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Paper I(i) Shear behaviour of an amorphous film with bubbles soap raft model
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Shear behaviour of an amorphous film with bubbles soap raft model D. Mazuyer, J.M. Georges and B. Carnbou
Wear and f r i c t i o n i n boundary regime a r e o f t e n governed by t h e mechanical b e h a v i o r o f v e r y t h i n l a y e r s s e p a r a t i n g t h e two s o l i d s i n s l i d i n g c o n t a c t . Sometimes, t h e s e l a y e r s a r e amorphous. we s i m u l a t e t h e s l i d i n g o f two homogeneous r e c t a n g u l a r Thanks t o a b u b b l e soap r a f t model, The t h i c k n e s s and t h e l e n g t h o f t h i s l a y e r a r e c r y s t a l l i n e r a f t s s e p a r a t e ? by an amorphous l a y e r . vsriable. The k i n e m a t i c f i e l d o f t h e p l a s t i c flow k i t h l a r g e displacements is experimentally E x p e r i m e c t a l r e s u l t s show a determine? i n t h e same time as t h e r e c o r d c f t h e t a n g e n t i a l s t r e s s . A t large dependence o f t h e s l i d i n g s t r e s s w i t h t h e r a t i o t h i c k n e s s - l e n g t h of t h e amorphous l a y e r . r a t i o , s o l i d mechanical t h e o r i e s e x p l a i n t h e b e h a v i o r , b u t a t small r a t i o , t h e e f f e c t o f m i c r o s c c p i c i n s t a b i l i t i e s i n t h e amorphous l a y e r i s dominant. 1 INTRODUCTION
2 EXPERIMENTAL PROCEDURE
I n t h e boundary l u b r i c a t i o n regime, a t h i n discontinuous film separates the sliding s u r f a c e s . T h i s f i l m is c r e a t e d by a d h e s i o n and packing o f p r o d x t s coming from t h e s u r f a c e s i n c o n t a c t and t h e r e a c t i o n w i t h an anti-wear In t h i s case, t h e mechanical a d d i t i v e 111. a c t i o n o f f r i c t i o n makes t h e s e f i l m s amorphcus (21. In addition, the tribochemical f i l m s undergo h i g h p r e s s u r e , i n some cases high t e m p e r a t u r e and h i g h s h e a r r a t e s i n t h e c o n t a c t . These e f f e c t s a r e r e s p o n s i b l e f o r a d u a l i t y between a b r i t t l e b e h a v i o u r and a d u c t i l e one i n wear o f t h e f i l m by d e l a m i n a t i o n 121. Two mechanical p r o p e r t i e s , t h e compressive and shearing strengths are essertial and I t i s w e l l known t h a t i n t h e s t a t i c related. situation the p l a s t i c yield strength of a layer compressed between two r i g i d s o l i d s depends cn t h r e e f a c t o r s : p l a s t i c p r o p e r t i e s of t h e l a y e r , r a t i o of t hick n es s t o l en g t h c a l l e d t h e H i l l nurrber 131 and a d h e r e n c e between t h e l a y e r and t h e s u b s t r a t e s 121. Concerning the shearing behaviour, m i c r o s l i d i n g e x p e r i m e n t s show t h a t d i f f e r e n t t r i b o c h e m i c a l f i l m s have a l m c s t t h e same e l a s t i c properties, however t h e i r anti-wear p r o p e r t i e s We a r e l o o k i n g i f t h e s e are different 14). p r o p e r t i e s are n o t more r e l a t e d t o a p l a s t i c o r a v i s c o p l a s t i c behaviour. I n o r d e r t o understard t h e s h e a r i n g p r o c e s s i n a d u c t i l e amorphous f i l m i n l a r g e d e f o r m a t i o n , we u s e a bubble soap r a f t model w i t h which Bragg v i s u a l i z e d d e f e c t s i n m a t e r i a l s 151, 161, 171. Bubble r a f t s p r o v i d e B u s e f u l two d i m e n s i o n a l model f o r t h e s t u d y o f solids. An attractive repulsive potential between p a r t i c l e s i s due t o t h e s u r f a c e t e n s i o r . o f t h e soap s o l u t i o r . and t h e p r e s s u r e i n s i d e t h e b u b b l e s and g o v e r n s t h e i r b e h w i o u r 181. With a uniform s i z e o f b u b b l e s , we g e t a c r y s t a l l i n e l a t t i c e ; w i t h two d i f f e r e n t s i z e s a p p r o p r i e t a l y mixed an amorphous s t r u c t u r e . The aim o f t h i s p a p e r is t o s t u d y t h e e v o l u t i o n of t h e p l a s t i c flow for a l a r g e r a n g e of t h e t h i c k n e s s t o l e n g t h r a t i o o f t h e l a y e r .
The e x p e r i m e n t a l c o n f i g u r a t i o n i s shown i n f i g . 1 . A i r a t c o n s t a n t p r e s s u r e is blown througk s h o r t c a p i l l a r i e s t o produce b u b b l e s a t t h e a soap solutior. (surface surface of N.m-’ ). If o n l y a t e n s i o n : b;”= 6 . 9 c a p i l l a r y i s used w e o b t a i n a r a f t w i t h E c r y s t a l l i n e s t r u c t u r e . With two c a p i l l a r i e s , we a f t e r mixing. The o b t a i n an amorphous r a f t , c r y s t a l l i n e s t r u c t u r e is made o f 2 . 3 ma d i a m e t e r bubbles, and t h e amorphous s t r u c t u r e is a m i x t u r e (45-55) o f 2 . 3 a r d 1 . 6 mm d i a m e t e r I t is known t h a t t h e bubbles r e s p e c t i v e l y . mechanical p r o p e r t i e s f o r an amorphous l a y e r do n o t v a r y w i t h t h e r a t i o o f bubble s i z e s over a W e c r e a t e two r e c t a n g u l a r l a r g e r a n g e 191. c r y s t a l l i n e rafts which a d h e r e t o two p a r a l l e l , a2fi, ( f i g . 1). They a r e a b o u t frames twelve b u b b l e s t h i c k . We s e p a r a t e t h e two c r y s t a l l i n e r a f t s by an amorphous l a y e r whose T h e frame l e n g t h and t h i c k n e s s a r e v a r i e d . WzA,is t h e n moved a t a c o n s t a n t low speed (1 m m / s ) , p a r a l l e l t o t h e frame ~,fi,c a u s i n g a s h e a r where t h e d i s p l a c e m e n t i s imposed. During t h e t e s t ( a b o u t 20 s e c c n d s ) , t h e d i a m e t e r c f t h e howevtr t h e b u t b l e is decreasing very slowly, v a r i a t i o n i s t o o small t o a f f e c t t h e mechanical b e h a v i o u r . The s t i f f n e s s G f t h e amorptous l a y e r is much lower t h a n t h a t o f t h e c r y s t a l l i n e l a y e r s o t h e whole d e f o r m a t i o n due t o t h e 1101, impose? d i s p l a c e a e n t o c c u r s i n t h e amorphous film. During t h e e x p e r i m e n t , the tangertial f o r c e i s t r a n s m i t t e d t o t h e f i x e d frame End c o n t i n L o u s l y r e c o r d e d . By f i l m i n g t h e experiment on a v i d e o t a p e system and by photographing t h e t e s t a t r e g u l a r two s e c o n d s time i n t e r v a l s , w e c z n determine t h e d i s p l a c e m e n t s i n t h e an.orphcus l a y e r with a r e s o l u t i o n corresponding t o l e z s t h a n h a l f a bubble. W e neglect a l l the effects c,f t h e roughness. The boundary c o n d i t i o n s a r e determined by t h e c r y s t a l l i n e r a f t s ( a d h e r e n c e of the layer t o the s u b s t r a t e s ) .
a,n,
4
Figure 1 : Bubble raft used to simulate a sliding process. The displacement of the frameol,qcauses a shearing of the amorphous layer. The tangential force F is transmitted and measured on the frame IX,~.
5
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3 RESULTS A N D LISCUSSION
We c a l l h t h e t h i c k n e s s o f t h e l a y e r and L its length. 3.1 R e s u l t s for a g i v e r . l a y e r
The m e c h a n i c a l b e h a v i o u r of a n amorphous is governed by local raft of bubbles displa.cemer.ts of t h e b u b b l e s r e s u l t i n g from two s i m u l t a n e o c j p - o c e s s e s d e s c r i b e d by Argcn 1101. F i r s t , t h e r e i s what Argon c a l l s a d i f f u s e E h e a r c a u s i n g t h e r o t a t i o n of c l u s t e r s of 6 b u b b l e s . Around t h e s e r o t a t . i n g g,roups, t h e r e are s m a l l a m p l i t u d e t r a r s l a t i o n e of l i n e s o f b L b b l e s . two locel t r a n s formations create These a e c h a r : i c a l i n s t a b i l i t i e s whick a r e r e s p o n s i b l e for a p l a s t i c s t r a i n . In order t o describe the we define a two behaviour in sliding, d i m e n s i o n a l s h e i r stress 3 E i v e n by t h e r a t i o of t h e measured t a n g e n t i a l force F t.o t h e 1engt.h L o f t h e l a y e r . ., W e assc,ciate t o t h i s s t r e s s a two-dimensiocal d i s t o r t i o n a l s t r c i n = U/h where U is t h e . r e l a t i v e d i s p l a c e m e n t o f t h e t w c , frames and ciz fi2 e n d h t h e t h i c k n e s s of t h e layer.
p
e f f e c t i v e e l a s t i c s h e a r mcdulus G = 1 , 2 lO-'N/m which is j n agreemer.t w i t h t h e E x p e r i m e n t a l v a l u e s c a l c u l a t e d from a n i n d e r . t a t i o r t e s t 191. DLring t h i s p e r i o d , w e observe f e w local movements o f t h e b L b b l e s . Then, t h e stress i s s t i l l i n c r e a s i n g niore g r a d u a l l y , i n proportion t o t h e defamation and r e a c h e s a const.ar?t l i m i t i n g v a l u e TI. Ar0ur.d t h i s mean b a l u e , we c o t i c e small p e r t u r b a t i o n s due t o t h e numerous mechar:ical i n s t a b i l i t i e s . For e v e r y t e s t , we h a v e o b s e r v e d t h a t f o r a g i v e n H i l l number, F is p r o p c r t i o n a l t o L , and w e o b t a i n a unique 1 i m i t . i n g stress fl =: F l / L . 3.2
c f t h e Kill Influence b e h z v i o w of t h e l a y e r
number
cn
the
F i g u r e 3 shc,ws t h e e v o l u t i o n c 8 f t h e s l i d i n g stress u i t h H i l l numtser. I t is i n t e r e s t i n g t o o b s e r v e t h z t e x t r e m a o c c u r f o r s i m r l e v a l u e s of h/L ( r e s p e c t i v e l y 0.15, 0 . 2 5 , 0 . 5 ) . E x p e r i m e n t s made w i t h c i f f e r e n t s i z e s of b u b t l e s g i v e t h e sarre c u r v e .
a,n,
Hill Number h/L
Shear strain
v=
U/h
F i g u r e 2 : M e c h a n i c a l b e h a v i o u r of t h e s l i d i n g airorphous l a y e r . A : t h e t a n g e n t i a l stress Z is increEsing i n porportion t o the deformation. B : t h e t a n g e n t i a l stress f is reaching a l i m i t i n g value c h a r a c t e r i s t i c fror. the p l a s t i c f l o w . F i g u r e 2 shows a t y p i c 2 1 r e c o r d c f t h e f o r c e F v e r s u s d i s p l a c e m e r t U. A t f i r $ t , t h e stress i n c r e a s e s i n p r o p o r t i o r . t o t h e s t r a i n . I n s F i t e cf l i n e a r i t y t h i s s t r , a i n i s n o t c o m p l e t e l y r e v e r s i b l e , however i t is p o s s i b l e t o d e f i n e E "pseudo" e l a s t i c s h e a r nloduluz c h a r a c t e r i s t i c Then we o b t a i n a n o f small s t r 2 i n s b e h a v i o u r .
F i g u r e 7 : I n f l u e n c e o f t h e H i l l numter on t h e Po i s t h e l i m i t i n g t a r g e n t i a l p l a s t i c flow. stress f o r h/L = 1. ( A ) : e x p e r j ment a1 p o i n t s .
W e c a n d i s t i n g u i s h two r e g i o n s i n t h e c u r v e ( f i g . 3 ) : a first f o r h/L greater t h a n 0 , 2 5 acd E s e c o n d f o r k / L less t h a n 0 . 2 5 . We l a t e r show the plastic t h a t i f h/L is greater t h a n C.2E;, beha.viour c a n t e e x p l a i n e d t y t h e continuum the s o l i d s m e c h a n i c s t u t a t h/L l e s s t h a n C.25, d e f e c t s i n t h e arrc'rptous l a y e r a r e d o m i n a n t . 3.2.1
-
h/L
>
0,25
WE v i s u z l i z e i n f i g u r e 4, t h e displacement: a t t h e b e g i n n i n g of t h e l i n e a r i n c r e a s e . field, of t h e s t r e s s a c c o r d i n g t o t h e sk.ear s t r a i n ( f i g u r e 4 A ) and when t h e s t r e s s r e a c h e s i t s l i m i t (figure 4E). These pictures are cbtained by nhotographing t h e experiment, a t regular time
6
,IA
Shear strain
= U/h
Figure 4 : Description on the evolution of the kinematic field of the plastic flow, by superimposition of two successive pictures of the layer, parallel to the curve sliding stress versus shear deformation. (A) : the tangential stress begins to increase slowly in proportion to the displacement. The shear band 2 appears between two lateral zones 1 and 3 respectively locked relative to the framew,qand the frame ciz f$ (B) : the limiting stress ilis reached. The thickness of the shear band increases while the thickness of the two lateral zones (1, 3 ) remains constant at five bubbles, the slip line becomes horizontal.
7
intervals (every two seconds) and by for two superimposing the photographs c o n s e c u t i v e times. Thanks t o t h i s v i s u a l i z a t i o n , we c a n d e f i n e t h r e e z o n e s i n t h e amorphous layer, when t h e t a n g e n t i a l stress b e g i n s t o to the increase zlowly in proportion displacerrent. The f i r s t zone is c l o s e t o t h e f i x e d frame o,O, .Most of t h e bubbles are h e r e s t a t i o n n a r y and some o f them h a v e v e r y s m a l l disp1acement.s. If w e r e g a r d t h e an:orphous l a y e r i n a r e f e r e m e moving w i t h t h e s p e e d of t h e lower c r y s t a l l i n e r a f t , w e o b s e r v e a z o n e c l o s e t o t h e m o b i l e c r y s t a l l i n e r a f t where t h e b u b b l e s d o n ' t move r e l a t i v e t o t h e moving frawe. Between t,hese two r e g i o n s , t h e r e is a m i d d l e zone wl-.ich is a t o u t t e n b c b b l e s t h i c k f o r e v e r y H i l l number. Here, t h e b u b b l e s h a v e d i s p l a c e m e n t of l a r g e amplituc'e r e l a t i v e t o b o t h t h e f i x e d acd These m o t i o c s a r e numerous the mcsving frame. We zlso n o t i c e that the and d i s c r d e r e d . d i r e c t i o n caf t h e bznd depenc's c.n t h e K i l l number and c o r r r s p o c d s t o , a s l i p l i n e . The s h e a r b a r d , is l o c a l i s e d where t h e p l a s t i c d e f o r m a t i o r . h/L 0.5 and h i t s c r o s s e s t h e whole l a y e r i f one o f t h e c r y s t a l l i n e r a f t s wher. h/L < 0.5. These d i f f e r e n t c o n f i g u r a t i o n s a r e r e l a t e d t o differences i n mechanical behaviour. Further in the t e s t , wher. t h e s l i d i n g s t r e E s reack-,es i t s limit, we f i n d a g a i n t h e t h r e e z o c e s t h a t WE hzve j u s t d e s c r i b e d ( f i g u r e 4B) w i t h t h e s a r e c h a r a c t e r i s t i c s . B u t , t h e rr.iddle z.or,e i n c r e a s e s its thickness and its direction becomes k-orizontal, w h i l e t h e t h i c k n e s s of t h e two l a t e r a l z o c e s i s d e c r e a s i n g t o a minimwn v a l u e W e note t h a t aromc' t h e of abcut f i v e b u b t l e s . s i d e s c.f t h e l a y e r , t h e p a t h s c.f t h e b u b b l e s a r e not i n E s t r a i g h t l i n e but c i r c u l a r . Green 1111 o b s e r v e d s u c h pher.omena i n a p l a s t i c s h e a r e d j u n c t i o n f o r which t h e H i l l number is l e s s t h a n 1 . 4 7 and d i v i d e d t h e j u n c t i o n i n t o a p i d d l e zone ur.dergoing a p u r e s h e a r i n g s u r r o b n d e d . by two l a t e r a l z o n e s i n p u r e t o r s i o n . Frorr. our e x p e r i m e n t a l r e s u l t s c o m t i n e d with a s i m F l e mechanical approach based c n a Mohr d e z c r i p t i o c , w e c a n E x p l a i n t h e e v o l u t i o n f, c h x a c t e r i s t i c from t h e o f t h e stress p l a s t i c f l o w f o r kligh k i l l nurrber. We f i r s t d e f i n e , f o r t h e bubble r a f t s , a plasticity criterion t y : u tg'P+ c = r ,where c i s t h e c o k e s i o n of t h e K/m f o r o u r bLbble scap rr.ateria1 ( c = 17.8 r a f t s ) and (r is a normzl stress a s s o c i a t e d k i t h a r e c e s s a r y r e o r g a n i z i n g of t h e b u t b l e s i n s i d e t h e s l i d i n g l a y e r i n l a r g e displacerr.ents. \re represer.t the stress f i e l d by t h e t e n s o r ( 5 1 7 ) a z s u m i n g t h a t t h e stress f i e l d is T 9% uniform i n t h e l a y e r e x c e p t a t i t s l a t e r a l s i d e s . D i s p l a c e m e n t s i n t h e r.ormal d i r e c t i o n a r e r.ot p o s s i b l e b e c a u s e of t h e r i g i d i t y o f t h e rafts 191 so the aecharical crystalline i n s t a b i l i t i e s c a n o n l y c c c u r w i t h a n i n c r e a s e of t h e ncrrnal stress q, Depending c n w h e t h e r h j L is g r e a t e r t h a n 0 . 5 o r n o t t h e s h e a r band c r o s s e s t h e whole l a y e r or h i t s t h e f i x e d crystalline r a f t .
>
.
3.2.1.1.
h/L
>
0.5
W e call h, the thickness of the shear b a n d , d h , i s a normal e l e m e n t a r y d i s p l a c e m e c t due As h t o t h e s l i d i n g i n s i d e t h e p l a s t i c zone. r e m a i n s c o n s t a n t t h i s small d i s p l a c e m e n t must b e e q u i l i b r a t e d by t h e e l a s t i c d e f o r m a t i o n of t h e two l a t e r a l z o n e s . T h i s d i s p l a c e m e n t t h e n c a u s e s a small v a r i a t i o n of t h e normal stress g i v e n by[dh,/(h-h,)].E = do;, We c h e c k e d t h a t h , d o e s n ' t depend o n t h e H i l l number, s o t h i s e q u z t i o n shows that a;, d e c r e a s e s i f h i n c r e a s e s a n d v i c e v e r s a . The g e o m e t r i c a l r e p r e s e n t a t i o n o f t h e stress tensor with the evolution of f , ( f i g . 5 A'- B ' ) shows t h a t a n i n c r e a s e of q, i s r e l a t e d t o a d e c r e a s e of t h e s l i d i n g stress , which p r o v e s t h a t i f h/L > 0.5, t l v a r i e s i n t h e same way as h/L r a t i o . 3.2.1.2.
0.25
<
h/L
<
0.5
The s h e a r band h i t s t h e u p p e r c r y s t a l l i n e raft, t h e r e f o r e s i n c e t h e c r y s t a l l i n e r a f t is much more r i g i d t h a n t h e amorphous s t r u c t u r e , we impose a s l i d i n g i n t h e h o r i z o r . t a 1 d i r e c t i o n . anymore and a The s i d e s a r e c o t f r e e normal stress waZ c a n a p p e a r . The d e s c r i p t i o n by a Mohr r e p r e s e n t a t i o n i n F i g . 5 C ' shows t h a t as t h e s l i d i n g d i r e c t i o n is h o r i z o n t a l , P, must is t h e n reach the p l a s t i c i t y criterion. g r e a t e r t h a n f o r H i l l number more t h a n 0.5. From continLurr1 s o l i d m e c h e n i c s , we c a n f i n d a g a i n t h e e x p e r i m e n t a l c u r v e c f t h e e v o l u t i o n of t h e f o r H i l l number great,er t h a n l i m i t i n g stress 0.25.
el
3.2.2.
h/L
<
0.25
By coL*nting t h e i n s t a b i l i t i e s i n t h e layer,we p l o t the distribution of i n s t a b i l i t i e s i n t h e amorpkous f i l m ( F i g . 6 ) . This curve confirms t h e preceeding r e s u l t s particularly the existence of f o r h/L > 0 . 2 5 , two l o c k e d z o n e s r e s p e c t i v e l y c l o s e t o t h e f i x e d and t h e moving frame z u r r o u n d i n g a m i d d l e zone where t h e b u b b l e s h a v e d i s o r d e r e d and numerc.us movetxer.ts. But i f h/L is less t h a n 0 . 2 5 we o b s e r v e a change i n t h e d i s t r i b u t i o n o f t h e i n s t a b i l i t i e s : t h e d i s t r i b u t i o n is not centered a r o u n d x = h / 2 anymore and t h e number o f i n s t a b i l i t i e s d o e s r t ' t s t o p i n c r e a s i n g frorr. t h e f i x e d f r a m e t o t h e moving frame. These numerous n:icroscopic i n s t a b i l i t i e s are r e s p o n s i b l e f o r bour.dary e f f e c t s ( l a r g e d e f o r m a t i o c o f t h e s i d e s ) which d i d n ' t o c c u r f o r h i g h H i l l number. We c a n n o t e x p l a i n t h e m e c h a n i c a l b e h a v i o u r w i t h t h e t h e o r y u s e d f o r H i l l number g r e z t e r than 0.25. F o r small H i l l number, t h e t h i c k n e s s o f t h e l a y e r c o r r e s p o n d s t o t h e dimensions o f t h e c l u s t e r s of b u b b l e s a r o u n d which t h e l o c a l i n s t a b i l i t i e s o c c u r . T h e r e f o r e , t h e e f f e c t s due t o t h e s e m o t i o n s a r e d o m i n a n t and t h e p l a s t i c f l o w i s n o t o n l y d e p e n d a n t o n t h e H i l l number also on the thickness h. anymore but c
8
'
I
A'
B'
I
C'
I i
Hill N u m b e r h / L
Figure 5 : Mohr representation of the stress tensor inside the layer for different configurations to describe the evolution of Z L for high Hill numbers ( A ' , B', C'). ( A ' ) : h/L = 1 : Reference configuration. (B') : 0.5 < h/L < 1 : the normal stress increases and elbecomes smaller than for h/L = 1 . (C') : 0 . 2 5 < h/L C O . 5 : the normal stress appears, the slip line is horizontal then 'E is greater than 1 for 0.5< h/L< 1 .
9
In addition, for large Hill number the deformztior. is localised in a shear tar.d with a privilegied direction : this process is related to a minimization of the er.ergy. For low Hill number, the thickness is too small and the slip line is necessarily horizontal, so there is ar increase of the deformation energy, which can explain the steep increase of the limiting shear stress if h/L 0.25.
<
i
*2 .1
0 0
.1
.2 .3 .4
.5 .6 .7
.8
.9
1
x/h
-
Figure 6 : Distribution of instabilities in amorphous layer for h/L = 0.5 ( ) ard h/L = 0.25 t---*). x/h = C : bcundary between the layer arZ fixed crystalline raft. x/b = 1 : boundary between the layer and moving crystalline raft.
the for the the
4 COICLUSION
With the bubble soap raft mcdel, we could determine the shear behaviour of an amorphous layer adi-ering to two rectangular crystalline rafts. For large disFlacements the tangential stress reaches a ccnstant limiting value z, , which chatacterizes the plastic flow of the layer and dependE on the Hill nunber. F o r large Hill nun,ber (h/L > 0.25) a mechanical apprcach based on the cbservatior. of the kinematic field cf the plastic flow can explain the behaviour of the layer. But for small Hill nLmber, the macrosccpic behsviour of the amorphous film is gcvernc-d by physic21 effects due to the motions of small clusters of five cr six bubbles. 5 ACKNOhLEDGMENTS We are grateful to G. during this work.
MEILLE for his help
Bibliography Ph. KAPSA, "Etude microscopique et macroscopique de l'usure en regime de lubrification limite", Th&se d'Etat 8219, Universitd Claude Bernard Lyon, p. 54 (1982). J.M. GEORGES, J.M. MARTIN, "Quelques relations entre les structures et les propridtes mdcaniques des films de lubrification limite", Eurotrib 85, V O ~ . 11, 5.2.1, p. 8-10. HILL, "Mathematical theory of plasticity", Claredcn Press, Oxford, p. 226-235 (1950). A. TONCK, Ph. KAPSA, J. SABOT, "Mechanical behavior of tribochemical films under a cyclic tangential load flat-contact", Trans. ASME, vol. 108, p. 117-122 (1986). L. BRAGG, J.F. NYE, " A dynamical model of a crystal structure I", Proc. Roy. SOC., A190, p. 474 (1947). L. BRAGG, W.M. LOMER, "A dynamical model of a crystal structure II", Proc. Roy. SOC., A.196, p. 171 (1949). W.M. LOMER, "A dynamical mcdel of a crystal structure III", Prcc. Roy. SOC., A. 196, p. 182 (1949). A.S. ARGON, L.T. SHI, "Simulation of plastic flow and distribLted shear relaxation by means of the amorphous Bragg bubble raft", Conf. Proc. Met. SOC. AIME, p. 279 (1982). J.M. GEORCES, G. MEILLE, J.L. LOUBE?', A.M. TOLEN, "Bubble raft mcdel for indentation with adhesion", Nature, vol. 320, p. 342-344 (1986). A.S. AF-GON, H.V. KUO, "Plastic flok in disordered bcbble raft", Materials science and Engineering, p. 107 (1979). A.P. GREEN, "The plastic yielding of metal junctions due to comtBined shear and pressbre", Jourr a1 of the Mechsnical Physics of solids, vol. 2, p. 202 (1954).
Shear behaviour of an amorphous film with bubbles soap raft model D. Mazuyer, J.M. Georges and B. Carnbou
Wear and f r i c t i o n i n boundary regime a r e o f t e n governed by t h e mechanical b e h a v i o r o f v e r y t h i n l a y e r s s e p a r a t i n g t h e two s o l i d s i n s l i d i n g c o n t a c t . Sometimes, t h e s e l a y e r s a r e amorphous. we s i m u l a t e t h e s l i d i n g o f two homogeneous r e c t a n g u l a r Thanks t o a b u b b l e soap r a f t model, The t h i c k n e s s and t h e l e n g t h o f t h i s l a y e r a r e c r y s t a l l i n e r a f t s s e p a r a t e ? by an amorphous l a y e r . vsriable. The k i n e m a t i c f i e l d o f t h e p l a s t i c flow k i t h l a r g e displacements is experimentally E x p e r i m e c t a l r e s u l t s show a determine? i n t h e same time as t h e r e c o r d c f t h e t a n g e n t i a l s t r e s s . A t large dependence o f t h e s l i d i n g s t r e s s w i t h t h e r a t i o t h i c k n e s s - l e n g t h of t h e amorphous l a y e r . r a t i o , s o l i d mechanical t h e o r i e s e x p l a i n t h e b e h a v i o r , b u t a t small r a t i o , t h e e f f e c t o f m i c r o s c c p i c i n s t a b i l i t i e s i n t h e amorphous l a y e r i s dominant. 1 INTRODUCTION
2 EXPERIMENTAL PROCEDURE
I n t h e boundary l u b r i c a t i o n regime, a t h i n discontinuous film separates the sliding s u r f a c e s . T h i s f i l m is c r e a t e d by a d h e s i o n and packing o f p r o d x t s coming from t h e s u r f a c e s i n c o n t a c t and t h e r e a c t i o n w i t h an anti-wear In t h i s case, t h e mechanical a d d i t i v e 111. a c t i o n o f f r i c t i o n makes t h e s e f i l m s amorphcus (21. In addition, the tribochemical f i l m s undergo h i g h p r e s s u r e , i n some cases high t e m p e r a t u r e and h i g h s h e a r r a t e s i n t h e c o n t a c t . These e f f e c t s a r e r e s p o n s i b l e f o r a d u a l i t y between a b r i t t l e b e h a v i o u r and a d u c t i l e one i n wear o f t h e f i l m by d e l a m i n a t i o n 121. Two mechanical p r o p e r t i e s , t h e compressive and shearing strengths are essertial and I t i s w e l l known t h a t i n t h e s t a t i c related. situation the p l a s t i c yield strength of a layer compressed between two r i g i d s o l i d s depends cn t h r e e f a c t o r s : p l a s t i c p r o p e r t i e s of t h e l a y e r , r a t i o of t hick n es s t o l en g t h c a l l e d t h e H i l l nurrber 131 and a d h e r e n c e between t h e l a y e r and t h e s u b s t r a t e s 121. Concerning the shearing behaviour, m i c r o s l i d i n g e x p e r i m e n t s show t h a t d i f f e r e n t t r i b o c h e m i c a l f i l m s have a l m c s t t h e same e l a s t i c properties, however t h e i r anti-wear p r o p e r t i e s We a r e l o o k i n g i f t h e s e are different 14). p r o p e r t i e s are n o t more r e l a t e d t o a p l a s t i c o r a v i s c o p l a s t i c behaviour. I n o r d e r t o understard t h e s h e a r i n g p r o c e s s i n a d u c t i l e amorphous f i l m i n l a r g e d e f o r m a t i o n , we u s e a bubble soap r a f t model w i t h which Bragg v i s u a l i z e d d e f e c t s i n m a t e r i a l s 151, 161, 171. Bubble r a f t s p r o v i d e B u s e f u l two d i m e n s i o n a l model f o r t h e s t u d y o f solids. An attractive repulsive potential between p a r t i c l e s i s due t o t h e s u r f a c e t e n s i o r . o f t h e soap s o l u t i o r . and t h e p r e s s u r e i n s i d e t h e b u b b l e s and g o v e r n s t h e i r b e h w i o u r 181. With a uniform s i z e o f b u b b l e s , we g e t a c r y s t a l l i n e l a t t i c e ; w i t h two d i f f e r e n t s i z e s a p p r o p r i e t a l y mixed an amorphous s t r u c t u r e . The aim o f t h i s p a p e r is t o s t u d y t h e e v o l u t i o n of t h e p l a s t i c flow for a l a r g e r a n g e of t h e t h i c k n e s s t o l e n g t h r a t i o o f t h e l a y e r .
The e x p e r i m e n t a l c o n f i g u r a t i o n i s shown i n f i g . 1 . A i r a t c o n s t a n t p r e s s u r e is blown througk s h o r t c a p i l l a r i e s t o produce b u b b l e s a t t h e a soap solutior. (surface surface of N.m-’ ). If o n l y a t e n s i o n : b;”= 6 . 9 c a p i l l a r y i s used w e o b t a i n a r a f t w i t h E c r y s t a l l i n e s t r u c t u r e . With two c a p i l l a r i e s , we a f t e r mixing. The o b t a i n an amorphous r a f t , c r y s t a l l i n e s t r u c t u r e is made o f 2 . 3 ma d i a m e t e r bubbles, and t h e amorphous s t r u c t u r e is a m i x t u r e (45-55) o f 2 . 3 a r d 1 . 6 mm d i a m e t e r I t is known t h a t t h e bubbles r e s p e c t i v e l y . mechanical p r o p e r t i e s f o r an amorphous l a y e r do n o t v a r y w i t h t h e r a t i o o f bubble s i z e s over a W e c r e a t e two r e c t a n g u l a r l a r g e r a n g e 191. c r y s t a l l i n e rafts which a d h e r e t o two p a r a l l e l , a2fi, ( f i g . 1). They a r e a b o u t frames twelve b u b b l e s t h i c k . We s e p a r a t e t h e two c r y s t a l l i n e r a f t s by an amorphous l a y e r whose T h e frame l e n g t h and t h i c k n e s s a r e v a r i e d . WzA,is t h e n moved a t a c o n s t a n t low speed (1 m m / s ) , p a r a l l e l t o t h e frame ~,fi,c a u s i n g a s h e a r where t h e d i s p l a c e m e n t i s imposed. During t h e t e s t ( a b o u t 20 s e c c n d s ) , t h e d i a m e t e r c f t h e howevtr t h e b u t b l e is decreasing very slowly, v a r i a t i o n i s t o o small t o a f f e c t t h e mechanical b e h a v i o u r . The s t i f f n e s s G f t h e amorptous l a y e r is much lower t h a n t h a t o f t h e c r y s t a l l i n e l a y e r s o t h e whole d e f o r m a t i o n due t o t h e 1101, impose? d i s p l a c e a e n t o c c u r s i n t h e amorphous film. During t h e e x p e r i m e n t , the tangertial f o r c e i s t r a n s m i t t e d t o t h e f i x e d frame End c o n t i n L o u s l y r e c o r d e d . By f i l m i n g t h e experiment on a v i d e o t a p e system and by photographing t h e t e s t a t r e g u l a r two s e c o n d s time i n t e r v a l s , w e c z n determine t h e d i s p l a c e m e n t s i n t h e an.orphcus l a y e r with a r e s o l u t i o n corresponding t o l e z s t h a n h a l f a bubble. W e neglect a l l the effects c,f t h e roughness. The boundary c o n d i t i o n s a r e determined by t h e c r y s t a l l i n e r a f t s ( a d h e r e n c e of the layer t o the s u b s t r a t e s ) .
a,n,
4
Figure 1 : Bubble raft used to simulate a sliding process. The displacement of the frameol,qcauses a shearing of the amorphous layer. The tangential force F is transmitted and measured on the frame IX,~.
5
-
3 RESULTS A N D LISCUSSION
We c a l l h t h e t h i c k n e s s o f t h e l a y e r and L its length. 3.1 R e s u l t s for a g i v e r . l a y e r
The m e c h a n i c a l b e h a v i o u r of a n amorphous is governed by local raft of bubbles displa.cemer.ts of t h e b u b b l e s r e s u l t i n g from two s i m u l t a n e o c j p - o c e s s e s d e s c r i b e d by Argcn 1101. F i r s t , t h e r e i s what Argon c a l l s a d i f f u s e E h e a r c a u s i n g t h e r o t a t i o n of c l u s t e r s of 6 b u b b l e s . Around t h e s e r o t a t . i n g g,roups, t h e r e are s m a l l a m p l i t u d e t r a r s l a t i o n e of l i n e s o f b L b b l e s . two locel t r a n s formations create These a e c h a r : i c a l i n s t a b i l i t i e s whick a r e r e s p o n s i b l e for a p l a s t i c s t r a i n . In order t o describe the we define a two behaviour in sliding, d i m e n s i o n a l s h e i r stress 3 E i v e n by t h e r a t i o of t h e measured t a n g e n t i a l force F t.o t h e 1engt.h L o f t h e l a y e r . ., W e assc,ciate t o t h i s s t r e s s a two-dimensiocal d i s t o r t i o n a l s t r c i n = U/h where U is t h e . r e l a t i v e d i s p l a c e m e n t o f t h e t w c , frames and ciz fi2 e n d h t h e t h i c k n e s s of t h e layer.
p
e f f e c t i v e e l a s t i c s h e a r mcdulus G = 1 , 2 lO-'N/m which is j n agreemer.t w i t h t h e E x p e r i m e n t a l v a l u e s c a l c u l a t e d from a n i n d e r . t a t i o r t e s t 191. DLring t h i s p e r i o d , w e observe f e w local movements o f t h e b L b b l e s . Then, t h e stress i s s t i l l i n c r e a s i n g niore g r a d u a l l y , i n proportion t o t h e defamation and r e a c h e s a const.ar?t l i m i t i n g v a l u e TI. Ar0ur.d t h i s mean b a l u e , we c o t i c e small p e r t u r b a t i o n s due t o t h e numerous mechar:ical i n s t a b i l i t i e s . For e v e r y t e s t , we h a v e o b s e r v e d t h a t f o r a g i v e n H i l l number, F is p r o p c r t i o n a l t o L , and w e o b t a i n a unique 1 i m i t . i n g stress fl =: F l / L . 3.2
c f t h e Kill Influence b e h z v i o w of t h e l a y e r
number
cn
the
F i g u r e 3 shc,ws t h e e v o l u t i o n c 8 f t h e s l i d i n g stress u i t h H i l l numtser. I t is i n t e r e s t i n g t o o b s e r v e t h z t e x t r e m a o c c u r f o r s i m r l e v a l u e s of h/L ( r e s p e c t i v e l y 0.15, 0 . 2 5 , 0 . 5 ) . E x p e r i m e n t s made w i t h c i f f e r e n t s i z e s of b u b t l e s g i v e t h e sarre c u r v e .
a,n,
Hill Number h/L
Shear strain
v=
U/h
F i g u r e 2 : M e c h a n i c a l b e h a v i o u r of t h e s l i d i n g airorphous l a y e r . A : t h e t a n g e n t i a l stress Z is increEsing i n porportion t o the deformation. B : t h e t a n g e n t i a l stress f is reaching a l i m i t i n g value c h a r a c t e r i s t i c fror. the p l a s t i c f l o w . F i g u r e 2 shows a t y p i c 2 1 r e c o r d c f t h e f o r c e F v e r s u s d i s p l a c e m e r t U. A t f i r $ t , t h e stress i n c r e a s e s i n p r o p o r t i o r . t o t h e s t r a i n . I n s F i t e cf l i n e a r i t y t h i s s t r , a i n i s n o t c o m p l e t e l y r e v e r s i b l e , however i t is p o s s i b l e t o d e f i n e E "pseudo" e l a s t i c s h e a r nloduluz c h a r a c t e r i s t i c Then we o b t a i n a n o f small s t r 2 i n s b e h a v i o u r .
F i g u r e 7 : I n f l u e n c e o f t h e H i l l numter on t h e Po i s t h e l i m i t i n g t a r g e n t i a l p l a s t i c flow. stress f o r h/L = 1. ( A ) : e x p e r j ment a1 p o i n t s .
W e c a n d i s t i n g u i s h two r e g i o n s i n t h e c u r v e ( f i g . 3 ) : a first f o r h/L greater t h a n 0 , 2 5 acd E s e c o n d f o r k / L less t h a n 0 . 2 5 . We l a t e r show the plastic t h a t i f h/L is greater t h a n C.2E;, beha.viour c a n t e e x p l a i n e d t y t h e continuum the s o l i d s m e c h a n i c s t u t a t h/L l e s s t h a n C.25, d e f e c t s i n t h e arrc'rptous l a y e r a r e d o m i n a n t . 3.2.1
-
h/L
>
0,25
WE v i s u z l i z e i n f i g u r e 4, t h e displacement: a t t h e b e g i n n i n g of t h e l i n e a r i n c r e a s e . field, of t h e s t r e s s a c c o r d i n g t o t h e sk.ear s t r a i n ( f i g u r e 4 A ) and when t h e s t r e s s r e a c h e s i t s l i m i t (figure 4E). These pictures are cbtained by nhotographing t h e experiment, a t regular time
6
,IA
Shear strain
= U/h
Figure 4 : Description on the evolution of the kinematic field of the plastic flow, by superimposition of two successive pictures of the layer, parallel to the curve sliding stress versus shear deformation. (A) : the tangential stress begins to increase slowly in proportion to the displacement. The shear band 2 appears between two lateral zones 1 and 3 respectively locked relative to the framew,qand the frame ciz f$ (B) : the limiting stress ilis reached. The thickness of the shear band increases while the thickness of the two lateral zones (1, 3 ) remains constant at five bubbles, the slip line becomes horizontal.
7
intervals (every two seconds) and by for two superimposing the photographs c o n s e c u t i v e times. Thanks t o t h i s v i s u a l i z a t i o n , we c a n d e f i n e t h r e e z o n e s i n t h e amorphous layer, when t h e t a n g e n t i a l stress b e g i n s t o to the increase zlowly in proportion displacerrent. The f i r s t zone is c l o s e t o t h e f i x e d frame o,O, .Most of t h e bubbles are h e r e s t a t i o n n a r y and some o f them h a v e v e r y s m a l l disp1acement.s. If w e r e g a r d t h e an:orphous l a y e r i n a r e f e r e m e moving w i t h t h e s p e e d of t h e lower c r y s t a l l i n e r a f t , w e o b s e r v e a z o n e c l o s e t o t h e m o b i l e c r y s t a l l i n e r a f t where t h e b u b b l e s d o n ' t move r e l a t i v e t o t h e moving frawe. Between t,hese two r e g i o n s , t h e r e is a m i d d l e zone wl-.ich is a t o u t t e n b c b b l e s t h i c k f o r e v e r y H i l l number. Here, t h e b u b b l e s h a v e d i s p l a c e m e n t of l a r g e amplituc'e r e l a t i v e t o b o t h t h e f i x e d acd These m o t i o c s a r e numerous the mcsving frame. We zlso n o t i c e that the and d i s c r d e r e d . d i r e c t i o n caf t h e bznd depenc's c.n t h e K i l l number and c o r r r s p o c d s t o , a s l i p l i n e . The s h e a r b a r d , is l o c a l i s e d where t h e p l a s t i c d e f o r m a t i o r . h/L 0.5 and h i t s c r o s s e s t h e whole l a y e r i f one o f t h e c r y s t a l l i n e r a f t s wher. h/L < 0.5. These d i f f e r e n t c o n f i g u r a t i o n s a r e r e l a t e d t o differences i n mechanical behaviour. Further in the t e s t , wher. t h e s l i d i n g s t r e E s reack-,es i t s limit, we f i n d a g a i n t h e t h r e e z o c e s t h a t WE hzve j u s t d e s c r i b e d ( f i g u r e 4B) w i t h t h e s a r e c h a r a c t e r i s t i c s . B u t , t h e rr.iddle z.or,e i n c r e a s e s its thickness and its direction becomes k-orizontal, w h i l e t h e t h i c k n e s s of t h e two l a t e r a l z o c e s i s d e c r e a s i n g t o a minimwn v a l u e W e note t h a t aromc' t h e of abcut f i v e b u b t l e s . s i d e s c.f t h e l a y e r , t h e p a t h s c.f t h e b u b b l e s a r e not i n E s t r a i g h t l i n e but c i r c u l a r . Green 1111 o b s e r v e d s u c h pher.omena i n a p l a s t i c s h e a r e d j u n c t i o n f o r which t h e H i l l number is l e s s t h a n 1 . 4 7 and d i v i d e d t h e j u n c t i o n i n t o a p i d d l e zone ur.dergoing a p u r e s h e a r i n g s u r r o b n d e d . by two l a t e r a l z o n e s i n p u r e t o r s i o n . Frorr. our e x p e r i m e n t a l r e s u l t s c o m t i n e d with a s i m F l e mechanical approach based c n a Mohr d e z c r i p t i o c , w e c a n E x p l a i n t h e e v o l u t i o n f, c h x a c t e r i s t i c from t h e o f t h e stress p l a s t i c f l o w f o r kligh k i l l nurrber. We f i r s t d e f i n e , f o r t h e bubble r a f t s , a plasticity criterion t y : u tg'P+ c = r ,where c i s t h e c o k e s i o n of t h e K/m f o r o u r bLbble scap rr.ateria1 ( c = 17.8 r a f t s ) and (r is a normzl stress a s s o c i a t e d k i t h a r e c e s s a r y r e o r g a n i z i n g of t h e b u t b l e s i n s i d e t h e s l i d i n g l a y e r i n l a r g e displacerr.ents. \re represer.t the stress f i e l d by t h e t e n s o r ( 5 1 7 ) a z s u m i n g t h a t t h e stress f i e l d is T 9% uniform i n t h e l a y e r e x c e p t a t i t s l a t e r a l s i d e s . D i s p l a c e m e n t s i n t h e r.ormal d i r e c t i o n a r e r.ot p o s s i b l e b e c a u s e of t h e r i g i d i t y o f t h e rafts 191 so the aecharical crystalline i n s t a b i l i t i e s c a n o n l y c c c u r w i t h a n i n c r e a s e of t h e ncrrnal stress q, Depending c n w h e t h e r h j L is g r e a t e r t h a n 0 . 5 o r n o t t h e s h e a r band c r o s s e s t h e whole l a y e r or h i t s t h e f i x e d crystalline r a f t .
>
.
3.2.1.1.
h/L
>
0.5
W e call h, the thickness of the shear b a n d , d h , i s a normal e l e m e n t a r y d i s p l a c e m e c t due As h t o t h e s l i d i n g i n s i d e t h e p l a s t i c zone. r e m a i n s c o n s t a n t t h i s small d i s p l a c e m e n t must b e e q u i l i b r a t e d by t h e e l a s t i c d e f o r m a t i o n of t h e two l a t e r a l z o n e s . T h i s d i s p l a c e m e n t t h e n c a u s e s a small v a r i a t i o n of t h e normal stress g i v e n by[dh,/(h-h,)].E = do;, We c h e c k e d t h a t h , d o e s n ' t depend o n t h e H i l l number, s o t h i s e q u z t i o n shows that a;, d e c r e a s e s i f h i n c r e a s e s a n d v i c e v e r s a . The g e o m e t r i c a l r e p r e s e n t a t i o n o f t h e stress tensor with the evolution of f , ( f i g . 5 A'- B ' ) shows t h a t a n i n c r e a s e of q, i s r e l a t e d t o a d e c r e a s e of t h e s l i d i n g stress , which p r o v e s t h a t i f h/L > 0.5, t l v a r i e s i n t h e same way as h/L r a t i o . 3.2.1.2.
0.25
<
h/L
<
0.5
The s h e a r band h i t s t h e u p p e r c r y s t a l l i n e raft, t h e r e f o r e s i n c e t h e c r y s t a l l i n e r a f t is much more r i g i d t h a n t h e amorphous s t r u c t u r e , we impose a s l i d i n g i n t h e h o r i z o r . t a 1 d i r e c t i o n . anymore and a The s i d e s a r e c o t f r e e normal stress waZ c a n a p p e a r . The d e s c r i p t i o n by a Mohr r e p r e s e n t a t i o n i n F i g . 5 C ' shows t h a t as t h e s l i d i n g d i r e c t i o n is h o r i z o n t a l , P, must is t h e n reach the p l a s t i c i t y criterion. g r e a t e r t h a n f o r H i l l number more t h a n 0.5. From continLurr1 s o l i d m e c h e n i c s , we c a n f i n d a g a i n t h e e x p e r i m e n t a l c u r v e c f t h e e v o l u t i o n of t h e f o r H i l l number great,er t h a n l i m i t i n g stress 0.25.
el
3.2.2.
h/L
<
0.25
By coL*nting t h e i n s t a b i l i t i e s i n t h e layer,we p l o t the distribution of i n s t a b i l i t i e s i n t h e amorpkous f i l m ( F i g . 6 ) . This curve confirms t h e preceeding r e s u l t s particularly the existence of f o r h/L > 0 . 2 5 , two l o c k e d z o n e s r e s p e c t i v e l y c l o s e t o t h e f i x e d and t h e moving frame z u r r o u n d i n g a m i d d l e zone where t h e b u b b l e s h a v e d i s o r d e r e d and numerc.us movetxer.ts. But i f h/L is less t h a n 0 . 2 5 we o b s e r v e a change i n t h e d i s t r i b u t i o n o f t h e i n s t a b i l i t i e s : t h e d i s t r i b u t i o n is not centered a r o u n d x = h / 2 anymore and t h e number o f i n s t a b i l i t i e s d o e s r t ' t s t o p i n c r e a s i n g frorr. t h e f i x e d f r a m e t o t h e moving frame. These numerous n:icroscopic i n s t a b i l i t i e s are r e s p o n s i b l e f o r bour.dary e f f e c t s ( l a r g e d e f o r m a t i o c o f t h e s i d e s ) which d i d n ' t o c c u r f o r h i g h H i l l number. We c a n n o t e x p l a i n t h e m e c h a n i c a l b e h a v i o u r w i t h t h e t h e o r y u s e d f o r H i l l number g r e z t e r than 0.25. F o r small H i l l number, t h e t h i c k n e s s o f t h e l a y e r c o r r e s p o n d s t o t h e dimensions o f t h e c l u s t e r s of b u b b l e s a r o u n d which t h e l o c a l i n s t a b i l i t i e s o c c u r . T h e r e f o r e , t h e e f f e c t s due t o t h e s e m o t i o n s a r e d o m i n a n t and t h e p l a s t i c f l o w i s n o t o n l y d e p e n d a n t o n t h e H i l l number also on the thickness h. anymore but c
8
'
I
A'
B'
I
C'
I i
Hill N u m b e r h / L
Figure 5 : Mohr representation of the stress tensor inside the layer for different configurations to describe the evolution of Z L for high Hill numbers ( A ' , B', C'). ( A ' ) : h/L = 1 : Reference configuration. (B') : 0.5 < h/L < 1 : the normal stress increases and elbecomes smaller than for h/L = 1 . (C') : 0 . 2 5 < h/L C O . 5 : the normal stress appears, the slip line is horizontal then 'E is greater than 1 for 0.5< h/L< 1 .
9
In addition, for large Hill number the deformztior. is localised in a shear tar.d with a privilegied direction : this process is related to a minimization of the er.ergy. For low Hill number, the thickness is too small and the slip line is necessarily horizontal, so there is ar increase of the deformation energy, which can explain the steep increase of the limiting shear stress if h/L 0.25.
<
i
*2 .1
0 0
.1
.2 .3 .4
.5 .6 .7
.8
.9
1
x/h
-
Figure 6 : Distribution of instabilities in amorphous layer for h/L = 0.5 ( ) ard h/L = 0.25 t---*). x/h = C : bcundary between the layer arZ fixed crystalline raft. x/b = 1 : boundary between the layer and moving crystalline raft.
the for the the
4 COICLUSION
With the bubble soap raft mcdel, we could determine the shear behaviour of an amorphous layer adi-ering to two rectangular crystalline rafts. For large disFlacements the tangential stress reaches a ccnstant limiting value z, , which chatacterizes the plastic flow of the layer and dependE on the Hill nunber. F o r large Hill nun,ber (h/L > 0.25) a mechanical apprcach based on the cbservatior. of the kinematic field cf the plastic flow can explain the behaviour of the layer. But for small Hill nLmber, the macrosccpic behsviour of the amorphous film is gcvernc-d by physic21 effects due to the motions of small clusters of five cr six bubbles. 5 ACKNOhLEDGMENTS We are grateful to G. during this work.
MEILLE for his help
Bibliography Ph. KAPSA, "Etude microscopique et macroscopique de l'usure en regime de lubrification limite", Th&se d'Etat 8219, Universitd Claude Bernard Lyon, p. 54 (1982). J.M. GEORGES, J.M. MARTIN, "Quelques relations entre les structures et les propridtes mdcaniques des films de lubrification limite", Eurotrib 85, V O ~ . 11, 5.2.1, p. 8-10. HILL, "Mathematical theory of plasticity", Claredcn Press, Oxford, p. 226-235 (1950). A. TONCK, Ph. KAPSA, J. SABOT, "Mechanical behavior of tribochemical films under a cyclic tangential load flat-contact", Trans. ASME, vol. 108, p. 117-122 (1986). L. BRAGG, J.F. NYE, " A dynamical model of a crystal structure I", Proc. Roy. SOC., A190, p. 474 (1947). L. BRAGG, W.M. LOMER, "A dynamical model of a crystal structure II", Proc. Roy. SOC., A.196, p. 171 (1949). W.M. LOMER, "A dynamical mcdel of a crystal structure III", Prcc. Roy. SOC., A. 196, p. 182 (1949). A.S. ARGON, L.T. SHI, "Simulation of plastic flow and distribLted shear relaxation by means of the amorphous Bragg bubble raft", Conf. Proc. Met. SOC. AIME, p. 279 (1982). J.M. GEORCES, G. MEILLE, J.L. LOUBE?', A.M. TOLEN, "Bubble raft mcdel for indentation with adhesion", Nature, vol. 320, p. 342-344 (1986). A.S. AF-GON, H.V. KUO, "Plastic flok in disordered bcbble raft", Materials science and Engineering, p. 107 (1979). A.P. GREEN, "The plastic yielding of metal junctions due to comtBined shear and pressbre", Jourr a1 of the Mechsnical Physics of solids, vol. 2, p. 202 (1954).