Contact Deformation and Static Friction of Polymers - American

1 Contact Deformation and Static Friction of Polymers Influences of Viscoelasticity and Adhesion Horst Czichos

Downloaded by 80.82.77.83 on November 2, 2017 | http://pubs.acs.org Publication Date: September 12, 1985 | doi: 10.1021/bk-1985-0287.ch001

Bundesanstalt für Materialprüfung, Federal Institute for Materials Research and Testing, Berlin-Dahlem, Federal Republic of Germany

In using a newly developed tribometer which allows the microscopic "in-situ" investigation of i n t e r f a c i a l t r i b o l o g i c a l processes through an optically transparent counterbody, contact deformation and static f r i c t i o n of thermoplastic polymers were studied. Examination of the contact deformation of polymers under load revealed that the contact deformation displacement is mainly influenced by the viscoelastic properties of the polymers. By measuring relaxed moduli and relaxation times the Hertzian theory of contacting bodies could be extended leading to a contact deformation formula which includes viscoelastic effects. For the onset of motion of polymer/polymer sliding pairs the static f r i c t i o n appears to be connected with i n t e r f a c i a l adhesion. When data of the surface energy of the examined polymers were used, the experimentally determined f r i c t i o n values could be related to the adhesion energies of the different p o l ymer/polymer sliding pairs.

A l l t r i b o l o g i c a l p r o c e s s e s , i . e . f r i c t i o n a n d wear p r o c e s s e s o f i n t e r a c t i n g m a t e r i a l s s t a r t w i t h two e l e m e n t a ry steps: (i)

The f o r m a t i o n o f t h e c o n t a c t a r e a u n d e r a g i v e n load.

(ii)

The i n i t i a t i o n o f r e l a t i v e m o t i o n between t h e c o n t a c t i n g b o d i e s by o v e r c o m i n g s t a t i c f r i c tion. 0097-6156/ 85/ 0287-O003S07.00/ 0 © 1985 American Chemical Society

Lee; Polymer Wear and Its Control ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

4

P O L Y M E R W E A R A N D ITS

CONTROL


The b a s i s f o r t h e c h a r a c t e r i z a t i o n o f t h e c o n t a c t b e h a v i o u r o f c u r v e d b o d i e s i s g i v e n by t h e w e l l known c l a s s i c a l Hertzian theory (1). Because t h i s t h e o r y b a s i c a l l y applies only to p e r f e c t l y e l a s t i c materials with i d e a l smooth s u r f a c e s , some a d d i t i o n a l f a c t o r s must be c o n s i d ered i n d i s c u s s i n g the contact behaviour of m a t e r i a l s l i k e polymers. J o h n s o n , K e n d a l l and R o b e r t s (2) e x t e n d e d t h e H e r t z i a n theory t a k i n g i n t o account the molecular a t t r a c t i o n o f i n t e r f a c i a l a d h e s i o n between c o n t a c t i n g b o d i e s . T h e i r a n a l y s i s w h i c h has b e e n e x p e r i m e n t a l l y confirmed f o r e l a s t o m e r - g l a s s c o n t a c t s (2, 3, 4 ) , shows t h a t due to the e x i s t e n c e of molecular a t t r a c t i o n f o r c e s contact a r e a and e l a s t i c d i s p l a c e m e n t a r e l a r g e r t h a n v a l u e s d e duced from the H e r t z i a n t h e o r y . In a d d i t i o n , the i n f l u e n c e o f v i s c o e l a s t i c i t y on c o n t a c t d e f o r m a t i o n has b e e n s t u d i e d , see e . g . (5 t o 1 2 ) . These t h e o r e t i c a l a n a l y s e s m a i n l y c o n c e r n i n g the c o n t a c t between a r i g i d i n d e n t e r and a v i s c o e l a s t i c h a l f - s p a c e i n d i c a t e a l s o t h a t c o n t a c t a r e a and c o n t a c t d i s p l a c e m e n t s a r e l a r g e r f o r v i s c o e l a s t i c m a t e r i a l s t h a n f o r p u r e e l a s t i c mate r i a l s and t h a t a d i f f e r e n c e b e t w e e n l o a d i n g and u n l o a d i n g i s t o be e x p e c t e d w i t h v i s c o e l a s t i c m a t e r i a l s (13) . I n t h i s p a p e r ( i ) c o n t a c t d e f o r m a t i o n and ( i i ) i n i t i a l f r i c t i o n a r e s t u d i e d e x p e r i m e n t a l l y and a t t e m p t s a r e made t o c o r r e l a t e t h e e x p e r i m e n t a l r e s u l t s w i t h b u l k and s u r f a c e p r o p e r t i e s o f t h e p o l y m e r s i n v e s t i g a t e d . EXPERIMENTAL I n t h e e x p e r i m e n t a l i n v e s t i g a t i o n s a b a l l - o n - d i s c and a p i n - o n - d i s c c o n f i g u r a t i o n were u s e d c o n s i s t i n g o f an op t i c a l l y transparent d i s c (glass or o p t i c a l l y transparent p o l y m e r ) and a p o l y m e r s p e c i m e n . For the contact d e f o r m a t i o n i n v e s t i g a t i o n s , c o m p l i a n c e and c o n t a c t s i z e were m e a s u r e d by means o f an i n d u c t i v e d i s p l a c e m e n t transdu c e r and an o p t i c a l m i c r o s c o p e c o n n e c t e d w i t h a TV s e t respectively. The a c c u r a c y o f t h e s e measurements was b e t t e r t h a n + 0.1 μπι f o r t h e d i s p l a c e m e n t measurements and b e t t e r t h a n ± 1 % f o r t h e d e t e r m i n a t i o n o f t h e c o n t a c t diameter. In F i g u r e 1 the arrangement o f the main p a r t s o f t h e t e s t e q u i p m e n t i s shown s c h e m a t i c a l l y . F i g u r e 2 shows a p h o t o g r a p h o f t h e e x p e r i m e n t a l a p p a r a tus. The t e s t s y s t e m o p e r a t e d u n d e r c o n t r o l l e d c o n d i t i o n s o f a m b i e n t t e m p e r a t u r e (T = 23 °C) and e n v i r o n mental atmospheric humidity (50% r e l . h u m i d i t y ) . For the f r i c t i o n experiments with the help of a d r i v i n g u n i t s u p p o r t e d by an a i r b e a r i n g and a l o a d i n g d e v i c e , s l i d i n g m o t i o n and l o a d c o u l d be a p p l i e d . The n o r m a l f o r c e and t h e f r i c t i o n a l f o r c e a c t i n g on t h e



1. C Z I C H O S

Contact Deformation and Static Friction of Polymers

F i g u r e 1. S c h e m a t i c a r r a n g e m e n t configuration.

Figure

2.

o f experimental

Photograph o f t h e t r i b o m e t e r .


5

6


CONTROL

p o l y m e r s p e c i m e n were d e t e c t e d by a two-component f o r c e t r a n s d u c e r (14) and t h e s i g n a l s o f t h i s f o r c e t r a n s d u c e r were f e d t o an e l e c t r o n i c d i v i d e r w h i c h c a l c u l a t e d d i r e c t l y the f r i c t i o n c o e f f i c i e n t . Normal l o a d , t a n g e n t i a l f r i c t i o n a l f o r c e , and f r i c t i o n c o e f f i c i e n t were r e c o r d e d on a t h r e e - c h a n n e l r e c o r d e r . A block diagram of t h i s t r i b o m e t e r i s g i v e n i n F i g u r e 3. D e t a i l s of the e x p e r i mental equipment, the m a t e r i a l s s t u d i e d , the p r e p a r a t i o n o f t h e s p e c i m e n s and t h e p e r f o r m a n c e o f t h e t e s t s a r e g i v e n elsewhere (15).


CONTACT DEFORMATION The e x p e r i m e n t s were p e r f o r m e d w i t h mers: - Polyoxymethylene - P o l y a m i d e 66

(PA

- Polypropylene

the

following poly

(POM) 66)

(PP)

- Polytetrafluoroethylene

(PTFE)

A l l m a t e r i a l s belong to the c l a s s of s e m i - c r y s t a l l i n e thermoplastic polymers. C h a r a c t e r i s t i c appearances of s p h e r u l i t i c m i c r o s t r u c t u r e s o f t h e p o l y m e r s a r e shown i n F i g u r e s 4 and 5 f o r t h e e x a m p l e s o f POM and PA66. The c o n t a c t d e f o r m a t i o n o f t h e s e thermoplastic poly mers was s t u d i e d e x p e r i m e n t a l l y by p r e s s i n g p o l y m e r i c b a l l s (of 4 mm d i a m e t e r ) w i t h c o n t i n u o u s l y i n c r e a s i n g l o a d (0.6 N/s) a g a i n s t an o p t i c a l l y smooth g l a s s s u r f a c e and m e a s u r i n g b o t h c o n t a c t d e f o r m a t i o n d i s p l a c e m e n t and c o n t a c t s i z e u n d e r l o a d as d e s c r i b e d above (see F i g u r e 1). The p o l y m e r b a l l s had a mean p e a k - t o - v a l l e y r o u g h n e s s o f R 0.6 - 1.0 μπι and a c . l . a . r o u g h n e s s o f R « 0.2-0.3 ζ a μιη. 0

T y p i c a l p l o t s of the v a r i a t i o n of contact diameter and d i s p l a c e m e n t i n c o m p r e s s i o n f o r i n c r e a s i n g and de c r e a s i n g l o a d s f o r one o f t h e p o l y m e r s i n v e s t i g a t e d a r e shown i n F i g u r e 6 t o g e t h e r w i t h t h e o r e t i c a l c u r v e s c a l c u l a t e d w i t h t h e w e l l known H e r t z i a n f o r m u l a s : ι

a

3

=

R

1

1 /3

/3

73 F

4 Ε

1

16

Ε

12

R

N

2

1

where Ε V

1

R:

0

f

:

Poisson s

Ball

1

1

V

" 2

1

ratio;

E.

0

: Elastic

modulus;

radius


1. C Z I C H O S

1


Tn bo me ter Test system

Driving device


Force transducer] Loading device J

Divider

>

3 Channel Recorder

iDiscriminatorl Signât

Figure

Figure

3.

4.

Block

feedback

diagram

Microstructure

of the

tribometer.

of polyoxymethylene.



8

P O L Y M E R W E A R A N D ITS C O N T R O L

Figure

5.

M i c r o s t r u c t u r e of polyamide

F i g u r e 6. Contact diameter displacement curves.

and

contact

66.

deformation



1. C Z I C H O S


9

In a d d i t i o n t o t h e continuous l o a d i n g and u n l o a d i n g p r o c e d u r e , a s t e p w i s e mode o f l o a d i n g was a l s o a p p l i e d . I n F i g u r e 7, c o n t a c t d e f o r m a t i o n d i s p l a c e m e n t s f o r b o t h c o n t i n u o u s a n d s t e p w i s e l o a d i n g a r e drawn f o r one o f t h e polymers i n v e s t i g a t e d . The p l o t s o f F i g u r e 6 a n d t h e c o r r e s p o n d i n g graphs f o r t h e o t h e r polymers s t u d i e d i n d i c a t e a r e s i d u a l d e f o r m a t i o n b e h a v i o u r w h i c h may be d i s cussed i n connection with t h ee l a s t o p l a s t i c hardness o f the m a t e r i a l s . I n a d d i t i o n i t i s obvious t h a t large d i f f e r e n c e s between e x p e r i m e n t a l c o n t a c t d e f o r m a t i o n d a t a and t h e o r e t i c a l v a l u e s c a l c u l a t e d o n t h e b a s i s o f t h e c l a s s i c a l H e r t z i a n t h e o r y e x i s t so t h a t b e s i d e s t h e p r o p e r t i e s and o p e r a t i n g v a r i a b l e s o f t h e H e r t z i a n t h e o r y some o t h e r i n f l u e n c i n g f a c t o r s must be c o n s i d e r e d . G e n e r a l l y s p e a k i n g , t h e f o l l o w i n g m a i n phenomena may i n f l u e n c e t h e c o n t a c t d e f o r m a t i o n b e h a v i o u r o f p o l y mers : 1.

Elastic

deformation

processes

2.

Viscoelastic

deformation

processes

3.

Viscoplastic

deformation

processes

4.

Strain hardening/softening effects

5.

Interfacial

adhesion

F i r s t , thep o s s i b l e influence o f i n t e r f a c i a l adhesion was s t u d i e d b y p e r f o r m i n g l o a d i n g - u n l o a d i n g t e s t s w i t h u n l u b r i c a t e d and l u b r i c a t e d specimens. I f i n t e r f a c i a l adh e s i o n e x e r t s a dominant i n f l u e n c e on c o n t a c t deformat i o n b e h a v i o u r , t h i s i n f l u e n c e s h o u l d be s i g n i f i c a n t l y reduced by a l u b r i c a n t . ( F o r example, J o h n s o n , K e n d a l l and R o b e r t s (2) f o u n d t h a t i f t h e c o n t a c t o f r u b b e r s p h e r e s was immersed i n a s o l u t i o n o f s o d i u m d o d e c y l s u l f a t e , t h e i n f l u e n c e o f i n t e r f a c i a l adhesion vanishes and t h e r e s u l t s o f c o n t a c t r a d i u s measurements a g r e e d e x a c t l y with the Hertz t h e o r y ) . From t h e r e s u l t s shown i n F i g u r e 8 i t c a n be s e e n t h a t t h e c o n t a c t d e f o r m a t i o n b e h a v i o u r i s n o t s i g n i f i c a n t l y changed through a l u b r i c a n t f o r the experimental c o n d i t i o n s o f t h i s i n v e s t i g a t i o n . A l t h o u g h no c o m p a r a t i v e c o n t a c t d i a m e t e r measurements were performed, (because t h e l u b r i c a n t p e r m i t t e d a c l e a r m i c r o s c o p i c d i s t i n c t i o n o f t h e c o n t a c t boundary) from t h e r e s u l t s o f F i g u r e 8, i t i s o b v i o u s t h a t i n t e r f a c i a l a d h e s i o n a p p e a r s n o t t o be a d o m i n a n t i n f l u e n c i n g f a c t o r f o r t h e c o n t a c t deformation displacement o f t h e polymers studied. 1

A Model f o r t h e c o n t a c t Deformation

o f polymers

In o r d e r t o e x p l a i n t h e e x p e r i m e n t a l l y observed c o n t a c t d e f o r m a t i o n b e h a v i o u r a s shown i n F i g u r e 6, a s i m p l e r h e o l o g i c a l model w h i c h c o m b i n e s e l a s t i c , v i s c o e l a s t i c



CONTROL

Polypropylene stepwise loading Omin duration at every load step)

140pm 120-

100-


80-60 QJ Ε eu

continuous loading (0.6 N/s)

-1.40 Hertzian theory

0

2

4

6

)

10 Load F

12

14

16

18 Ν 20

N

F i g u r e 7. D i s p l a c e m e n t stepwise loading.

curves

f o r c o n t i n u o u s and

lubricated minerol oil. SAE10.q= 52 mPos (23*0 PTFE

Load F M

F i g u r e 8. D i s p l a c e m e n t c u r v e s f o r u n l u b r i c a t e d a n d l u b r i c a t e d polymer-glass contacts.


1. C Z I C H O S

11


and v i s c o p l a s t i c i n f l u e n c e s i s c o n s i d e r e d , s e e e . g . (16, 1 7 ) , F i g u r e 9. A s i l l u s t r a t e d i n t h e u p p e r p a r t o f F i g u r e 9, w i t h t h e h e l p o f a f o u r - p a r a m e t e r o r B u r g e r model t h e d e f o r m a t i o n b e h a v i o u r o f p o l y m e r s may be a p p r o x i m a t e d t h r o u g h a c o m b i n a t i o n o f s p r i n g s a n d dampers. The s p r i n g E c h a r a c t e r i z e s t h e pure e l a s t i c s t a t e o f t h e polymer. I n a d d i t i o n , a t i m e - d e p e n d e n t v i s c o e l a s t i c component d e s c r i b e d by t h e s o - c a l l e d V o i g t - K e l v i n c o n f i g u r a t i o n , i . e . t h e c o m b i n a t i o n o f s p r i n g Ε a n d damper Yf comes into action. F u r t h e r , a v i s c o p l l s t i c f l o w component may e x i s t , m o d e l l e d b y t h e damper o f v i s c o s i t y ^ · Q


0

C o n s i d e r now a p o l y m e r u n d e r t h e i n s t a n t a n e o u s a c t i o n o f a c o n s t a n t u n i a x i a l s t r e s s 6* . A s s u m i n g t h a t t h e d e f o r m a t i o n b e h a v i o u r may be d e s c r i b e d b y t h e m o d e l shown i n F i g u r e 9, we may h y p o t h e s i z e t h a t t h e r e s u l t i n g deformation £ , . i s t h e sum o f total Ί

(i)

elastic

deformation Ε

Ί

el (ii)

viscoplastic

(iii) v i s c o e l a s t i c

(Ε : e l a s t i c ο

modulus)

deformation

(y :

deformation £

( E : r e l a x e d modulus)

Q

"viscosity")

r

T h e s e d e f o r m a t i o n components a n d t h e i r sum a r e p l o t t e d s c h e m a t i c a l l y i n F i g u r e 9. I f t h e a s y m p t o t e f o r t h e r e sulting £ . . - c u r v e i s c o n s t r u c t e d , t h e r e l a x e d modulus E may be d e t e r m i n e d f r o m t h e i n t e r s e c t i o n o f t h i s asymp tote with the £ -axis. I n a d d i t i o n , t h e r e l a x a t i o n time T * c a n be e s t i m a t e d a s i l l u s t r a t e d i n F i g u r e 9 Although t h e d e f o r m a t i o n m o d e l shown i n F i g u r e 9 i s e x t r e m e l y s i m p l i f i e d a n d n e g l e c t s some i m p o r t a n t f a c t o r s ( e . g . t h e f a c t , t h a t i n s t e a d o f a s i n g l e r e l a x a t i o n time a compli c a t e d r e l a x a t i o n s p e c t r u m may be v a l i d ) , t h e m o d e l may serve as a b a s i s f o r t h e d i s c u s s i o n o f t h e contact de formation behaviour o f polymers. r

Discussion o f theExperimentally observed Deformation Behaviour

Contact

In o r d e r t o study t h e i n f l u e n c e o f v i s c o e l a s t i c i t y on t h e experimentally observed contact deformation behaviour, i n a n a l o g y t o t h e model i l l u s t r a t e d i n F i g u r e 9 t h e d i s p l a c e ments u n d e r g i v e n l o a d a s f u n c t i o n o f t i m e were d e t e r mined. A t y p i c a l r e s u l t f o r one o f t h e p o l y m e r s i n v e s t i g a t e d i s shown i n F i g u r e 10. T h e s e d i a g r a m s were d e t e r mined f o r a l l f o u r polymers and t h e r e l a x e d m o d u l i Ε o f t h e p o l y m e r s a n d t h e i r n o m i n a l r e l a x a t i o n t i m e s T were e s t i m a t e d i n a s i m i l a r manner a s i l l u s t r a t e d i n F i g u r e 9. The r e s u l t s c o m p i l e d i n T a b l e I i n d i c a t e t h a t b o t h r e l a x e d moduli and r e l a x a t i o n times v a r y c o n s i d e r a b l y w i t h i n the range o f t h e a p p l i e d e x p e r i m e n t a l c o n d i t i o n s . -




''total = (Έ '-ο

Ίο

• t (1 - e - τ ) ) σ f

0

L

f

t - τ - β - ' -0.368

F i g u r e 9. F o u r - p a r a m e t e r o r B u r g e r model f o r t h e deformation behaviour o f polymers.

Polypropylene

°0

1

2

3 4 time t

F i g u r e 10. V a r i a t i o n o f c o n t a c t ment a s f u n c t i o n o f t i m e .

5 •

6

7 3 8 1Q

S

deformation displace


1. C Z I C H O S


13


In a d d i t i o n t o t h e s t u d y o f v i s c o e l a s t i c i t y a n d t h e 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 o f r e l a x e d moduli and r e l a x a t i o n time, p o s s i b l e i r r e v e r s i b l e v i s c o p l a s t i c contact d e f o r m a t i o n e f f e c t s were a l s o s t u d i e d . A f t e r a c e r t a i n l o a d i n g t i m e , t h e l o a d was t o t a l l y r e l e a s e d a n d t h e r e c o v e r y o f t h e p o l y m e r s was m e a s u r e d . The r e s u l t s o f F i g u r e 11 show no v i s c o p l a s t i c e f f e c t s f o r b o t h PA66 a n d POM, a m i n o r i n f l u e n c e f o r PP b u t a c l e a r e f f e c t o f v i s c o p l a s t i c i t y f o r P T F E . (Note t h a t t h e r e c o v e r y c u r v e s o f F i g u r e 11 a r e g o v e r n e d b y much h i g h e r r e l a x a t i o n t i m e s than t h o s e determined under an a p p l i e d l o a d and l i s t e d i n T a b l e I . T h i s a g a i n i n d i c a t e s t h e dependence o f r e l a x a t i o n times o r r e l a x a t i o n s p e c t r a on o p e r a t i n g c o n d i tions) . The r e s u l t s o f F i g u r e 11 i n d i c a t e t h a t t h e p o l y m e r s s t u d i e d were s u b j e c t t o d i f f e r e n t m i c r o s t r u c t u r a l d e f o r m a t i o n mechanisms. I n t h i s c o n n e c t i o n i t must be b o r n e i n m i n d t h a t t h e maximum n o m i n a l d e f o r m a t i o n o f POM a n d PA66 was o n l y 1% w h e r e a s PP a n d PTFE were d e f o r m e d up t o 1.8% a n d 3.3% r e s p e c t i v e l y . T h e r e f o r e i t may be assumed t h a t f o r POM a n d PA66 o n l y a n i n s t a n t a n e o u s l y r e v e r s i b l e d e f o r m a t i o n o f t h e amorphous m a t r i x o f t h e s p h e r u l i t i c m i c r o s t r u c t u r e o c c u r r e d (18) w h e r e a s f o r PP a n d PTFE some i r r e v e r s i b l e effects, l i k e interlamellar shearing or r e o r i e n t a t i o n o f t h e l a m e l l a e may have t a k e n p l a c e . With the determination o f v i s c o p l a s t i c displacements f r o m F i g u r e 11 a n d t h e d a t a o f v i s c o e l a s t i c i t y c o m p i l e d i n T a b l e I we a r e now i n a p o s i t i o n t o d e r i v e o n t h e b a s i s o f t h e m o d e l i l l u s t r a t e d i n F i g u r e 9, a s i m p l e c o n t a c t d e f o r m a t i o n f o r m u l a i f we assume t h a t i n t e r f a c i a l a d h e s i o n ( s e e F i g u r e 8) i s n e g l i g i b l e . As c o m p i l e d i n F i g u r e 12, i t i s assumed t h a t t h e t o t a l c o n t a c t d e f o r m a t i o n d i s p l a c e m e n t i s t h e sum o f a n e l a s t i c , a v i s c o e l a s t i c and a v i s c o p l a s t i c d e f o r m a t i o n component. Forthe f i r s t component t h e H e r t z i a n t h e o r y i s assumed t o be v a l i d a n d f o r t h e s e c o n d component i n a n a l o g y t o t h e model d e s c r i b e d i n F i g u r e 9, t h e e l a s t i c modulus Ε i s r e p l a c e d b y a t e r m w h i c h i n c l u d e s t h e r e l a x e d modulus Ε a n d t h e r e l a x a t i o n timeT". I f t h e v i s c o p l a s t i c component c a n be n e g l e c t e d , t h e c o n t a c t d e f o r m a t i o n f o r m u l a , g i v e n i n F i g u r e 12, r e sults. I t must be p o i n t e d o u t t h a t t h e f o r m u l a g i v e n i n F i g u r e 12 i s e x t r e m e l y s i m p l i f i e d b e c a u s e i t h a s b e e n d e r i v e d i n a n a l o g y t o t h e model i l l u s t r a t e d i n F i g u r e 9 u n d e r t h e a s s u m p t i o n s o f a l i n e a r s u p e r p o s i t i o n o f d e f o r m a t i o n com ponents and t h e a c t i o n o f an i n s t a n t a n e o u s c o n s t a n t l o a d . B e c a u s e o f d i f f e r e n c e s between t h e s e i d e a l i z e d assumptions and t h e a c t u a l e x p e r i m e n t a l c o n d i t i o n s ( e . g . c o n t i n u o u s loading instead o f instantaneous loading) t h e formula s h o u l d be c o n s i d e r e d o n l y a s a f i r s t r o u g h a p p r o x i m a t i o n .



Table

I.

Polymer

M a t e r i a l parameters o f polymers

Elastic modulus E [N/mm ] 2

Q


POM

3200

PA 66

3000

PP

1600

PTFE

600

Relaxed modulus E Elastic modulus E_

Load F [N] N

r

Relaxation time r[s]

0

2 8 20

0.56 0.58 0.61

25-9

2 8 20

0.54 0.60 0.65

18^6

2 8 20

0.13 0.22 0.49

27

2 8 20

0.21 0.41 0.57

52tl5

ts

Load : F= 20 Ν Loading time: 1 min N

PTFE

0

3

6

9 12 15 18 Time after deloading t

F i g u r e 11. R e c o v e r y c u r v e s a f t e r of polymers.

21

24

27 min 30

contact deformation



1. C Z I C H O S


15

The s i m p l i f i e d f o r m u l a g i v e n i n F i g u r e 12 i s valid o n l y f o r t h e l o a d i n g mode o f c o n t a c t d e f o r m a t i o n . F o r t h e u n l o a d i n g mode ( i . e . F d e c r e a s i n g ) f u r t h e r a s p e c t s must be c o n s i d e r e d . I f a f t e r a c e r t a i n l o a d i n g time t * the load i s l o w e r e d , t h e e l a s t i c component 6 , i n s t a n t a n e o u s l y drops t o t h e displacement c o n n e c t l a w i t h the lower l o a d o r s t r e s s l e v e l . F o rthe v i s c o e l a s t i c component, however, t h e w h o l e l o a d i n g - u n l o a d i n g h i s t o r y must be t a k e n i n t o c o n s i d e r a t i o n : On t h e one h a n d , t h e v i s c o e l a s t i c component S reduces t o the corresponding l o w e r l o a d o r s t r e s s l e v e ï i n a d e l a y e d manner d e p e n d i n g on t h e r e l a x a t i o n t i m e . On t h e o t h e r h a n d , a t t h e l o w e r l o a d o r s t r e s s l e v e l t h e v i s c o e l a s t i c component i n c r e a s e s a g a i n because o f i t s dependence on t i m e . In addition, a p o s s i b l y r e m a i n i n g i r r e v e r s i b l e v i s c o p l a s t i c component must a l s o be p a i d r e g a r d . Summing up a l l t h e s e components, a contact deformation formula f o r t h e unloading mode r e s u l t s w h i c h i s a l s o g i v e n i n F i g u r e 12. ( I t s h o u l d be m e n t i o n e d t h a t f o r t h e sake o f s i m p l i c i t y t h e same r e l a x a t i o n t i m e T * i s assumed t o be v a l i d f o r b o t h t h e l o a d i n g a n d u n l o a d i n g mode. T h e o r e t i c a l l y , i t s h o u l d be d i s t i n g u i s h e d between a " r e t a r d a t i o n t i m e " f o r t h e l o a d i n g mode a n d a " r e l a x a t i o n t i m e " f o r t h e u n l o a d i n g mode.) I n F i g u r e s 13 t o 16 t h e e x p e r i m e n t a l l y d e t e r m i n e d contact deformation displacement values together with curves c a l c u l a t e d with t h e deformation formulas o f F i g u r e 12 a n d t h e H e r t z i a n f o r m u l a a r e p l o t t e d . In the d i s p l a c e m e n t - l o a d g r a p h s f o r POM a n d PA66 o n l y e l a s t i c a n d v i s c o e l a s t i c components were t a k e n i n t o c o n s i d e r a t i o n w h e r e a s f o r PP a n d PTFE a l s o v i s c o p l a s t i c components ( s e e F i g u r e 11) were i n c l u d e d . I t c a n be s e e n t h a t a r e a s o n a b l y good c o r r e l a t i o n b e t w e e n e x p e r i m e n t a l d a t a a n d t h e o r e t i c a l l y d e t e r m i n e d c u r v e s r e s u l t s , so t h a t t h e f o r m u l a s c o m p i l e d i n F i g u r e 12 may be u s e d f o r t h e a p p r o x i m a t i v e estimation o f t h econtact deformation behaviour o f curved thermoplastic polymeric materials. STATIC

FRICTION

In a d d i t i o n t o t h e i n v e s t i g a t i o n s on c o n t a c t d e f o r m a t i o n , the i n i t i a l stage o f f r i c t i o n has been s t u d i e d (19). Whereas i n r e c e n t y e a r s s e v e r a l i n v e s t i g a t i o n s have b e e n p e r f o r m e d o n p o l y m e r / m e t a l s l i d i n g p a i r s (a c o m p r e h e n s i v e r e v i e w was p u b l i s h e d r e c e n t l y b y B r i s c o e (20) r e l a t i v e l y l i t t l e work h a s b e e n done t o s t u d y t h e t r i b o l o g i c a l b e h a v i o u r o f polymers s l i d i n g a g a i n s t polymers. In the f r i c t i o n investigations a pin-on-disc configuration with t h e f o l l o w i n g p o l y m e r s was u s e d ; t h e p o l y m e r p i n s were c u t w i t h a m i c r o t o m e i n o r d e r t o o b t a i n a f l a t smooth polymer s u r f a c e a t t h e b e g i n n i n g o f t h e t e s t s :



Contact deformation = f (elasticity, viscoelasticity, viscoplasticity) s

total

=

— g - ^

(1

δ

Γ

ο

)

r

where E : Relaxed modulus


r

τ

: Relaxation time

If & = 0 , for a ballon-plate contact it follows 9 n r9d-e«/r)=l u

5

t

o

t

a

l

Γ

"

M

[l6 E i ' Rj

F

N

+

L l 6 E

;

r

R j

FN ''' 2

Unloading mode (t > t* ; t*: Loading time) : total N Nmax'

ft

(F

t#)=6

(F

)

(F

Nmax, *

t)+5 (F V

+

t

-(t-t»)

)

f r ^Nmax/**

"*r

( Ρ

Ν/*> - * v

^Nmax^J

e

T

F i g u r e 12. S i m p l i f i e d f o r m u l a s f o r t h e c o n t a c t d e f o r m a t i o n d i s p l a c e m e n t s o f polymer b a l l s .

Experimental data:

*« : load increasing ο : load decreasing

Load F

N

F i g u r e 13. E x p e r i m e n t a l l y d e t e r m i n e d c o n t a c t d e f o r m a t i o n d i s p l a c e m e n t a n d c a l c u l a t e d c u r v e s f o r POM.



CZICHOS


Load F

•

N

F i g u r e 14. E x p e r i m e n t a l l y d e t e r m i n e d c o n t a c t d é f o r m a t i o n d i s p l a c e m e n t a n d c a l c u l a t e d c u r v e s f o r PA66.

0

2

4

6

8 10 Load F N

12

14

16

18 Ν 20

· -

F i g u r e 15. E x p e r i m e n t a l l y d e t e r m i n e d c o n t a c t d é f o r m a t i o n d i s p l a c e m e n t a n d c a l c u l a t e d c u r v e s f o r PP.



18 Pin

Polyoxymethylene

materials:

(POM)

P o l y a m i d e 66

(PA66)

Polypropylene

(PP)

Polytetrafluoroethylene Disc

Acrylonitrile-styrene

materials:

Polystyrene

(PTFE) (ANS)

(PS)

Polymethyl methacrylate

(PMMA)

The s u r f a c e r o u g h n e s s d a t a o f t h e p o l y m e r p i n s were R c= 0.3 ο ο

^

0.2 0,1 50

100

150 200 250 Sliding distance s

300 •

350

400 pm 450

F i g u r e 17. F r i c t i o n o f p o l y m e r / p o l y m e r p a i r s POM, PA66, PP a g a i n s t A N S ) .

(PTFE,


P O L Y M E R W E A R A N D ITS C O N T R O L p-0108-η F=10N;V = 2.410"Vs;S = 4 5 0 p m T = 2 3 ° C ^ = 5 0 % r e l . h u m i d i t y N

0.7-

x

;

a

Φ2

Disc: PS

Pin :PTFE POM PA66 PP

0.6-

1 0.5-

1 0.40.3-


S

a ο "o

Κ

£0,20.10

50

100

150 200 Sliding distance s

250

^

300

350

F i g u r e 18. F r i c t i o n o f p o l y m e r / p o l y m e r POM, PA66, PP a g a i n s t P S ) .

400

pairs

p

m

450

(PTFE,

ρ—ΦΊΡ8—-j

F=10N;V = 2.41u^Ws;S = 450pm î 23°C ip=507oreLhumidity

J^P 0.7-

N

x

:

aS

:

Φ2

Disc: PMMA

Pin:PÏFE POM PA66 PP

0.6-

|o.5^0,4ο

10.3"ο

0.1i

0

V 50

100

150 200 250 Sliding distance s

300 ^

F i g u r e 19. F r i c t i o n o f p o l y m e r / p o l y m e r POM, PA66, PP a g a i n s t PMMA).

350

400

pairs

p

m

450

(PTFE,


1. C Z I C H O S


21


adhesion computed b y E r h a r d (21) a n d t h e e x p e r i m e n t a l l y d e t e r m i n e d f r i c t i o n a l work, Ε . The f r i c t i o n a l work, Ε , was d e t e r m i n e d b y i n t e g r a t i n g t h e m e a s u r e d f r i c t i o n a l f o r c e s p l o t t e d i n F i g u r e s 17 t o 19 o v e r t h e s l i d i n g d i s t a n c e . The r e s u l t s a r e t h a t t h e d a t a f o r PTFE s l i d i n g a g a i n s t t h e t h r e e d i s c m a t e r i a l s a r e l o w e s t and t h e d a t a f o r t h e PA66 p i n s , s l i d i n g a g a i n s t t h e t h r e e disc materials aregenerally highest. I n F i g u r e 20 t h e e x p e r i m e n t a l l y d e t e r m i n e d f r i c t i o n a l work i s p l o t t e d a s a f u n c t i o n o f t h e c a l c u l a t e d work o f a d h e s i o n . On a l o g a r i t h m i c s c a l e t h e f r i c t i o n a l work i n c r e a s e s w i t h t h e work o f a d h e s i o n a n d t h e e x p e r i m e n t a l d a t a seem t o be i n a r e a s o n a b l y good c o r r e l a t i o n w i t h t h e t h e o r e t i c a l l y calculated values. (An e x c e p t i o n p e r h a p s may be t h e c o m b i n a t i o n o f PP s l i d i n g a g a i n s t P S ) . I t a p p e a r s that f o r t h e f l a t - t o - f l a t conformai contact o f the p o l ymer c o m b i n a t i o n s s t u d i e d , t h e a d h e s i o n component a t t h e i n t e r f a c e may be t h e d o m i n a n t f a c t o r g o v e r n i n g t h e i n i t i a l f r i c t i o n a l behaviour o f polymer-polymer s l i d i n g p a i r s . In c o n t r a s t , i n t h e c a s e o f t h e c o n t a c t d e f o r m a t i o n d i s placement o f b a l l - t o - f l a t c o u n t e r f o r m a l c o n t a c t s d i s c u s s e d above no e f f e c t o f a d h e s i o n was f o u n d a s compared w i t h t h e i n f l u e n c e o f t h e (bulk) v i s c o e l a s t i c p r o p e r t i e s o f t h e materials. ( T h i s may be due t o e l a s t i c r e l i e f f o r c e s w h i c h may b u r s t a d h e s i v e j u n c t i o n s d u r i n g t h e l o a d i n g unloading contact deformation cycles.) CONCLUSIONS From t h e i n v e s t i g a t i o n s o f ( i ) c o n t a c t d e f o r m a t i o n a n d ( i i ) i n i t i a l s t a t i c f r i c t i o n o f t h e r m o p l a s t i c polymers t h e f o l l o w i n g c o n c l u s i o n s may be drawn: a)

The c o n t a c t d e f o r m a t i o n b e h a v i o u r o f c o u n t e r f o r m a l polymer/glass contacts i s considerably influenced by t h e ( b u l k ) v i s c o e l a s t i c p r o p e r t i e s o f t h e p o l y m e r s .

b)

The c o n t a c t d e f o r m a t i o n d i s p l a c e m e n t f o r t h e u n l o a d i n g mode i s c o n s i d e r a b l y h i g h e r t h a n f o r t h e l o a d i n g mode.

c) The c o n t a c t d e f o r m a t i o n d i s p l a c e m e n t i n l o a d i n g / u n l o a d i n g c y c l e s c a n be e s t i m a t e d w i t h a s i m p l i f i e d extended H e r t z i a n formula i n c l u d i n g r e l a x a t i o n c h a r a c t e r i s t i c s o f t h e polymers. d)

The s t a t i c f r i c t i o n a t t h e o n s e t o f m o t i o n o f c o n formal polymer/polymer s l i d i n g p a i r s i s i n f l u e n c e d mainly by i n t e r f a c i a l adhesion.

e) A c o r r e l a t i o n between e x p e r i m e n t a l l y m e a s u r e d f r i c t i o n a l work a n d t h e o r e t i c a l l y e s t i m a t e d work o f a d h e s i o n was f o u n d .


22


Table I I . Pin

Work o f f r i c t i o n Disc

Frictional work E I Ν mm] F


PTFE

PP

POM

PA 66

and adhesion o f polymers Work of adhesion Δ γ =7

1

+7

2

-7

ANS

0.328

54.7

PS

0.343

55.4

PMMA

0.324

53.6

ANS

0.795

73.0

PS

1.215

75.3

PMMA

0.730

73.0

ANS

0.619

84.5

PS

0.760

81.0

PMMA

0.850

87.3

ANS

0.987

88.0

PS

0.864

83.1

PMMA

1.128

92.1

1

2

10 1

6

Nmm/mm ] 2

Nmml

F i g u r e 20. C o r r e l a t i o n work o f a d h e s i o n .

between f r i c t i o n a l

work a n d


1.

CZICHOS


23

Acknowledgments I t h a n k e n g i n e e r s Mr. N. K e l l i n g a n d Mr. J . S c h w e n z i e n o f B A M - L a b o r a t o r y "Wear P r o t e c t i o n , T r i b o m e t r y a n d T r i b o p h y s i c s " f o r having s k i l l f u l l y performed the ex p e r i m e n t s , a s w e l l a s D r . P. F e i n l e f o r h e l p f u l d i s c u s s i o n s a n d D i p l . - P h y s . H. T i s c h e r o f t h e BAM-Document a t i o n S e r v i c e on T r i b o l o g y f o r p r o v i d i n g background literature material. F i n a n c i a l s u p p o r t from t h e Deut s c h e F o r s c h u n g s g e m e i n s c h a f t (DFG) i s g r e a t f u l l y a c k n o w l edged.


Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. J. 9. 10. 11. 12.

13. 14.

Gladwell, G. M. L . "Contact Problems in the C l a s s i cal Theory of E l a s t i c i t y " , Sijthoff and Noordhoff, Alphen aan den Rijn, 1980. Johnson, K. L.; Kendall, K . ; Roberts, A. D. "Surface Energy and the Contact of Elastic Solids", Proc. R. Soc. London, Ser. A 324 (1971) 301. Barquins, M . ; Courtel, R. "Rubber F r i c t i o n and the Rheology of Viscoelastic Contact", Wear 32 (1975) 133. Barquins, M. "Adhesive Contact and Kinetics of Adherence between a Rigid Sphere and an Elastomeric Solid", Int. J. Adhesion and Adhesives 3 (1983) 71. Lee, Ε. H . ; Radok, J. R. M. "The Contact Problem for Viscoelastic Bodies", Trans. ASME (J. Appl. Mech.) 27 (1960) 438. Ting, T. C. T. "The Contact Stresses between a Rigid Indenter and a Viscoelastic Half-Space", Trans. ASME (J. Appl. Mech.) 33 (1966) 845. Comninou, M. "Contact between Viscoelastic Bodies", Trans. ASME (J. Appl. Mech.) 43 (1976) 630. Kravchuk, A. S. "On the Problem for Linearly and Nonlinearly Elastic Bodies of Finite Dimensions", Appl. Math. Mech. 41 (1977) 320. Chow, T. S. "Deformational Contact and F r i c t i o n on Viscoelastic Substrates", Wear 51 (1978) 355. Graham, G. A. C. "Viscoelastic Contact Problems with Friction", Int. J. Eng. S c i . 18 (1980) 191. Sabin, G. C. W.; Graham, G. A. C. "The Normal Aging Viscoelastic Contact Problem", Int. J. Eng. S c i . 18 (1980) 751. Aksel, N . ; Buggisch, H. "On the Problem of Contact between a Rigid Ball and a Viscoelastic Half-Space" (in German), Z. Angew. Math, u. Mech. (ZAMM) 62 (1982) 101. Johnson, K. L . "Adhesion at the Contact of Solids", i n : "Theoretical and Applied Mechanics"; Koiter, W. T., E d . ; North Holland, Amsterdam, 1977; 133. Mittmann, H . - U . ; Czaika, N . ; Czichos, H. "A New De vice for Simultaneous Measurement of F r i c t i o n Force, Normal Force and F r i c t i o n Coefficient", Wear 31 (1975) 179. Lee; Polymer Wear and Its Control ACS Symposium Series; American Chemical Society: Washington, DC, 1985.


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15.

16.


17. 18. 19. 20. 21.

Czichos, H . ; Feinle, P. "Tribological Behaviour of Thermoplastic, F i l l e d , and Glass-Fibre- Reinforced Polymers - Contact Deformation, F r i c t i o n and Wear, Surface Investigations", BAM Research Report 83, July 1982, 105 p. (Bundesanstalt fur Materialprüfung, Berlin-Dahlem) (in German). Becker, G. W., Schreuer, E . "Deformation Mechanics and Relaxation Behaviour", i n : "Structure and Physi c a l Behaviour of Polymers" (in German); Nitsche, R. and Wolf, Κ. Α . , Eds.; Springer, Berlin/Göttingen/Heidelberg, 1962; 331 f f . Ehrenstein, G. W. "Polymeric Materials" (in German); Hanser, München, 1978; 104 f f . Höhn, W.; Jungnickel, B.-J. "Structure and Relaxation Behaviour of Compressed Polyoxymethylene" (in German), Colloid and Polym. S c i . 260 (1982) 1093. Czichos, H. "Influence of Adhesive and Abrasive Mechanisms on the Tribological Behaviour of Thermoplastic Polymers", Wear 88 (1983) 27. Briscoe, B . J . "Wear of Polymers: An Essay on Fundamental Aspects", Tribology Int. 14 (1981) 231. Erhard, G. "Sliding F r i c t i o n of Polymer-Polymer Pairs" (in German), Jahrbuch VDI-Ges. Werkstofftechnik (Düsseldorf), 1981; 7.

DISCUSSION Q u e s t i o n b y P r o f e s s o r S. B a h a d u r , Iowa S t a t e U n i v e r s i t y : The v i s c o e l a s t i c e f f e c t i n d e f o r m a t i o n r e p o r t e d b y y o u i s s i m i l a r t o what h a s b e e n known f o r a l o n g t i m e i n wood. As s u c h t h e d e f o r m a t i o n v a l u e s i n l o a d i n g a n d unloading are d i f f e r e n t . Have y o u g i v e n a n y c o n s i d e r a t i o n how t h e s e r e s u l t s c o u l d be f a c t o r e d i n t o t h e a d h e s i o n t h e o r y o f f r i c t i o n w h i c h h a s a s o n e o f i t s components the r e a l area o f contact? Answer: T h e a i m o f t h e d e f o r m a t i o n e x p e r i m e n t s was t o study t h e i n f l u e n c e s o f p o s s i b l e v i s c o e l a s t i c and v i s c o p l a s t i c e f f e c t s on the H e r t z i a n c o n t a c t behaviour o f curved thermoplastic polymers. Because these experiments were p e r f o r m e d o n l y w i t h n o r m a l l o a d s we have n o t a t tempted t o e x t e n d t h e s e r e s u l t s t o s l i d i n g H e r t z i a n contacts. I n c o n t r a s t , t h e s t a t i c f r i c t i o n experiments r e p o r t e d i n t h i s p a p e r were p e r f o r m e d w i t h f l a t p o l y m e r polymer c o n t a c t s ( p i n - o n - d i s c c o n f i g u r a t i o n ) i n o r d e r t o keep i n these experiments t h e m e c h a n i c a l s t r e s s s i t u a t i o n as simple as p o s s i b l e . Question b y P r o f e s s o r D. Dowson, U n i v e r s i t y o f L e e d s : How was t h e l u b r i c a n t a p p l i e d i n t h e d e f o r m a t i o n studies? D i d t h e author observe any l u b r i c a n t entrapment w i t h i n t h e H e r t z i a n c o n t a c t zone i n t h e n o r m a l a p p r o a c h p r o c e s s ? Many a u t h o r s have r e p o r t e d t r a p p e d p o o l s o f l u b r i c a n t i n


1.

CZICHOS


25

t h e c e n t r e o f H e r t z i a n c o n t a c t s when e l a s t i c s p h e r e s a r e loaded against f l a t surfaces i n t h epresence o f l i q u i d s .


Answer: The l u b r i c a n t was a p p l i e d a s a t h i n b o u n d a r y film to thedisc surface. An e n t r a p m e n t o f l u b r i c a n t was n o t v i s i b l e i n o u r i n - s i t u m i c r o s c o p i c o b s e r v a t i o n s of t h e c o n t a c t zone. Question by Dr. B r i s c o e , I m p e r i a l C o l l e g e , London: I f e e l t h a t s u r f a c e f r e e e n e r g y i s n o t a good p a r a m e t e r to c o r r e l a t e w i t h s u r f a c e f r i c t i o n . Surface f r e e energy may be r e l a t e d t o c o h e s i v e e n e r g y d e n s i t y . There i s t o o much d a t a w h i c h c o n t r a d i c t s t h i s , e . g . (1) s t a t i c / d y n a m i c f r i c t i o n , (2) s p i n e f f e c t s , (3) t e m p e r a t u r e a n d v e l o c i t y dependence o f f r i c t i o n . Answer: I n c o n s i d e r i n g t h e r e s u l t s o f t h e f r i c t i o n ex p e r i m e n t s i t must be b o r n e i n m i n d t h a t o n l y t h e i n i t i a l s t a t i c f r i c t i o n , with a s l i d i n g d i s t a n c e o f l e s s than 500 μπι a n d n e g l i g i b l e i n f l u e n c e s o f v e l o c i t y a n d tempe r a t u r e was s t u d i e d . Because the m i c r o s c o p i c i n - s i t u ob servations o f thecontact interface revealed that the i n i t i a l s l i d i n g motion occured a t t h e a c t u a l i n t e r f a c e (without i n d i c a t i o n o f deformation o r m a t e r i a l t r a n s f e r effects) i t i s hypothesized that i n i t i a l s t a t i c f r i c t i o n i s t h e r e s i s t a n c e t o overcome i n t e r f a c i a l adhesion as c h a r a c t e r i z e d by t h e Dupré e q u a t i o n . Question b y Dr. R.C. C a v e s t r i , C o p e l a n d C o r p o r a t i o n , Sidney, Ohio: Have y o u c o n s i d e r e d t h e e f f e c t o f t h e d e n s i t y o f the polymers and p o s s i b l e c o r r e l a t i o n w i t h the s c a t t e r o f p i n - o n - d i s c data? Answer: The q u e s t i o n o f t h e i n f l u e n c e s o f p o l y m e r d e n s i t i e s was n o t c o n s i d e r e d i n t h i s p a p e r b u t i n a n o t h e r r e s e a r c h r e p o r t (see r e f . (15) ) . Q u e s t i o n b y Dr. D.H. B u c k l e y , NASA L e w i s R e s e a r c h C e n t e r , Cleveland: Don't t h e m o i s t u r e (50% r e l . h u m i d i t y ) o f y o u r e x p e r i m e n t a l e n v i r o n m e n t a l t e r t h e work o f a d h e s i o n and s h o u l d y o u e x p e c t t h e r e f o r e a c o r r e l a t i o n b e t w e e n f r i c t i o n a n d work o f a d h e s i o n ? Answer: The i n f l u e n c e o f m o i s t u r e a d s o r b e d b y t h e p o l y m e r s may i n d e e d have a n i n f l u e n c e o n t h e work o f adhesion. However, i n t h i s s t u d y o n l y t h e r a n k i n g o f data and n o t t h e i r a b s o l u t e v a l u e s i s c o n s i d e r e d . I t i s p o s s i b l e t h a t t h e e f f e c t o f m o i s t u r e i s one p o s s i b l e reason f o r theexperimentally observed s c a t t e r o f data. Question b y Dr. Y o u t i Kuo, X e r o x Company: What i s t h e r o l e o f shear i n your experiments and a n a l y s i s ? Answer: B e c a u s e t h e d e f o r m a t i o n e x p e r i m e n t s where p e r formed o n l y w i t h normal l o a d s and the c o n t a c t d e f o r m a t i o n



26

d i s p l a c e m e n t p e r p e n d i c u l a r t o t h e c o n t a c t i n t e r f a c e was e x p e r i m e n t a l l y m e a s u r e d , t h e r o l e o f s h e a r was n o t s t u d ied e x p l i c i t e l y .


R E C E I V E D January 23, 1985


Contact Deformation and Static Friction of Polymers - American

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