Introduction

    Rheology models express the way tribological conditions
translate to shear stress of the lubricant and friction force
on the interacting surfaces. Due to the complexity of the lubricant
rheology, the friction coefficient is usually obtained
experimentally either under the same operating conditions
or by curve fitting in a properly chosen friction map. The current
study aims at determining the rheological parameters of
a lubricant based on friction measurements carried out on
a commercial, readily available ball-on-disc machine. They
can then be implemented in power loss prediction methods
that utilize state of the art thermo-elastohydrodynamic numerical
models considering the non-Newtonian lubricant
behaviour and the dependency on pressure and temperature
of the lubricant properties.
Lubricants are commonly macro-molecular chains that
behave like polymers in elastohydrodynamic lubrication.
These chains follow a Newtonian linear law during steady
state or shear rates close to zero. However, they deviate
greatly under conditions of high pressure and high shear rate
such as those in a typical gear mesh or a rolling element bearing.
Under those conditions, the maximum friction typically
peaks and reaches a plateau at around 0.08, far less compared
to that a linear law would predict. Evans and Johnson
(Ref.1) using the extended rheology equation by Johnson
and Tevaarwerk (Ref.2) classified the behavior of the lubricant
into four distinct regions, which indicate if it stems from
the non-linear viscous or linear-elastic regime. The classification
is based on the Deborah number defined as the ratio of
the relaxation time of the lubricant to the time needed to pass
through the contact. When it becomes greater than unity,
which is typical for EHD contacts, the traction curve (friction
coefficient over slide-to-roll ratio) is linear-elastic at first and
then non-linear with a potential peak. The extended rheology
equation uses a hyperbolic sinus function which is attributed
to the studies of Eyring (Ref.3) on polymers. The sinh()
function is used to model the thermal activation theory of a
molecule which defines the amount of work a molecule must
perform to jump from an energy well to the next.
This aspect is known to be affected by both temperature
and pressure and hence it is reasonable to expect a similar
dependence in lubricants as well. Indeed, there are many
different models proposed, such as those by Johnson and
Tevaarwerk (Ref.2), Houpert et al (Ref.4), and Mihailidis and
Panagiotidis (Ref.5). Some contain both parameters while others omit temperature in favour of a simpler formulation.

    Friction would still rise with the increase of shear rate even
if the thermal influence on viscosity were negligible, due to
the term. Limiting shear stress introduces a threshold to the
maximum shear stress that a material could sustain before
actually deforming as a “plastic” one. The theory was first proposed
in 1960 by Smith (Ref.6), although hinted in a previous
work of Petrusevich in 1951 (Ref. 7). The flow mechanism of
thermally activated zones and viscous flow has been shown
in polymers to give its place to a different one in shear stress
above G/30, where G is the shear modulus. The new mechanism
is the formation of a shear band. A straightforward
separation of the thermal effects due to shearing is almost
impossible (Johnson and Greenwood (Ref.8)). Experiments
by Bair and Winer (Ref.9) in low shear rates but very high
pressures in an isothermal disc machine showed a clear and
distinct maximum of the friction coefficient indicating a limiting
shear stress. Further calculations and later microflow
images of shear bands have been presented by Bair (Ref.10).
On the other hand, there have been additional phenomena
observed, such as wall slip—especially in dissimilar, interfacing
materials (Guo et al (Ref.11)) that may also contribute
to the reduction of the friction. Despite that, in steel on steel
friction these phenomena have been only observed under
extreme sliding, thus the limiting shear stress can arguably
be considered the most probable explanation.

    Various models for implementing in simulation the theory
of the limiting shear stress have been proposed. Initially, Bair
(Ref.12) attempted an analytic approach in order to develop
a parameter that would contain physical properties such as
bond strength of the molecular structure. The calculation of
such parameter is quite difficult and in practice it was experimentally
obtained. In fact, this is an issue, since the temperature
effect on the oil viscosity under high pressure is very difficult
to isolate and calculate outside of an EHL contact. The
second issue relates to the fact that an EHL contact is not at
constant pressure overall, which in turn means that the total
friction force is a sum over the contact area that includes both
thermally activated zones and shear bands. Temperature
and pressure have a strong impact on the shear band formation
by affecting the shear modulus. Houpert (Ref. 4) presented
in his dissertation an exponential model concerning
the effect of temperature on the limiting shear stress. Wang
and Zhang in 1987 (Ref.13) also utilized an exponential law,
which was later modified by Hsiao and Hamrock in 1992 (Ref.14). Roshetov and Gryazon, as mentioned by Wikstro?m
and Ho?glund (Ref.15), presented an equation that includes
first degree factors concerning pressure and temperature,
which are multiplied. Kleemola and Lehtovaara in 2008
(Ref.16) presented two models, a simplified and a more complex
multi-parametric one including both second order and
exponential laws.

     The present study incorporates a simple
relationship to model the influence of temperature and pressure
without exponential function that could generate instability
in a solver.
The present study describes an experimental-analytical
procedure that has been developed, in order determine the
Eyring stress and the limiting shear stress of a given oil as
functions of temperature and pressure. Special care has been
taken to use commercially available equipment and standard
experimental procedures. These data could then be used as
input to any EHL model that would require them in order to
calculate the friction coefficient accurately. As an example, the
rheological parameters of the FVA (Forschungsvereinigung
Antriebstechnik) reference oil Nr. 4 are determined.

Conclusions

    The method, outlined in the present study,
achieved to extract the rheological parameters
needed to describe the oil behavior in elastohydrodynamic
contacts. The Eyring and the limiting
shear stress, as well as the factors considering the
temperature and pressure influence, are obtained
by evaluating the friction coefficient measurements
conducted under nearly isothermal, fullyflooded
EHL conditions.

    Based on the friction coefficient over slide-toroll
ratio measurements, obtained from 18 test
runs following the proposed procedure, the rheology
parameters for the FVA4 reference oil were
extracted. They can be used in advanced EHL
models to calculate the friction coefficient.

    As a preliminary validation of the method,
these parameters were then fed in a thermo-EHL
solver and the friction coefficient was calculated.
The results showed very good agreement with
measurements carried out under conditions outside
the range of those used to extract the rheology
parameters. A final validation incorporating
experiments on a two-disk machine is on the way.

    Summing up, the proposed method allows the
use of a commercial, readily available test rig with
automated process and minimum oil requirements
to extract rheology parameters needed for
advanced thermos-EHL simulation models and
for conditions commonly observed in rolling element
bearing and gears.