13 Theory of Rubber Abrasion by a Line Contact S. W. Zhang
1
Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, MI 48109
On the basis of the investigation of rubber abrasion by small-scale tearing on a line contact, a theory has been presented. According to the physical processes of rubber abrasion and its mathematic description proposed, a general process of rubber abrasion can be described by using a simple wear curve. A wear equation obtained describes the basic correlation among the material property, running condition and wear characteristic. It is pointed out that the number of revolution to trasform the wear state from unsteady to steady can be considered as a criterion of rubber abrasion. An approximate relationship between the state criterion and tensile rupture ratio has been obtained. These theoretical results agree qualitatively with experimental observation. Rubber a b r a s i o n i s an e x t r e m e l y complex p r o c e s s . R e c e n t l y , a s i m p l i f i e d t e s t method has been d e v e l o p e d , by c h o o s i n g a l i n e c o n t a c t ( i n p r a c t i c e a r a z o r b l a d e ) as t h e a b r a d i n g a p p a r a t u s ( 1 ) . A number o f e x p e r i m e n t a l s t u d i e s about t h i s mode o f wear have been c a r r i e d out ( 1 - 4 ) . Champ, S o u t h e r n , and Thomas proposed a s i m p l e t h e o r y r e l a t i n g the rubber a b r a s i o n i n s t e a d y s t a t e t o t h e c r a c k growth p r o p e r t y o f rubber ( 1_,2) . However, a complete e x p l a n a t i o n o f t h e f o r m a t i o n and development o f t h e a b r a s i o n p a t t e r n has n o t y e t been g i v e n . I n t h i s p a p e r , a t h e o r y c o n c e r n i n g t h e mechanism o f rubber a b r a s i o n i s presented. I t i s s u p p o r t e d by some e x p e r i m e n t a l o b s e r v a t i o n i n d i f f e r e n t s t a g e s o f a b r a s i o n . These e x p e r i m e n t a l r e s u l t s were o b t a i n e d f o r s e v e r a l e l a s t o m e r s , u s i n g a b l a d e abrader as d e s c r i b e d by Southern and Thomas (2) and a s c a n n i n g e l e c t r o n microscope Ç5,_6). P h y s i c a l P r o c e s s e s o f Rubber A b r a s i o n As a b l a d e moves r e l a t i v e t o t h e r u b b e r s u r f a c e , under t h e a c t i o n o f a n o r m a l f o r c e , a b r a s i v e wear o f rubber o c c u r s . A c c o r d i n g t o some of the e x p e r i m e n t a l o b s e r v a t i o n s and t h e o r e t i c a l a n a l y s e s , t h e p r o cesses o f d r y ( p a r t i c u l a t e ) a b r a s i v e wear might be c o n s i d e r e d t o be 'Current address: Department of Mechanical Engineering, East China Petroleum Institute, Dongying, Shangdong, China
0097-6156/85/0287-0189S06.00/0 © 1985 American Chemical Society
190
POLYMER WEAR AND
ITS
CONTROL
comprised o f t h e f o l l o w i n g p e r i o d i c a l l y r e p e a t i n g p r o c e s s e s . (a) The b l a d e deforms and t e a r s the r u b b e r s u r f a c e under the a c t i o n o f the normal f o r c e and t h e f r i c t i o n a l f o r c e . Consequently, c r a c k growth appears and a t o r n tongue of rubber forms ( F i g u r e l,a). (b) As the b l a d e moves, the tongue of rubber i s bent backwards; t h e r e f o r e , a p a r t o f t h e rubber s u r f a c e i s p r o t e c t e d from t h e s c r a p i n g a c t i o n o f b l a d e a t the r e a r . T h i s p r o c e s s i s s i m i l a r t o u s i n g a n e e d l e o r a s m a l l hemisphere of 1 mm i n diameter t o s c r a t c h the rubber s u r f a c e , as d e s c r i b e d by Schallamach (7-J9) · (c) Under t h e t e n s i o n caused by the f r i c t i o n a l f o r c e , the t i p of tongue i s r u p t u r e d . Then, the r e m a i n i n g p a r t of the tongue i s released (Figure l , b ) . (d) Along w i t h the g r a d u a l c r a c k growth and t h e r u p t u r e of the t o n gue t i p , r i d g e s a r e formed, l e a d i n g t o a sawtooth p r o f i l e on the rubber s u r f a c e . Such a p h o t o g r a p h can be found i n Reference 2. As seen, the a b r a s i o n p r o c e s s e s o f rubber are i n v o l v e d not o n l y t h e c r a c k growth v i a m e c h a n i c a l f a t i g u e , but a l s o t h e r u p t u r e of t o n gue t i p r e s u l t i n g from t e n s i l e s t r e s s . Moreover, t h e l a t t e r i s the p r o x i m a t e cause of the l o s s o f m a t e r i a l . Mathematic D e s c r i p t i o n of the P r o c e s s e s o f Rubber A b r a s i o n A p h y s i c a l model of rubber a b r a s i o n i n unsteady s t a t e and the c o r r e s ponding p r o c e s s e s o f tongue r u p t u r e d a r e shown i n F i g u r e s 2 and 3 r e s p e c t i v e l y . As seen from F i g u r e 3, tanO = — ± Δχ
χ
= -— Δχ 4Δχ ί
= ... = 2
, ΣΔχ
(1)
±
then Ay
-ΓΔχ^ηθ
±
(i=l,2,...)
(2)
For a g i v e n m a t e r i a l and c o n s t a n t f r i c t i o n a l f o r c e , the a n g l e θ can be c o n s i d e r e d as a c o n s t a n t a p p r o x i m a t e l y . Thus, ΑΎ±&ζΣΑχ
±
(3)
However, i n l i g h t o f the p h y s i c a l p r o c e s s e s s t a t e d above, the t h i c k ness o f tongue i s unchangeable once steady s t a t e has been reached. At t h i s p o i n t , assuming t h a t Ν r e p r e s e n t s the number of r e v o l u t i o n s t o a t t a i n s t e a d y s t a t e , the r u p t u r e t h i c k n e s s of tongue can be dedu ced from F i g u r e 2 : Ay
N
- N.rSinO
(4)
A p p a r e n t l y , the magnitude of t h e r u p t u r e t h i c k n e s s of tongue has a maximum w h i c h i s g i v e n by
I n o r d e r t o d e s c r i b e the p r o c e s s o f the r u p t u r e of tongue t i p p r e c i s e l y , a t e c h n i c a l t e r m , t e n s i l e r u p t u r e r a t i o , i s i n t r o d u c e d and d e f i n e d as a r a t i o of t h e r u p t u r e l e n g t h of the tongue t i p t o the o r i g i n a l l e n g t h o f the tongue ( F i g u r e 2 ) . Thus,
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Rubber Abrasion by a Line Contact
Figure 1. Formation of ridge from f i l l e d NBR,W = 1.5kJ/in (10Ox): (top) tongue formed; (bottom) tongue ruptured.
P O L Y M E R W E A R A N D ITS
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6
K
Δ
/1.
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(6)
As seen from E q u a t i o n s ( 3 ) , (5) and ( 6 ) , t h e t e n s i l e r u p t u r e r a t i o can be r o u g h l y c o n s i d e r e d t o be d i r e c t l y p r o p o r t i o n a l t o t h e f r i c t i o n a l f o r c e and i n v e r s e l y p r o p o r t i o n a l t o t h e t e n s i l e s t r e n g t h : t ocl/%
(7)
K
T h e r e f o r e , i t i s supposed t o be a c o n s t a n t f o r a g i v e n m a t e r i a l and c o n s t a n t f r i c t i o n a l f o r c e . Under t h e circumtances,Δχ^ i s d i r e c t l y i n p r o p o r t i o n t o 1^; t h e l e n g t h 1^, which m a i n l y r e s u l t s from c r a c k growth, i n c r e a s e s w i t h t h e number o f p a s s e s o f t h e b l a d e o n l y . Moreover, t h e c r a c k growth p e r r e v o l u t i o n ( p a s s ) , r , i s n e a r l y a constant. Hence, t h e p r o c e s s e s o f r u p t u r e o f t h e tongue t i p can be d e s c r i b e d as f o l l o w s : Since, i-1 l i = i r - ]ΓΔχ.£, Δ χ ο = 0, i=l t h e n , a f t e r t h e 1 s t pass ( i = 1 ) , Δχι = ι ε 1
= r £ ,
κ
K
a f t e r t h e 2nd p a s s ( i = 2 ) , 2 Δχ
2
= 1 £ 2
= ( 2r -Δχ )
K
1
t
=
K
2r £
-
K
t,
r
K
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l
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r
- [ i r - ( Ax±
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-
[(i-l)
+
+ ... + Δ χ _ ) ] £ 1
tl-l>ji-2)
]
r
&
1
κ
2
+ [(i,Tl)(i-2> + ( j - l ) ( i - 2 ) ( i - 3 ) ] 2! 3!
r £
3
( 8 )
K
Thus, Ax
±
= (i)
( i - 1,2,3,...)
(9)
A c c o r d i n g t o t h e a n a l y s i s above, t h e sum o f r u p t u r e tongue a f t e r Ν r e v o l u t i o n s w i l l be g i v e n by
1
=
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+
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+
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= Nr£ f (N, £ ) , K
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0
κ
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K
ε ) - i Ν steady s t a t e Oh t h e b a s i s o f t h e p h y s i c a l p r o c e s s e s and mathematic d e s c r i p t i o n of rubber a b r a s i o n s t a t e d above, a t h e o r e t i c a l r e l a t i o n s h i p between Ν and £ can be o b t a i n e d t h r o u g h t h e use of n u m e r i c a l c a l c u l a t i o n under t h e c o n d i t i o n of Δ χ ^ = r , as shown i n F i g u r e 11. More o v e r , i t can be a p p r o x i m a t e l y r e p r e s e n t e d as f o l l o w s K
Ν = 1.28£
D e 9
(30)
E v i d e n t l y , the h i g h e r t h e t e n s i l e r u p t u r e r a t i o , t h e lower t h e v a l u e o f t h e s t a t e c r i t e r i o n i s . Thus, t h i s s t a t e c r i t e r i o n can be a p p l i e d t o e s t i m a t e t h e wear c h a r a c t e r i s t i c s of d i f f e r e n t e l a s t o m e r s under s i m i l a r r u n n i n g c o n d i t i o n s . Conclusions The p h y s i c a l p r o c e s s d u r i n g dry a b r a s i v e wear i s a g r a d u a l t e a r i n g l e a d i n g t o n o t o n l y the c r a c k growth but the r u p t u r e of tongue t i p as w e l l . I t can be d e s c r i b e d m a t h e m a t i c a l l y t h r o u g h the medium of i n t r o d u c i n g a concept o f t e n s i l e r u p t u r e r a t i o . A wear c u r v e can be used t o d e s c r i b e t h e g e n e r a l p r o c e s s of rubber a b r a s i o n , i n w h i c h i t i s d i v i d e d i n t o t h r e e r e g i o n s : unsteady, s t e a d y and damage s t a g e . The wear e q u a t i o n o f r u b b e r a b r a s i o n 4i s t e a d y s t a t e r e v e a l s t h e b a s i c c o r r e l a t i o n among the m a t e r i a l p r o p e r t y , r u n n i n g c o n d i t i o n and wear c h a r a c t e r i s t i c . The wear r a t e i n c r e a s e s w i t h an i n c r e a s e i n the f r i c t i o n a l f o r c e , however, i t i s i n v e r s e l y p r o p o r t i o n a l t o the t e n s i l e s t r e n g t h . The c h a r a c t e r i s t i c f u n c t i o n , ^ ( N , £ ) , i s a c h a r a c t e r i z i n g f a c t o r o f r u b b e r a b r a s i o n i n unsteady s t a t e . I t s v a l u e i n c r e a s e s w i t h an i n c r e a s e i n t h e number o f r e v o l u t i o n s and t e n s i l e r u p t u r e r a t i o . However, i t approaches u n i t y as a l i m i t i n the u n s t e a d y - s t a t e p r o c e s s of wear. Hence, a s t e a d y s t a t e i s reached i f once Ij (N,6jç) ~ 1· K
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Rubber Abrasion by a Line Contact
U ( N,6 ) k
F i g u r e 10. C h a r a c t e r i s t i c f u n c t i o n p l o t t e d a g a i n s t t e n s i l e r u p t u r e r a t i o : (A) i = 20 r e v ; (B) i = 40 r e v ; (C) i = 60 r e v .
F i g u r e 11. S t a t e c r i t e r i o n o f rubber a b r a s i o n p l o t t e d a g a i n s t t e n s i l e r u p t u r e r a t i o ( r = 1.0).
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P O L Y M E R W E A R A N D ITS C O N T R O L
The number o f r e v o l u t i o n s t r a n s f o r m e d t h e wear s t a t e from un s t e a d y t o s t e a d y can be regarded as a s t a t e c r i t e r i o n o f rubber a b r a s i o n t o e s t i m a t e t h e wear c h a r a c t e r i s t i c s o f r u b b e r under i d e n t i c a l r u n n i n g c o n d i t i o n s . I t was found t o be p r o p o r t i o n a l t o a nega t i v e exponent o f t h e t e n s i l e r u p t u r e r a t i o . I t i s c o n c l u d e d t h a t t h e t h e o r y proposed can be a p p l i e d t o c l a r i f y t h e phenomena and p r o c e s s e s o f rubber a b r a s i o n i n d i f f e r e n t s t a g e s o f wear by a l i n e c o n t a c t . Ackn owled gmen t s The a u t h o r i s i n d e b t e d t o Mr. J . A. H a r t w e l l f o r e x p e r i m e n t a l a s s i s t a n c e , t o Mr. II. F. B a i f o r computer a s s i s t a n c e , and t o P r o f e s s o r A. N. Gent f o r h i s encouragement and a d v i c e a t the I n s t i t u t e o f Polymer S c i e n c e , The U n i v e r s i t y o f Akron. A l s o , t h e a u t h o r i s g r a t e f u l t o P r o f e s s o r K. C. Ludema f o r h i s much h e l p f u l comments on t h e d r a f t m a n u s c r i p t a t The U n i v e r s i t y o f M i c h i g a n . Legend o f Symbols A B_ F Κ Ν
l i n e a r wear r a t e crack-growth constant f r i c t i o n a l force per width coefficient s t a t e c r i t e r i o n , t h e number o f r e v o l u t i o n s t o t r a n s f o r m t h e wear s t a t e from unsteady t o s t e a d y R r a d i u s o f t e s t i n g rubber wheel S_ spacing of ridges S average s p a c i n g o f r i d g e s V|j sum o f abraded volume o f a tongue a f t e r Ν r e v o l u t i o n s W f r i c t i o n a l work c p r o p o r t i o n a l i t y constant h w i d t h o f tongue ( t e s t i n g r u b b e r wheel) i number o f r e v o l u t i o n s ( p a s s e s ) 1 l e n g t h o f tongue r c r a c k growth p e r r e v o l u t i o n t time ΔΧ r u p t u r e l e n g t h o f tongue Ay r u p t u r e t h i c k n e s s o f tongue
