FRICTION I. THE

IN AREA S.

OF

POLYMER CONTACT

MATERIALS WHEN

ONLY

NORMAL

LOADS

ACT

UDC 6'78:539.62

]3. Ainbinder

The author calculates the area of contact between two rough flat surfaces on the supposition that the normal law of distribution holds for the heights of the roughnesses. He determines the nominal stress at which the deformations of the roughnesses can be regarded as independent. He shows that in real cases of contact, the deformation of the roughnesses of polymer surfaces can be regarded as elastic.

In dry or boundary friction, the frictional forces are determined by the resistance to rupture of adhesion bonds arising on parts of surfaces in pressure contact, by the deformation resistance of the roughnesses in pressure interaction without adhesion, and finally, by the deformation resistance of the intermediate medium. As a rule, for surfaces of the cleanness used in bearings, the second and third components of the frictional force are much less than the first. The rupture resistance of the adhesion bonds is proportional to the area. of contact over which they arise; consequently, to calculate the frictional forces we must know the area of pressure contact between the rubbing bodies. In general, for the same normal load, the area. of contact in the immobile state will be different from that in motion [I, 2]. If the deformations of the surface roughnesses are elastic, the difference between the areas of contact will be small [i]. In polymer materials, as we shall show, the surface roughnesses are mainly deformed elastically, and thus the area. of contact can be determined on tl~.e supposition that only normal loads act. At present, two methods are used to calculate the areas of contact. The first is based on the use of the so-called bearing surface, which is constructed from experimental profilograms of the friction surfaces. Later, several hypotheses were advanced on the form of the bearing surface curve and the character of the deformation, and a formula was derived for the area. of contact for a given load. This method was developed by Kragel'skii [3] and Demkin [4]. The second method of calculation is based on the distribution function of the roughnesses on the contact surface on the hypothesis that the roughnesses are of the same shape. This method of calculation was apparently first suggested by Kragel'skii [5] and'was further developed by Greenwood et al. [6, 7]. The accuracy of both methods of calculation is the same, but the second method leads to clearer and simpler equations, and therefore we shall consider it in more detail. We shall suppose that there is a contact between of the roughnesses on the rough surface have a normal ical segments with a mean radius r.

a smooth and a rough surface. Suppose that the heights distribution, q~(r), and that the roughnesses are spher-

Let us denote by d the distance between the average under a given load. Then the probability that roughnesses surface will be

plane of the rough surface and the surface plane with height y will come into contact with the plane

c~

Prob (au>d) = f (~(b,)dbJ

O)

d

Institute of P o l y m e r Mechanics, Academy of Sciences of the Latvian SSR, Riga. T r a n s l a t e d from Mekhanika P o l i m e r o v , No. 4, pp. 654-660, July-August, 1972. Original articl~ sumbitted March 3, 1971. © 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 ~est 17th Street, New York, N. Y. iO011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

565

L e t the t o t a l n u m b e r of r o u g h n e s s e s on the r o u g h s u r f a c e b e N = m0Snom w h e r e m 0 i s the n u m b e r of r o u g h n e s s e s p e r u n i t a r e a of c o n t a c t and Snom i s the n o m i n a l a r e a of c o n t a c t . T h e n the e x p e c t e d v a l u e of the n u m b e r of r o u g h n e s s e s c o m i n g into c o n t a c t w i l l b e oo

(2) d

We shall suppose that the contact is elastic and that the relation between the area of contact and the forces at the contact due to approach for a single spherical roughness is represented by the functions S=fl(a);

P=/2(cz).

(3)

The f o r m of f u n c t i o n s (3) d e p e n d s on the c o n d i t i o n s at the c o n t a c t s u r f a c e . The w e l l - k n o w n f o r m u l a s of H e r t z w e r e d e r i v e d on the a s s u m p t i o n t h a t the f r i c t i o n at the c o n t a c t s u r f a c e i s e q u a l to z e r o . A p p a r e n t l y , in c a s e s of d r y f r i c t i o n it i s m o r e c o r r e c t to s u p p o s e t h a t the c o n t a c t s u r f a c e s a r e in c o m p l e t e c o h e s i o n . On t h i s a s s u m p t i o n , the c o n t a c t p r o b l e m w a s d i s c u s s e d by M a s s a k o v s k i i [8]. F o r the c a s e of p e n e t r a t i o n of a r i g i d s p h e r e into an e l a s t i c s u r f a c e he d e r i v e d the f o r m u l a 4 in ( 3 _ 4v) ~(2[ .( - ~2~t+)~ ) '2 -

1 ] a~

p - - _ _

(4)

3 (3~ + 1) d0r

w h e r e u i s the P o i s s o n r a t i o , / ~ and X a r e the Lam~ c o n s t a n t s , d o i s a w e a k function of the P o i s s o n r a t i o , d0a v = 0.245, a i s the r a d i u s of the a r e a of c o n t a c t , and r i s the r a d i u s of the s p h e r e . The a n a l o g o u s f o r m u l a of H e r t z is

P=

4Ea3

3(1-v~)r

(5)

C a l c u l a t i o n s by E q s . (4) and (5) g a v e the f o l l o w i n g r e s u l t s [a 4 and a 5 a r e the r a d i i of the a r e a of c o n t a c t a c c o r d i n g to E q s . (4) and (5)]: if u = 0.3, a 5 / a 4 = 1.14; if u = 0.35, a J a 4 = 1.10; if u = 0.4, a J a 4 = 1.07. T h u s f o r p o l y m e r m a t e r i a l s in w h i c h the P o i s s o n r a t i o i s g r e a t e r than 0.35, the e r r o r due to n e g l e c t of the f r i c t i o n a l f o r c e s in c a l c u l a t i n g the a r e a of c o n t a c t w i l l not e x c e e d 10%. T h e r e f o r e we s h a l l in f u t u r e u s e the o r d i n a r y H e r t z f o r m u l a s [9], w h i c h c a n be w r i t t e n a s

h (o~)=nr~;

(6)

f~ (o0 = 4 E~rV~o~v-,,

w h e r e 1 / E ' = ( 1 - v l 2 ) / E 1 + ( 1 - u 2 2 ) / E 2 , w h e r e ul, Et, u2, and E 2 a r e the d e f o r m a t i o n c h a r a c t e r i s t i c s s u r f a c e s in c o n t a c t , r i s the r a d i u s of a s p h e r i c a l r o u g h n e s s , and o~ i s the d e g r e e of a p p r o a c h .

of the

U s i n g Eq. (6), l e t us w r i t e down an e x p r e s s i o n f o r the e x p e c t e d v a l u e of the a r e a of a s i n g l e c o n t a c t and the f o r c e on a s i n g l e c o n t a c t , a s f o l l o w s : co

M {S~}

~r f (y-d)~(y)@ d oo

;

(7)

f q~(,y)@ d

M{Ri}_4

f (j-d)'/,~(tj)@

Ur'/: ~

f ~(y)@

d

566

(8)

Then the total a r e a of c o n t a c t f o r the given d e g r e e of a p p r o a c h and the total load will be equal to co

(9) d

ee

Pd=M{h}M{R~} =43 NE'r'/f

d,.

(10)

d

Let us introduce

the notation

F~,(h)

w h e r e o" is the r m s deviation.

f(z-~),,e 2dz; z=±" h=L. ~

1

= V~-~ ~

'

~

'

Then E q s . (9) and (10) b e c o m e

Sh : p

171oSflDIIl~F

(11)

CfF 1 ( h ) "

4 I~= ~ m~,Snomr "d'~F3 2 ([z).

(12)

o

The function Fl(h ) reduces to a tabulated ated by numerical integration. By Eqs. (ii) and (12), the mean

integral and a Laplace

integral.

The function F3/2(h) can be evalu-

areas of contact are Sh n

Savh=--=nra

Fl(h) - -

Fo(h)

(13)

The ratio Fl(h)/F0(h ) i n c r e a s e s as h d e c r e a s e s . Thus the m e a n a r e a of c o n t a c t i n c r e a s e s m o n o t o n i c ally with the d e g r e e of a p p r o a c h . H o w e v e r , in the r e a l i s t i c r a n g e of h, the i n c r e a s e in the c o n t a c t a r e a is small. The m e a n f o r c e is equal to Ph n

4 3

t~lf%lj~/-

F3/, (h) J Fo(h)

~ --_ ~ ,

(14)

The m e a n s t r e s s on the a r e a s of c o n t a c t is P R_R=__4E~(° ),/2 F3/2(h) & = sh 3~ 7 p~(h)

(i5)

The ratio F3/2(h)/F1(h ) v a r i e s o v e r the r a n g e 0.54 <- F3/2(h)/Fl(h ) ~ 1.14. Within the w o r k i n g load rmage it v a r i e s m u c h l e s s , as we shall show l a t e r . T h e r e f o r e e x p r e s s i o n (15) c a n be c a l l e d the e l a s t i c h a r d n e s s of the p o l y m e r m a t e r i a l . In Table 1 we give the v a l u e s of H e f o r v a r i o u s p o l y m e r m a t e r i a l s . The c a l c u l a t i o n s w e r e p e r f o r m e d f o r s u r f a c e s of the sixth and eighth c l a s s e s of c l e a n n e s s f o r t h r e e types of c o n t a c t : i) rigid plane with rough e l a s t i c s u r f a c e ; ii) two r o u g h e l a s t i c s u r f a c e s , f o r which [6] ff = 9 - ~ and iii) an e l a s t i c plane with a rough e l a s t i c s u r f a c e . The value of H e was c a l c u l a t e d f r o m the f o r m u l a

As we s e e f r o m Table 1, the e l a s t i c h a r d n e s s is m u c h l e s s than the o r d i n a w h a r d n e s s of p o l y m e r m a t e r i a I s , and t h e r e f o r e a c a l c u l a t i o n of the a r e a of c o n t a c t f r o m the o r d i n a r y h a r d n e s s [10, 11] wilI i n c u r large errors.

567

TABLE 1 Rigidplane - rough [Elasticplane-rou~t~Two rough elastic "Aasti6surface [elastic surface L suffaees _ _

Material*

6th class l 8th class[ 6th class 8th class 6th class[ 8th class Steel Aluminum Viniplast Polycapmlactam T~fion PE'ND PMMA K-17-2 Polyformaldehyde Polycarbonate Rubber

9,9.103 3,7.102 2,3" 102 1,37.102 1,56.102 5,75" 102 7,40. l02 7,40. l0s 3,70- 102 0,26

7,5.103 2,8.102 1,7.102 1,0- 102 1,2.102 4,4.102 5,6.102 5,6.102 2,9- l0s 0,2

7,7- 103 5,85.103

2,54.102 1,76.102 1,23.102 1,36.102 5,16.102 3,80.102 3,80.102 2,56.102

1,93.102 1,34.102 0,93.102 1,04.102 3,90.102 2,90.102 2,90.102 1,95.102

23,4. lOs 5,8.103 2,34.102 1,35' 102 0,82" 102 0,97.102 3,32. lO2 4,41 • lO2 4,41 • lO2 2,34. lO2 0,16

17,6.103 4,4. IO3 1,76.102 1,02.102 0,61 • 102 0,73" 102 2,48" 102 3,32.102 3,32,102 1,76" 102

0,12

* P o l y m e r m a t e r i a l s p a i r e d with s t e e l .

We s h a l l show that of the t h r e e q u a n t i t i e s c h a r a c t e r i z i n g the s u r f a c e g e o m e t r y , m0, r, and o', two a r e i n d e p e n d e n t . By (11), the r e l a t i v e c o n t a c t a r e a is

Sh Srel = Sno-----m=nmmoFt (h) whenh=0,

Sre 1 = 0 . 5 , F l ( h ) = 0 . 4 .

(17)

Hence 0,4 ro

(18)

In p r a c t i c e it is m o r e c o n v e n i e n t to take the c o n s t a n t s m 0 and o- as i n d e p e n d e n t , as they a r e m o r e e a s i l y d e t e r m i n e d f r o m p r o f i l o g r a m s than the m e a n r a d i u s r. U s i n g Eq. (18), we get Srel = 1.25F1 (h) .

(19)

T h u s the r e l a t i v e c o n t a c t a r e a is a f u n c t i o n of the r e l a t i v e d e g r e e of a p p r o a c h only. Note that the a r e a of c o n t a c t d e p e n d s only w e a k l y on the n o m i n a l c o n t a c t a r e a and the n o m i n a l s t r e s s . T h i s follows f r o m E q . (15), which c a n be r e w r i t t e n as S/~ = 2.35

PhFi (h) O

(20)

~/2

[71 ( .~. ) E3/2(h) H e r e Ph is a g i v e n q u a n t i t y , and the r a t i o Fi(h)/F3/2(h), a s a l r e a d y m e n t i o n e d , d e p e n d s weakly on the d e g r e e of a p p r o a c h . T h u s Sh d e p e n d s m a i n l y on the a p p l i e d f o r c e . TABLE 2

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,6 2,8 3,o

568

Fo

FI

Fa/2

0,5 0,46 0,42 0,38 0,35 0,32 0,28 0,25 0,16 0,I 1 0,08 0,055 0,035

0,4 0,34 0,31 0,26 0,22 0,19 0,14 0,13 0,08 0,06 0,04 0,027 0,015 0,008 0,005 0,003 0,0016 0,0004

0,43 0,4 0,33 0,3 0,24 0,218 0,196 0,16 0,l 0,06 0,045 0,022 0,014 0,0084 0,0042 0,0018 0,0006 0,00023

0,025

0,015 0,005 0,0025 0,0017

We c a n u s e the above f o r m u l a s in o u r c a l c u l a t i o n s as long as o u r i n i t i a l a s s u m p t i o n that the d e f o r m a t i o n s of i n d i v i d u a l r o u g h n e s s e s a r e i n d e p e n d e n t r e m a i n s v a l i d . But at high enough d e g r e e s of a p p r o a c h this a s s u m p t i o n is not j u s t i f i e d . C o n s i d e r a t i o n of the p a t t e r n of the s t a t e of s t r e s s a r o u n d an e l a s tic c o n t a c t [12] r e v e a l s that the s t r e s s e s b e c o m e quite s m a l l at one d i a m e t e r d i s t a n c e f r o m the c e n t e r of the r o u g h n e s s . T h e n we c a n a s s u m e that with an e q u i p r o b a b l e d i s t r i b u t i o n of r o u g h n e s s e s on the c o n t a c t s u r face, the d i s t a n c e b e t w e e n the c e n t e r s of the r o u g h n e s s e s m u s t be at l e a s t two d i a m e t e r s . The r e l a t i v e c o n t a c t a r e a m u s t be l e s s than 0.2. In this c a s e , f r o m Eq. (19) we get F l ( h ) = 0.16 and h = 0.64. P u t t i n g (18) into (12), f o r the n o m i n a l load we get P h / S n o m = 0.53E'(cr/r) 1/2 F3/2(h ). When h = 0.64, F3/2(h ) = 0.19, and c o n s e q u e n t l y P h / S n o m = 0.102E'(o-/r)i/2-< 0.2H e. T h i s v a l u e is about o n e - f i f t h of the e l a s t i c h a r d n e s s of p o l y m e r s [see (16)]. At high n o m i n a l l o a d s the above f o r m u l a s w i l l give an e r r o r which i n c r e a s e s as h b e c o m e s s m a l l e r . In

most cases, the nominal

load during friction of polymers

is less than 0.2H e.

Table 2 lists values of F0, Fi, and F3/2 as functions of h. With the aid of this table and for given values of P, m0, 0~, Snom, and E', we can easily calculate the contact area, the degree of approach, and the mean stress by means of the above formulas. Note that in the overwhelming majority of cases, the relative nominal load is such that h -< 1.6; and when h = 1.6, Ph/Snom = 0.02H e. Here 1.05 - F3/2(h)/Fi(h ) ~ 1.14, and consequently in cases of practical importance we can assume that the elastic hardness is equal[ to H e = 0.45E'(~/r)I/2 (within 2-3%). This was the formula used to calculate the tables. Then the area of contact can be calculated from the formula Sh = P/He, and then from the value of Sre I with the aid of Table 2 we can determine the relative degree of approach h and the other parameters of the contact area. At some value of the relative load, forced highly elastic or plastic deformation of the surfaces in contact may begin. It is known that in the deformation of an individual roughness, plastic deformation arises when the mean pressure on the contact surface is equal to 1.06~ S (where ~S is the yield point) [i]. Yield begins under the contact surface at a distance of about 0.Sa. Thus the zone of yield is surrounded by elastically deformed mate rial. The degree of approach at the onset of yield in the contact of a rigid plane with an elastic rough surface [9] will be equal toe S = 6.25r (l-v2) 2 (~s/E') 2. In the other cases of contact listed in Table I, the degree of approach at the onset of yield will be greater, and so this case can be taken as the nominal one. After attainment of the mean limiting stress, elasticoplastic deformation of the contact begins. However, experiments [13] reveal that the Hertz formula will be usable in practice up to a value of O-av equal to (1.5-2)~S, and developed plastic yield begins at O-av N 3~S. mer

Thus, assuming that the limiting value is (~av = 1.5O's, we get o~/im/r = 14(1-v2) 2 (~s/E) 2. For polymaterials the ratio o-s/E is about 0.05, and, assuming that v = 0.33, we get c~lim/r = 0.032.

For polymer materials, the values of Hmax/r for various surface cleanness classes [14] are as follows (here Hma x is the maximum height of the roughnesses): class 5, 0.025; class 6, 0.0125; class 7, 0.0066; class 8, 0.0033. Thus, even for comparatively rough surface treatment, with complete deformations of the roughnesses we do not attain values of ~/r for which deviations from the elastic contact case are appreciable. Consequently in all real cases the contact can be regarded as elastic. LITERATURE i. 2. 3. 4. 5. 6. 7. 8. 9. i0. II. 12. 13. 14.

CITED

F. Bowden and D. Tabor, Friction and Lubrication of Solids, Oxford University Press. S.B. Ainbinder, Izv. Akad. Nauk SSSR, OTN, No. 6, 172 (1962). I.V. Kragel'skii, Friction and Wear [in Russian], Moscow (1968). I.B. Demkin, Contact of Rough Surfaces [in Russian], Moscow (1970). I.V. Kragel'skii, Izv. Akad. Nauk SSSR, OTN, No. I0, 1621 (1948). Y. Greenwood and Y. Gripp, Applied Mechanics Series E (USA), No. 4, 7 (1964). Y. Greenwood and Y. Williamson, Proc. Roy. Soc., 295_, 300 (1966). V.I. Massakovskii, PM, 27_, 418 (1963). H. Hertz, Journ. J. Reine u. Angew. Mathem., 92_, 150 (1881). K. Shooter and D. Tabor, Proc. Phys. Soc. Ser. B, No. 9, 393 (1957). M. Pascoe and L. Tabor, in: Mechanical Engineering [Russian translation], Moscow (1955), p. 8. A.N. Dinnik, Selected Works [in Russian], Vol. I, Kiev (1952). S.B. Ainbinder and A. Ya. Loginova, Mekh. Polim., No. 6, 995 (1971). V.A. Belyi, A. I. Sviridenok, and M. M. Petrokovets, Mekh. Polim., No. I, 172 (1970).

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