Computational mean particle erosion model

~~.

WEAR E LS EV I E R

w,..ar 214 t It~9~,j 64-73

Computational mean particle erosion model Dan/an Chen a.*, M. Sarumi b. S.T.S. AI-Hassani b * Mech,olic~ and Ahttertal~ Science Rewarch Centre. :Vint, h , I'tlivt,r~ity. Nim,,ho. Zll~jhmt~ 31521 I. Pe,ph".s R~'lmhlil ol'('himt 1' I)t.]ltlt-Iin(.ttl ,q Ah'~hanical I:'nk,im'crtnk,. I 'niversi/~ O/ ,thm~'hexter ht.~tilult" Ol SHen~e tlt/d 7~'('h/toh~tly. PO Bo~ 88. ;I,hltll'ht'Ml't; AIO0 I QD. I,'K Rccei'.cd 4 March 191)7: accepted 2 Scplember 1997

Abstract A general impact friction model ,aith application to solid particle erosion is presented. The mudel treats the I'rictiorJcoefficient at a function .,ff the impact process and the initial impact angle. The proposed fricti.,m model for mean particle erosion hat a s/grill/cant influence on the tangeutial ,.xork done b v the abrasive, and has Ic~,~,influeuce on the normal work. The characteristic impact angle at which the erosion reaches max/man1 can be determined by numerical calculation. The computational mean particle erosion mt~del it effective in simula'ing the eras/an rate ,brained h,. exwriments. ,' 1998 Elsevier Science S.A. K¢,vtt,,nh: l-to,ion: Wear: I:ric|ion: Ohliquc impacl

!. I n t n ~ l u c t i o n Brittle and ductile target materials exihibil different mechanisms (11 erosion. The erosion of brittle nlateri;.tls is ch;,lracterized by an erosion rate that increases with increasing angle of impacit and hat a nlaximum at normal impact. A n interesting erosion pllenomenon for ductile material is that Ihe maximum depth of the deformed zone and the nlaxinlgln rate of wear do not t'n.'cur at the same angle of impact. Several models were presented in the literature to account lot this phenomenon. These ,.*,ere recently reviewed attd cornpared by Ellermaa I II. A nticro-cutting model v, as lirst proposed by Finnie 121 and a kinetic energy loss ntodel was presented by Bitter 13 I. Recently. Magnce 141 suggested a generalized law of eros/tin, which is bated on the works of Finnie 121 and Bitter 131- These models a~,sumed that the erosion el'l~2cl of a particle stream results from the supcrpt)sititnt of the cl'ti:ct of a ,.ingle particle f - r ~dmilar particles possessing identical cro,,ion ability. Tberelore the) are mean panicle erosion model,,. Ho````e',er. the c~veflicient of friction '.~.hich plays the fundamental nile in these itltldels remains I(i be clarilied, Recent advances in the stud) of friction in oblique impact 15.61 pn~vide a possibilil) of s o b ing this important problent. In this paper, a general impact friclion nlodel '.~.ilh application t , solid panicle erosion of ductile materials is pre,.entcd. "l'hi., dot.tribe,, the friction c(~.'l'licient as a function of (",l~'lc,l~.lldltlg

aLIIhor

(~M.LIfv4S,'I~S;blq.IM) , II~lS I![%'~lcr '~fJencc S:'~ All right, rc,cr'.¢d I*ll 511i143-1f.481 ,~7 ii1(1" l a ,~

the impact process and the initial impact angle. A computational mean particle erosion model is developed on the basis of the impact dynamics involving a complex friction model. The proposed filet/on model has sign/l/cant inlluence on the tangenli=:l work done by the abrasive and the characteristic impact angle at which the erosion reaches maximum is determined by numerical calculation. Comparisons between predicted and experimental results show that the computational mean particle erosion model is reasonable.

2. A friction model with a p p l i c a t i o n to solid particle erosion The coeflicient of friction in impact wear plays the same fundanlental n)lc as in surface wear by a counter body. The fatig,te durability of the deformed volume depends on the coefficient of friction. "Ploughing" and "micro-cutting" are also dependent on the coeflicient of friction. In this section we propose a general impact friction model which will be used in ~olving the equations of motion of an erosion panicle as it impinges upon a surface.

2. I. A gelteral iotpa,'t lH¢'tion mo~h'l Ralncr and Styller I 7 ] assumed dutt the friction coefficient has a ct)ns|inlt value during one impact. Th :y carried out a series of nleasurenlenls of die friction coefticient for glancing impact ~``ith beads and river sand on steel, mirror glass.

PMMA and vulcanizate. Recently. Kopylov and Popov 151 indicated that the impact angle has the largest influence on the coeflieiem of viscous and dry impactive friction, while the material and the impact vek~ity have less influence. Lewis and Rogers [6] studied forces during oblique impact of a steel sphere onto a steel plate, h was shown that the friction coefficient rises to a "plateau" value and then declines m zero as contact is lost. The plateau value was found to be independent of impact velocity but is a bilinear function of impact angle. By using least squares curve lifting, the equation for the plateau level p. of the lrietion coefficient ~as found to be ,, fO.(X)4480 for 0<0_<41) ° /z "=.~% 0.179 for O>4l) °

dH / ~'t':"~-='J

0

Id,,,l

dv

j _<0and ~,<.*

dv ~ < O a n d ~>bt*

J

(2.1)

where O is the impact ;,ngle measured away from the normal. If the variation of friction coefficient with impact angle measured by Ramer and Styller 171 is considered as the hehaviour of the plateau level of the friction ct~zfficicm described by Lewis and Rogers 16 I. their results are substantially similar. Based on the friction phenomemm described by Lewis and Rogers I 6 ] and a series of measurements of the friction c,~llicient given by Ramer and Styller {7 I- we propose a general impact friction model as fifllows. The impact friction coefficient is a function of the impact process and the initial impact angle. In proportion to the rate of variation of the normal velocity component of the projectile. the friction coefficient rises to a plateau value and then declines to zero according to the rate of ,~ariation of the normal displacement of the projectile. The plateau value is a bilinear function of the impact angle, the turning point of which is delined as the critical parameter dependent on the impact condition including the influence of material and impact velocity. This c~,n be expressed mathematically as follows:

,.Id,: !1 ,1 d, ,, I

/

/

12.2)

,:>,,

P":" 2 - ~ "

"==/( .... )

';"'

/.t~:= fz,,/.~ i 1 . ~'. ~ !t

{2.4)

where H is the hardness of the worn material and the function F(T./~O is defined as

m,

ra2

t:ig. I. t a; Sk¢lchof Irietion¢l~¢ffi~:i~:uttin|,: hi~to~ as expre~.~dby Eq. 2.2 ). qh i Sketchof the t ariutiun of the t aluc ol Irietion ¢oel)icicnl plateau ~tilh Hllp~lC[ ;lllgle as expres,ed b'. Eq. 12.3).

/"{ u ) = I --

In( I + u } - u In(I-u-')

fi)r

' 7~

>tan,~,,,>

I

(2.5)

where/1.(t) is the friction cocflicienL v(t) and y ( t ) are the normal veh~:ity componem and normal displacement of the projectile respectively, t is time. and ~,, is the initial impact angle measurud from the horizon. Eqs. {2.2) and ( 2.3 ) are sketched in Fig. I ( a ) and (b). Constants c~ and c, in Eq. ( 2.2 ) characterize the rise and decline of the friction ctmfficient with lime I. Theoretically the critical value p.,* could be estimated u.,,ing Eq. ( 2.4 ). given by Heilmann and Rigncy 18 I. where r and z,, arc the average surface shear stress and shear ,,trength respectively. Practically. the experimental value of /.L,* is required. The critical impact angle ~,,, is an important parameter of the impact fiiction mexlel. Brach [91 derived the critical coefliciem of friction/z, from sliding to roiling which was given. [br a solid sphere, as fifllows:

2 cot ~,, P'~ -

7( I +e)

(2.8)

D. ('hen r/aL / Wrar 214 t.l¢J98) 64-Z;

66

where the angular velocity of projectile is omitted, e is the rz,tio of the rebounding to the inith, I normal velocity comramem. Assuming thai the critical value/,t,.* can be expressed by Eq. ( 2.8 I. the range of the critical impact angle ¢e,~ is derived and given by Eq. ( 2.5 ). ~,. can be obtained by using an iienuive numerical simulation of the oblique impact of a particle upon a surface, described in Section 3. The determined critical angle aqk decreases with increasing critical value of the coeflicicnt of friction #<*. which is consistent with the series of experimental data given by Ramer and Siyller 17 I. 2.2. A ./Hctiolt molh'l for mean partich" eroshm

The micro-cutting erosion model was lirst presented by Finnie 121. This model was lbnnulated from the study of deli~nnation caused by an individual particle. The volume of material Q removed by a single abrasive grain of mass m. velocity V,, and impact angle o4, was given by Q = p--~kffk(sin 2 m , -

t for tan m,_< '

()=

fi:r tan tl,,> 6

t 23) (2.10)

where d~and k are the nuio of length to depth of tl'.c scratch lonned and the ratio of vertical to horizontal force component respectively, p is a constant plastic IIow stress. It was assumed that the maximum erosion occurs, when tanot,, = k / 6 . which delined a critical impact angle ~*.. The critical angle a< is the impact angle at which the horizontal veh~ity comptment has just h,.'co:ne zero v,hen the particle leaves the body. i.e. the impact angle above which the residual tangential speed of the panicle equals zero. Finnie [ 21 indicated that an ,~ clali value of/,. very close to 2. for angular abrasive grains was shown and or wa~, deduced to be 18.5 °. If Eq. (2.8) is adopted to e.-.tilnate the critical angle ~.. the rcsuh is that <~, = 15.9 ~when / z = 0.5 t k = 2 ) and e = 1.0. Panicle impacts were lirst categnrized according to deformation and cutting wear by Bitter 13 I. Rccemly. Magnee 14 I presented a generalized law of erosion, which was also based on the works of Finnie 121 and Bitter 131. The residual tangcnti;,I speed of a panicle V, was given by 141 as V. cost~. I - s i n l n m , )

I

v,=

'il

I

a,,< 2n

°

12.111

Finnie and other author's assumption, we carried out a series of numerical simulations l~,)r some erosion tests using corresponding constant friction coeflicients ranging from 0.51 to 0.096. Calculated results show that the residual tangential speed of the particle with impact ;,ngle larger than tbe critical ;ingle delined by tan i ( k / 6 ) is not close to zero. In fact. these basic models all assumed that the effect of the particle stream results from the superposition of single particle effects and the particles are similar and posses~ identical erosion ability. Therefore. these models ;ire actually mean particle erosion models and should be described by mean characteristic parameters. Winter and Hutchings I I01 identilied the two regimes of delormation as ploughing and cutting. Ploughing was not favoured when the particle rolled over instead of sliding along the surface. Rolling caused the cutting edge of the particle to penetrate deeply into the metal surface instead of performing a scooping action. Naim and Bahadur [ I I ] studied the erosion mechanisms by repetitive impacts at the same location. It was shown that unlike the case of normal impact..'l significant ;nnounl (if erosion occurs because of the fragmentation of the crater lip by the sliding action of the particle in the impact direction. The crater and surface provide evidence of intense shear deformation and the severely deformed layers seem to have irregular edges. These experimental facts prompted us to suppose that the shear slrength equals the hardness ~ff the worn material at the turning point of the bilinear function of the plateau level vs. the impact angle, i.e. /z~.* = 1.0. The extreme situation is shown in Fig. 2la) and t b). where the constants c~ and ~'_~in Eq. (2.21 approach inlinity and o4,~ is between 8.| ° and 15.9 °. It is necessary to emphasize that the critical angle ~,,~ in Fig. 21 b) is not the critical angle ~ delined by Finnie 12J and Magnee 14 l. The impact angle at which the wear reaches maximum should be determined by numerical simulation of the inean panicle impact-based erosion and could not be simply considered as the critical impact angle o4,¢or ot~.

3. A simulation of m e a n particle impact-based erosion

Impact dynalnics involving a complex friction inodel must bc described by numerical calculation. The colnputational inean parliclc inlpacl erosion model was developed on this basis. 3. I. Mod~/i~'d .~imuhaion q l Uw ohlique i m p . c / ~ ¢ a hard .~7~lwre agaillxt a ductile solid

t~,,_> 2n

~ here

n

2 tau ~(#,!{}l

(-:.121

filr lilling the cliaracleri~tic impact angle at ~hich Ihc ~;,car reache~ a nla,xilnuiil. ¢~periluenlall)./~ was chosen as rlingin~ Iroip, 1.9{~ to 10,4 i #, = t).51 - II.ilgO ). In order Io certil)

Hmchings et al. 1121 developed a simple and cfl'cctive numerical sinmlation for the oblique impact of a hard sphere ;,gainst a ductile solid. The coeflicien( of friction was assumed to be a constant during the impact process and for any impact atlgle, In order to couple the general impact friction model, the governing equations were inodilied as fiillows,

D, (7ten el ¢11,I Wear 214 ¢109~1 t)J-73

07

y

|:i~. 3. (ice,metric ,hape (q(fle ohli%lc impact of a hard ,phcrc of radius r a! ~t~ehl~il~ i.' il~;llll~l il duclih." ,olid.

(

I dg I I i-

~

d/~ dt =

vd2

Fig, 2. (u) Sketch of friction cooltiglon! tinl¢ IliMOf~ li,r Ill¢illl par(Jolt ¢r(lsiOll. l h ) Sketch of the varialion of tile +altle (ll Iriclion ¢()e[li,.'iClllpl;lle,m with inlDil¢| allgle I~srmean pilrdcle erosion.

The geometric ligure o f the oblique impact of a hard sphere o f radius r at a veloeily V again.',l a duclile solid is shown in

t~

..

dv ~//" _<0 and/I.->//. ~:

0

I V

--('~

¢'=)

dv

-

,*

'1

| _._ ~ ~' )

-~ > 0 dl

.... > ......

(2.3')

Fig. 3. ~ H'rrir sin (31-' dx ~-/=u

(3.1)

.-.. dv -~t=r

13.2)

Equations o f horizontal and vertical motions, arc ~*=tan dl¢

m'~t

r L~/5

d.

--/'~'ITH ( r ~'-'.~"-' )

t<-I*

"i

r

Itl-~l = . ~

( 3.3 ) sin((~+/3) -p./~ ¢os(¢~4-B)

WH()""- - y

"- )

~.. - I5 cos((~ +,(31--#/5 s i n ( ~ + 3 1

I>1"

I<_!* t>l*

(3.4)

I(r/u)

H is the hardness of (he worn material./=i: is the time at which the sphere w i l l not remain in contact with the emire surface o f the craler. Eq. ( 3 . 1 ) E q s . ( 3 . 2 ) and (3.4) and Eqs. (2.2) and (2.3) can he soh ed exactly for the quantities r..rl r). y l ! ) . . ( / ) . r( I ) and gt(t) using a tourth-order Runga-Kutla inlcLzralor under the lolhwcing initial conditions: / = O . . t ( 0 ) = O,

I). ('hen vt al. I It'cur 21411~18~ 64-73

h~

Tilblc I Parameter, in the friction model used ttl produce c.tlr~cs I and 2 in Fig.41 a ) lind I b)

200 ~ , (a) I

Cur~¢3

E,~p,:rtwxn al [ 211

160

('tlrX ¢ I ('ilrxc -~ ,

,

,,

,-.

1.0 l.li

1.0 I.n

c r i l i c a l impact angle ~l,~ in Eq. ~2.3) is d e t e r m i n e d using ilerali,,e nunlerical calculations to satisfy Eq. ( 2.8 ).

.~' 100

Fig. 4 1 a l and ( b ) show the variations o f the rebound ~ elocity lind angle with the initial impact angle for the oblique impact of a steel sphere o f 4.75 m m nldius tit a velocity of 200 Ill s ' Ohio a steel plate o f 3.0 G P a hardness. The parameters in the friction model used to produce curves I and 2 in Fig. 4 ( a ) anti I b ) are listed in Table I. where p.y++= O. 179 and (L I arc experimental values given by Lewis and Rogers 161 and Rather and Styller 171 respeclively. +r,,<=52.0 ° lind 69.1.)++ are determined by iterative

6O 40 20 0 0

20

10

30

40

50

60

70

80

n u m e r i c a l sinltilations. C u r v e 3 w a s c:llculated using a c o n slant f r i c t i o n coel'lic~.ent o f 0 . 0 5 lind is s h o w n lilting with

90

experimental data in Fig. -1( a ) and ( b L These experimental data were given by Hutchings et al. I 12 I.

Initial ImpactAngled),:grcc,I 45

1~,~ 52 .(I 6till

y( O ) = r . . ( 0 ) = I~;,cos~+.. I'( 0 ) = V,,siml. lind p.( 0 ) = O. T h e

\, \ ,

120

#a l). 17t) I h I 0.1 171

h is seen I'roul Fig. -i( a ) and ~b ) that the i u l l u e n c e o f the f r i c t i o n m o d e l on the o b l i q u e i m p a c t o f a ~,phere w i t h a phite is o b v i o u s . W e shall s h o w that the f r i c t i o n m o d e l is v e r y i n l [ n l r l a n t Ior the m e a n particle impact-based erosion.

f

ib)

+t

('u,,c ~ •

35

• ('

I , p e r i s h , II_'I

,/

d,

3.2. Sinllthtlioll ~/'ln¢tlll pal'tit'le ilnpa~ "t-I~as~'d erosion Bitter 1 31 presented three equations expressing erosion.

3O

T h e x o h u l l e r e n l o x e d by d ¢ l b r n l a l i o n is !,4'#1 !.',, sitl +l+,,-k ~ i :

25

II/ll =

-

.fj

I 3.5

)

T h e x o ] u m e r e m o v e d by c u t t i n g is

20

~_Jl, (I.,.hl

+t,,- ~

,~,,,,,,,,,,i5

,} (

,t' ........

,,x ,in+~ - , t . ~ "

,~,7,,,7,,

)

{lll~-fl+'

o

. . . .

o/,"

t+,:<< ..... .~,,,., ......... -~ "+1 ,!+

/ /

13.6) 1"he ltltal x o i u n i e is It', = IVl>+ Wv

ol 0

10

20

50 60 huliallmp~ct~gl¢ll~grcc,l 30

40

70

80

90

13.7)

,,,,here

LI =Ii.Ij142H.I TH/IJ.,i I I.lg .l. ~.lJ (',,mparl.,m bciv.ccn ~M.ul.,_lcd and C'.l+~'r,mcniM'.aziali,,n ¢,1

/, . = I H ) 1 5 5 t t "

'( glp.,

tiler¢l*hnmd ~¢hwti~ ~ttil tIIclllllI.iilllip*tL'l,lII~It" h,r d qc¢l .piloT,:.1475

111111~adlu, ,:l .I xc[,'~ ;1~ o! 21MIul - ' illlplll~lll~ L+bliqu¢l]. ,role a qc¢l plak' ill ~ () { ]|'.l hklldIl¢,- t b I ("oltlr~dN,~ql ,It...'l'~'.CCII~'dicUldiCd *did C\PCFilIICIIIkl]

X'+Ifl'Itlltll''tlhcr¢l~lUliddllglcxt'lllllh'fllillhlllllil'dl~'t'lliglchll 'IM'+"¢l~phcrk" LIt ~ 7'g IllIll Iddltl..It .i +¢hKlI~+ol 20(I IIi. ' llllplli~lll~ obhqtl¢l+x ,into a ,reel pl.++¢.,I :~0 ( ;I+.1h+l:dllC,,

I-,,, ().2~

£'=::

I:', p,/~tl) " ' tt

I l

a{..

)' "[;'

69

D. Chert et al. I Wear 214 t 19q,% gul 73

cos a , =2c6V~/. (sin a , ) ~/-" ..,.

p , is the density of abrasive and H is the hardness of worn

m~terial. E.,. v,,. E,, and v,, are Y o u n g ' s m ~ u l u s and Poiss o f t s ratio of worn material and abrasive respectively, and g is the acceleration due to gravity..[~ and & are the energy needed to remove a unit volume of material from the body by deformation wear and cutting wear respectively, and M is the mass of eroding panicles. Recently. Eq. (3.6) was modilied by Magnee [41 in line with Finnie's model [ 2 ] as follows: W ....

M

IV,-'(cos a )2 - V ~]

2&t

'

"

( 3.8 )

' !

where V, was given by Eqs. 12.111 and 12.121. However. the critical impact angle a , was considered as the angle. determined experimentally, at which the wear reaches maximum and was attributed to the friction coefficient given by Eq. (2.~2). For the erosion tests of AISI 316l. b~, SiO_, of I).5 mm radius at velocity 50 m s ~given by Magnee 14 I. variations of the residu:d tangential speed of the panicle V, with the impact angle a . given by Eq. ( 2. I I ) are compared in Fig. 5 with that calculated using different friction models. The parameters in the friction models used to produce curves 2 - 4 in Fig. 5 are listed in Table 2. Fig. 5 shows that a constant friction coefficient of 0 . t 6 ( n = 1.251 given by Magnee 141 cannot make the critical impact angle a~ { ~ = 45 ° ). determined experimentally, meet the postulated condition, i.e. "'the critical angle is the angle at and above which the residual tangential speed of the particle equals z e r o " . It is also shown that adopting the proposed friction model for mean panicle erosion is a reasonable approach to solving this key problem. T h e essence of Eqs. ! 3.5 ) and ( 3.6 ) or Eq. ( 3.8 ) is that a certain amount of the impact energy is assigned It) '; olumc removal by deformation wear and cutting wear respectively. The computational mean particle erosion mt~.lel can bc expressed as EI~

7, mH

Z,,Ar ,.n_rt

I:c

T, ,,.b

E.mA. ,,./J

T=T, +T, E=ED+E

( 3. t) )

(3.10) (3. i l )

v

13.12)

where T,. T, and T are tangential, normal and total works done by the abrasive rer pectively. Eu. Ec and E are delorming. cutting and total erosion rates rcspectkely. Numcri,:al simulations show that the frictio.q modeJ has the largest influence on the tangential work T,, and has less influence on the normal work 7",.

. --

\

.... -

.....

\

Cu~el Cu~e

1 2

Cu~e3 Cu~e 4

............ ,,

x 2

~ ~o .2-

-

oi o

10

'.,

\.

20

30

40

50

60

70

80

90

Initial Impact Angle (Degrees)

Fig. 5. Ct,:npari,,onsof calculated '.arialions of the rc,.iduallangcnlial:,pecd ol 1he pilrli¢lc~ ith impact angle Ior the cm~.ionof AISI 316L tl.. SiO.. ll.5 111111.~11~1", chK'il} Ill 5{) nl s u ~ h l g I~t I. ( 2. I I I.

' g i v e n b', M a g n e t

[ 4 I. ( ' u r ~ e I v.a~ c a i c u h t t e d

Table 2 Pilr~lnlch.'r~ill friclion inodt.'b,tl~cdIo produce ¢tlr'~¢~2. ~ lind4 in Fig 5

Cur'.c 2 ('tlr~c 3 ('ur~c 4

,ul t I = n. i fi for all illlpact~tllgl¢~ I.n 12.5 1.0 I .o 12.5 infinil¢

].n inlillil¢

Using the parameters for curve 3 in Table 2. calculated normal, tangential, and total works done by the abrasive durin,:_ the impact process as a function of the impact angle are shown in Fig. 6t a). while the corresponding erosion rotes derived with ~b=60.O J mm ' and l l = 125.11J mm ' 141 are shown in Fig. 61 b). Meguid et al. [ 131 carried out a series of erosion experiments on mild steel, pure copper and aluminium alloy. Using the parameters listed in Table 3. variations of the calculated tangential, normal and total works done by the abrasive with the impact angle are shown in Fig. 71 a). Fig. 81 a) and Fig. 9( a ) h~r the oblique impact of a steel ~,hot of 11.2 mm radius a t a v e l o c i t y o f 148m s ~~:gainststeel.copperandaluminium respectively. Corresponding comparisons be,ween calculated and experimental erosion rates are shown in Fig. 7( b~. Fig. g ( b ) and Fig. 9 ( b ) . Fig. I0( a ) and ( b ) provide another example to illustrate the effectiveness of the computational mean particle erosion

D. Chcn el al. / Weur 214 t I qq,% f~4-73

2.C

- , .... (o)

Table 3 Paramelers used in ealculalion*,(o pn,ducc ,.:ur',es in Figs. ( 71 -( tl )

:

--T, i o.- T , T'!

(Jmlll II ~J inm ~1 I(]P;I) /

t.5 / /

/ 1.0

/ .... -...

0.5

\',\

"-.. 0.0

t . 10

0

.

. . 20

.

1(I.5

I.(I

('oppcr AI

I.n I.n

111.5 inliniw I 0 . 5 i.()

1.0

124.8

74.1

1.55

inlinilc 1.0

I2(LI 39.1

89.0 27.o

(I.85 o.62

/

"-\\

/

-0.5

I.I)

/.

. N

Steel

. . 30

~ 50

40

60

.... 70

r

80

90

model. An annealed a l u m i n i u m alloy was worn by a round quartz sand of 0.25 m m radius and a velocity of 82 m s i which was given by Ellermaa [ I I. The parameters used in the calculations are listed in Table 4. The proposed friction model plays a fundamental role in the m e a n particle erosion model. T h e c h a r a c t e r i s t i c impact angle at which the w e a r reaches m a x i m u m can be d e t e r m i n e d by numerical simulation, which in general is not the critical impacl angle a,. or tl,.,. It should he noted that .Q and ~b are not material constants and dependent on the w e a r condition. In fact. -Q could be estimated as follows:

ImualImpactAngle(I)t'gtee~t 16

. . . .

,

f

,

.tl= ! V , KD

'

(b)

14

l h e r e l b r e . . ( l o t V ' - ' "' where n is the velocity exponent. Because the erosion rate E,, o f material at normal incidence is proportional to the exponent o f the particle velocity E~. which can range from 2.2 ~ 3.0. therefore .(.) I ~;, "'-.V,, ~). As for tb. it is smaller than .(l for genenfl ductile nmtelial. H o w e v e r . the exact prediction of .(2 and d~ remains to be ,;olved Iheoretically.

i0

4. C o n d u s i t m s

-

0

10

20

30

40

I

!

I:l)

i

50

60

70

80

90

ItzlpJ~l *'~rt~2]¢II)¢grt:¢sp

Is~ h t .a i VariallOll, ol ca(cu(aled langcnl/.d. II~)rlll;llkllld IOl;llv,,,rk doll¢ h~ l|1¢ .tbra,i~c x~i||1 the mlp.lc1.111~]¢t~lr Ihc cro,ion IcM-ol AISI t lid h~.

SIt) .. n~ Illlll. ill "=¢hK'll~ill 2111tl, d. ~1'~¢II h~ ~l;l~ll(C [~]. {b} ('(,nlpk/lis,,ll, h~'1"¢.~.'¢llc./lcul,Jtcd and ¢lix'tllt|¢nl.d ct,l.~Olltale, IL,I er~,,i,,ll It,l, i)1 ..'*.|SI 11 I h.x Nit)2 I r.tdlu, ~ illnl Litit "~L'IAK'II), I ¢~ ill, ' '.'1'~.'11h~ Ma~,lcc 141

I. A general inlpaCl IYiction nlodel is proposed, which can characterize the rise ai|d decline o f the friction cocfliciem wilh time and describes the variation of the plateau level o f the friction coeflicient with the impact ;ingle. The tun)ing point o f the bilinear function o f the plateau level vs. Ihc impac! angle is determined by iterative numerical calculation for the oblique irnpact o f a s p h o ical particle on a surface. 2. A friction model for m e a n particle erosion is presented. which reflects the m e a n characteristic o f worn material: sex en:ly deft.tuned and irregular edges. The m a x i m u m o f the plateau level o f the friction coefficient is assumed to be 1.0 and the corrcsptmding crilical impact angle is small. while the rise and decline of the friction coefficient with time could approach inlinity. 3. The nunlerical calculation o f the oblique impact of a hard sphere against a ductile solid is modilied in order t~ couple the general impact friction model. It is shown that the

71

D. ( ' h e n ~'/ +d. / W~,ar 214 I Iq=~Y,I f~4-7.~

(a)[ - - . ~ - T :

........

(a) i - -

T,

i 3 /

J

/

/ /- ....

/

\

//

\

/

/

/

/

\,\

"\

/,!

.

/ /

",,,,

/

/

/

/

,,

/ /

10

20

,,

",,, \

30 40 50 60 70 Initial lmpa¢l angle (Degrees=

--E ...

I

El)

. . . .

/

80

90

0

10

20

30

40

50

60

70

80

90

lnmal Impacl :m~k"
i ++ o

0.08

/- . . . . . . , /' /

-

0.08 ,,\ \k

•

/'

0.06

\,,

/

0.06

\,, \

•

/

,,"

0,O4

0.04

/ /

/

"\

2

\

/ / /" J

0

10

20

0.02

/

----

/

30 40 50 60 70 Inltiallmpa~lan~l¢(Dc~r¢¢~)

80

90

20

E ED k

30 40 50 60 Imlial lnlpact angle (D¢~r~'csl

70

80

90

Fig. 7. (a) V~iriatin. t,l cal¢ulatcd t.lll~ClttJ;ll+ nOrll1~l] ~llld tuhll +~rk ~|l+ll¢

Fi B. ~. l;t) VarJ++tilln l,I c;iIculalvd l;III~cIltl;ll. IIt+FIIt~II ;Ind hal;It ~+llrk d,m¢

h ) t l w ~lbrilhjt. C V. jth Ih...' jlllPkl+l Itll~lu+ fi,r tIle t+l'diguc jlllp;l+..'t t~I ,t¢¢1 Qlt,I w i t h a radjun t d 0.2 rain ill ;I .+¢h>¢jt_x o l Ik 14 k i l l s I :l~+lillnt ~1 m i l d ,Ic+-i x~Jth a hafdncns i l l I+.~/:; (]l)~l. t b l ('tllllp;IrJstllln I~s:I~¢¢II ~.++l]¢tlJillCd alld CxPCFjlIICIIL;II ¢ro~+il}n l'[llCn IDF ....fosjlill ICsls o1' Illlld McC] Pl~ >t¢,..'l shill x'.tlll ~1f+ldJtl~ ill' ()+-i i n l l l ;1| il x¢l~)cJl} ,it + 148 Ill s I ,~Jvcn h ) Ml:~tlJd cl ;iI. I I .t J.

h} Ih¢ :lbl'~lsjxC ~.t jib till: jITIPacl ilIl~lC Ibr the ohlh.lll¢ jlllpIi+21 ol ,reel stll,I ,.'+jIll ~1r~tditls I,llj+ +1HIIII Ill ~1xeh..'jty of 14X I l l , ' ~l."-':lmsl I1Ur,' t r i p p e r ~sjlh ;I h;lrdllCSS l)f ().n.g (;I);I. { h i ('Ol11p~lrl~ons hl~lx+¢ci1 cilI+lllillcet ,tlld ¢xperlIllk'lltill CI+LlsjLIll l-iltL's fur ¢'l'(P*jlln ICsl "++I l l IIlID..* coPPer b.x ,feel ,hIll x+jill ~1 nldJu+ oJ+(I.-~ IIIlll ill +1 x c h ; ¢ i l } n f 14~ In s I ~jX~'ll I+'~ .~.lC~tlJd ¢1 ;If J t.:t I.

i

72

D. ('hen et al. / W e a r 214, VVP,I 0-1-73

4

.......

,

.... ,

• :.

.,

. ,..

, .......

0.6

(++)r--- t'~

[--T;i

~_~+~

__]

0.5

//

3

/ /'

0.4

/

/ /

,,/""'~ -~',,\

2

1

/

/'. / ," "\

,

/

/, "\

/

---=0.3

. . . . . . . . .

//

\

/

X

] t/

,

................ "l ,/

++ 0.2 ~-/

\.

/

~\ '

i/ '~'\

\

,//'

/ /

X

\ \ \

/

.

o.o t .... i 0 10

i ....... ;, ., . . . . . . . . . 20 30 40 50 60 70 80 InitiM Impact angle (Degrees)

90 0

10

20

30

40

+.

,+0~

60

70

80

90

,/~-.....~

/

120 I

0.08 /

l

......................

140 o. 10 f

\

/

/

'\ \

//

',.. '.

-

50

Inithd Impact angle (Dcgrccn)

lOO l"

+o~

/ /

,\

.w/ 60

..a

40

0.02

I. D

i

• ?:....... }

o.oo

:

+

i

EL)

i ....

.-

•

i,

27'Y'.'_'2']

20

. ,\

...

I

""\\

°l 10

20

3o

40

' 50

60

. . . . 70

t 80

90

inlttal ilnpa+l ,.tr,gic t Ik.gl¢c,I I;ig t). (+i] '~.+.ifi;tilO[lltl ¢.dcu[aK-d lal1~Cztliktl. IIIIFIIILII;llld h+hzl x~llr~, dtmc h) tIle abra,ixe ++ith illzpad an~le for the +~bhqu¢ iltlp+ld t~1 sled ,l+~l xsilh ;j rJJ.u~ ,,I 0 2 Into at ~1~¢h~.il} ~+~ i4N III, ; ~l+~ailtsl ~111ahll;mlimll ;llhL~ v+lth a hurdt~¢,, I,t Ikh2 (il'+l I hl (oltlPari,llns I~.l~¢t.'n ~'kll~tl~itct) ~lnd Cs,l~...rl,llcllt+zl ?[I,.hlll ~ut~'- lot ct,l,i,ll) test, ol ~lllllllllllli) ~11,,%b) sIt'el ~h,,~ v+ilh a f..dlus,,t I} 2 I1111~~ll ~t ~¢l~l~lt% I+1 14~ II~, I gi~¢,~ hx P'.|cL,uid ¢1 ;d. I }31.

I

0

. . . . .

10

20

30

i.

ii

40

50

,i..

60

:

70

. . . . ,, 80

. 90

Inhial Impact angle (Dcgtc¢~l Ig~:. ](l. ( ;i I V;iri;=ti~m of ¢al,,:uhllcd t;mgcntial, nornml and h~tal t+(+rk done by the abru,ix¢ ~ith the in)Pa~l ~1111-'1•~.~'l)r the I+bliquc impact olqua~z with ~1rLtditl, ~,1 ().'15 II1111~lt kl xch+¢il} t)f ,~3 Ill s I ;l~Llifi~t .111;dumil~ium allo+~ %%ith kl hdl'tlllk'nn I,I (1.2)7 (;l}~t. ( b ) ('~mIpari,onn hCl~Cl..ll and C\l'~.'(llllCll| Cx'osioll F~IICNI'llf CrtP+iOl} II.'Mst~rLIn ~llulltillilll|l ~IIID) h'v qtl~lrl/ v. ilh a r~Idius of ~)...25 mm ..it a ,.¢loch x of t+2 m :, ~ gi,.cn h.', |~licrlllaa I I I.

calculated

I,1.('h,-++c! +ikI W<,ur214 119<,+,'+~f+4-"3 Tahl,,: 4 Paralncter~+. I+t>rquart/xs, alunzinJUllt,

in calculatitm~,u',cdto producecur+es

in Fig. 10(u) and (h+ +r,~

,'+

+',

It


(Jn|lll ")

(Jlnnl

I.(1

I I.O

JnlJl'ljltP

hllinilc

qS.()

17.n

4.

5.

6.

7.

Acknowledgements T h e a u t h o r s w i s h to t h a n k the N a t i o n a l

I~ :~

II

:1

l(;Pal 0.237

influence o f the friction rondel on the oblique impact of a sphere with a plate is signilicant. Numerical calculations show that a constant I'ricdon c o c f licienl ranging from 0.51 to 0.0%. given by some authors. cannot make the critical impact angle, determined experimentally, meet the condition postul:lted by them. i.e. "'the critical angle is the angle at and above which the residual tangenti:tl speed of the pardcle equals z e r o " . Il is shown lhat the friction model for me;|n particle crosion has the greatest influence on the tangential work done h v the abrasive during the impact process,, and has less influence on the normal work. The characteristic impact angle at which the erosion reaches maximum can he determined by namerical simulation, which in general is not the critical impact angle at which the friction coeflJciem reaches maximum. The computational mean particle erosion model is effective in explaining erosion rate obtained by experiments. However, the exact prediction of the wear condition dependent energ 7 needed to remove a unit vohlm¢ of material from the body hy deformation and culling wear remains Io he ~oh'ed theoretically.

77

Foundation

N a t u r a l Science

o f C h i n a and Sci. and T e c h . F u n d s o f C h i n a

Academy of Engineering Physics (CAEP) for their support.

References i I I R.R.R. EIIcrmaa. Earl,ion predicll*,n ,f pure m~m,I, ;md,.:arh.,n .,tcei,. Wear I (+2-164 t I'.F)31 I ! 14- I 122. 121 I. Finni,..'. Ero,&m tff ~.urluce,, h~ ,.tllid panicl,.:,.. Wear 3 I 1%01 ~71o3. 131 J.(;.A. Bitter. A ,tad?' i~t er~,,,ion phcnomcmm. We;ir h ( 1:)631 16019l). I J,I A. Ma~nc,,:. (;¢ncralb, cd law olcr¢~,.icm: appliCalion Io ,.ariou,, alh~T,, =lllcl inlcrlllclalic~.Wcar I~I-IN3 ( lCF}f; b 5(x)-51n. I~l Y R. Kop~ hw+ 5.P. P,~pov. A .,uldy tff lYiction in oblique impact. So~. J. Friction Wear I I * I*~X): :~5-59. 161 A.I). I.ct~ i,. R.J. Ro;cr~. I~xpcrimemal and numerical ~ludy of f*+rcc~ durhlg l+hliqtle,IIIpLI+I.StttJiId Vlhralitm 12~ I I()N~ ) 41)3--I.I 2. J7J +";B. R;itstcr.I+.I+ SL,,Iler.(.'haraclcrJ,,l~:s ¢+I"h++paclfriction;Jnd +=,cur of p=+l.,,lncric n~alcri;d+ Wear 73 ¢ 19H I I 213-234. I,X! p. Hcilma~::. t;.A. Rignc'.. The energy ha,,ed m,~dcl t+f friction and it, ~lpp]ic;iti,,n It, co;~tcd ,).,Ictus. ~,Vc;ir 72 119~1 ) It)5-217. 191 R.M. Br,lch+ Impact ,.l',nanfic, with application to solid Irarticle cro,Jtnt. Inl. J. Impact Enp. 7, I¢)NNI .~7-.~3. IIOI R.IL ',Vinlcr. 1.~,,1.Hatching,=. Solid particle erosion studio, u~.ing ,,h=~tc angular parliclc,. Wear 2t; 11974) 1~1-194. I I t I M. N;=im. S. Bahadur. The ,,i,-'mlicanc'.: {~1"cro,,imt parmrtcter and the i11¢¢ltilllJSnl+ of cro~itnl hi nm.-'le p;lrtich: hmpact~. Wear ¢)4 111)~44) 2 Iq--23_~. 1121 I.M+ HutchinL..,s. N.H. M++~.'n+illan. D.(;. Ri,.:kcrh,,. Further ,,tudic', ++l the uhli..lUC inlp+lcl (ff a hard ~phc+,cag;iin~,l a ,Ju,.:til=" ,olid. hit. J. ML.¢h. ~ci. 2~+ I I~J~l I 6~tJ-046. I 131 Y,.A Mc+uid.",V.J~+ht.+++l+,+~.T.+,AI-Ha,.+ani..++~4ulllpl,,:impacl.:rosion ~1 ductile mclals I+,.x.,ph...rJcal parlich+.,,. Prl>=.:. 1'71h Machine "[tP,~l. l),..',ign and Rc,~.'~tr,:h Conf.. Hal,~cd. N.,:'.t Y,~rk. VO'TP,.p. 661.