EL5EV1ER P0wder 7echn0109y 99 ( 1998 ) 154-162

< • 1/• 1 :)/;1 ~ 1~/111 • 1));11 •; 1 ;1 ••1)• ~1/111~ ~111 • •1• 1 1

A the0ret1ca1 m0de1 f0r the 5t1ck/60unce 6ehav10ur 0f adhe51ve, e1a5t1c-p1a5t1c 5phere5

C011n 7h0rnt0n ~*, 2em1n N1n9 6 "C1v11 and Mechan1ca1 En91neer1n9 D1v1510n. 5ch001 0f En91neer1n9 and App11ed 5c1ence, A5t0n Un1ver51ty. 81rm1n9ham 84 7E7, UK

"P01ymer5 am/C01101d5 6r0up. Cavend15h La60rat0ry. Un1ver510~ 0f Cam6rh19e. Can,6r1d9e C83 0HE. UK

Rece1ved 23 June 1497; rece1ved 1n rev15ed f0rm 7 May 1998: accepted 9 June 1998

A65tract

7he paper c0n51der5 the n0rma1 1mpact 0f e1a5t1c-perfect1y p1a5t1c 5phere5, w1th and w1th0ut 1nterface adhe510n, and pre5ent5 an ana1yt1ca1 501ut10n t6r the c0eff1c1ent 0f re5t1tut10n wh1ch 15 expre55ed 1n term5 0f the 1mpact ve10c1ty, the cr1t1ca1 5t1ck1n9 ve10c1ty and the ve10c1ty 6e10w wh1ch the 1nteract10n 15 a55umed t0 6e e1a5t1c. • 1998 E15ev1er 5c1ence 5.A. A11 r19ht5 re5erved.

Keyw0rd5: 1mpact; Re5t1tut10n: Adhe510n: C0ntact mechan1c5

1.1ntr0duct10n

Numer1ca1 51mu1at10n5 0f 9a5-part1c1e f10w5 are n0w c0mm0n1y app11ed t0 many 1ndu5tr1a1 pr061em5. H0wever, whether the 51mu1at10n5 track 1nd1v1dua1 part1c1e5 0r c0n51der the part1c1e pha5e t0 6e a c0nt1nuum, the way 1n wh1ch 1nter- part1c1e and part1c1e-wa11 c0111510n5 are m0de11ed ha5 a 519- n1f1cant 1nf1uence 0n the f10w pred1ct10n5. 7he centra1 155ue wh1ch ha5 t0 6e addre55ed 15 whether part1c1e5 5t1ck 0r re60und up0n c0111510n and, 1n the ca5e 0f re60und, the chan9e5 that 0ccur 1n 60th the 11near and r0tat10na1 k1net1c ener9y, Neverthe1e55, many re5earcher5 a55ume a c0n5tant c0eff1c1ent 0f re5t1tut10n and, 1n area5 5uch a5 111trat10n and part1c1e dep051t10n 5tud1e5, 1t 15 0ften a55umed that part1c1e5 wh1ch c0me 1nt0 c0ntact w1th each 0ther 0r w1th a c0nta1n1n9 5urface d0 n0t re60und.

7he 06ject1ve 0f the w0rk pre5ented here 15 t0 06ta1n an ana1yt1ca1 501ut10n f0r the c0eff1c1ent 0f re5t1tut10n 6a5ed 0n the c0ncept5 0f the0ret1ca1 c0ntact mechan1c5. We f1r5t 0f aU c0n51der the ca5e 0f e1a5t1c-perfect1y p1a5t1c 5phere5 w1th n0 1nter1ace ener9y and 06ta1n a ve10c1ty dependent c0eff1c1ent 0f re5t1tut10n wh1ch, at h19h 1mpact ve10c1t1e5, 15 1nver5e1y pr0p0rt10na1 t0 the ve10c1ty ra15ed t0 the p0wer 0.25. We then exam1ne the ca5e 0fe1a5t1c-adhe51ve 5phere5 and der1ve e4ua- t10n5 f0r the cr1t1ca1 5t1ck1n9 ve10c1ty 6e10w wh1ch re60und d0e5 n0t 0ccur and the c0eff1c1ent 0f re5t1tut10n f0r ve10c1t1e5

* C0rre5p0nd1n9 auth0r. Fax: + 44-121-333-3389; E-ma11: c.th0mt0n• a5t0n.ac.uk

a60ve the 5t1ck1n9 ve10c1ty. F1na11y, 6y a55um1n9 that the w0rk d1551pated 6y p1a5t1c w0rk and the w0rk d1551pated due t0 1nter1~ace adhe510n are add1t1ve, we 06ta1n a 9enera1 501u- t10n f0r the c0eff1c1ent 0f re5t1tut10n f0r adhe51ve, e1a5t1c- p1a5t1c 5phere5.

2. E1a5t1c-perfect1y p1a5t1c 5phere5

F0r tw0 c0ntact1n9 5phere5 0frad11 R, and e1a5t1c pr0pert1e5 E, and v, ( 1 = 1, 2) the Hert21an pre55ure d15tr16ut10n 0ver the c0ntact area 0f rad1u5 a 15

p ( r ) = - - 1 (1 ) 2 ,ra 2 ""

7he c0ntact n0rma1 t6rce P and c0ntact rad1u5 a are 91ven 6y

4 ~ t . ~ . . (2) P=~E:~R*~ ~

( 3PR* ~ ~ a - 1 4---~, 1 (3)

where 0¢ 15 the re1at1ve appr0ach 0f the tw0 part1c1e centr01d5 and

a~---R*a (4)

1n the a60ve e4uat10n5

1 1--v~ 1--v~ E "~-~= E~---~ + E , (5)

0032-5910/98/$ - 5ee fr0nt matter • 1998 E15ev1er 5c1ence 5.A. A11 r19ht5 re5erved. P11 5 0 0 3 2 - 5 9 1 0 ( 98 ) 00099-0

C. 7h0rnt0n. 2 N1n9 / P0wder 7echn0109Y 99 (1998) 154-162 155

1 1 1 ~=--+~ (6) R* R1 R2

1f the re1at1ve 1mpact ve10c1ty V 15 ju5t 1ar9e en0u9h t0 1n1t1ate y1e1d 1n 0ne 0f the 5phere5 then, u51n9 E45. (2) and (4) we may wr1te

~y

1 . , f 8E*a~ 9~m~V/= Pd0t=•15R,, "

0

(7)

where V r, wh1ch we def1ne a5 the y1e1d ve10c1ty, 15 the re1at1ve 1mpact ve10c1ty 6e10w wh1ch the 1nteract10n 6ehav10ur 15 a55umed t0 6e e1a5t1c, a.,, 15 the c0ntact rad1u5 when y1e1d 0ccur5 and m* 15 re1ated t0 the part1c1e ma55e5 m; 6y the e4uat10n

1 1 1 ~ = ~ + ~ (8) 111:1: 1111 111 2

We n0w def1ne a ~11m1t1n9 c0ntact pre55ure" p: =p,, when a = a:,. U51n9 E45. ( 1 ) and ( 3 ),

2E*a,. P: = rrR*" (9)

C0m61n1n9 E45. { 7) and (9), we 06ta1n

2 ~1-~ 1 / 2 ~ 1 / 2

v: ~ 1 k ~ 1 p~-~=3.194[E;,:.,m , ) (10)

1n the ca5e 0f a 5phere 0f den51ty p 1mpact1n9 w1th a p1ane 5urface, R* =R, m* =m and E4. (10) reduce5 t0

•" ( . k 2 E , 1 ~ 5 0 ] . -~.5-p] (11)

wh1ch wa5 0r191na11y 06ta1ned 6y Dav1e5 [ 1 ]. 1n 0rder t0 m0de1 the p05t-•y1e1d• 6ehav10ur, 1t 15 nece55ary

t0 make 50me 51mp11fy1n9 a55umpt10n5.1f p1a5t1c def0rmat10n 0ccur5 we a55ume a Hert21an pre55ure d15tr16ut10n w1th a cut- 0ff c0rre5p0nd1n9 t0 the 11m1t1n9 c0ntact pre55ure p: def1ned 6y E4. (9). After y1e1d, the n0rma1 f0rce 15 91ven 6y

c1 p

P=P~.-27r11P(r)-p: 1rdr (12)

Jl ,

11

where P¢ 15 the e4u1va1ent e1a5t1c f0rce 91ven 6y E4. (3} wh1ch w0u1d re5u1t 1n the 5ame t0ta1 c0ntact area and ap 15 the rad1u5 0f the c0ntact area 0ver wh1ch a un1f0rm pre55ure 15 a55umed, a5 1nd1cated 1n F19. 1.1nte9rat1n9 E4. (12) we 06ta1n

P=~n~a~p,..+P~. 1- (13)

C0n51der1n9 the c0nd1t10n5 at y1e1d, p: may 6e def1ned 1n term5 0f the n0rma1 f0rce Pr and c0ntact rad1u5 ay a5

Hert2

1

a J

py

L a ~ 1

r a *1 F19. 1. N0rma1 tract10n d15tr16ut10n f0r e1a5t1c-perfect1y p1a5t1c 5phere5.

3P, P: = 22ra~ (14)

0r, acc0rd1n9 t0 F19. 1,

p:-- ~ 1 - (15) 2 rm ~

7he c0ntact rad1u5 a 15 06ta1ned fr0m

3R*P,. a = ~ {16)

4E*

Hence, c0m61n1n9 E45. ( 14)-(16) we f1nd that

[ ( 1- a,, = ~ 0ra-~=a~,+a-~, • a 1 . J • a 1

(17)

5U65t1tUt1n9 E45. (16) and ( 17 ) 1nt0 E4. (13) We 06ta1n

P=P: +7rp:(a" -c: , ) (18)

5u65t1tut1n9 E4. (4), the 16rce-d15p1acement re1at10n5h1p dur1n9 p1a5t1c 10ad1n9 15 91ven a5

P=P: +~rrp: R*(0t-0t, ) (19)

wh1ch 15 11near, a5 5h0wn 1n F19.2.

A pJ . . . . . . . . . . . . . . . . . . . . . . . 1

Y

F19.2. F0rce--d15p1acen1ent re1at10n5h1p f0r e1a5t1c-perfect1y p1a5t1c 5phere5.

156 C 7h0rnt0n, 2. N1n9 / P0wder 7echn0109y 99 (1998) 154-162

1f p1a5t1c def0rmat10n 0ccur5 dur1n9 the 10ad1n9 5ta9e the c0ntact curvature dur1n9 un10ad1n9 15 1/Rp* < 1/R* due t0 permanent def0rmat10n 0f the c0ntact 5urface5. Dur1n9 un10ad1n9 the f0rce-d15p1acement 6ehav10ur 15 a55umed t0 6e e1a5t1c and 15 pr0v1ded 6y the Hert21an e4uat10n5 6ut w1th a curvature 1/Rp* c0rre5p0nd1n9 t0 the p01nt 0f max1mum c0mpre5510n. At the p01nt 0f un10ad1n9, the c0ntact area deve10ped 6y the actua1 max1mum n0rma1 f0rce P* and reduced curvature 1/Rp* 15 the 5ame a5 that wh1ch w0u1d 6e 9enerated 6y an e4u1va1ent e1a5t1c f0rce P¢* and a c0ntact curvature 1/R*. Hence, fr0m E4. (16),

Rp*P*=R*P~.* (20)

where

p~.,= 4E,R,1 ,.v, "~ /~a (21)

1t can 6¢ 5een fr0m F19, 2 that the 11near p1a5t1c 10ad1n9 curve 15 tan9ent1a1 t0 the Hert21an curve at the y1e1d p01nt and, when extended, 1nter5ect5 the vert1ca1 ax15 at P,, <0. Fr0m E45, ( 18 ) and (9) the p1a5t1c c0ntact 5t1ffne5515 def1ned a5

k5 -¢rR*p~ - 2E*a: (22)

7heref0re,

a*= P*-P•• "rrR*p~ ( 23 )

and

P~ = P: - 2E*a ; 0t, ( 24 )

5U65t1tUt1n9 E45. ( 3 ) and (4),

e.--- - 5" (25)

and, hence,

c~*~ 2P*+P, 2 vrR*p, ( 26 )

0t*:= 2P*+P, 2 vr*p: ( 27 )

C0m61n1n9 (20), ( 21 ) and (26) 1ead5 t0

R t . - 4 E * ( 2 P * + P, ~ " - 3 P * 2wp, ~] (28)

and, dur1n9 e1a5t1c un10ad1n9,

4 , 1/ ~ P= "~E" R1,* :(ct-at1, ) •- (29)

where % 15 def1ned 1n F19.2, Numer1ca151mu1at10n5,1n wh1ch the c0ntact 1nteract10n5 were 6a5ed 0n the a60ve the0ret1ca1 c0n51derat10n5, have 6een perf0rmed and F19. 3111u5trate5 the re5u1t5 06ta1ned f0r d1fferent 1mpact ve10c1t1e5. 7he f19ure 5h0w5 h0w the n0rma1 c0ntact f0rce-d15p1acement re5p0n5e

30

2 E

20

0

0 0

,.4 13

8 8 6

8 8 Q

8 r~ 8

8 D 8

•/ °

a ~ • m

100 200 300 4 0 0

n0rma1 d15p1acement ( nm ) F19. 3. Effect 0f 1mpact ve10c1ty 0n the n0rma1 c0ntact f0rce-d15p1acement re5p0n5e f0r e1a5t1c-perfect1y p1a5t1c 5phere5.

15 affected 6y the 5ever1ty 0f the 1mpact. When the 1mpact ve10c1ty 15 9reater than the y1e1d ve10c1ty, the un10ad1n9 5t1ff- ne55 1ncrea5e5 w1th 1ncrea5e 1n 1mpact ve10c1ty. C0n5e- 4uent1y, the rat10 0f w0rk d0ne dur1n9 e1a5t1c rec0very t0 the w0rk d0ne dur1n9 c0mpre5510n depend5 0n the ma9n1tude 0f the 1mpact ve10c1ty and theref0re the c0eff1c1ent 0fre5t1tut10n 15 ve10c1ty dependent, a5 dem0n5trated 1n the next 5ect10n.

3. C0eff1c1ent 0f re5t1tut10n

At 1mpact ve10c1t1e5 V, < V~ n0 p1a5t1c def0rmat10n 0ccur5 and, 19n0r1n9 ener9y 1055e5 due t0 e1a5t1c wave m0t10n 1n the tw0 60d1e5, the c0eff1c1ent 0f re5t1tut10n e = 1.0. 1f the 1mpact ve10c1ty V, > 1/.,, the re60und k1net1c ener9y 15 e4ua1 t0 the w0rk d0ne dur1n9 e1a5t1c rec0very. 7hu5,

1 , , ~ ;,,, v , -- (30)

where (/:92

( a * - a p ) = (31) R p*

Hence, u51n9 E45. (27) :md( 28 )

1 , , 3P *2 ~m~V,: = 10E.a-~------~-~-~. (32)

7he c0eff1c1ent 0f re5t1tut10n 15 def1ned a5 e = Vr/V,. 7hu5.

3P*•" e 2 - . 5E*a*m* V1" (33)

7he 1n1t1a1 k1net1c ener9y 15 e4ua1 t0 the w0rk d0ne dur1n9 dece11erat10n 0f the part1c1e5. 7heref0re, fr0m 1n5pect10n 0f F19. 2,

1 . , 2 1 "~m V1 "= "~Pya~ + ~ (Py+P*) (a* -ay ) (34)

C. 7h0rnt0n, 2. N1n9 / P0wder 7echn0109y 99 (1998) 154-162 157

U51n9 E45. (4), (14) and (26)

1 , ., 2 1 ~m V~= ~Pyay+ ~(Py+P*)(t~*-ay) (35)

7heref0re,

1 . ., 2 p.2~p# 5 Pyt~y 2•n•R*py

~m V,~=- + (36)

U51n9 E45. (2) and (22) we 06ta1n

1 . . 2 p*2 p~ ~m V1 -= ]Pyay+ 21rR*py (37)

Fr0m wh1ch

P*~=2E*ay[~*V12~m 1 , ,~ - -~ m "V:) (38)

5u65t1tut1n9 E4. (38) 1nt0 E4. (33) the c0eff1c1ent 0f re5t1- tut10n can 6e 06ta1ned fr0m

:

e = 5a* 6~,~11 J (39)

U51n9 the 5u65t1tut10n5

3P~, (40) a~ = 2"n•py

and

2P*+P•, a * 2 = 2 ~p.~

we 06ta1n

e••• ~ - ( P•• •/•:•[ 1 (V~1 = 2P*~-Py) 1- 61,~1J J

C0m61n1n9 E45. (2), (7) and (38)

(41)

(42)

(43)

7heref0re, the 9enera1 expre5510n f0r the c0eff1c1ent 0f re5t1tut10n 15

e = ( 1- ~ ] j (44)

× . v, 6

w1th the y1e1d ve10c1ty Vy def1ned 6y E4. (10) 0r, 1n the 5pec1a1 ca5e 0f a 5phere 1mpact1n9 a p1ane 5urface, 6y E4. (12). E4. (44) 5at15f1e5 the c0nd1t10n e = 1.0 when E = V~. At h19h ve10c1t1e5, (VffV1) 2 ~0 and E4. (44) 6ec0me5

1.0

0.8 e4u.

.~, t~

0.7•

e,.,

.0 ¢J

0.6•

0.5 0 - c0mputer 51mu1ated data " ~

0.4 . . . . . . . . = . . . . . . . . 10 100

n0rma115ed Ve10C1ty ( V / V y )

F19. 4. C0eff1c1ent 0f re5t1tut10n f0r e1a5t1c-perfect1y p1a5t1c 5phere5.

Vv + 2~f6 (45) •

7ak1n9 V~ >> Vy we then 06ta1n

e=~--~--] [ - ~ ] ~V~] =1.185 (46)

F0r the ca5e 0f a 5phere 1mpact1n9 a p13ne 5urface we may 5u65t1tute 1n E4. (46) f0r Vy u51n9 E4. ( 11 ) t0 06ta1n

1 5 •118

e=1.324[ EP.~,40) (V,)- •/4 (47)

Wh1Ch Wa5 91Ven 6y 7h0rnt0n and N1n9 [2]. J0hn50n [3] pr0V1ded a 51m11ar expre5510n t0 E4. (47) eXCept that the prefaCt0r Wa5 1.72 a5 a re5U1t 0f a55Um1n9 that the p135t1C n0rma1 C0ntaCt 5t1ffne55 Wa5 tW1Ce that 91Ven 6y E4. (22).

F19.4 5h0W5 the dependence 0fthe C0eff1C1ent 0fre5t1tUt10n 0n the 1mpaCt Ve10C1ty aCC0rd1n9 t0 the 9enera1 expre5510n (44). 5Uper1mp05ed 0n the f19Ure 15 the pred1Ct10n aCC0rd1n9 t0 E4. (46) and the numer1Ca1 re5U1t5 06ta1ned 6y 7h0rnt0n and N1n9 [ 2 ]. 1t Can 6e 5een that E4. (46) 15 0n1y 5at15faCt0ry f0r V> 10V:.

4. Adhe51ve e1a5t1c 5phere5

F0r adhe51ve e1a5t1c 5phere5 the the0ry 0f J0hn50n et a1. [4], c0mm0n1y kn0wn a5 JKR the0ry, 15 a55umed. J0hn50n [ 5] pr0v1ded the f0110w1n9 re1at10n5h1p 6etween the c0ntact f0rce and re13t1ve appr0ach

01 0.: [, ( , ) : ] 32/-~ ~+2+21+~ (48)

158 C 7h0mt0n, 2. N1n9/P0wder 7echn0109y 99 (1998) 154-162

w1th

a,.= ~ ] =1,16k-~;-~1 where/••15 the 1nterface ener9y and

3 P, = : ~ r R *

(49)

(50)

F19. 5 5h0w5 the f0rce d15p1acement re1at10n5h1p 1n d1men- 510n1e55 f0rm a5 def1ned 6y E4. (48).

7he f0110w1n9 exp1anat10n 0f the 5t1ck/60unce cr1ter10n wa5 pr0v1ded 6y Pr0fe550r K.L. J0hn50n 1n an 0ra1 c0ntr1- 6ut10n t0 a meet1n9 0f the 1n5t1tute 0f Phy51c5 (7r160109Y 6r0up) 0n •Adhe51ve F0rce5 1n P0wder F10w5• at the Un1- ver51ty 0f 5urrey 1n 5eptem6er 1985.

Acc0rd1n9 t0 JKR the0ry, when tw0 c0111d1n9 5urface5 c0me 1nt0 c0ntact the n0rma1 16rce 6etween the tw0 60d1e5 w1111mmed1ate1y dr0p t0 a certa1n va1ue, P = - 8P~/9 (p01nt A 1n F19. 5), due t0 Van der Waa15 attract1ve f0rce5. 7he ve10c1ty 0f the 5phere 15 then reduced 9radua11y and part 0f the 1n1t1a1 k1net1c ener9y 15 rad1ated 1nt0 the 5u65trate a5 e1a5t1c wave5. When the c0ntact f0rce reache5 a max1mum va1ue (5ay p01nt 8 1n F19. 5) the part1c1e ve10c1ty ha5 6een reduced t0 2er0 and the 1nc0m1n9 5ta9e 15 c0mp1ete. 1n the rec0very 5ta9e the 5t0red e1a5t1c ener9y 15 re1ea5ed and c0nverted 1nt0 k1net1c ener9y and the part1c1e m0ve5 1n the 0pp051te d1rec- t10n. A11 the w0rk d0ne dur1n9 the 10ad1n9 5ta9e ha5 6een rec0vered when p01nt A 15 reached dur1n9 the rec0very 5ta9e. H0wever, at th15 p01nt, when a = 0 , the 5phere rema1n5 adhered t0 the tar9et and further w0rk 15 re4u1red t0 5eparate the 5urface5. A5 5h0wn 1n F19.5, 5eparat10n 0ccur5 at p01nt F and hence the w0rk re4u1red t0 6reak the c0ntact W, 15 91ven 6y the area under the curve f0r - 1 < t~/m < 0. J0hn50n 5u9- 9e5ted that a 5uff1c1ent1y accurate expre5510n f0r W, wa5

[1~R.*.'~ ~1~ ,

H0wever, E4. ( 51 ) may 6e 1nte9rated 6etween the 11m1t5 0 and - m~ 10 06ta1n

P/Pc

8

-1 ff./¢xf

F 1

F19. 5. F0rce~115p1acement re1at10n5h1p f0r adhe51ve e1a5t1c 5phere5.

4 112 5 , 4 1/(, V (1 .18] (F•R (54)

F0r a 5phere 1mpact1n9 a f1at 5urface, R* = R and m* = m, 1ead1n9 t0

V~=1~84[(F/R)~] ~ " ~ 0 • E * " (55)

1f V, > V, then 60unce 0ccur5 and we may rewr1te E4. (53) a5

• v,1 ~ v,~ (56)

wh1ch 15 e4u1va1ent t0 E4. (14) 1n Ref. [61 and fr0m wh1ch we def1ne the c0eff1c1ent 0f re5t1tut10n 6y

(57)

F19.6 111u5trate5 h0w the c0eff1c1ent 0f re5t1tut10n var1e5 w1th 1mpact ve10c1ty. When the 1mpact ve10c1ty 15 ten t1me5 h19her than the cr1t1ca1 5t1ckm9 ve10c1ty the c0eff1c1ent 0f re5t1tut10n 15 0.995.

¢~1 f . .. 1/~

,, •E•••1 (52)

wh1ch wa5 a150 06ta1ned, 111 a d1fferent manner, 6y J0hn50n and P0110ck [ 61.

Ne91ect1n9 ener9y 1055e5 due t0 e1a5t1c wave pr0pa9at10n, the 0n1y w0rk d1551pated dur1n9 a c0111510n 15 the w0rk d0ne 1n 5eparat1n9 the 5urf11ce5 W,. 7heref0re, we may wr1te

1 . , 1 . , ~.r~v,-- ~.r~v,~=W. (53)

1f the re60und ve10c1ty Vr=0 then the 1mpact ve10c1ty V, = V, the cr1t1ca1 ve10c1ty 6e10w wh1ch 5t1ck1n9 0ccur5 and fr0m E45. (52) and (53) we 06ta1n the 5t1ck1n9 cr1ter10n

1 0 - - - -

0.8

~" 0 .6 ,

r~, ~ 0.4.

0.2•

0.0 10

n0rma115ed ve10c1ty (WV5)

F19. 6. C0eff1c1ent 0f re5t1tut10n f0r adhe51ve e1a5t1c 5phere5.

C. 7h0rnt0n. 2. N1n9 / P0wder 7echn0109y 99 (1998) 154-162 159

5. E1a5t1c-perfect1y p1a5t1c adhe51ve 5phere5

E1a5t1c-p1a5t1c 1mpact 0f adhe51ve 5phere5 wa5 5tud1ed 6y R09er5 and Reed [ 7 ] wh0 ne91ected any effect5 0f adhe510n dur1n9 the 10ad1n9 5ta9e. 7hey a150 a55umed that the def0r- mat10n due t01mpact take5 the f0rm 0fa centra1 area 0f p1a5t1c def0rmat10n 5urr0unded 6y an annu1u5 0fe1a5t1c def0rmat10n. 7h15 a55umpt10n, h0wever, 15 n0t 5upp0rted 6y the re5u1t5 0f f1n1te e1ement ana1y5e5, e.9., Hardy et a1. [ 8]. Hardy et a1. [8] 5h0wed that the c0ntact pre55ure d15tr16ut10n chan9e5 fr0m the e1a5t1c e111pt1ca1 5hape t0 an e55ent1a11y un1f0rm pre55ure d15tr16ut10n a5 the 10ad 1ncrea5e5. P1a5t1c def0rma- t10n wa5 1n1t1ated 6e10w the c0ntact 5urface 1n the centre 0f the c0ntact area and the p1a5t1c def0rmat10n 20ne 5pread unt11, when the c0ntact f0rce wa5 ca. 51x t1me5 the c0ntact f0rce at 1n1t1a1 y1e1d, 1t reached the c0ntact 5urface at the per1meter 0f the c0ntact area. W1th further 10ad1n9 the p1a5t1c def0rmat10n 20ne c0nt1nued t0 en1ar9e 6ut there rema1ned an e1a5t1c re910n at the 5urface 1n the centre 0fthe c0ntact area. 7he per515tence 0f an •e1a5t1c enc1ave• 1n the centre 0f the c0ntact area dur1n9 e1a5t0-p1a5t1c def0rmat10n wa5 a150 065erved 6y 51nc1a1r et a1. [9]. 1n the ana1y515 pre5ented 6e10w, un11ke R09er5 and Reed [ 7 ], we acc0unt f0r adhe510n effect5 dur1n9 the 10ad1n9 5ta9e and, a1th0u9h we u5e E4. (17) a5 an appr0x1mat10n, th15 d0e5 n0t 1mp1y a centra1 area 0f p1a5t1c def0rmat10n 5ur- r0unded 6y an annu1u5 0f e1a5t1c def0rmat10n. E4. (17) wa5 06ta1ned 51mp1y fr0m the a55umed 9e0metry 0f the c0ntact pre55ure d15tr16ut10n f0r a n0n-adhe51ve e1a5t0-p1a5t1c 5phere.

A5 dem0n5trated 6y Mau915 and P0110ck [10], adhe510n 519n1f1cant1y affect5 the 5u65urface 5tre55 d15tr16ut10n 1ead1n9 t0 p1a5t1c def0rmat10n near the per1meter 0f the c0ntact area. 7h15 re5u1t51n a truncat10n 0f the JKR ten511e c0ntact pre55ure d15tr16ut10n 0ver an 0uter annu1u5 0f the c0ntact area and w111 affect the c0nd1t10n5 f0r part1c1e5 t0 5t1ck t0 each 0ther. F1ch- man and Pnue11 [ 111 06ta1ned an expre5510n f0r the w0rk d1551pated due t0 the adhe510n 1nduced p1a5t1c def0rmat10n 6ut the c0mp1ex1ty prec1ude5 1t5 u5e here. H0wever, very 5ma11 ve10c1t1e5 are n0rma11y 5uff1c1ent t01n1t1ate p1a5t1c y1e1d 6e10w the centre 0f the c0ntact area and when th15 ve10c1ty 1nduced p1a5t1c def0rmat10n 20ne 5pread5 t0 reach the 5urface at the c0ntact per1meter 1t5 effect w111 d0m1nate 0ver the adhe510n 1nduced p1a5t1c def0rmat10n annu1u5 wh1ch reduce5 1n th1ckne55 w1th app11ed 10ad1n9. C0n5e4uent1y, we ne91ect the adhe510n 1nduced p1a5t1c def0rmat10n at the c0ntact per1meter 1n the ana1y515 pre5ented 6e10w.

Acc0rd1n9 t0 JKR the0ry, the n0rma1 pre55ure d15tr16ut10n 0ver the c0ntact area, 111u5trated 1n F19.7, 15 91ven 6y

2 E * a r 2]1/2

~ , r a 1 L

7he rad1u5 0f the area 0f c0ntact 15 def1ned 6y

a p JKR

F19. 7. N0rma1 tract10n d15tr16ut 5phere5.

[ 3R*P, ,~1,.~ °-U2F)

py1

a 1 n f0r adhe51ve, e1a5t1c-perfect1y p1a5t1c

(59)

where

P~ =P+2Pc+(4PP~+4P~. ) ~/2 (60)

15 the effect1ve Hert21an f0rce wh1ch w0u1d pr0duce the 5ame c0ntact area, P 15 the app11ed f0rce and P~ 15 the pu11-0ff f0rce def1ned 6y E4. (50). 7he re1at1ve appr0ach and c0ntact f0rce may a150 6e expre55ed [ 3] a5

a 2 (2,n.Fa~ ~/2

a = ~---~ -1 - - ~ 1 (61)

4E*a 3 P= ~ (8¢rFE*a5) 1/2 (62)

3R*

1n 0rder t0 m0de1 e1a5t1c-perfect1y p1a5t1c 5phere5 w1th adhe510n, we a9a1n a55ume a 11m1t1n9 c0ntact pre55ure py, a5 5h0wn 1n F19.7.7he app11ed f0rce dur1n9 p1a5t1c def0rmat10n, P~,, 15 06ta1ned fr0m

~t p

P0 = P - 2 ~ f [p ( r ) -p r ]/•dr 11

(63)

wh1ch 1ead5 t0

PP=P- 3R "----~- a: ] J

a••1 j +=a~,p,

(64)

w1th P def1ned 6y E4. (62). U51n9 E4. (58). the 11m1t1n9 c0ntact pre55ure 15 91ven 6y

2E*a~ ~ ( 2 F E * ~ ~/2 (65) PY- •trR* ,n~a-~ ~

]60 C. 7h0rnt0n, 2 N1n9 / P0wder 7echn0109Y 99 (1998) 154-162

Unf0mmate1y, f0r the ca5e 0f adhe51ve 5phere5, E4. (17) 15 n0t exact1y c0rrect 6ut we neverthe1e55 a55ume 1t t0 6e a rea50na61e appr0x1mat10n 1n 0rder t0 51mp11fy E4. (64) and 06ta1n

pp= 4E*a3 -ay(8¢rFE*a) 1/2 +¢rpy(a2--a~) (66) 3R*

D1fferent1at1n9 E45. (66) and (61),

da 2a

(67)

(68)

8y c0m61n1n9 F,45. (67) and (68) and u51n9 5u1ta61e 5u6- 5t1tut10n5 1t can 6e 5h0wn that the c0ntact 5t1ffne55 dur1n9 p1a5t1c def0rmat10n 15 91ven 6y

dPp_ 3~n~R*p~,~-2£*a.~, . ~ (69)

Dur1n9 e1a5t1c 10ad1n9 the c0ntact 5t1ffne55 wa5 91ven 6y 7h0mt0n and Y1n [ 12] a5

d.P.Pde =2E*a [a~,~,/2 (70)

where

,:,E, (71)

5u65t1tut1n9 E45. (59) and (71) 1nt0 E : ~.)) 91ve5

"~¢dP-2E*a~ 3~-3~]3VrP-~1~ V~-~ j (72)

DUr1n9 e1a5t1C reC0very, 1t wa5 5h0wn 6y N1n9 [ 13] that the C0ntaCt 5t1ffne5515 pr0v1ded 6y

d~aa -- 2E*a 3-~1~- ~ (73)

where

a.~ff1 3 R r * R t r 4 E *

P,,=P+ 2P~r~ V/~4PP~, +4P~, ~, Rp*= R*P,*

P * + ~

and

(74)

(75)

(76)

3 :0: P - , . - ~ =FR p (77)

where P~* 15 the va1ue 0f the e4u1va1ent Hert21~ f0rce,91ven 6y E4. (60), fr0m wh1ch un10ad1n9 c0mmenced. F19. 8111u5- trate5 the f0rm 0f the f0rce-d15p1acement re1at10n5h1p f0r e1a5t1c-p1a5t1c 10ad1n9 and e1a5t1c un10ad1n9, a5 06ta1ned fr0m numer1ca1 51mu1at10n5 6a5ed 0n the a60ve the0ret1ca1 m0de1. 1t can 6e 5een that, n0t 0n1y d0e5 the un10ad1n9 5t1ff- ne55 1ncrea5e w1th 1ncrea5e 1n 1mpact ve10c1ty 6ut a150, the pu11-0ff f0rce re4u1red t0 0verc0me the adhe510n 1ncrea5e5 w1th 1mpact ve10c1ty due t0 the 10ca11y 1ncrea5ed rad1u5 0f curvature, a51nd1cated 6y E4. (77).

8y c0m61n1n9 the a60ve the0ry w1th Newt0n•5 1aw5 0f m0t10n, N1n9 [ 13 ] 501ved the 1mpact pr061em numer1ca11y. F19. 9 5h0w5 the effect 0f 1nterface ener9y 0n the c0eff1c1ent 0f re5t1tut10n. F0r the ca5e 111u5trated, the y1e1d ve10c1ty 15 0.62 m 5- ~. 1t can 6e 5een that the c0eff1c1ent 0f re5t1tut10n 15 5en51t1ve t0 the va1ue 0f 1nterface ener9y at ve10c1t1e5 6e10w the y1e1d ve10c1ty 6ut 15 n0t 519n1f1cant1y dependent 0n the 1nterface ener9y when the 1mpact ve10c1ty 15 9reater than the y1e1d ve10c1ty. 7he effect 0fthe 11m1t1n9 c0ntact pre55ure (and hence y1e1d ve10c1ty) 0n the c0eff1c1ent 0fre5t1tut10n 15 5h0wn 1n F19. 10 f0r an 1nterface ener9y 0f 0.2 J m-2. When the 11m1t1n9 c0ntact pre55ure py = 3.04 6Pa the y1e1d ve10c1ty Vy = 0.62 m 5- * 15 m0re than an 0rder 0f ma9n1tude 9reater than the 5t1ck1n9 ve10c1ty V, = 0.016 m 5- ~ and the max1mum va1ue 0f the c0eff1c1ent 0f re5t1tut10n e = 1.0. When the 11m1- t1n9 c0ntact pre55ure 15 reduced t0 py ---- 1.0 6Pa , the 5t1Ck1n9 Ve10C1ty 15 n0t affected 6Ut the y1e1d Ve10C1ty 15 redUCed t0 Vy ff1 0.045 m 5- ' and the max1mUm Va1Ue 0f e = 0.9. 1f the 11m1t1n9 C0ntaCt pre55Ure 15 redUCed t0 p y - 0.5 6Pa then p1a5t1C def0rmat10n 0CCUr5 at 2er0 10ad, 1.e., Vy = 0, and the max1mUm Va1Ue 0fe = 0.5. F0r th15 Ca5e, a5 5h0Wn 1n F19. 10, the fact that the y1e1d Ve10C1ty 151e55 than the 5t1Ck1n9 Ve10C1ty pred1Cted f0r e1a5t1C-adhe51Ve 5phere5 6y E4. (54) re5U1t51n a h19her 5t1Ck1n9 Ve10C1ty: V, = 0.032 m 5- ~. 1n a11 Ca5e5, 1f

2 30 ~L.

u

• ~ 20

~ ,0

- 1 0

0

1D 8mm m m m

m m m

m

: 60.." : . j ; 9

. ~ / : 0 1 .J" : ; •

- • • , • • . •

5 0 1 0 0 1 5 0 2 0 0 2 5 0

n0rma1 d15p1acement (nm ) F19. 8. Effect 0f 1mpact ve10c1ty 0n the n0rma1 c0ntact f0rce-d15p1acement re5p0n5e f0r adhe51ve, e1a5t1c-perfect1y p1a5t1c 5phere5.

C. 7h0rnt0n, 2. N1n9 / P0wder 7echn0109y 99 (1998) 154-162 161

~ J

1.0

0.8

0.6

0.4

0.2

0.0 .01

• ~ . . . . . • . . . . . . . . 1 . . . . . . . . 1

.1 1 10 00

1mpact ve10c1ty (m/5) F19.9. Effect 0f 1nterface ener9y 0n the c0eff1c1ent 0f re5t1tut10n ( 0 : F = 0.2 J m "-, 0 : F = 0.4 J m -2).

1.0

0 . 8 •

.,-1

. - 0.6

f12

"~a 0.4,

¢1

0.2

0.0 .01 .1 1 10 100

1mpact ve10c1ty (m/5) F19. 10. Effect 0f 11m1t1n9 c0ntact pre55ure 0n the c0eff1c1ent 0f re5t1tut10n (0 : p ) = 3.04 6Pa, 0 : p: = 1.0 6Pa, 11: p ) = 0.5 6Pa) .

the 1mpact ve10c1ty V1 > 10Vy the re5u1t5 f0110w a p0wer 1aw re1at10n5h1p w1th an exp0nent 0f - 1/4.

Due t0 the c0mp1ex1ty 0f the e4uat10n5 wh1ch def1ne the 6ehav10ur 0f adhe51ve e1a5t1c-perfect1y p1a5t1c 5phere51t ha5 n0t 6een p055161e t0 06ta1n an ana1yt1ca1 501ut10n 1n the way that th15 wa5 06ta1ned f0r n0n-adhe51ve 5phere5. H0wever, an ana1yt1ca1501ut10n can 6e 06ta1ned 6y a55um1n9 that the w0rk

d1551pated due t0 p1a5t1c def0rmat10n and the w0rk d1551pated due t0 adhe51ve rupture are add1t1ve. 7hat 15 t0 5ay

(1-e••) = ( 1 - e~) + ( 1 - e,~) (78)

where ep 15 the c0eff1c1ent 0f re5t1tut10n due t0 p1a5t1c def0r- mat10n 91ven 6y E4. (44) and e. 15 the c0eff1c1ent 0f re5t1tu-

162 •

1.0

0.0•

9 ] 0.6•

0.4•

0.2

0.0

C. 7h0rnt0n, 2 N1n9 1 P0wder 7echn0109Y 99 (1998) 154-162

1 10 100

n0rma115ed ve10c1ty (VNy)

F19, 11. 7he0ret1ca1 pred1ct10n 0f the ve10c1ty dependent c0eff1c1ent 0f fr1ct10n.

t10n due t0 adhe51ve rupture 91ven 6y E4. (57). 7h15 1ead5 t0 the f0110w1n9 e4uat10n5: f0r V1 < V,

e=0 (79)

f0r V, < E < V,

e= 1- (80)

and f0r V1 > Vy

6 e 2 ~

1-6. V11 .J (81)

1 1 X

7he 5m00th curve5 5uper1mp05ed 0n F19, 10 were 06ta1ned u51n9 the a60ve e4uat10n5 and, a5 can 6e 5een, a9ree we11 w1th the numer1ca11y 51mu1ated re5u1t5 06ta1ned 6y N1n9 [ 13 ]. F1na11y, the 501ut10n t0 the a60ve e4uat10n515111u5trated 1n F19. 11 6y p10tt1n9 the c0eff1c1ent 0fre5t1tut10n a9a1n5t the n0rma115ed ve10c1ty (V1/Vy) f0r d1fferent rat105 0f (V~/V~).

6. C0nc1u510n5

An ana1yt1ca1 501ut10n f0r the c0eff1c1ent 0f re5t1tut10n f0r n0rma1 1mpact 0f adhe51ve, e1a5t1c-perfect1y p1a5t1c 5phere5

• ~::: :::: :/~ :1~, ::J1~ ~:: 11:/:~:~ •11 ~11~111~ {/1~ ~ 1•1•11••1f• 1~11 ~ 1•/• 1

ha5 6een 06ta1ned. 7he 501ut10n 15 expre55ed1n t e~5 0f the 1mpact ve10c1ty, the cr1t1ca15t1ck1n9 ve10c1ty ~ d the ve10c1ty 6e10w wh1ch the 1nteract10n 15 a55umed t0 6e e1a5t1c. 1fthe 1mpact ve10c1ty 15 1e55 than 0r e4ua1 t0 the 5t1ck1n9 ve10c1ty the part1c1e5 rema1n adhered t0 each 0ther and e - 0 . 1f the 1mpact ve10c1ty 15 9reater than the 5t1ck1n9 ve10c1ty then, a5 the 1mpact ve10c1ty 1ncrea5e5, the c0eff1c1ent 0f re5t1tut10n 1ncrea5e5 at a decrea51n9 rate t0 a max1mum va1ue and then decrea5e5 at an 1ncrea51n9 rate unt11, at 1mpact ve10c1t1e5 m0re than ten t1me5 the y1e1d ve10c1ty, the c0eff1c1ent 0f re5t1tut10n 15 a funct10n 0f the 1mpact ve10c1ty ra15ed t0 the p0wer - 1/4. 7he max1mum va1ue 0f the c0eff1c1ent 0f re5t1tut10n decrea5e5 w1th 1ncrea5e 1n the rat10 0f the 5t1ck1n9 ve10c1ty t0 the y1e1d ve10c1ty.

7he ana1yt1ca1 501ut10n pr0v1ded 6y E45. (79)-(81 ) can ea511y 6e 1nc0rp0rated 1nt0 numer1ca1 51mu1at10n5 0f 9a5- part1c1e f10w5 t0 pr0v1de var1a61e c0eff1c1ent5 0f re5t1tut10n wh1ch are ve10c1ty dependent and hence w111 1ead t0 m0re rea115t1c d15tr16ut10n5 0f ener9y d1551pat10n 1n c0mp1ex f10w f1e1d5. 7he e4uat10n5 are expre55ed 1n term5 0f parameter5 wh1ch may 6e determ1ned 6y 51mp1e 1mpact exper1ment5 rather than 1n term5 0f mater1a1 pr0pert1e5 5uch a5 the y1e1d 5tre55 and 1nterface ener9y wh1ch are d1ff1cu1t t0 re11a61y a5certa1n f0r many mater1a15.

Ackn0w1ed9ement5

7he w0rk rep0rted a60ve wa5 part 0f a pr0ject wh1ch wa5 f1nanc1a11y 5upp0rted 6y Nuc1ear E1ectr1c p1c.

Reference5

[101 [111 1121 1131

[ 1 ] R,M. Dav1e5, Pr0c. R. 50c. L0nd0n A 197 (1949) 416. [21 C. 7h0rnt0n, 2. N1n9, Pr0c. 15t 1nt. Part1c1e 7echn0109y F0rum, 11

(1994) 14. [ 31 K.L. J0hn50n, C0ntact Mechan1c5, Cam6r1d9e Un1ver51ty Pre55,1985,

p. 363. [4] K.L J0hn50n, K. Kenda11, A.D. R06ert5, Pr0c. R. 50c. L0nd0n A 324

( 1971 ) 301. [51 K.L, J0hn50n, 1n: W.7. K01ter (Ed.), 7he0ret1ca1 and App11ed

Mechan1c5, N0rth-H011and, 1976, p. 133. [ 6 ] K.L, J0hn50n, H .M. P0110ck, J. Adhe510n 5c1.7echn01.8 (1994) 1323. [71 L.N, R09er5, J, Reed, J. Phy5. D 17 (1984) 677. [81 C. Hardy, C.N. 8ar0net, 6.V. 70rd10n, 1nt. J. Numer1ca1 Meth0d51n

En91neer1n9 3 ( 1971 ) 451, [91 6.8.51nc1a1r, P.5. F011an56ee, K.L. J0hn50n, 1nt. J. 5011d5 5tructure5

21 (1985) 865. D. Mau915, H.M. P0110ck, Acta Meta11.32 (1984) 1323. M. F1chman, D. Pnue11, J. App1. Mech. 52 (1985) 105. C. 7h0rnt0n, K.K. Y1n, P0wder 7echn01.65 ( 1991 ) 153. 2, N1n9, PhD 7he515, A5t0n Un1ver51ty, 1995,