.=../r`4 t,. . i I

ukni,,iar srab ch.mical.T.chnologv unnrFiiv
ca8arin An B' Dnrpropdlo\śk, .19.005, Ukaine
Mathematical model of fiiction wearPloc€ss on the examPle
of polymer composites on the basis of phenol-formaldehyde resins
sunmaly Madtń
i.al cxPrśson allowmg cil.u]aijon of t.nPcrahĘ
6nńor,.nc Nis lourd Ę uscofsinilrńtyiltÓql and dińq\i s
n\.is ThśtcmPcr.forc*i sitcrionoIibnsion 4timańon |f&ijon Gbń
sn() wcarl ]inc nrod.rhis bco clibonicd in,r on s basn onc c rcircuLir
T valuP ofa|y clem.nibeńs ł pa of fri.lion pajl Thc ondihon of uscn'].
n$s ofa noJorh b.ihg famiriarN h frc.hani.al and thcmal.h adois(is
(T) m rhc
ofa FolyńdcÓńFÓŚibas
It
is known
d.fir
ihcrcl[b,hiy
and
wellaŚ i
dunbilityof
itnal ńodd,
thc rrrc
cxploir.tion cfiicicn.y oI ryindusbnl
quipment Malrunciion or r ction pan hads io prcma.
turc weat oI al cqlipmcnt' (o tcchnolo$' śri.tncŚ
taĘ\8l$sio , io.xccedi'rB arclB)l eve'Elsl to Uffea
rion paiB
$rciy
{o.k .on,r,iions so, thc rr(tion
wcłI k o|! of ihc mct imPo ant fcihtrcs of lhc t y.
bologi. objcd Nan.lr ii d.rincs m nly thc cffnrcrci
,,r hv*icnc or
fri.tion pans
Thc PoLymcr .oDPosjlcs it !5cd
reli^bilit_v and salery ol
An ability io p'.dict wci. ol thc r!
jc.
Siven cxFroiiins.ondiriorr providcs in optimum.ho
of mat.rials and rpeilt dcsign ot the I .tion pair To
h.nuti.al modch arc
ńaierńlcxpcnscs fq th. ktho| paitłdeśixnl2l
dcsoibc ihc fri.6oń Nca!
of a Polyner comPosiic młc.jil T
inc lhcmołluduaiion.onccpl for sohd bo.ly stcngth
developed by s. N' jurkov ei ł1 l]l
b
'hc
^(o.di]rg
a
kircti. modc]
lymer .omPositq
łi.tioi
.o|..Pt, thc dci't€graiiol oI mat
dcv.]oFing tlrclno adivatioń F(rś5(1)
sribcd bysu.h f!Mańeńi.l caUłiill
illit
j5
d.
.=../r'4
Whcrc r .... li''. oIl;s'łryl,li,ł łi)linŚt' |
lTd (lł. l in, Iollt! i\), \
rP]|ir.lIl
yńd|falau',Lh łdiM,i.ł łll'f! lŁ.lł' dśłq]'.
'!
illr
.fn ,a'ł'],6
łŁ\l
ńoĘ ł.q!.ntly nl|t, ''łlmłlal;olś
RT hil nńin!|\ !n!!, 'fuł'v']l
arc inacasc ilr rclńbilit-Yo'ihc ń'
in ih€ir working fP UrG RcJusing thc noblc lrcbls al
lo},s' b!i$ iI s]rołsuPply, lctd(casc michńcs' Pri.c
Ps5pcdive usins.f p.l),m$ m
th! gÓgFcj i'r oll
kno}łcd8c
ńcir Physi.o m.d'a|i.al Prop.łies
'bour
rd criboration
of thcr drici.!cn$
w. .hosc
po
t14it1d by nr natr t\ nt tu rt t
Frcń thc PoiŃ oI vlcw ol kimb. thcoly ihc
Lays is onsidcrcd m a body iha
fl t1t
numbcr of sbucture
E']nation
(] )
su]li(
.kmef6 kin.ii.s unih
a$uńc5
Thcy
rhai
rl) are.on*ant Howcvc. drrc ^tci.l
\b
poind ovc! xhrch ihc.on in.y d
unlńitcd jncr.asc in tcnrPcraŃc r
ros of rhc lorn ofa body or irs inr.gntL H.rcc, ths!
ń sonc boundary tonprnturc, .
givcn m elial' abÓv.lvl'jĆh dr. ql k'r (l) lolN th!
i,r
RĆally,
rh..rP..c|..s havc show
(1)itńnes$ry to usc ilr lolloqngc(]ul
lhe cquJńo|
t,. .
to
T
o' lu'}i]i,v 4l
''l' uill! r.DłIxrrft
B I kh a. ltsihh
whdĆ.
i
I
t ailholl n1 }]E ńax,kl ljfbns fun1|ś lo iłdńi!.
brrk a| lnłlralfukl t ltdh|s al allrpłs E r rcllll a|
bmk al n d\r.lw.a,rcdi.ts l 5 l.
]lndn$
polymeG.
mrsr b. bker inio ar
ńÓd.l th. \'eari$ Prme
Ir
dro kij'etiĆ
ad(lcs. And so, itis
noncnt whc|
bad was ipplied and finish onc with scpiation ol.pr
bjnParń.lc simili'rolcrsdunbililyplaysresislan.cto
nÓt FÓŚible
b fn
rh.
śał6ńc
at thc
i'. hmc oibnn.h ofre ain lveight oI a nab}ił]
Thcrcfor. in ihc cquation (2) lor we
nc.csrry toput l/lwhcrc tdcscri
ply l ńot iłs re.ipill
.al' ]hen equation (2) aŚumes th. IÓl]o\'in3 fÓ!ń]
rvcar
łt
whcrc] ł/ ' Utrl .o'!'łl l:!''
soi''r otr dŁńJe .,łł !4,{lit
s''
\ /
'/rlr
L.r!s
the
li
nolc r'', froń
.q!ahol
i
l
(2) is a
1.l
af
iłłklkh,
mjnjm'1vilucof
fcłimc l'hcn clcmcntaly disi. tqm lion
is 3o i13 on
(3)
thc flicrion
' lo lhc worLing
śrathc! s.[lihve
condiiions and .rn va!y ,n a widc ransc 16l. Thc
phyliĆÓ ńaih.ńżńcal ńodcling alon8 w h lhcsjmilaF
iiy and dimoBions arilyzir8 rhcory mrc us.d io dc
ycloP in analyti.łl relaiions delining the lempPRfure
withh i ftictionzone lol the polyner ÓmPsite mai.
lial bas.d on Phcnd n'ńald.hyde rrsin |7l
bńPc!ab!.
which
jn thc casc oI flichon
ńon ol sPa.c a|d
Pro(ss Thn
hnc.ooldnutcs:
whseary,. rmc.eld!nd$.fi hrd!,a
is
Lnt.
t|
ź )\]r
=|l'; tit
tr
ł'
_ mnałoldndiily lą|ni wis}rdIumst
u łi]k itd t] Ę l}t h t atnnl's plt}1 lÓwn)
Iol nll ^
llt |ńlkr !4rd,. thun[a$ti|] rqnbł łulaŚe
ll'] ls),
wj|hu
tal łl] łdtl:k Pałnidrl&
body and how thc fjrchon Pafs
$da..
rlr) = T. = nrrr, &r,
ihe sufa.c teńpcmfurc is łePl lhc same durinlj ih.
.ase (of the static bmpcratu.c licld)
tleat Jlow lhrough
a
lunłi,ń ofhnc,łi(1)
body surfa.Ć dĆfinld ż\
l(r)'
Tllo sińpl.{.a5c $h.
=
it.
nowdoesnolĆhaĘą4'(T) =łI - dlf'
Rclłiion of thc.ońVedjo| hełexchJ|g.bchvcc|
thcbodys rla.e ańd outscnlnonmcnt is dcńncd so lo
*e!P the .ÓnsbŃ heat flow valuc
thjs
_
lhoc'o...
/ną*! ł .
1'dn ).ttblhr
'trl'lo/
r]'\!iĄ l
rłlbr.nl ltuplnhn,T-
^ND
.llrc
lr rc, qu 1 l I
h/łl i'{r r.-
_ lolptn|n ' ol r bal|
BAslc ŃNsTIc EaUATIoN
ccnem1solutionof ('i5an jnI ł.5ctoIill!oi'blc
Pi iculirslutions Thc cxad sol!
.ou]d bg IÓ!nd only iIsomcsimPlili. ions irc iPPlicd,
inE ai d'e
h.m orousc,look.
nain valus Jnd invśh8ihnE tlrffIcliho|s
tutdi
l]Ui|ldh t,ilr! raxflitirńl łął|iinl
h l hańłłnrir. |t hrdł) o| ,l!l N|łllłnnkful nl
b
ÓI the
int€rad lvjth outcr.nvńolnent (boundaryconditjÓn)
m!$ be knovn. Follol ing iypes of
iions could bc apPli!d to dlc fricńon PaiĄ
- T.mperature disEibltion irons drc body s a..
dcfincd as a flidion ofhnc 1;1r) = f(r). tn i spdal
-
.an not di5rilsuEh, in gcncnl, p. icular fciturcs rd
.ÓmńÓn onG GPpli.i . lo nuny Phcnomcfutr)' snni
lariiy thŃIy PLovid€s th. .ommol
l3l lt cstab]khes relatiÓN bdlvlen cvcriń!ńral dah
dircdly .otrmded wirh ihe b'5i. $!łn'\' ind n)
cla Iiśtheir .ońńon feiturcs' Priclicilly' nr m h*
a funś n i.alanalysjs of hc pro.6sśthcconrtlc\ dimcnsiÓńle$ lariablG .ould bc casily uscd inścid of Pri
Lelańons of thes. valJbls arc dcscribcd b], thc
Fou.cr drffc.cniiil equaiion
g\dE
rical shape
]
duin€ onc ad of me.hanical iŃei
ev.rv łtomśodllati.tr ln parall.l
l,/inequatnm G)shossao (alvaluewhcndisinrcgn'
ti. il8oing wi'h vcry 1'Ę.sPccd, and wh.n hPem'
turc or/andslreŚ esohiBh that timinE or an aĆtiva
tiolkineticdesc ption arc Mt app
t.mpelable
To solvc ihe equłion (5) lhe Iixt n.n.nfś l.ńPcń
torcdisribnrion(asa a*condibon)aswclltuscoDcL
i' lł.,'dl ltr,J'ś
Wc arc opdating Lvitlr Ćomphxe
lor common ViĆv Óf thĆ iń ńl eq!łtńrŚ sollrn
.oDld be prsnicd in thc lom ol critc.r is i dcpcr.
d.rce of atcń i-Iundions ol som e
l}e scnmal lacros ihar influcncc rhc frictiof .ould
bc deslibcd by Io]lowiĘ doPenden!.s, whcro: M
nros, |
lenBth, r
tin! aM T r.mporatry. is a
-
Tnemal chatactthtiĆi frhtion l.mpetaiul. de
pcndi,r! on rhc quintity orhc cv
ą - M ' L: r.' hcaLbanłcr (frcn lhc su!fa.c) .oeffl
rr hc {n8inccnnE propshc. oI
śoring.aPaĆity . = LŻ' rŻ' T l
PhyŚi.o neh i.al prpeńies of
d'_"
olilEłc
a(cPi ant Iivc azs
t is k|own l3l'
6se ol unknown quaniiry (,D or ^s
quations d or
cedjńanumbd ('),(]l _'ł ind€Pe
on thc btric of it lve s.r dE Paęńeter5 T, r,' y' ., ll qual
seliilly 1' €!Jatińs Płańetc of oths sizcs to zc'o
on d'! basicofcxPĘsD| (3)' wc'cciv.lhc fo]]ow.
it c PoŚible to
in
=L:.M.r',(n1
k|/ń':), Ul6ńat bcndine s@nArh 6| = M r:r T' (in
Ml'a), ulhńat ońprslon stenlth o. = M L! I' (nr
=M. Ll(Ę kg/ń1
dni.mgcomcbyoIthcf
.tionfa.cI-Madobasicsample size L (in.m), t
rcuEhnp$ Mn'.of rhc
MPa); dcNity
l)
tuictinB$trra.esL(i'r.m).s.me.h.racteristic rdshrpe
orrherniuBab paiE .onhdinB ar
Frn6ońtyP.:slldinEfrictiń,n'lling'ictńnańda
.ombimtion oI ih6c typ$.
time and tui.|.
-ThcFriction
compl& dtrcria can be rcmiv.d
mad'rnati.al dĆkription
basd only on the analylis ÓI
the
ÓI
Theory ol
d
o1di
diń!
łł|rca|
dinensions
thesc cquatiotrs (hmlgh svfbÓlŚ
noi only usi'\E
thc gcom.kicil pamnrctcE oI f crion pii.loidirS
spe.d modc or oporario'E, thcrno physical and
PhysiĆo m!Ćhankal Ćh^laĆt.rhti
js
Eiven' it posibl. b dciormij'e ienpdaturc ń i zon.
af
qPlcs
u'. ph.nÓńena and .łn
1f
\9h&| - uodllń
Lett
r=M
. r re .l' ,v,r , rr' ,
L
r,lliąl frnsijłd.n (i,s
similarity rneory
lim cd lubricaho., wc sh.ll onsdcr a dclintion of
tcmpffafurc nodĆ of oPcntions of fictjon pans worknr8 without 8leasiĘ The function (7) imludes 12 viri
E']uating Paramchs at aPPrcP łtc unib of dmcn.
sio'B' lve shallmeive 5 gloups ol ihe clułiions tyith
thc
should he
]2 ł -
3 of th!
dińĆs
ts'en Parańctffs ol cxPle$ion (7) l t śPośibleto ńakś
non thc abovc ncnlion.d Palamctgs ln (7) a nuńbc! of
thc dlnc6ion1e$
ind io atbibutc thcń to thc
basic function. not 'clrhoN
applyms rhc analysis of rhc ihcorJ, of
:la:r-_.!-flne'
ńn NF*L } o ińśŚ'. ;ńii]'.a i', _tl"
tiÓn Ófo hea|baŃfor d,eflicien
dińcnslonl.ss fon hc -bansfc. ocffi.lcnt ind
.hd...hńzcs inloBifu of prŃs of hcat cxchangc by
the
.ontact belween
a
body sullac€
I
and cnmronment'
Ł
'='r,,'
l ",'''',,. " Ó, - l]
!__':l
APPlying lo othe! 9z$ the rhedry of dińensiÓĄś, we
,l: l'="
Haling dccidcd o1 th. systcńs oi rhe .qlatiÓN
(]9} (23) (tne lhfuly ol dete!ńiń
swq it oncc), w. shall
rNiv.
the rollowing
h.anirBs or
.I
l, I
.,,,
.= 1,
) -:,
"=
"='
Having Jdd.cŚcd ihcm toexPr6sions (9) (13)and
nc in8sofP amele$,n'eŚhall
rc..i!c th rhc dimcnsionlss .orc
Havińg cxPresed dle a]latiol (29) ln thc iom norc
Ónv.nicni lor Usei we re.eilc (ąuation (30))'
T.Li']3 intÓ 3ccoun', thal somc p łnetere of e]Llti{'r
(]0) can b! t}e sańc for modPl .nd real ob]ed, tlF Valucs
of łponenh in rhc aPprcPriaic d
tions lvill b. equil to zem łnd th
el.ment o1cqułiion (30) s'illb<ońe onc' FoI *amPlq
wh€n ol and ó. valu* aF th.sat. iol mod.l an,l rea]
Óhrcdlhccxpon.nrĘ= 0and clcnrot (d
'/o,l =r
lf m the modĆl borh n criil and n
and nlio Óf gsmcbi.il P anete
rionof thedcpcndene(30)Ćanb!ttanłornrcd'ind sill
.,,.,,,,i,,*,,.i
qt
subshfohng ihc found
on
D.p.ńden.e of lenPeBfo
The
g.n€nl doP.ŃenĆ.
,-r',:l1,tll
,"
"
re on
Óf
a zonc
I
sqthg.was develoPod^ń hcńJti.ilfrodcLbimd
th.similiritythory rrd kinch. modclofthc rri.riof
wcar TheńÓdcl ]ct sl.ulal. th.l'
Pińs Polymcr ńadc of .omPGites
P amPleBand ba$cdih olth.composihs
we beliovĆ th modd Usc let 9V.
rials in .omPalison wilh a rcal fri.tion Ncal
fadosanalvzed
iĆńFtlabt ń
I
'
HavinS bansfomcd cxpcrimental dara it is possblc
to debrnin c nU mcrica] meJ ni lgs .I fłdo! r ind cx.
ponenh łl ind b lor (3] ) with rh. h
łerno\ł
.rPcrińc
o'
:r!rl4rer. ł!!
I
DUe ro rhc ihcory
o'sinilińiyand.nłlysis oidimen.
2'
"osnovy hibologir
Gd chichjl]adz.A !!' Nłlka
sions wc hivc had an oPPo UnilJ, to e*łblish a de8re.
of dependen.e o1tcnPłallE .n rhe fa.b6 ańalyzcd
The siI of funĆliÓn (7) sl'Ólld be defŃed by exFeńene,
oblious kjnd ofexPgiń.nhny invcstigahd
dlFrndeńce oficńPsafurc on loading and onsPccd of
sLlinB Ól Polyń.ń. ońPosihs on a basis ih. phcnol.
(29) is an
To
łnswfl thc
PurPose
lh. l.m'
hav. bc.n clGcd up ir tlL sanplPs
a
s!i..
s
of
hiction Ther*Ulb of
r' "-'f
,=,,""'".l
1,""'-"'
l L r'
q I l;t l;l
l
2mr'chcmn
z (ccmatry],
conl
]Vlar, ]9Ź