Tribology data handbook

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VIII
Component Performance
and Design Data

Copyright © 1997 CRC Press, LLC.

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58 Fundamentals of Lubrication
Elastohydrodynamic

Michael M. Khonsari and D. Y. Hua
CONTENTS

Nomenclature.................................................................................................................................611
Geometry of Contact....................................................................................................................613
Dry Contact....................................................................................................................................614
Elastohydrodynamic Line Contact..............................................................................................616
Elastohydrodynamic Elliptical Contact.......................................................................................621
Starvation........................................................................................................................................625
Thermal Correction.......................................................................................................................625
Partial-Film EHL............................................................................................................................627
Traction............................................................................................................................................627
Examples.........................................................................................................................................630
References.......................................................................................................................................636

NOMENCLATURE


0-8493-3904-9/97/$0.00+$.50
1997 by CRC Press LLC 611


612 Tribology Data Handbook



Fundamentals of Elastohydrodynamic Lubrication 613

GEOMETRY OF CONTACT

A general Hertzian contact between two bodies is shown in Figure 1.1 Two principal planes are
used to characterize the geometry at the point of contact. Rxl, Ryl, and Rx2, Ry2 are principal radii
for body 1 and body 2, respectively. In general, the principal planes of body 1 and body 2 may
not coincide. However, for most engineering machine elements, the principal radii Rxl and Rx2,
as well as Ry1, and Ry2 lie in the same plane. In this chapter, the following equivalent radii and
equivalent modulus of elasticity are introduced.

FIGURE 1 Geometry of elliptical contact.1

The equivalent radius in x direction is

and the equivalent radius in y direction is




where “+” and “-” represent convex and concave of the surface 2, respectively. Then, the curvature
sum in x and y direction is defined as

The equivalent elastic modulus is

The above equations are valid for the general case of an elliptical contact as formed between two ellip-
soids with aligned principal axes, two crowned cylinders, or two cylinders that cross at right angle. The
elliptical contact can be reduced to two special cases:

Circular contact — when Rx1 = Ryl = R1 and Rx2 = Ry2 = R2, i.e., contact between two spheres.
In this case, R = 1/(1/R1 + 1/R2).
Line contact — both Ryl and Ry2 are infinity. Then, Ry → ∞ and the curvature sum R = Rx.
(cf. Figure 2).

FIGURE 2 Line contact: (a) nonconformal; (b) conformal; (c) equivalent elastic cylinder and rigid
surface.

DRY CONTACT
LINE CONTACT
Two cylinders pressed against one another under a normal load will produce a plane rectangular con-
tact area. If the cylinders are unequal, the contact area is not truly rectangular. Nevertheless, the plane
contact is a reasonable assumption. Under a normal load, w, the “contact patch” will have width of
2b. In the absence of lubricant, the normal load is parabolically distributed over this area. The half-
width of contact and the maximum Hertzian contact pressure are functions of the load per unit
length, the equivalent curvature radius, R, and the equivalent elastic modulus, E. The




Hertzian predictions of mean pressure, the magnitude and location of the maximum shear
stress, as well as the normal approach of the centers are listed in Table 1.

CIRCULAR CONTACT
The contact between two spheres forms a circular region whose diameter is 2a. The radius of
the contact and the maximum pressure in terms of the load, radii of the spheres, and elastici-
ty modulus are given in Table 1 along with mean pressure, maximum shear stress, maximum
tensile stress, and the normal approach of the center.

ELLIPTICAL CONTACT
The geometry of an elliptical contact is shown in Figure 1. The elliptic parameter k is defined
as the ratio of the ellipse semimajor axis a to that of semiminor axis b. In general, the ellipti-
cal parameter requires solving the first and the second elliptical integrals. The approximation
of the elliptical parameter and the integrals can be used to simplify the expression which is
related to the radius ratio.2 The definition and the approximation equations are listed in Table
2. These approximations are valid for the range of 1 ≤ Ry/Rx≤ 100, or 1 ≤ κ ≤ 18.




The radii of the elliptical contact, a and b, as well as the maximum Hertzian pressure, pH, are
functions of several parameters such as load, equivalent radius of the bodies, and the elasticity mod-
ulus, as well as the elliptic parameter and the elliptic integral. The appropriate equations are listed in
Table 1. The contact deformation at the center of the contact is also provided in Table 1.

ELASTOHYDRODYNAMIC LINE CONTACT
FILM SHAPE AND PRESSURE DISTRIBUTION
A typical film shape and pressure distribution of elastohydrodynamic lubrication (EHL) is shown
in Figure 3. Generally, EHL pressure distribution closely resembles the dry Hertzian contact with
the major exception of a pressure build-up in the inlet region and a pressure spike in the exit region.
Existence of the sharp pressure spike accompanied by a film constriction at the exit region are
important characteristics of the elastohydrodynamic lubrication regime.
Several trends in EHL may be noted. First, increasing speed or decreasing load tend to increase
the magnitude of pressure spike and move its location towards the inlet region. Under very heavy
loading, the pressure spike tends to decrease and eventually vanish, i.e., the pressure profile
approaches that of the dry Hertzian. In EHL applications, both the maximum Hertzian contact
pressure and the pressure spike are important parameters. Although the pressure spike is very nar-
row, its occurrence is very important since it may produce high subsurface stresses that directly
affect the rolling element bearing fatigue life.
The minimum film thickness at the film constriction compared to surface roughness dictates
whether the lubrication film is thick enough to protect the surfaces. The central film thickness
(essentially the parallel central region) is also a useful parameter in engineering design. The film
thickness is reduced by starvation of the lubricant and by inlet heating as discussed in sections on
“Starvation” and “Thermal Correction.”
The appropriate EHL equations can be conveniently grouped in terms of the following dimen-
sionless parameters:


FIGURE 3 © 1997shapePress, pressure distribution of line contact.
Copyright Film CRC and LLC.



where the viscosity–pressure coefficient is defined as

In nonconformal contacts, it is important to include the variation of viscosity with pressure.
There are two general relationship. The Barus viscosity–pressure relation is

The typical values of viscosity-pressure coefficient a for several lubricants are listed in Table
3.3 The other relation due to Roelands4 is given below:

The typical value for z is 0.6, S0 is 1.1 and a is 5.1 × 10-9
The EHL formulae reported in this chapter are based on Barus’ equation unless otherwise spec-
ified.

REGIMES OF FLUID FILM LUBRICATION
Many expressions for evaluating EHL film thickness are available in the literature. These are
obtained using curve fitting techniques to the numerical solutions of the governing equations that
involve the Reynolds equation coupled with surface deformation. These expressions, however, only
apply to a particular range of operation conditions and cannot be extrapolated into different regimes.
It is, therefore, necessary to define the regimes for appropriate usage of the film thickness expressions.
Referring to Figure 4, the following regimes may be defined:5
• Rigid-isoviscous, load is not high enough to produce either an appreciable viscosity change or elas-
tic deformation of contact surfaces
• Rigid-viscous, significant viscosity increase occurs due to high pressure but the elastic deformation
of contact surfaces is negligible
• Elastic-isoviscous, elastic deformation of contact surfaces is quite large compared to the film thick-
ness but the viscosity change due to pressure is negligible
• Elastic-viscous,6 viscosity changes due to pressure and elastic deformation of contact surfaces
play important roles. This is the regime of “full” EHL

FILM THICKNESS FORMULAE
The following dimensionless groups conveniently categorize the appropriate regime:




Film thickness formulae for the above-mentioned regimes are listed in Table 4.

PRESSURE SPIKE FORMULAE
Pressure spike amplitude and its locations are also determined by curve fitting the results of numeri-
cal simulations. Data which were used in curve fitting covered a wide range of operating parameters
with dimensionless load W varying from 0.2045 × 10-4, dimensionless speed U varying from 0.1 × 10-11
tp 5.0 × 10-11, and values of dimensionless materials parameter G of 2504, 5007, and 7511. One
must check to make certain that these restrictions are satisfied for a given application.
The pressure spike magnitude and its location are determined from the following expressions,7




FIGURE 4 Lubrication regimes of line contact.4 (From Roelands, D.J.A., Correlational
Aspects of the Velocity-Temperature-Pressure Relationship of Lubrication Oils, Druk, V.R.B.,
Groningen, Netherlands, 1966.)

Pressure spike location is

The center of pressure (the location of the center of pressure indicates the position at
which the resulting force acts) is given by:

Another form of minimum film thickness expression is also available,7

In dimensional form where w is the load-per-width, minimum film thickness is




The central film thickness is

ELASTOHYDRODYNAMIC ELLIPTICAL CONTACT
The characteristic film shape and pressure distribution of an elliptical EHL is similar to that of
the line contact. Some typical pressure and film thickness profiles predicted by the EHL theo-
ry are shown in Figure 5.8 The maximum Hertzian contact pressure, pressure spike, and mini-
mum film thickness, as well as central film thickness are of interest.

FIGURE 5 Typical contour plot of film thickness (left) and pressure profile (right) for a cir-
cular contact.8

In order to show the different regimes of lubrication problems, the dimensionless parameters
defined in Equations 5 to 9 are used. The four regimes of rigid-isoviscous, rigid-viscous, elastic-iso-
viscous and elastic-viscous are illustrated in Figure 6.9
FILM THICKNESS FORMULAE
To determine the appropriate regime, the following dimensionless parameter groups are defined as:

Film thickness formulae in these different regimes are summarized in Table 4 and Table 5.
The minimum film thickness for more general consideration of the velocity vector is:14




FIGURE 6 Lubrication regimes of elliptical contact.8 (a) k = 111; (b) k = 1; (c) k = 3; (d) k = 6.

where




FIGURE 6 (Continued)

and




where, u and v are mean velocities in x and y direction, respectively; θ = tan-1 (u/ν). If pure
rolling or pure sliding exists, θ = 0 and ν = 0.

STARVATION
Reduction of film thickness due to starvation for a line contact is shown in Figure 7. For
starved circular contacts, the film thickness formula is:15,16

where subscript s refers to starved boundary condition; subscript F denotes flooded contact m
is the dimensionless distance of the inlet meniscus from the center of the contact; m* is the
dimensionless inlet distance required for achieving flooded conditions:

D, n, and c for different regimes are listed in Table 6.

FIGURE 7 Influence of starvation on film thickness predicted by numerical simulation.
Parameters hstarved and hflooded refer to the starved film thicknesses, respectively. The distance from
the inlet meniscus to the edge of Hertzian boundary is denoted by Xj.19

THERMAL CORRECTION
For a line contact, film thickness reduction due to viscous heating of the lubricant at the con-
junction inlet can be estimated by a thermal correction factor as




where the thermal correction factor Ct is17

where

ur is rolling velocity, m/s; S is slide-roll ratio; Kf is the thermal conductivity of the lubricant,
W/(m ⋅ K).
Reduction of film thickness due to inlet shear heating can be estimated from Figure 8,18
which is based on the following empirical viscosity–temperature relation.

FIGURE 8 Thermal correction factor. Parameter µo denotes the viscosity under the ambient con-
dition and Kf is the lubricant thermal conductivity. With a known temperature-viscosity coefficient, β,
the dimensionless thermal parameter, Lm, and the thermal reduction factor, φf, are easily evaluated.18,19

Parameter L* is simply




PARTIAL-FILM EHL
Figure 919 illustrates full-film and partial-film elastohydrodynamic lubrication. Partial-film EHL
is the regime where average film thickness becomes less than three times the composite sur-
face roughness, h < 3σ. For determining partial-film EHL performance, surface roughness
parameters required for each surface include: (1) σ, root mean square of surface roughness; (2)
surface roughness height distribution function; (3) λ0.5x, λ0.5y, 50% correlation lengths of sur-
face roughness in x and y directions; (4) autocorrelation function of roughness.

FIGURE 9 Full-film and partial-film lubrication.19

Typical contact area patterns for oriented rough surfaces are shown in Figure 10.20
Parameter γ is used to describe the surface pattern of the roughness.

where λ0.5x and λ0.5y are correlation lengths at which the autocorrelation function of the profile
is 50% of the value at the origin. The autocorrelation function is a measure of the wave length
structure of a surface profile, defined as follows:

where λ is the correlation length; δ is the height function along the x direction; and Rx(λ) is
the autocorrelation function in the x direction.
The surface roughness correction factor is defined as

Effect of surface roughness on the average film thickness of EHL contacts under pure
rolling condition is shown in Figure 1121 where Λ is film parameter, Λ = hsmooth/σ.

TRACTION
In EHL, as in all lubrication mechanisms, surface traction is present. In pure rolling, the rolling
traction is FR. When sliding occurs, a sliding traction, FS, will be present. The total traction force
on faster and slower surfaces will be

where “+” is for the faster surface and “-” is for the slower surface.




FIGURE 10 Contact pattern of oriented rough surfaces: left, transverse (γ < 1); center,
isotropic (γ = 1); and right, longitudinal (γ = 1).19,20

FIGURE 11 Effect of surface roughness on film thickness.19,21 PH/E = 0.003; pure rolling; G
= 3333; σ/R = 1.8 × 10-5.

Typical traction curves measured experimentally at various mean contact pressures are shown
in Figure 12. Rolling traction is much smaller than sliding traction, except for pure rolling. In
the low-slip region, traction increases almost linearly as slip increases. If the lubricant is assumed
to behave as a Newtonian fluid, this linear trend persists over large slips. However, experimental
measurements show that the traction curve rises linearly from pure rolling (zero traction) and
reaches a plateau at a certain slip ratio in the so-called nonlinear isothermal region shown in
Figure 12. In this region, the linearly viscous (Newtonian) constitutive equation for the lubricant
is no longer valid. In the so-called thermal region, traction tends to drop with increasing slip.
This trend can only be predicted if the model properly incorporates non-Newtonian effects with
thermal consideration. One example of the traction coefficient predicted, using Bair-Winer’s
constitutive equation22 with its comparison to experimental data, is shown in Figure 13.23. The
interested reader may refer to References 23 and 24 for the details of the formulation of the
governing equations for generalized non-Newtonian formulation including thermal effects and




FIGURE 12 Experimental traction curve under various mean contact pressures, illustrating
the linear, nonlinear isothermal, and thermal traction regimes.19

numerical solution technique. The effects of load, speed, and inlet temperature on traction
coefficient curves are illustrated in Figure 14. These trends are important in predicting the
trend of traction under various operating conditions. For example, increasing the mean con-
tact pressure tends to increase the traction coefficient, whereas increasing speed results in a
reduction of friction.

FIGURE 13 Comparison of thermoelastohydrodynamic traction coefficient using the Bair-
Winer’s constitutive equation and experimental results (W = 5.5185 × 10-5, U1 = 2.8 m/s, G =
5152, τo = 1.4 × 107 N/m2, β = 0.05).23 The experimental results are taken from a research
report published by Zhang et al. at the Twente University of Technology 1983. (From
Khonsari, M.M. and Hua, D.Y., J. Tricol., Trans. ASME, 116(1),37–46, 1994. With permission.)




FIGURE 14 Effects of load, speed, and inlet temperature on the traction curve.

EXAMPLES
LINE CONTACT
Consider a cylindrical roller of 40 mm diameter and 30 mm length contacting a cylinder of 120
mm diameter which rotates at 1000 rpm. The load on the roller is 3000 N. The viscosity of the
lubricant at ambient pressure and room temperature is 0.04 N ⋅ s/m2. The pressure viscosity
coefficient is 2.1 × 10-8m2/N. The two surfaces are steel with an elastic modulus of 2.08 × 1011
N/m2 and Poisson ratio of 0.3.

Geometry of contact
From Equation 1, the equivalent radius is

The equivalent elastic modulus is defined by Equation 4. As the material is the same for the
two surfaces,

For pure rolling, the rolling velocity is




Dry contact
From Table 1, the half-width of Hertzian contact is

Maximum Hertzian contact pressure is

Mean contact pressure is

Maximum shear stress is

The location of τmax is at x = 0 and z = 1.02 × 10-4m (refer to Table 1).

Regime of lubrication
Refer to Table 4 and Equations 5 through 9. Calculating the dimensionless parameters yields
the following results:

Dimensionless velocity

Dimensionless material parameter

Dimensionless load

To determine the regime of lubrication, from Table 4 the dimensionless viscosity parameter is




The dimensionless elasticity parameter is

From Figure 4, this is within the regime of elastic-viscous and the dimensionless film thick-
ness parameter is

Film thickness
From Table 4, the minimum film thickness is

In dimensional form, we get the film thickness as

If the alternative equation (20) is used, the minimum film thickness is

and from Equation 22, the central film thickness is

Starvation effect
Assuming the distance from inlet oil meniscus to inlet edge of Hertzian boundary, xi is 2b,

From Figure 7, the reduction of film thickness is about 0.8.

Pressure spike
From Equation 17, the dimensionless pressure spike amplitude is

The dimensional pressure spike is

Dimensionless distance of the spike from the center of Hertzian contact by Equation 18 is




The dimensional distance from the center of the pressure to the center of Hertzian contact is

Consider the same rolling velocity and load, but with slip of 0.15 between two surfaces.
Estimate the thermal reduction in the film thickness. Assuming β = 0.05 and Kf = 0.12 W/(m
⋅ K), from Equation 37

Then using Equation 34, thermal correction factor Ct is

ELLIPTICAL CONTACT
Consider a steel roller of 40 mm diameter with a 50 mm crown radius (surface 1) contact with
80 mm diameter steel cylinder (surface 2). Rotation speed of the roller is 1500 rpm and the
cylinder is 1000 rpm. The load is 50 N. Viscosity of the lubricant is 0.028 N ⋅ s/m2. The visco-
pressure parameter is 1.45 × 10-8 m2/N. Equivalent elastic modulus for steel is 2.3 × 1011 N/m2.

Geometry of contact
Radii of the two surfaces are:

Velocities of the two surfaces are:

Rolling velocity is

From Equations 1 and 2, the equivalent radii are




From Equation 3, the curvature sum in x and y direction is,

From Table 2, the elliptic parameter is

Dry contact
From Table 2, the second kind of elliptic integral is

From Table 1, the elliptic contact radius is:

From the definition of the elliptic parameter in Table 2

The maximum Hertzian contact pressure is

the mean pressure is

Regime of lubrication
Appropriate dimensionless parameters are:

Dimensionless velocity

Dimensionless material parameter

Dimensionless load




To determine the regime of lubrication (cf. Table 4), the dimensionless viscosity parameter is

The dimensionless elasticity parameter is

From Figure 6 (d), it is in the elastic-viscous regime.

Film thickness
From Table 4, the dimensionless minimum film thickness parameter is

From Table 4, the dimensionless minimum film thickness is

In dimensional form the minimum film thickness is

The dimensionless central film thickness parameter is

The dimensionless central film thickness is

In dimensional form, the central film thickness is

Starvation effect
From Equations 31 and 32 and Table 6, m* for minimum film thickness is




Assuming dimensionless inlet distance m = 1.5, the reduction of minimum film thickness is

m* for the central film thickness is

Reduction of the central film thickness for m = 1.5 is

REFERENCES

1. Hamrock, B.J. and Dowson, D., Minimum Film Thickness in Elliptical Contacts for Different
Regimes of Fluid Film Lubricants, NASA Tech. Pap., No. 1342, 1978.
2. Brewe, D.E. and Hamrock, B.J., Simplified solution of elliptical contact deformation between two
elastic solids, J. Lubr. Technol. Trans. ASME, 99(4), 485–487, 1977.
3. Jones, W.R., Johnson, R.L., Sanborn, D.M., and Winer, W.O., Viscosity-pressure measurements of
several lubricants to 5.5 × 108 N/m2 (8 × 104 psi) and 149°C (300°F), Trans. ASLE, 18(4),
249–262, 1975.
4. Roelands, D.J.A., Correlational aspects of the viscosity-temperature-pressure relationship of lubri-
cating oils, Druk, V.R.B., Groningen, Netherlands, 1966.
5. Hooke, C.J., The elastohydrodynamic lubrication of heavily loaded contacts, J. Mech. Eng. Sci., 19(4),
149–156, 1977.
6. Dowson, D. and Higginson, G.R., Elastohydrodynamic Lubrication, Pergamon Press, Oxford, 1977.
7. Pan, P. and Hamrock, B.J., Simple formulae for performance parameters used in elastohydrody-
namically lubricated line contacts, J. Tribol., Trans. ASME, 111(2), 246–251, 1989.
8. Venner, C.H., Multilevel Solution of the EHL Line and Point Contact Problems, Ph.D. thesis,
University of Twente, Enschede, Netherlands, ISBN 90-9003974-0, 1991.
9. Esfahamian, M. and Hamrock, B.J., Fluid-film lubrication regimes revisited, STLE Tribol. Trans.,
34(4), 618–632, 1991.
10. Brewe, D.E., Hamrock, B.J., and Taylor, C.M., Effects of geometry on hydrodynamic film thick-
ness, J. Lubr. Technol., Trans. ASME, 101(2), 231–239, 1979.
11. Jeng, Y.R., Hamrock, B.J., and Brewe, D.E., Piezoviscous effects in nonconformal contacts lubri-
cated hydrodynamically, ASLE Trans., 30(4), 452–464, 1987.
12. Hamrock, B.J. and Dowson, D., Elastohydrodynamic lubrication of elliptical contacts for materi-
als of low elastic modulus, I. Fully flooded conjunctions, J. Lubr. Technol., Trans. ASME, 100(2),
236–245, 1978.
13. Hamrock, B.J. and Dowson, D., Isothermal elastohydrodynamic lubrication of point contacts, III.
Fully flooded results, J. Lubr. Technol., Trans. ASME, 99(2), 264–276, 1977.
14. Chittenden, R.J. et al., Theoretical analysis of isothermal EHL concentrated contacts: I and II,
Proc. R. Soc., London, Ser. A, 387, 245–294, 1985.




15. Hamrock, B.J. and Dowson, D., Isothermal elastohydrodynamic lubrication of point contacts, IV.
Starvation results, J. Lubr. Technol., Trans. ASME, 99(1), 15–23, 1977.
16. Hamrock, B.J. and Dowson, D., Elastohydrodynamic lubrication of elliptical contacts for materi-
als of low elastic modulus, II. Starved conjunctions, J. Lubr. Technol., Trans. ASME, 101(1),
92–98,1979.
17. Gupta, P.K. et al., Visco-elastic effects in Mil-L-7808 type lubricant, I. Analytical formulation,
STLE Tribol. Trans., 34(4), 608–617, 1991.
18. Cheng, H.S., Calculation of elastohydrodynamic film thickness in high-speed rolling and sliding
contacts, Rep. No. MTI-67TR24, Mechanical Technology, Latham, NY, 1967.
19. Cheng, H.S., Elastohydrodynamic lubrication, CRC Handbook of Lubrication, Vol. 2, CRC Press,
1984, 139–162.
20. Patir, N. and Cheng, H.S., Effect of surface roughness on the central film thickness in EHD con-
tacts, Elastohydrodynamic and Related Topics, Proc. 5th Leeds-Lyon Symp. Tribology, Institution of
Mechanical Engineers, London, 1978, 15–21.
21. Patir, N. and Cheng, H.S., An average flow model for determining effects of three dimensional
roughness on partial hydrodynamic lubrication, J. Lubr. Technol., Trans. ASME, 100(1), 12–17,
1978.
22. Bair, S. and Winer, W.O., A rheological model for EHL contacts based on primary laboratory data,
J. Lubr. Technol., Trans. ASME, 101, 258–265, 1979.
23. Khonsari, M.M. and Hua, D.Y., Thermal elastohydrodynamic analysis using a generalized non-
Newtonian formulation with application to Bair-Winer constitutive equation, J. Tribol., Trans.
ASME, 116(1), 37–46, 1994.
24. Khonsari, M.M. and Hua, D. Y. Generalized non-Newtonian elastohydrodynamic lubrication,
Tribol. Int., 26, 45–411, 1994.


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