/
WEAR ELSEVIER
Wear 198(1996) 307312
Viscoelastic effects in lubricated contacts Irina Goryaehevaa, FarshidSadeghib, Gang Xu
b
institutefor Problemsin Mechanics,RussianAcademyof Sciences, Moscow,i17526,Russia t, Schoolof MechanicalEngintering,PurdueUniversity,WestLafayene,IN 47907, USA
Received12December1995;accepted3 April1996
Abstract The effect of viscoelasticlayer on pressure, filmthickness and friction coefficientin lubricatedcontacts has been investigated.The model consists of two elastic rollers coated with viscoelastic layers and separatedby a thin film of lubricant. The results indicate thd under low velocity conditions the friction coefficientmonotonicallyreduces to a minimum and then increasesas the speed increases.The results also indicatethat for low velocitiesthe contact characteristicsare dominatedby the propertiesof the viscoelasticlayer and the elastic substrateand as the speed increases, the viscoelasticlayer effect becomes negligible.
Keywords:Viscoelasticlayer;Maxwell model; Frictioncoefficient;Lubricatedcontacts
1, Introduction Surface nonhomogeneities, such as boundary layer, coatings, etc., play a major role in normally loaded contacting bodies with relative rolling/sliding motion. The mechanical properties of the surface layer are distinctly different from the bulk material and the layer thickness is usually comparable to the size of wear particles. The layer properties have a significant influence on internal stresses and surface failure. A number of investigators [ 14] etc. have developed theoretical models to calculate pressure and internal stresses for multilayered elastic bodies in contact. These studies have demonstrated that the internal stresses depend on geometrical and mechanical characteristics of the surface layer, However, inelastic ccating~ on elastic substmte have been investigated less rigorously. Kalker [5], Braat and Kalker [6], Goryacheva and Sadeghi [7] and Goryacheva eL al. [8] investigated rolling/sliding contact of viscoelastic layered bodies. The results illustrated that due to the inelasticity of the surface layer the contact characteristics and internal stresses depend on the sliding velocity. Goryacheva and Sadeghi [7 ] showed that the effect of viscoelastic layer on friction and stresses is more significant under low velocity conditions. In the present study, the effect of viscoelastic layer on pressure, film thickness and coefficient of friction in lubricated contacts is investigated, Hydrodynamic and elastohydrodynamic lubrication has been studied extensively by a number of investigators (see [9,10] ). These studies in general concentrated on pressure and film thickness and dem00431648196/$15.00O 1996ElsevierScience$.A. All rightsreserved PilS0043.1648(96)07206.7
onstrated that the Newtonian fluid model predicts satisfactory film thickness between contacting bodies. However, the Newtonian fluid model fails to predict friction and po'uer loss similar to experimental results at high loads and low velocities. In order to predict friction results correlating with experiments, researchers have introduced thermal effects or nonNewtonian fluid behavior or both into their studies [ 1115]. The Eyring fluid model [ 14], the nonlinear Maxwell model [ 13] and others have been used to describe the lubricant shear behaviour at high loads. Recently, Elsharkawy and Hararoek [ 16] investigated the effect of elastic coating on the pressure and the film thickness on EHL of multilayered elastic bodies. In this study, the combined effect of the viscoelastic layer bonded to elasdc cylinders and a th.:n film of lubricant is investigated, The Maxwell model is t,~e.d to describe the mechanical properties of the layer. The i n f i u ~ ~.:e..~,.'.'*vry and mechanical properties of a thin viscoelastic layer on contact stress, film thickness and friction coefficient is analyzed for various operating conditions.
2, Statement of the problem Fig. 1 illustrates a schematic of the contact between two layered rotating cylinders, separated by a thin film of lubri. cant, The (ri,03 coordinate system is fixed on each cylinder and rotates with angular velocity w~ (i= 1,2 for upper and lower cylinder respectively), The (x,y) coordinate system is
L Goryachevaetal./ Wear198(1996)307312
308
du, Hi
vi~oela~ic llyct
H,@
dx ~EP÷ A dx
W
(4)
where V~ and V2 axe the linear velocities of the cylinder surfaces. The Reynolds equation is used to describe the twodimensionS flow of a thin lubricating film between two surfaces moving with velocities It, and V2as shown in Fig. 1.
rz
d ha dp ~(~)=6(V,
dh + V2)~
(5)
where h(x) is the film thickness and p(x) is the pressure. The variation of viscosity with pressure is taken into account using the Barus [ 17] model: g = ~e~
Fig. !. Schematico f ~ layeredcylindersin lubricatedcontact. fixed in the plane such that the yaxis coincides with the line between the centers of the cylinders (Fig. 1). The shape functions of the cylinders aref~(x)  +_x2/2Ri+gi. The relationships between the moving (r~,O~)and the fixed (x,y) coordinate systems axe
x= ri cos( O~+ oJit)
(1)
y = r~ sin( 0~+coit) +yo.i i 1,2 where Yo.t, Yo.z axe the coordinates of the centers of the cylinders. The substrate of the layered cylindrical rollers is considered m be two.dimensional, isotropic, homogeneous and linear elastic. The displacement gradient for the substrate is:
x2_ b2
h(x) =h(b) +    ~ + (ui(x) +u2(x))  (u,(b) +u2(b)) + (w,(x) +w2(x)) (w,(b) +w2(b))
(7)
where IlR= IIR t + I/R 2. The displacements wi(x ), ui(x) are given by Eq. (2) and Eq. (4). The boundary conditions for the Reynolds Eq. (5) axe: p(~) =p(b)=0
d_.p
=0
(8)
x=b
The force equilibrium condition within the contact region is given by b
b
dw_.j.= 2(1v~) f pCs) ds dx ~tEi ,t xs
(6)
go is the absolute viscosity at ambient pressure and temperature, a is the pressureviscosity coefficient of the lubricant. Within the contact region, the thickness of the film can be expressed by
(2)
f p(x) dx=F
(9)
 m
In this study, the layers axe modeled as a onedimensional Maxwell (viscoelastic) solid which is represented by a spring of modulus ,~ in series with a dashpot of viscosity ~. The Maxwell solid exhibits a steady creep under a constant stress. The onedimensional timedependent relationship between the normal pressurep, which is assumed to be uniform across the viscoelastic layer thickness (H/), and the normal displacement ui is used to describe the layer compliance in the normal direction.
dui Hi ,Hidp
(3)
To simplify the analysis, it is assumed that the mechanical properties of the layers at the upper and lower cylinders axe the same. However, the method developed in this study can be used to consider the gen(aal ease of different mechanical properties of layers. Using F.q. (1) and the total derivative relationship, F_,q.(3) can be written as
Eqs. (2), (4)(9) are used to define the pressurep(x), the film profile h(x), the elastic wix) and viscoelastic ui(x) displacements of the bounding surfaces.
2.i. Filmprofile and contactpressureanalysis In this investigation, pressure and film profiles axe obtained for a constant viscosity and a variable viscosity relationship according to Eq. (6). In the case of constant viscosity, the system of Eqs. (2)(9) can be reduced to one equation only, function of film thickness. Integrating Eq. (2) and Eq. (4) and using boundary condition Eq. (8) we obtain b 1 1 Ht //2
u(x)u(b).~ f (~(~+~)[ ( x  s )  Ixs, ] (H, +//2)
"1 [sgn(xs)  1]~, J ~ds
(1o)
L OoryacMmetaLIWear198(1996)JO7M2
The system of F,qs. (I 6), (18) and (19) has be~nus~ to define the function fi(x) and the two parameters b and p. Nondimensional contact pressure/~(x) is obtained by integrating Eq. (5),
b
w(x)w(b)="~
309
[(bs)Inlbs[ oo
dp
 (xs) l n l x  s l ] ~ d s

(11)
,

6bs
P(O='T]oh3(s)
Os
(20)
where
u(x) =ui(x) +u2(x)
(12)
w(x) = wt(x) +w2(x)
(13)
Substituting Eq. (10) and Eq. (11) in Eq. (7) and using the integral form of the Reynolds F_x1. (5) we obtain b
h(x) =h(b) + ~ +
jfk"
L i t
,.h(s)h*
os
(l+,
where h* is the film thickness when dp/dx = O. The kernel in Eq. (14) is:
"
"
.
11)
b2~2
oo<~
(21)
Parameters 8 and e are defined to satisfy the conditions
h'(eO) =lz'(e+O)
 ( x  s ) lnlxsll~~,~[t~2jttx j  l x  s l ]
IHi+H2"  ~ t sgntx s)

h(O =~p +8
/~(eO) =h(e+0)
f2 k(x,s,b) = 6g(Vi + V2)'~g[ (b s) In(bs)
l/S,n~L.
Newton's method was used to solve the system of Eqs. (16), (18) and (19). In the numerical analysis, the infinite integrals in the region (.oo,1) were reduced to the finite integrals in the region (e,1) and the integrals in the region (  ®,e) were calculated analytically using the following function:
(15)
The system of equations has been reduced to the linear algebraic equations which are solved by the Gauss modification method using the condition number evaluation.
2.2. RoUingfrictionand traction analysis Eqs. (14) and (15) in the nondimensional form are given by ~2 1
.P/~2 1) + ~ f : ( , , s , / ~ ) ~ ~(0 I +=(~=
ds
The asymmetry of the pressure distribution within the contact area generates the rolling resistance couple M given by
(16,
b all
M= t xp(x) dx
(22)
I/
m
/~(~,s,/7,) = 4.~.FS[(1 s) In(1 s)  (~s) lnl ~sll  " ' 'Z'~ 1 [(~s)
In lubricatedcontacts the traction resistance due to the film of lubricant for the upper and lower cylinders is given by
l~sl]
b
b
[email protected]
?S
+ ~ [sgn(~ S)  11
(17)
°

I
j ~/t ,
,m
(24)
 er .
Then the rolling resistance coefliciem and the traction coefficients are given by
(,s)
L=~M
(19)
~ =~
^
(2s)
T
_
h(x) I.
b
_
6b2 [ h ( x )  l .
1
rv,v2
. . . . b
Here parameter [J= (HmH:312H indicates the difference in layer thickness and H= (Hi +//2)/2 is the average layer thickness. Combining Eqs. (5), (8) and (9) and nondimensionalizing gives
,
In the nondimensionalform they are given by
(26)
310
L Goryachevaetal. I Wear198('996):707312
3b'~s[~ d~ * : T J h'
(27)
12
i~ 10 _
I
_
_
I
.~bs f h l dr:+~bs f d~
(2s)
±7.!=T 7!5
i
3, Results and discussion The results of this investigation are presented as a function of five nondimensional parameters. The nondimensional parameter C ! =4ptt/~R characterizes the relative fluid film and layer viscosities. H is the layer thickness which has been assumed to be the same on the upper and lower cylinders. The relative elastic modulus of the layer and substrate is characterized by X = ARI2E*H. The Sommerfeld number, S, describes the average velocity effect and y denotes the difference in velocities. The nondimensional load is given by
.4
.
.3
2
.I
0
1
Fig. 3. Pressureresult~ias a functionof $ommeffeldnumberforFffi104, .~=1. and C,l(Iy2)ffi5e8: (a) S~le5; (b) S5e5. (c) $=1e3. 2
....
.
li
?=v/z*R.
Fig. 2 depicts the pressure and film thickness results for the conditions when C , = 5 × 1 0 s and C1=0 at S= 2 × 10 s. The figures damonstrate the influence of relative viscosity parameter C1 on pressure and film '..hickness. Please note that when C, = 0 it is the condition of elastic layer on elastic substrate. The results indicate that the maximum pressure is nearly the same; however, when viscoelastic effects are included (Ci =5 x 10s), the pressure distribution becomes less symmetrical. Fig. 2 also depicts that the film thickness is nearly constant in the contact when C~=5×10 s. Figs. 3 and 4 illustrate contact pressure and film thickness as a function of the Sommerfeld number. The results show that for low velocity conditions (S= 1 X 105), the film thickness within the contact zone is nearly constant and the pressure distribution is similar to the case without the lubricating film [7]. For increasing values of velocity (5= t X 10 3) the pressure is distributed over a large area
'"",.4....... g
i
.2
.i X/5
0
I
Fig.2. PressureandfilmthicknessresultsforF 104 ~ = i, andS= 2e 5: (a),(b), Ci/(ly z) = 5e8; (a'),(b'), CI0.
i,
't xlb Fig.4. Filmthizknessresultsas a functionof $ommerfeldnumberfor/~= 10't, ~= I, and CJ(I yz)=5e8: (a) S= ie5; (b) S=5e5; (c) S=le3.
and the film shape develops features corresponding to hydrodynamic regime. The maximum contact pressure also decreasesand moves toward the exit of_thecontact as velocity increases. The contact exit location b strongly depends on velocity for low values of velocity. Under low velocity conditions, when Ct increases the exit location approaches the axis of symmetry of the cylinder. For high velocity conditions, the exit location is nearly the same for all values of Ct. A comparison of the minimum film thickness and maximum indentation of the cylinder in the viscoelastic layer is shown in Fig. 5. The figure indicates that under low velocity conditions, hm~nis much lower than the maximum indentation. However, as the Sommerfeldnumber increases, the minimum film thickness is much larger than the indentation. This illustrates that at low Sommerfeld values the viscoelastic layer is the major contributor to the contact characteristics. It is to be noted that for increasing values of S the minimum film thickness increases and does not depend on viscosity 7/ of the Maxwell solid, Minimum film thickness and maximum indentation increase when parameter A decreases.
31!
L Goryachevaetal. i Weor198(I996)307312
]/. o~I~i
Z
oo~!
~
s Fig.5. Minimumfilmthl~knessandmaximumdisplacementof viscoelastic layeras a funcdonof Sommeffeldnumberfor F= 104: (a),(a'), A= I, C, =0; (b).(b'), ,~= l. C~$e 8; (c),(c+).A=0.5.C~= 5e8.
,:A
+"
\\
// \
!,o \\//
.
°~(i
.
.
.
o~flOl
~
,
0.0G015
~
s Fig. 7. kolling friction coefficient (j';) and traction coe~cicat ~) as a function of" SommeffeM number for #= I0'; (a).(a'), ,~t, C,~,O; (b),(b'), ,~= t, Ci =So 8; (c).(c'). A= 0.5. Ci =5e8.
,.
,!
o:o25
,i this
oms
Co' ) +
YCe
Fig, 6. Variable viscosity effect on pressure and film lhickness for CI "le9, C 2 = 2 e  ii, and S='/e6: {a) viscosityconstant;{b) viscositychanges with pre~,~em,
Fig. 6 illustrates the effect of variable viscosity (Eq. (6)) on pressure and film thickness profile. The results indicate Lhatwhen viscosity is allowed to vary as a function of pressure the film thickness is larger than the case when vis~os;ty i~ consL'mt and the film thickness exhibits a small reduction near me exit. The pressure profile is negligibly higher, Fig. 7 depicts the coefficient of rolling friction f.fr) and traction coefficient ~ ) as a function of the Sommerfeld number for different values of layer viscosities, Ci. The lesulL~indicate that for high values of CI = 5 × 1O s the rolling friction coefficient nonmonotonically reduces as the Sommerfeld number increases. Hear $z =5 x 10 9 the friction coefficient reaches its minimum and then increases as the speed increases. Please note as shown in Fig. 2, when CI =5 X i0 ~ the pressure distribution has the largest offset relative to the center of the contact (X=O) at S= 1 × 10 5, and at S  5 × 10 5 the smallest offset. When CI is large and S is small the maximum deflection into the viscoelastic layer is the largest (Fig, 5), When CI = 1 x 10 "s, the coefficient of rollin[; friction again reduces and reaches its minimum near S 2 × 10 9 and i~mreascsas S increases. When CI = 0, the contact is an elastic layer on elastic substrate. In this case,
(c)
O.GOI
G (¢}
~ ~ :
+,~
,
0~I
, ,
o~t ¥
Fig, g. IZ~llingfrictioncoeffi¢tent.~)and Ir~tion coefficient~) as a run.ion of slidi.g velechyfor F=I0 +, A= I: (a),(~'), C,=5e8; (b),(b'), C, = le8; (c),(c'), CI =0. the coefficientofro]ling friction non.monotonical]yincreases as the speed increases. The traction coefficient is nearly the same for all values of Ci and increases as the speed increases. However, its magnitude in general is lower than the rolling friction coefficient. Fig. 8 illustrates the effect of moment of friction and traction coefficient as a function of sliding velocity. The results indicate that in general the fricdon coefficients arc nearly constant, however at y near 0.1 they monotonically increase with increasing sliding velocity.
4. Conclusion
The results indicate that due to a viscoelastic layer the rolling friction coefficientis a nonmonotonic function of the Sommerfeld number of velocity, For low velocities the friction coefficient reduces to a minimum and increases as the velocity increases. The analysis shows also that forlow velocities the contact characteristics are dominated by the properties of the viscoelastic layer and the elastic substrate: and as
312
!. Goryod~,vaet al. I Wear 198 (1996)307312
velocity increases the viscoelastic layer effect becomes negligible. The resultsare in goodqualitativeagreementwiththe wellknown experimental results of Stribeek. Therefore, when investigatinglubricatedcontactsthe specificpropertiesof the thin surface layer must be taken into account. Acknowledgements The authors would like to offer their deepest appreciation to the National Science Foundation for their support of this project. The authors would specially like to thank Dr. Join LarsenBasse for his encouragement and discussion during this endeavor. Appendix A. Nomenclature C1 E*
F /!. b ~_ b S Y
nondimensionalparameter,4p.H/~lR equivalentmodulus of elasticity [ ( 1 v2) / El + (I  t~2)/E2] i normal load film viscosity layer viscosity parameter exit point nondimensionalcoordinate,x/b nondimensionalcoordinate of the exit point, b/R Sommerfeld number,P,o(VIFV2)/F nondimensionalparameter,F/E*R nondimensionaldifference of velocities, 
(V~ V2)l(Vt + V2)
p(x) P h_(x) h~ P
~,,~
nondimensionalparameter,ARIE*(Ht + 1t2) contact pressure nondimensionalpressure,pR/F film thickness nondimensionalfilm thickness, h(b~)Ih(b) nondimensionalminimum film thickness, h~./R nondimensional film thickness at the exit point,
h(b)/R nondimensionalmaximum displacementof the viscoelastic layer surface, urnJ R
References
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