A Quantitative Friction-Based Approach of the Limiting Shear Stress Pressure and Temperature Dependence S.-N. Ndiaye, L. Martinie, D. Philippon, N. Devaux, P. Vergne1
Univ Lyon, INSA Lyon, CNRS, LaMCoS UMR5259, F-69621 Villeurbanne, France
DOI: 10.1007/s11249-017-0929-2
Abstract Friction in highly loaded lubricated contacts (where the pressure is greater than 1GPa) may present a plateau at intermediate slide-to-roll ratio, known in literature as the limiting shear stress (LSS) plateau. Its physical origins and its dependence to the operating conditions are still unclear, that is why predicting friction in such contacts still remains an issue. Apart from the nature of the lubricant, the two main parameters influencing the friction plateau value are pressure and temperature. Literature provides several empirical expressions of the LSS which either consider the pressure influence only, or both pressure and temperature but almost always with coupled terms. Therefore, the published LSS values derived from friction measurements can be considered as the macroscopic consequence of the influence of pressure and temperature but also of shear heating that occurs in sliding highly loaded contacts. In this paper, the contribution of each parameter was studied separately, i.e. through experiments accrued out under nominal isothermal conditions, but conducted at different temperatures and pressures on two lubricants: a synthetic ester (benzyl benzoate) and a turbine mineral oil. A new LSS model was derived, based only on the mechanical (i.e. shear) contribution to the LSS. Surprisingly, a simple linear dependence of the LSS with both pressure and temperature was found, revealing that the influence of each of the two parameters is decoupled from the other. As far as we know, this is the first time that such an uncoupled LSS model in pressure and temperature is reported. This work offers a better quantification of the response of lubricants submitted to very high pressure and high shear: it should help to improve friction prediction in highly-loaded lubricated contacts. Keywords Elastohydrodynamic lubrication (EHL); limiting shear stress (LSS); friction; traction; highly-loaded contacts, benzyl benzoate, turbine mineral oil
1
Corresponding author:
[email protected]
1
1. Introduction The earliest literature on film thickness prediction in elastohydrodynamic lubrication (EHL) provides models which showed acceptable agreement with experimental results (Foord et al. [1], Gohar & Cameron [2]). These models were based on the assumption of a Newtonian behavior of the lubricant and, for ease of numerical computations, viscosity and density variations versus pressure were described by simple empirical expressions.
Among them, the exponential viscosity-pressure
dependence (the so-called Barus expression [3]), and the Dowson & Higginson [4] model for density variations were widely used for qualitative predictions as they can be applied on a large panel of fluids and for wide ranges of pressure and temperature. Unlike film thickness, friction prediction has been and still is more problematic, especially when the contact pressure reaches significantly high values, equal or higher than 1 GPa. Under these circumstances, the lubricant’s behavior in the pressurized contact area is much more complex than at the contact inlet where film thickness is generated. Despite the fact that this issue has been debated during decades, there is no clear consensus today. The interested reader is referred to the review proposed by Berthe & Vergne [5] which, although dating back to 1990, show the diversity of the possible scenarios: it is clear that some have not yet been explored.
Indeed, friction
measurements may show several distinct regimes relatively to the slide-to-roll ratio (
) value,
dominated either by a viscous-, elastic- or plastic-like response of the lubricant as quoted by Bair & Winer [6]. Firstly a linear regime occurs at (very) low or intermediate
, followed by a shear-thinning region at low
, where the friction coefficient increase with
is lower than linear. Finally, and
mostly in the highly-loaded cases, the high contact pressures lead to a plateau regime which corresponds to the appearance of a limit in the response of the lubricant to the operating conditions. Even if the lubricant can already start experiencing self-heating in the plateau regime, an ultimate thermal regime may appear at medium-to-high
values, characterized by a friction reduction
originating from a strong thermal dissipation. The latter does no longer apply to the sole lubricant but extends to the contacting surfaces which heat up. This regime therefore involves a strong coupling between mechanisms occurring at the contact inlet and within the contact zone where friction is generated. It will be not considered in this paper as it induces effects which have an additional impact on the lubricant's behavior and can biased the LSS measurement. Several authors have already intended to predict the linear regime of a friction curve thanks to a pressure and temperature dependent Newtonian viscosity measured in rheological devices. Even if some authors have been able, from friction tests run at low rolling/sliding velocities, to derive apparent viscosities consistent with measured ones with high pressure viscometers (see e.g. Crook [7], Johnson & Cameron [8]), this approach failed for more severe operating conditions. In this latter case, the measured friction is much lower than the predicted Newtonian one. Moreover, the reason why a plateau regime occurs at high pressure and beyond a critical
is still poorly understood.
Most of the existing models capable to predict this regime are derived from friction measurements. They are based on the limiting shear stress concept, at first evoked by Smith [9] and macroscopically 2
defined as the mean shear stress in the contact area when the friction coefficient becomes rate (or
) independent. However, as quoted by Martinie & Vergne [10], the physical origin of the
occurrence of a LSS in highly loaded lubricating films remains unclear. Some authors like Evans & Johnson [11] and later Bair et al. [12] have suggested the formation of shear bands in the contact, even though no in situ evidence of shear banding has been reported. Yet, according to Chang [13], this shear banding mechanism can be triggered either thermally or mechanically, depending upon the experimental temperature and/or pressure conditions. Hence, it is fundamental to precisely control both parameters in order to be able to dissociate the contribution of each of them. Indeed, up to now no model based on a physical, well-identified, mechanism or on an independent characterization has been established and the LSS still must be otained from experimental friction results themselves.
Friction tests have therefore to be performed, as initiated by Johnson &
Tevaarwerk [14], Hirst & Moore [15] or Evans & Johnson [11], among many others. Nevertheless, these tests have or may have limitations as the limiting shear stress might depend, according to the literature, upon a large set of parameters (e.g. the lubricant’s nature and its loading history, pressure, temperature, surfaces wettability...) which can vary within a contact. In this context, the main parameters of a friction test, which are a priori input data and should thus be perfectly controlled, are the lubricant chemical composition, the normal load and the lubricant temperature. However, even the control of the two latter parameters is not trivial. On one side the normal load and the contact geometry lead to a variable pressure distribution over the contact area: the mean contact pressure, , equal to two-thirds of the Hertzian pressure,
, is generally used
as the relevant parameter. On the other side, under conditions combining high pressure and significant sliding rates, viscous heating of the lubricant makes difficult the determination of its actual temperature during a test, as explained by Habchi et al. [16]. Despite this lack of control on the contact settings, the macroscopic LSS of many lubricants has been measured by twin-disc experiments (e.g. in refs [11, 14-15]) as mentioned above and several models have been proposed to express its pressure and temperature dependence. A constant proportionality between the limiting (or peak) shear stress,
(or mean limiting shear stress ), and
pressure (or mean pressure ), was emphasized very early, for instance by Hirst & Moore (Figure 18 in [15]), so that the most widely used model in the literature writes: =
+
or
=
+
Equation (1)
Where -
is the limiting shear stress extrapolated to atmospheric pressure. Note that this parameter has no physical meaning since there is no valid reason why the LSS could happen in this condition,
-
Λ is a dimensionless parameter, specific to the considered lubricant and which may depend on temperature. Its typical value ranges from 0.05 to 0.12 according to Bair and Winer [6]. Moreover Höglund [17] has shown, in the range 293-353 K, that this parameter decreases moderately with temperature. 3
Others models from the literature (see for instance [18-23]) take into account both pressure and temperature influence but inevitably in a coupled way, due in particular to the shear heating of the lubricant which inherently occurs in the contact if no specific caution is taken. Therefore these models make impossible to determine the respective contributions of pressure and temperature independently of one another.
In this paper, a special attention was paid to reduce shear heating of the lubricant in order to uncouple the pressure and temperature contributions to the limiting shear stress. Ignoring or minimizing thermal effects should also help to focus on the mechanical origin of the LSS and to better characterize its dependence to pressure. The experimental approach is detailed first. In the second part, friction tests conducted under various operating conditions are presented: LSS variations with pressure and temperature are derived from appropriate friction curves. Several key points are afterward addressed in the last part of the paper. Results from friction experiments designed to minimize the undesirable viscous heating influence are discussed. The effects of pressure and temperature on the LSS found here are compared with the results from literature. This allows to come up with a new model of LSS, in which the fully uncoupled pressure and temperature influences are quantified.
2. Experimental approach 2.1. Lubricants properties Two lubricants were used in this experimental work: a newly characterized one and a second one already studied in previous works. The first fluid is benzyl benzoate (supplied by abcr GmbH (Karlsruhe, Germany), purity 99%, CAS 120-51-4), an organic compound of formula C6H5COOCH2C6H5 which is the ester of benzyl alcohol and benzoic acid.
Its rheological
characterization has been realized according to the same methodology as that described in detail in [24]. Viscosity measurements have been performed at atmospheric pressure for temperatures between 263 and 393 K in an Anton Paar Couette rheometer (Physica MCR 301). In the range of applied shear rates (10-1 - 10+3 s-1) the lubricant behavior is Newtonian and its viscosity-temperature dependence can be described by the Vogel-Fulcher-Tammann equation. A high-pressure fallingbody viscometer was then used to characterize the benzyl benzoate viscosity variations with pressure, up to 700 MPa: in these experiments, the temperature was varied from 313 to 403 K. The viscosity-pressure-temperature variations were regressed to a modified WLF-Yasutomi correlation which is capable to accurately reproduce (see Figure 1) the experimental viscosity variations, both at high and at ambient pressures [25]. In this expression, the viscosity at the glass transition was arbitrarily taken equal to 10+12 Pa.s, as mentioned in Table 1 where the values of the coefficients of this model are reported. 4
Figure 1. Viscosity-pressure dependence of benzyl benzoate at different temperatures.
The viscosity-pressure dependence at constant temperature is usually estimated by the so-called pressure-viscosity coefficient. Here, the reciprocal asymptotic isoviscous pressure,
∗
, has been
calculated from the modified WLF-Yasutomi model and some values obtained at four temperatures are given in Table 2: they will be used to estimate film thickness in the friction experiments.
Parameter
Units
Value
(°C)
400.19
-1
,
(GPa )
0.363
(GPa-1)
5.391
(-)
-0.399
(-)
16.00
(°C)
14.55
(°C)
-81.73
(Pa.s)
10+12
Table 1. Parameters of the modified WLF-Yasutomi correlation [25] for benzyl benzoate.
In a second step, the same experimental protocols as those developed for characterizing friction and then LSS of benzyl benzoate were applied to study an industrial lubricant of a different nature, a commercial turbine mineral oil (Shell T9), denoted TMO in the following. This should make possible to validate the experimental method described in the following and to confirm the findings on the 5
LSS variations with pressure and temperature obtained with benzyl benzoate. This turbine mineral oil had already been studied and the models, derived from rheological measurements performed on the specific batch used in this work, are described in Wheeler et al. [26]. As for benzyl benzoate, some rheological properties of TMO at different temperatures are included in Table 2.
Benzyl benzoate Viscosity Temperature
Turbine mineral oil (TMO)
Pressure-viscosity coefficient
Viscosity
∗
Pressure-viscosity coefficient
(K)
(mPa.s)
(GPa-1)
(mPa.s)
(GPa-1)
293
10.0
15.0
19.6
23.6
313
5.1
12.0
8.6
19.2
333
3.1
9.9
4.7
16.2
353
2.1
8.4
3.0
13.9
∗
Table 2. Rheological properties of the two lubricants at four temperatures calculated from the modified WLF-Yasutomi correlations. 2.2. Tribometer and specimen Friction tests were carried out with a ball-on-disc tribometer in which the two specimen are independently driven by two motors equipped with optical encoders of 2.10+4 steps per revolution combined with a 8-bit discretization to produce with high accuracy any desired slide-to-roll ratio (
). The radius of the ball specimen (of 12.7 mm) and the radius of gyration of the disk (i.e. the
distance between its center of rotation and the center of the contact track) were systematically controlled after the mounting of a new specimen. For an entrainment velocity of 3 m/s and 1%, the relative uncertainty on
is lower than 6% and this value drops significantly when
=
increases. A steel (or tungsten carbide, WC) ball was used with either a steel or a sapphire disc (or a WC disc): their material properties are given in Table 3. The measured roughness of the specimen is of the order of a few nanometers for both the steel ball and the discs: !"&'(# $%## ~ 5 nm, !"&'(# *+", ~ 10 nm
&'( and !"%--.. *+", ~ 8 nm. In the case of WC specimen, the roughness is the same for the disc and
&'( the ball: !01 ~ 20 nm.
Normal and friction forces (2 and 23 in the following, respectively) are simultaneously measured with
a maximum uncertainty of 0.8% for the former and 0.3% for the latter, leading to a friction coefficient uncertainty of ∆567 ⁄567 = 1.1%. For the clarity in the presentation of the results, the friction forces obtained at
of the same intensity but with opposite signs have been systematically averaged.
Furthermore, some tests have been repeated, leading to a mean relative standard deviation (considering values obtained at
> 0 and
< 0) of 2.8% with a rather low dispersion. Inlet
temperature,
?
@
A
B
3
(kg/m ) (W/mK) (J/kgK)
(GPa)
(-)
Bearing Steel
210
0.30
7850
46
470
Sapphire
360
0.34
4000
40
750
Tungsten carbide
610
0.258
14850
29.3
194.7
Table 3. Mechanical and thermal properties of the specimen materials. 2.3. Experimental approach In addition to cover the largest ranges of contact pressure and temperature, the main objective was to carry out friction measurements under isothermal or, more realistically, nominal isothermal conditions, as explained previously. The last expression means that even if thermal effects must occur in any friction measurement, it is possible to minimize them and to make the results almost insensitive to a weak energy dissipation within the experimental volume of interest. Therefore the following experimental strategy was applied: - For each temperature, the entrainment velocity, C , was chosen at its lowest value to minimize inlet shear heating while insuring a sufficient film thickness to remain in a full separation regime and to prevent any direct contact between the specimen which could biased the measurements. - Therefore the film thickness thermal reduction coefficient proposed by Cheng [28], D E , was calculated prior any experiment.
Values under 0.96 led to the cancellation of the test, in
agreement with past works performed on the same test-rig which showed [29] and confirmed [30], on the basis of film thickness measurements, a reduction of the slope in the film thickness versus C curves, when D E exceeded this threshold. - The duration of each non-zero
step was limited to 10 s, which is long enough to reach a
steady state and to record the data over a significant time. The temperature of the inlet zone,
