POLYMER
MECHANIC
S
469
THE ROLE OF DIFFUSION OF AGGRESSIVE MEDIA IN THE FRACTURE OF RUBBER UNDER STRESS Yu. S. Zuev, t. I. G e l ' b e r g , and A. Z. B o r s h c h e v s k a y a Mekhanika P o l i m e r o v , Vol. 3, No. 4, pp. 7 0 8 - 7 1 2 , 1967 UDC 678.43.01:539.4.019:532.72 Formulas expressing the t i m e to rupture of rubbers in aggressive m e d i a were derived in t e r m s of the r a t e of diffusion of these m e d i a and the r a t e of d i s p l a c e m e n t of the l e a d i n g boundary of the cracking zone. The effect of increasing stress (due to a reduction in the l o a d - c a r r y i n g c a p a c i t y of the s p e c i m e n cross section) on the I o n g - t i m e strength (according to the Bailey principle) was t a k e n into account in these c a l c u l a t i o n s . The c a l c u l a t e d and e x p e r i m e n t a l data are in satisfactory a g r e e m e n t when there is a destructive a c t i o n of working m e d i a in the a b s e n c e of overstressing due to c r a c k formation and s w e l l i n g .
It was e s t a b l i s h e d [1] that a g g r e s s i v e m e d i a acting on r u b b e r u n d e r s t r e s s a e e e l e r a t e its r u p t u r e r e g a r d l e s s of w h e t h e r c r a c k i n g does o r does not take place. In e i t h e r e a s e r u p t u r e of the m a t e r i a l is o b v i o u s l y a s s o c i a t e d with the fact that r u b b e r s u r f a c e I a y e r s g r a d u a l l y lose t h e i r l o a d - c a r r y i n g capacity. If e r a c k s a r e f o r m e d i n the s u r f a e e I a y e r s , they c e a s e e a r r y i n g the load, which is then s p r e a d o v e r the r e d u e e d c r o s s s e c t i o n of the u n d a m a g e d s p e c i m e n [2]. If t h e r e i s no e r a c k f o r m a t i o n but only d e s t r u c t i o n o r s w e l l i n g of the r u b b e r , it m a y also be a s s u m e d that the affected s u r f a c e l a y e r s c a r r y no load. In t h e s e c i r c u m s t a n c e s , f r a c t u r e at a e o n s t a n t i n i t i a l s t r e s s and u n d e r the i n f l u e n c e of an a g g r e s s i v e m e d i u m m a y be f o r m a l l y r e g a r d e d as the action of f r a c t u r e in the a b s e n e e of a g g r e s s i v e m e d i a (i. e . , i n air) taking p l a c e at a c o n tinuously increasing mean nominal stress. The aim of the w o r k d e s e r i b e d i n this a r t i c l e was to c a l c u l a t e the t i m e to r u p t u r e (r a) of s t r e s s e d r u b b e r s p e e i m e n s i n an a g g r e s s i v e m e d i u m and then to c o m p a r e the r e s u l t s with e x p e r i m e n t a l data to r e v e a l any s i n g u l a r i t i e s of f r a c t u r e i n a g g r e s s i v e m e d i a and to e s t i m a t e q u a n t i t a t i v e l y the r o l e of n o n 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 , whose effect should be m o s t p r o n o u n c e d i n the p r e s e n c e of c r a c k s . The w o r k i n v o l v e d the following: 1) c a l c u l a t i o n of the r a t e of d e c r e a s e of the effective s p e c i m e n c r o s s s e c t i o n f r o m data on the r a t e of d i f f u s i o n of the a g g r e s s i v e m e d i u m p r o d u c i n g d e s t r u e t i o n or s w e l I i n g , i . e . , d e t e r m i n a t i o n of the t i m e d e p e n d e n c e of r e a l s t r e s s ; 2) e x p e r i m e n t a l d e t e r m i n a t i o n of the s t r e s s d e p e n d e n c e of the t i m e to r u p t u r e of r u b b e r i n a i r ; 3) a p p l i e a t i o n of the B a i l e y p r i n e i p l e [3], a e c o r d i n g to which a s p e c i m e n b r e a k s when the s u m of r e l a t i v e f r a c t u r e s b e c o m e s equal to 1.0. E x p e r i m e n t a l . T e s t s w e r e c a r r i e d out on r u b b e r s whose r u p t u r e i n a g g r e s s i v e m e d i a is d e t e r m i n e d m a i n l y by d e s t r u c t i o n p r o e e s s e s (butyl r u b b e r and S K F - 3 2 i n HNOa) , by c r a e k i n g (SKN-40 i n ozone), o r by s w e l l i n g (butyl r u b b e r i n CHaCOOH). The m e a s u r e m e n t s w e r e m a d e o n a n i n s t r u m e n t with a s p i r a l shaped l e v e r ; the t e s t e o n d i t i o n s w e r e such that r e -
d u c t i o n of the applied load c o m p e n s a t e d only those c h a n g e s i n the s p e c i m e n c r o s s s e c t i o n that w e r e a s s o c i a t e d with e l o n g a t i o n of the s p e c i m e n . T h e s e m e a s u r e m e n t s w e r e a c c o m p a n i e d by m i c r o s c o p i c
0
g? m ln 2
e
6
8
N
Effect of t i m e (t) o n t h e depth of the c r a c k i n g zone (curve 1) and effect of (t) 1/2 on the depth of diffusion of aggressive media (curves 2, 3, and 4). 1 ) S K N - 4 0 in ozone, cr = 50 kgf/cm 2, T = 30 ° C; 2) butyl r u b b e r i n CH3COOH , cr = 200 kgf/cm 3, T = 50 ° C; 3 ) b u tyl r u b b e r i n HNO3, cr = = 200 k g f / c m 2, T ---- 50 ° C; 4) S K F - 3 2 in HNO3, cr = = 40 kgf/cm 2, T = 40 ° C . d e t e r m i n a t i o n of the depth of p e n e t r a t i o n of a given liquid a g g r e s s i v e m e d i u m into the s p e c i m e n (the e x t e n t of which was i n d i c a t e d by a change i n c o l o r of the d e g e n e r a t e d l a y e r [4]); f o r s p e c i m e n s t e s t e d i n ozone, the depth of the edge c r a c k i n g zone was m e a s u r e d . When the diffusion of an a g g r e s s i v e m e d i u m led to s w e l l i n g , the t h i c k n e s s ~ of the diffused l a y e r on each side of the s p e c i m e n was c a l c u l a t e d f r o m the f o r m u l a ~ = e + c / 2 - d / 2 , w h e r e c i s the i n i t i a l s p e c i m e n t h i c k n e s s , d is the s p e c i m e n t h i c k n e s s a f t e r s w e l l i n g , and e is the e x p e r i m e n t a l l y d e t e r m i n e d t h i c k n e s s of the diffused and s w o l l e n l a y e r on one side of the s p e c i m e n . If no s w e l l i n g took p l a c e , c = d and, c o n s e q u e n t l y , ~ = e. The s t r e s s d e p e n d e n c e (at c o n s t a n t s t r e s s ) of the t i m e to r u p t u r e of r u b b e r in a i r is d e s c r i b e d by [4] • = B ~ -~ . . . .
(1)
F o r butyl r u b b e r , S K F - 3 2 , and SKN-40 we d e t e r m i n e d p a r a m e t e r s n and r (for a g i v e n or) i n a i r (see the table); t h e s e p a r a m e t e r s a r e r e q u i r e d f o r c a l c u l a t i n g ~'a i n a g g r e s s i v e m e d i a . Data on the k i n e t i c s of the d i f f u s i o n of a g g r e s s i v e m e d i a ( n i t r i c and a c e t i c acids) into butyl r u b b e r and S K F - 3 2 s p e c i m e n s a r e
4 70
ME KHANIKA PO LIMEROV
r e p r o d u c e d in the f i g u r e , which also shows data on the k i n e t i c s of the p r o p a g a t i o n of the c r a c k i n g zone f r o n t in SKN-40 s p e c i m e n s u n d e r the i n f l u e n c e of ozone. The diffusion c u r v e s a r e s a t i s f a c t o r i l y d e s c r i b e d by the known e q u a t i o n [5]
notes the rnaximum t i m e of action of v a r i a b l e s t r e s s c o r r e s p o n d i n g to the t i m e to r u p t u r e r) f r o m the B a i ley equation: trup
2Dat
~ =
(3 ................
(2)
where D is the diffusion coefficient, oz is the c o n c e n t r a t i o n of the m e d i u m , fl d e n o t e s the d e n s i t y of the r e a c t i o n c e n t e r s , and t is t i m e . In our e x p e r i m e n t s (as follows f r o m the figure) the t e r m 2Dd/fi = c o n s t = = k, i . e . , ~:=kt . . . .
(3)
The k i n e t i c s c u r v e r e p r e s e n t i n g the change in the depth of the c r a c k i n g zone ~' m a y be d e s c r i b e d by
The r e s u l t s o b t a i n e d w e r e u s e d i n the c a l c u l a t i o n of Ta. C a l c u l a t i o n s . The v a l u e of Ta for a s t r i p s p e c i m e n (on the a s s u m p t i o n that the p o r t i o n of the m a t e r i a l into which the aggressive medium has permeated has zero strength) was computed as follows. Starting from the rate of decrease in the specimen cross section [determined with the aid of Eqs. (3) and (4)], we found the time dependence of the real stress; this relation and Eq. (i) were then substituted into the Bailey equation, whose solution gave 7 a. Denoting the initial cross sectional area of the specimen by f0, the applied load by P0, and the stress by fro, we have, in the absence of aggressive media, Po
(5) - . .
trap
or°" ] f B[1 - (1 +v) --ff-
trup
(12) 0
S u b s t i t u t i n g v a I u e s of ~ f r o m (3), we have trup
f
B
t+n(l+y)
0
----7-J
~ ....
(7)
C o m p l e t e p e n e t r a t i o n of a s p e c i m e n by a diffusing a g g r e s s i v e m e d i u m will take place at ~ = b / 2 . We c o n s i d e r e d c a s e s of s p e c i m e n s f r a c t u r i n g long b e f o r e this m o m e n t , i . e . , when ~ << b / 2 , so that the t e r m 4~ z m a y be n e g l e c t e d . In t h e s e c i r c u m s t a n c e s Eq. (7) becomes f(~) = f o - 2 ( a + b ) ~
....
(8)
Since the d i f f u s i o n of an a g g r e s s i v e m e d i u m is a f u n c t i o n of t i m e , the s t r e s s cr i n a s p e c i m e n i n c r e a s e s with t i m e due to a r e d u c t i o n i n the s p e c i m e n c r o s s s e c t i o n a l a r e a as follows: o
P0 f(~)
dt=l.
(13)
If ~' f r o m (4) i s s u b s t i t u t e d , we o b t a i n trup
ao f B
0
2k't [l+n(1 +y}"-E"
] dt =
l'
(14)
I n t e g r a t i n g Eqs. (13) and (14), we c a l c u l a t e t r u p f r o m data on the r a t e of diffusion of an a g g r e s s i v e m e d i u m as follows: trup + , l + y , n 3 b
rup
B oo"
(15)
w h e r e r 0 i s the t i m e to r u p t u r e in a i r at a 0 = c o n s t ; for c a l c u l a t i n g t r u p f r o m data on the growth r a t e of the c r a c k i n g zone, we have
where a and b denote, respectively, the specimen width and thickness (i. e. , a > b). After an aggressive medium has diffused to a depth ~ we have f(~) = (a--2~) ( b - 2 ~ ) = a b - 2 ( a 4 - b ) ~ + 4 ~
(11)
w h e r e 7 = b / a . The i n t e g r a n d i s expanded into a s e r i e s and, n e g l e c t i n g t e r m s c o n t a i n i n g h i g h e r p o w e r s of ~, we o b t a i n
4 k ~/e (6)
dz=t . . . ,
0
and fo=ab,
(10) ....
As shown in the e x p e r i m e n t a l s e c t i o n , Tic-(t)] is d e s c r i b e d by Eq. (1). After s u b s t i t u t i n g (1), (6), (8), and (9) into Eq. (10), we obtain
(4)
~'=Ut ....
~0~-~0
l
dt
T[o(t)]
(9)
K the s t r e s s v a r y i n g with t i m e is t a k e n into a c c o u n t , it is p o s s i b l e to d e t e r m i n e r a = t r u p (where t r u p d e -
truv@ (l+y)n--fft2rup= oBon=to . . . .
(16)
E q u a t i o n (15) can be s o l v e d by a n u m e r i c a l method, and f r o m Eq. (16) we o b t a i n
tr"P=
V
1 "~o " ~ T-I A
t A
....
(17)
w h e r e A = ((1 + y)n) k ' / b . Sample c a l c u l a t i o n . The diffusion of a 30% HNO,t s o l u t i o n in a butyl r u b b e r s p e c i m e n was s t u d i e d u n d e r following c o n d i t i o n s : T = 50° C, cr = 200 k g f / c m 2, a = 0.3 cm, b = 3 . 7 . 10 -2 cm, y = 0.12, ~ = 9 - 10 -~ cm, and t = 1.8- l 0 s sec. U n d e r the s a m e c o n d i t i o n s i n a i r , n = 2.3 a n d % = 6. 1 0 3 s e c . F r o m Eq. (2) we have 2Da =k=~2=
l~
t
81 • 10-6= 4 . 5 . 1 0 ~8; 1,8.103
POLYMER
MECHANICS
471 Comparison of Experimental and Calculated Data on the Time to Rupture (ra) of Rubbers in Aggressive Media & Rubber-based material
Medium
Concentration, %
Stress ] tt
E~
1:o' sec
em 2 SKF-32 SKN-40 Butyl rubber Butyl rubber
HNO~
03
I
54
0.0005vol. CH~COOH / 30 HN03 l 30
substituting values of k, 7, and n into (15), we obtain 4 ]/4.5.10 -8. trup+ (1+0.12)2.3" 3 3.7- l0 -2 tru~/~=6" 103 or
trup+ 0.019t3/2rup= 6, I 0 ~ , w h e r e -ca = t r u p ~ 3 • 10 ~ s e c . The r e s u l t s of t h e s e and o t h e r s i m i l a r c a l c u l a t i o n s a r e shown in the t a b l e , w h e r e e x p e r i m e n t a l d a t a a r e also reproduced for comparison. In the c a s e of a d e s t r u c t i v e a c t i o n of the m e d i u m (butyl r u b b e r in HNOa) , when no c r a c k f o r m a t i o n t a k e s p l a c e , the r u p t u r e i s due to the a c t i o n of the m e a n n o m i n a l s t r e s s , and c a l c u l a t e d d a t a a r e in g o o d agreement with experimental results. F o r i n t e n s e s w e l l i n g t h e r e i s a c e r t a i n d e g r e e of d i s s i p a t i o n of l o c a l o v e r s t r e s s e s , and the a c t u a l r a t e of f r a c t u r e i s s l o w e r t h a n that o b t a i n e d by c a l c u l a t i o n s (butyl r u b b e r in a c e t i c acid). W h e n c r a c k i n g t a k e s p l a c e , the d e c i s i v e f a c t o r in the r a t e of f r a c t u r e i s not the r a t e of d i s p l a c e m e n t of the b o u n d a r y of the c r a c k i n g zone, i . e . , the i n c r e a s e in the m e a n n o m i n a l s t r e s s , but o v e r s t r e s s e s in t h e c r a c k tip r e g i o n s . T h e s e o v e r s t r e s s e s i n c r e a s e the r a t e of f r a c t u r e (i. e . , r e d u c e the t i m e to r u p t u r e ) by s e v e r a l o r d e r s of m a g n i t u d e (SKN-40 in ozone). The f a c t that 1-a f o r S K F - 3 2 in HNO~ w a s s m a l l e r than the c a l c u l a t e d v a l u e m a y be an i n d i r e c t i n d i c a t i o n that v i s u a l l y u n d e t e c t a b l e d e f e c t s a r e f o r m e d in t h i s m a terial tested under these conditions.
60 30 45 50
ga, f~C ] Exp, I Calc.
,0 18 I,5,o 5 Lo06],o2 , 39,o, 50 25 3,2.1o,oLolli9.6.11o ,oo
200 2 . 3 200 2,3
6.103 008'48,10212,5 IOs 6 - lOS t 0.12i 2 4 . I0~ [ 3 " 1 0 3
f
in r u b b e r s p e c i m e n s (which affect the l o n g - t i m e s t r e n g t h of t h e s e m a t e r i a l s ) w e r e u s e d to c a l c u l a t e t h e i r t i m e to r u p t u r e in v a r i o u s a g g r e s s i v e m e d i a . The c a l c u l a t e d v a l u e s a g r e e d with e x p e r i m e n t a l d a t a when s p e c i m e n s w e r e t e s t e d in m e d i a that have a d e s t r u c t i v e a c t i o n without p r o d u c i n g c r a c k s o r n o t i c e a b l e swelling. 2. The p a r t p l a y e d by c r a c k s (which a c c e l e r a t e the f r a c t u r e of r u b b e r s p e c i m e n s by s e v e r a l o r d e r s of m a g n i t u d e due to s t r e s s c o n c e n t r a t i o n in the c r a c k tip r e g i o n s ) and by s w e l l i n g (which d e l a y s f r a c t u r e due to the d i s s i p a t i o n of s t r e s s ) was e s t i m a t e d q u a n titatively. REFERENCES I. G. M. Bartenev and Yu. S. Zuev, Strength and F r a c t u r e of H i g h - P o l y m e r M a t e r i a l s [in R u s s i a n ] , M o s c o w - L e n i n g r a d , 1964. 2. Yu. S. Zuev and S. I. P r a v e d n i k o v a , ZhFKh, 32, 7, 1958. 3. Yu. S. Zuev, G. M. B a r t e n e v , and N. I. K i r s h e n s h t e i n , Kauchuk i r e z i n a , 9, 1964. 4. G. M. B a r t e n e v and L. S. B r y u k h a n o v a , Z h T F , 28, 287, 1958. 5. N. S. T i k h o m i r o v a , K. I. Z e r n o v a , and V. N. K o t r e l e v , P l a s t . m a s s y , 12, 40, 1962. 6. E. H. A n d r e w s and M. B r a d e n , J. P o l y m . S c i . , 55, 787, 1961.
CONCLUSIONS 1. D a t a on the r a t e of d i f f u s i o n of a g g r e s s i v e m e d i a and on the r a t e of d i s p l a c e m e n t of the c r a c k i n g zone
24 O c t o b e r 1966
Moscow