c t u r e was 300 m / s e e . With the c r a c k growing in a homogeneous m e d i u m a t a m o d e r a t e r a t e , the s t r e s s s y s t e m ahead of i t s tip is of h y d r o s t a t i c n a t u r e and the p l a s t i c e f f e c t s have difficulty in d e v e l o p i n g . In o u r c a s e the s t r e s s e d condition is c h a r a c t e r i z e d by the p r e s e n c e a h e a d of the c r a c k of a r e g i o n of i n c r e a s e d s h e a r s t r e s s which s t e a d i l y inc r e a s e s as the c r a c k a p p r o a c h e s the i n t e r f a c e . T h i s c r e a t e s f a v o r a b l e conditions f o r s t r e s s r e l a x a t i o n b e c a u s e of the p l a s t i c flow which e v e n t u a l l y i n c r e a s e s the work of f r a c t u r e . S t a r t i n g with l=10 m m the s t r e s s i n t e n s i t y c o e f f i c i e n t d e c r e a s e s ( F i g . 2b). F o r e x a m p l e , f o r l=5 m m K / K 0 d e c r e a s e s 0 . 8 2 t i m e s l e s s than at a d i s t a n c e l=20 r a m . A f t e r the c r a c k has c r o s s e d the i n t e r f a c e the s t r e s s i n t e n s i t y at its tip i n c r e a s e s . The c r a c k a p p r o a c h e s the i n t e r f a c e f r o m the side of the l a y e r with a l o w e r modulus of e l a s t i c i t y . At a d i s t a n c e l=5 m m the c o e f f i c i e n t K/K 0 shows a tendency to d e c r e a s e while at l> 10 mm it r e m a i n s c o n s t a n t f o r the e n t i r e d u r a t i o n of f i l m i n g . It should be noted that in this e a s e the i n t e r a c t i o n between c r a c k and i n t e r f a c e along the c r a c k p a t h r e s u l t s in a r e d u c t i o n in s t r e s s i n t e n s i t y at the c r a c k tip which, t o g e t h e r with a p o s s i b l e p l a s t i c r e l a x a t i o n a t the c r a c k tip, f a c i l i t a t e s r e d u c t i o n in the c r a c k p r o p a g a t i o n r a t e o r its s t a b i l i z a t i o n . F i g u r e 4 shows a s e r i e s of f r a m e s r e p r e s e n t i n g the propeg~tion of a c r a c k in the c o m p o s i t e m a t e r i a l with the r a t i o of e l a s t i c i t y moduli in the l a y e r E i / E 2 =100. By v a r y i n g the load it b e c a m e p o s s i b l e to obtain c r a c k s with i n i t i a l growth r a t e s of 250 and 350 m / s e e . In this c a s e an e l a s t i c i n t e r a c t i o n is o b s e r v e d a t about 18 m m f r o m the s u r f a c e of the l a y e r j o i n t . With the c r a c k a p p r o a c h i n g the r i g i d l a y e r the s t r e s s i n t e n s i t y c o e f f i c i e n t d e c r e a s e s ( F i g . 2c) a I o n g s i d e with the d e c r e a s i n g c r a c k p r o p a g a t i o n r a t e ( F i g . 5). At 5 . 5 m m f r o m the interfa'ee the c r a c k s t o p s . The s t r e s s i n t e n s i t y c o e f f i c i e n t at the point where the c r a c k s t o p p e d was 0.65 t i m e s l o w e r than the i n i t i a l v a l u e . With the i n i t i a l r a t e of c r a c k p r o p a g a t i o n r i s i n g to 300 m / s e e the c r a c k c r o s s e s the i n t e r f a c e . of p r o p a g a t i o n changes only i n s i g n i f i c a n t l y .

The r a t e

Thu s, an i n c r e a sing diffe r e n c e in l a y e r r i g i d i t y i n c r e a s e s the zone of i n t e r a c t i o n of the c r a c k w i t h the i n t e r face and r e d u c e s to a g r e a t e r extent the s t r e s s i n t e n s i t y at its tip. The e f f e c t i v e n e s s of i n t e r a c t i o n f r o m the f a i l u r e c o n t r o l viewpoint a l s o depends on the I n i t i a l c r a c k p r o p a g a t i o n r a t e . Slow c r a c k s a r e m o r e s e n s i t i v e to h e t e r o g e n e i t y . LITERATURE 1.

2. 3. 4. 5. 6. 7. 8.

CITED

A. R. Zak and M. L. W i l l i a m s , " C r a c k point s t r e s s s i n g u l a r i t i e s at a B i - m a t e r i a l s i n t e r f a c e , " J . A p p l . M e c h . , 3__O0,No. 2, 141-143 (1963). D. O. Swenson and C. A. Rau, " S t r e s s d i s t r i b u t i o n a r o u n d a c r a c k p e r p e n d i c u l a r to a n i n t e r f a e e , " I n t . J . F r a c . M e c h . , 6, No. 4, 357-365 (1970). G. B. Bodzhi, " F l a t s t a t i c p r o b l e m on a l o a d e d c r a c k ending o n a n i n t e r f a c e , " P r i k l . M e k h . , N o . 4 , 1 9 6 2O2 (1971). T . S. Cook and F . E r d o g a n , " S t r e s s e s in bonded m a t e r i a l s with a c r a c k p e r p e n d i c u l a r to the i n t e r f a c e , " Int. J . Eng. S c i . , 10, No 8, 677-697 (1972). P . O. Hilton and G. C. Sih, "A l a m i n a t e c o m p o s i t e with a c r a c k n o r m a l to t h e i n t e r f a c e , " Int. J. Solids F r a c . , 7, No. 7, 913-930 (1971). V. M. F i n k e l ' , Yu. A. B r u s e n t s o v , V. E. S e r e d a , and Yu. I. T y a p i n , " I n t e r a c t i o n of a f a s t c r a c k with s t r e s s w a v e s , " F i z . T v e r d . T e l a , No. 2, 125-128 (1974). G. R. Irwin, "The d y n a m i c s t r e s s d i s t r i b u t i o n s u r r o u n d i n g a running c r a c k , " P r o c . Soc. E x p t l . S t r e s s A n a l . , 1_66, No. 1, 121-128 (1958). W. B. B r s d l y and A. S. Koboyashi, "An i n v e s t i g a t i o n of p r o p a g a t i n g c r a c k by d y n a m i c s p h o t o e l a s t i c i t y , " E x p e r . M e c h . , 3, No. 10, 106-113 (1970).

1429