Application to Elastohydrodynamic Friction

CHAPTER THIRTEEN

Application to Elastohydrodynamic Friction 13.1 INTRODUCTION The terms traction and friction are used almost interchangeably, traction when a large value is required as in an infinitely variable transmission, traction drive, or skewed roller brake and friction when a low value is desired as for a gear tooth contact or a cam/follower. Over the course of the study of elastohydrodynamic lubrication (EHL), much effort has been directed toward understanding the mechanisms which produce the particular dependence of the traction force on the sliding velocity as characterized by a traction curve. A traction curve is a plot of traction force, average shear stress, τ, or friction (traction) coefficient, τ=p, versus sliding velocity, u1 2 u2 , slide-to-roll ratio, Σ 5 ðu1 2 u2 Þ=u, or effective shear rate, ðu1 2 u2 Þ=hc . There were multiple approaches to the problem. As early as 1959 Smith [1], in perhaps the first mention of a limiting shear stress, postulated that “the process in the contact zone for a predominantly rolling motion may involve the deformation of a plastic solid whose shear strength is independent of the rate of shear.” It is remarkable that this early assessment of the rheological response was so very accurate. As long ago as 1963, Crook [2] found that the decrease in traction with sliding speed past the maximum in the friction curve resulted from thermal softening and that both the thermal conductivity and the shear-dependent viscosity of the liquid at the pressures in the film were of extreme importance to a calculation in the thermal regime. The importance of shear-thinning and temperatureviscosity response to the thermal regime is still not fully realized today in the classical approach. Early attempts to explain traction curves employed ad hoc rheological models. In 1969, Allen et al. [3] proposed that the response may be regarded as Newtonian with pressure-dependence that follows two Arrhenius relations, Eq. (6.20), one with a large slope for low pressure High Pressure Rheology for Quantitative Elastohydrodynamics. DOI: https://doi.org/10.1016/B978-0-444-64156-4.00013-1

© 2019 Elsevier B.V. All rights reserved.

327

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and the second with a very small slope at high pressure, the viscosity being continuous at the transition pressure. This model continues in use, known as the composite or two-slope model, although there is no primary data to support it and the second slope likely results from the attempt to explain the elastic creep of the rollers by a reduction in viscosity. Viscoelasticity was invoked to explain traction curves in three separate schemes. In 1967, Fein [4] supposed that the time for which the liquid was exposed to pressure was less than the time required for the liquid to respond in compression to reach the equilibrium density. The volume relaxation resulted from the bulk viscosity, υ. See Section 2.2. The free volume would then be greater than the equilibrium value causing a reduction in viscosity as the contact transit time was decreased, that is, as velocity was increased. In 1971, Chow and Saibel [5] employed the single mode Maxwell model with an objective time derivative of the stress tensor as in Section 2.4.1. In 1970, Dyson [6] used the analogy between the behavior of viscoelastic liquids in oscillatory shear and in steady shear, Sections 2.4.1 and 8.3.2, to explain the linear and nonlinear portions of the traction curve. By far, the most often invoked explanation of traction behavior has been the notion that the steady shear constitutive response is reflected in the shape of the traction curve itself, the second rheological assumption of classical EHL, Section 1.5. The most influential work has been that of Johnson and Tevaarwerk [7] from 1977. They presented traction curves as the average shear stress, τ, versus shear rate which were accurately fitted by the sinh-law. τ 5 τ E sinh21 ðCE Σ Þ

(1.19)

Shear rate is related to slide-to-roll ratio by γ_ 5 Σu=h. The argument is then made that the constitutive behavior should have the same form with the same value for τ E known as the Eyring stress.

 _ _ τE 21 μγ 21 μγ η5 or τ 5 τ E sinh (8.2) sinh τE τE γ_ This hypothesis could have, and should have, been tested over the years using the pressure dependence of the viscosity of the liquids employed by Johnson and Tevaarwerk [7]. One of these liquids was the mineral oil, LVI260, for which the low-shear viscosity was modeled in Section 6.6.1 using the properties of Tables 4.14 and 6.2 with the result

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329

Figure 13.1 The sinh-law of Eq. (8.2) with τ E 5 2.9 MPa is plotted as the broken curve assuming a value for the low-shear viscosity of 4 kPa s. The same law averaged over the contact with the measured pressure dependence of viscosity is shown as the solid curve.

shown in Fig. 6.11. This mineral oil has been the subject [812] of many measurements and perhaps the greatest number of EHL analyses using the sinh law of equation (8.2). A survey of published values of τ E employed in Eq. (1.19) to describe traction curves indicates that for 80 C the value should be about 2.9 MPa. Eq. (8.2) is plotted for τ E 5 2.9 MPa in Fig. 13.1 as the dashed curve assuming an arbitrary value for the lowshear viscosity of 4 kPa s. The second rheological assumption of classical EHL can be tested with a calculation of a traction curve for LVI 260 at 80 C for Hertz pressure of pH 5 0.65 GPa in a circular contact and the measured pressure dependence, μðpÞ, as described in Fig. 6.11. As was done in Fig. 8.2, the traction can be calculated with reasonable accuracy with the assumption of Hertz pressure distribution when the film is thin.  1 p 5 pH 12r 2 2 (13.1) Here, r is the dimensionless contact radius, equal to one at the edge of the contact and zero at the center. For very thick films, the pressure distribution in a circular EHL contact will move toward the inlet and the

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Hertz pressure assumption obviously cannot be used. The average contact shear stress is computed from integration over the contact area, A. ð1 ð 1 τ5 τ ðp; γ_ ÞdA 5 2 τ ðp; γ_ Þrdr (13.2) A A 0 First, consider specifying the local shear stress by Eq. (8.2). The average shear stress calculated in this way is plotted in Fig. 13.1 as the solid curve. Obviously, the assumption that the constitutive behavior should have the same form as a traction curve is not generally valid, although it may be in special circumstances such as when the response is Newtonian throughout. In particular, the logarithmic regime, the straight line of the sinh-law occupying almost two decades of shear rate in Fig. 13.1, does not appear when integrated over the contact area. The reason for this can be understood from an investigation of the average contact shear stress for different regions of the contact. For this, it is useful to divide the contact into, say, five annular regionsÐ each occupying the area between ri and r ri11 . Regions of equal area, rii11 2πrdr, are not so helpful because the extreme pressure dependence of viscosity causes the shear stress to be insignificant in many of the Ð r outermost regions of the contact. Instead, regions of equal values of rii11 2πrμdr are chosen which gives dimensionless radii of 0, 0.14, 0.20, 0.27, 0.36, and 1. In Fig. 13.2, the shear stresses averaged over each region are plotted versus shear rate for the indicted regions. ð ri11 τ i;i11 5 2 τ ðp; γ_ Þrdr (13.3) ri

These are the traction curves that would be obtained for just the selected areas if such a measurement were experimentally possible. Assuming the response of Eq. (8.2), the logarithmic behavior would occur within the four central regions but the large outermost region is not logarithmic and this region occupies 87% of the contact area. Therefore, the traction curve which can be measured should not be expected to be logarithmic despite the logarithmic nature of the assumed response. A rheological shear response that does indeed result in a logarithmic traction curve has a limit to the shear stress that is proportional to the pressure as described in Chapter 10, Shear Localization, Slip and the Limiting Stress.

Average shear stress (MPa)

40 35 30

0.14 < r < 0.20

35 30

0.20 < r < 0.27

25 20 40 15 35

10 5

25 0 100

20 15 10 5 0 100

10,000 1000 Shear rate (1/s)

10,000 1000 Shear rate (1/s)

100,000

Average shear stress (MPa)

Average shear stress (MPa)

40

0.27 < r < 0.36 30 25 20 15 10 5

100,000

0 100

0.36 < r < 1 35 Average shear stress (MPa)

35 Average shear stress (MPa)

100,000

40

40

0 < r < 0.14

30 25 20 15 10 5 0 100

10,000 1000 Shear rate (1/s)

30 25 20 15 10 5

1000

10,000

Shear rate (1/s)

100,000

0 100

1000

10,000 Shear rate (1/s)

100,000

Figure 13.2 Traction curves for five regions of equal area averaged low-shear viscosity assuming the sinh-law of Eq. (8.2) with τ E 5 2.9 MPa.

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  _ Λp τ 5 min μγ;

(13.4)

The previous exercise has been repeated using this model with Λ 5 0:06 and the results are shown in Fig. 13.3. Again, the outermost region having the major portion of the contact area dominates the response as shown by Fig. 13.4. The friction calculated for the limiting stress model (13.4) is a better fit to the sinh-law representation of the traction curve than the Eyring equation (8.2). An arbitrary value for the low-shear viscosity of 55 kPa s was used for the dashed curve. The shear-thinning described by the Carreau equation (8.1) with the parameters in Table 8.1 for LVI260 will shift the solid curve in Fig. 13.4 to the left so that a smaller viscosity must be applied to the sinh-law for the dashed curve. In fact, the value of τ E which quantifies the slope of the logarithmic portion of a traction curve may be estimated from the pressure dependence of the low shear viscosity and Λ [13]. τE 5

2Λ αðp 5 pH Þ

(13.5)

For the example above, α evaluated at the Hertz pressure is 39 GPa21, which results in τ E 5 3.1 MPa, clearly a reasonable approximation. In Section 6.3 and Table 6.1, an inverse correlation was found for the pressure fragility, mp (or αg ), and the representative stress, τ E . The reason for this relationship may be understood from Eq. (13.5), if αðp 5 pH Þ depends closely upon the local pressureviscosity coefficient at the glass transition, αg . It seems that the outermost regions of the contact do influence the shape of the traction curve. However, much of the traction force is generated close to the contact center and it will be shown in Section 13.4 that the transition between traction regimes is determined by the conditions at the center.

13.2 A SIMPLE CALCULATION OF FRICTION FROM MATERIAL PROPERTIES The prediction of lubricated concentrated contact traction from the properties of the liquid lubricant has been a goal of elastohydrodynamics

35 30 0.14 < r < 0.20

25

30 25

0.20 < r < 0.27

40

20

10 5 1000 100 Shear rate (1/s)

10,000

15 10

30 25 20 15 10

5

5

0 10

0 1000 100 Shear rate (1/s)

0.27 < r < 0.36

35

15

0 10

20

10,000

10

40

40

35

35

100

1000 Shear rate (1/s)

10,000

0.36 < r < 1 0.14 < r < 0.14

30

Average shear stress (MPa)

Average shear stress (MPa)

Average shear stress (MPa)

40

35

Average shear stress (MPa)

Average shear stress (MPa)

40

25 20 15 10 5

30 25 20 15 10 5

0 10

100

1000

Shear rate (1/s)

10,000

0 10

100

1000 Shear rate (1/s)

10,000

Figure 13.3 Traction curves for five regions of equal area averaged low-shear viscosity assuming the limiting stress Eq. (13.4) with Λ 5 0.06.

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Figure 13.4 Newtonian response with stress limited to 0.06p, averaged over the contact with the measured pressure dependence of viscosity, is shown as the solid curve. The sinh-law of Eq. (8.2) with τ E 5 2.9 MPa is plotted as the broken curve assuming an arbitrary value for the low-shear viscosity of 55 kPa s.

for at least 50 years. Progress here has been slower than for film thickness and the accuracy of traction calculation at present does not approach that for film thickness. There do exist conditions and materials for which filmforming is Newtonian. A possible explanation for the slower progress on friction may be the near lack of conditions for which the Newtonian assumption yields reasonable predictions. The example for squalane in Figs. 1.6 and 12.6 indicate that, given a liquid with a large Newtonian limit, G, the film thickness in a large glass/steel contact can be predicted without considering shear-thinning. It was shown in Section 8.1, Fig. 8.2, that for a high molecular weight PAO, the Newtonian portion of a traction curve must reside at slide-roll ratio smaller than the usual experimental limit of resolution. It is seems to be impossible to extract the pressure dependence of lowshear viscosity from friction. Return to the example of LVI260 above, and assume Newtonian response for the same contact conditions. The shear stress at the contact center would reach to 10 MPa, which exceeds the Newtonian limit for most base oils, when the average shear stress is less than 1 MPa and the friction coefficient is only 0.002. When the average shear stress reaches to 10 MPa the stress at the contact center is

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335

Figure 13.5 An example of the viscosity extracted from a traction curve using the assumptions of classical EHL compared with the viscosity measured in a viscometer.

120 MPa. Average shear stress is a poor indication of the stress state in the contact and the situation is worse at the high pressures of quite ordinary steel/steel contacts. The pressureviscosity behavior extracted from a traction curve [14] with the second assumption of classical EHL is compared with the viscosity measured in a viscometer [15] in Fig. 13.5. The pressure dependence is entirely different, the pressure response derived from measured traction being slower than exponential. However, it may be possible to experimentally observe Newtonian traction at very low contact pressure and slide-roll ratio by selecting a liquid with the combination of relatively low pressure dependence of viscosity such as the polyol ester of Fig. 6.6 and with relatively large Newtonian limit such as the dicylohexyl methylpentane of Table 8.1. To this author’s knowledge this has not been done. Unless the pressure dependence is linear, the average viscosity in the EHL film is not equal to the viscosity at the average pressure of the contact.

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Figure 13.6 A simple isothermal calculation of friction from the measured properties of a liquid obtained for a circular contact and the reference liquid, PGLY.

A simple isothermal calculation of friction from the measured properties of a reference liquid can be obtained for a circular contact using Eqs. (13.1) and (13.2). To illustrate, the friction in the concentrated contact between a steel ball and a glass disc was measured at INSA, Lyon, for Hertz pressure, pH 5 0.50 GPa, and rolling velocity, u 5 0.013 m/s, at 80 C. These measured friction coefficients are plotted in Fig. 13.6. The lubricant is a polyglycol reference liquid, PGLY, with properties thoroughly characterized in reference [16]. The average shear stress in the contact of Fig. 13.6 was calculated from Eq. (13.2) with the shear stress computed from   _ ðp; γ_ Þ; Λp τ 5 min γη (13.6) where Λ is the pressure-limiting stress coefficient (see Chapter 10: Shear Localization, Slip and the Limiting Stress). The shear rate is approximated by γ_ 

Σl u hc

(13.7)

where Σl 5 ðu2 2 u1 Þ=u is defined as the slide-roll ratio calculated from the surface velocities, ui , within the contact patch. For traction analysis, a distinction must be made between this definition of slide-roll ratio and the value of Σ that is obtained using velocities measured distant from the contact or calculated from shaft rotation speeds as there may be a

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difference due to the elasticity of the roller material. Here we can ignore the elastic creep of the rollers as the slide-roll ratios are large and Λ can be made arbitrarily large to avoid truncating the stress. The single Newtonian Carreau form was used in reference [16] to describe the shear dependent viscosity with Carreau parameters, G 5 0:256 MPa and n 5 0:33. "

Þ
2 #ðn21 2 μγ_ ηðp; γ_ Þ 5 μ 11 G

(8.1)

The pressure dependence of the low-shear viscosity was calculated from the TaitDoolittle framework, Section 6.6.1, using parameters in Bair [16]. For many friction calculations, accurate predictions can only be made with film thickness, hC , that is either measured or is calculated from the real shear dependence like that above and not the Newtonian value, hN , from a classical formula such as from Eq. (1.18). For the friction prediction in Fig. 13.6, the film thickness was corrected using hc 5 hN =κ with the correction from Chapter 12, Application to Elastohydrodynamic Film Thickness. (  1 )3:6ð12nÞ1:7 μ0 u 110:2Σ κ 5 110:79 ð11Σ Þ (12.8) GhN The result of this friction prediction is plotted in Fig. 13.6 along with the prediction of the Newtonian assumption. The classical film thickness formula of Eq. (1.18) gives the central value to be 215 nm. The film thickness corrected for shear-dependent viscosity with Eq. (12.8) varies from 66 to 53 nm over the range of measured slide-roll ratios. Thus the film thickness correction is extremely important to the successful prediction of friction for this reference liquid in Fig. 13.6. There was no need to specify a value of Λ, since the shear stress does not reach to a significant fraction of the pressure for such strongly shear-dependent liquids in low pressure contacts. For the purposes of discussion, it may be useful to classify friction behavior at small slide-to-roll ratio into low, moderate, and high pressure regimes. The classification of the operating regime of a contact should depend on both the contact pressure and the dependence of the shear properties of the liquid on pressure. The low pressure regime was illustrated by the example above in Fig. 13.6. At intermediate pressures, the

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average shear stress, when plotted against the logarithm of sliding velocity, slide-roll ratio, or other measure of average shear rate, will show a somewhat straight portion that indicates logarithmic traction response which may be described by the Eq. (1.19), or, more appropriately, the approximation for large Σ, τ  τ E lnð2CE Þ 1 τ E ln Σ, where τ E is the slope of the logarithmic portion. At low pressures, the logarithmic portion often does not develop before thermal softening begins to affect the traction. At high pressures, the logarithmic portion does not develop at all and a traction curve on double linear axes will show a linear increasing regime and a plateau separated by a short transition. Another example of experimental traction for a viscous polyalphaolefin in the low pressure regime (pH 5 0:528 GPa) was given in Fig. 8.2 and the calculation of traction using the Carreau law was discussed in Section 8.1. For these conditions, the local shear stress remained a small fraction of the local pressure ð , 0:015pÞ and therefore the limit to the stress, Λp, had no effect for the usual values of Λ: The low pressure traction response of squalane is shown in Fig. 13.7 from Bair et al. [17] where measurement and calculation are successfully

Traction coefficient (τ/p)

0.03

Squalane in circular contact, pH = 0.57GPa T = 37ºC, U = 0.57m/s, h = 0.11μm

0.02

0.01

0 10–2

10–1

1

Slide roll ratio (Σ)

Figure 13.7 Low-pressure traction curve for squalane. From Bair S, McCabe C, Cummings PT. Calculation of viscous EHL traction for squalane using molecular simulation and rheometry. Tribol Lett 2002;13(4):251254.

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compared. Here, the shear stress was obtained from Eqs. (13.1), (13.6), and (13.6) using the single-Newtonian Carreau model (8.1) and the standard timetemperaturepressure shifting rule G 5 GR

ρT ρR TR

(8.17)

Again, the pressure dependence of viscosity was described by the Doolittle equation (12.9) with the relative volume provided by the Tait EOS (4.6). The value of Λwas estimated to be 0.075 from the traction coefficient along the plateau of a traction curve as shown in Fig. 10.1. For the largest slide-roll ratio, Σ 5 0:9, in Fig. 13.7, the maximum value of the ratio τ=p is 0.052 and occurs at the contact center, although the traction coefficient here is τ=p 5 0:035. Therefore the shear stress limit may be reached locally before the traction coefficient reaches to Λ. This is what has occurred in the intermediate pressure traction curve for squalane in Fig. 13.8 from [17]. Multiple traction calculations are shown in Fig. 13.8; Newtonian, Newtonian with a stress limit, Carreau, and Carreau with a stress limit. Comparison with measured traction indicates that in this regime, both shear-thinning and a stress limit are required for the best accuracy; however, the Newtonian viscosity with a stress limit reproduces the

Traction coefficient (τ/p)

0.06

Newtonian

Newtonian with limited stress

Carreau

Carreau with limited stress

0.04

0.02

0 10–4

Squalane in elliptical contact pH = 1.29GPa, T = 40ºC, U = 2m/s, h = 0.177μm

10–3

10–2 Slide roll ratio (Σ)

10–1

Figure 13.8 Intermediate-pressure traction curves for squalane. From Bair S, McCabe C, Cummings PT. Calculation of viscous EHL traction for squalane using molecular simulation and rheometry. Tribol Lett 2002;13(4):251254.

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High Pressure Rheology for Quantitative Elastohydrodynamics

Traction coefficient

0.08

Elliptical contact, PH = 1.93GPa, U = 2m/s, 40ºC

Viscous traction calculation

0.04 Squalane, h = 163 nm Dry contact

Linear gradient with ms = 40

0 10–4

10–3

10–2

10–1

Slide roll ratio (Σ)

Figure 13.9 Traction curve for squalane indicating the beginning of the highpressure regime. The low slide-roll behavior is similar to dry contact. From Bair S, Kotzalas M. The contribution of roller compliance to elastohydrodynamic traction. Tribol Trans 2006;49(2):218224.

logarithmic character well with τ E 5 11 MPa. Shear-thinning simply moves the traction curve to the right in a semilog representation. A somewhat larger contact pressure has been investigated for squalane in Fig. 13.9 from Bair and Kotzalas [18]. These data show some aspects of the high pressure regime. The traction coefficient at the largest slide-roll ratio has become relatively independent of Σ and, at low slide-roll ratio, the viscous calculation severely overestimates the stress. Values of dry traction (with unlubricated rollers) have been included on the plot in Fig. 13.9 and they are surprisingly similar to the full film traction. A curve is shown indicating a linear gradient of ms 5 40, which is a reasonable description of the behavior of the dry contact and the lubricated contact as well to Σ 5 5 3 1024 . This value of ms 5 40 may be calculated for a friction prediction as follows. Bair and Kotzalas [18] investigated the traction of both dry and lubricated contacts at very high pressures, 1.432.65 GPa, varying the shear modulus of the rollers, Gs , from 45 to 179 GPa and using two different traction machines of different geometry, twin-disc, and ball on-disc. For steel, Gs 5 80 GPa. They found that the initial traction gradient, ms 5 ð1=pÞðdτ=dΣÞΣ50 , in dry or lubricated contact, could be described by the very simple relation.

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Application to Elastohydrodynamic Friction

ms 5

Gs pH

(13.8)

Kalker [19] developed a theory to relate the velocity difference of a pair of contacting elastic bodies to the tangential traction transferred between them. Even for perfect adhesion on the contact patch where the surface velocities must match, the velocities distant from the contact will differ due to elastic compliance. In a traction (friction) measurement, the slide-roll ratio is calculated from the roller shaft rotational velocities. Bair and Kotzalas [18] compared the experimental traction gradient with the theory of Kalker and found the experimental gradient was roughly twothirds of theory over the experimental range. No explanation for the two-thirds factor has been found. It may therefore be assumed that the enhanced values of Σ at high pressure and low sliding are the result of elastic compliance of the rollers with perhaps an additional foundation compliance and, in any event, the value of ms in Fig. 13.9 follows from Eq. (13.8). An obvious method to correct a traction calculation at high pressure and low slide-roll ratio is to simply add the contribution to the slide-roll ratio of roller compliance to the contribution of viscous shear of the film to arrive at the total slide-roll ratio. For a circular contact Σ 5 Σ l ðτ Þ 1

3 τ 2 Gs

(13.9)

The broken curve in Fig. 13.9 is the Eq. (13.9) and is an obvious improvement over the purely viscous traction calculation. Clearly any attempt to derive an estimate of the low-shear viscosity from the linear slope of a traction curve will be thwarted by roller compliance at high pressure. The traction curves displayed in Fig. 10.1 for a polybutene are excellent examples of the high pressure regime of traction where a pronounced plateau appears following a linear increase in traction coefficient with Σ. This linear increase may be described from Eq. (13.8) simply by τ=p 5 ΣGs =pH . See Table 13.1 where the experimental and analytical gradients are compared. The examples of isothermal, pointcontact traction behavior, above, should not be used to set pressure ranges for the behavior of liquids in general. For a given liquid, an increase of temperature will increase the pressure at which a particular aspect of traction will be present. Fragile

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Table 13.1 The initial traction slope for the experimental traction curves of Fig. 10.1 for the polybutene, L100, at 50 C and 2.5 m/s compared with the calculation of Eq. (13.8) for elastic creep

pH =GPa Measured ms Gs =pH

1.93 39 40

2.52 32 31

3.31 26 24

liquids (Chapter 6: Correlations for the Temperature and Pressure and Composition Dependence of Low-Shear Viscosity) will display each aspect of traction behavior at reduced pressure compared with strong liquids. Pressure fragility is one of the most important properties influencing traction, although it does not exist in the classical study of EHL. The simple traction calculations described above have not considered the possibility of multiple transitions in the shear-thinning law (Section 8.6), the effect of the glass transition on the viscosity (Section 9.4), the influence of the liquid film on the pressure distribution (requires a full simulation), and thermal softening. The thermal traction response at larger slide-roll ratio where the temperature varies along and across the film will require a full EHL simulation and the descriptions of temperature dependence given in the preceding chapters should be helpful to that end. The sheardependence of viscosity is surprisingly important in the thermal regime.

13.3 TRACTION FROM MATERIAL PROPERTIES BY FULL SOLUTIONS FOR THE HYDRODYNAMICS AND ELASTICITY Friction is a topic for which full numerical solutions of the coupled Reynolds and elasticity equations have not proven useful until very recently. It is necessary to model the viscosity over a range of pressure for which the pressure dependence is usually greater-than-exponential and doing so has caused problems for numerical stability.

13.3.1 The isothermal solution The first full numerical simulation employing the pressure and shear dependence of viscosity that can be measured in viscometers appeared in 2007 [20]. Results for the film thickness from that full numerical simulation were introduced in Chapter 12, Application to Elastohydrodynamic

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343

Film Thickness, where it was clear from Fig. 12.7 that the shear dependence influenced the thickness. Liu and coworkers [20] achieved a full solution to the isothermal problem using a generalized Reynolds equation (12.10) and (12.11), presented in the last chapter. Friction curves from that solution are shown in Fig. 13.10. The subject of the analysis was a

Figure 13.10 Comparisons of measured with calculated traction coefficient for sliding at entrainment speeds of: (A) 0.03 m/s and (B) 0.13 m/s for the PAO of Fig. 12.7 using the same pressure and shear dependence of viscosity. Reproduced from Liu Y, Wang QJ, Bair S, Vergne P. A quantitative solution for the full shear-thinning EHL point contact problem including traction. Tribol Lett 2007;28(2):171181.

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high viscosity polyalphaolefin, PAO-650 with shear dependence described by the single Newtonian modified Carreau equation (8.26) with properties listed in Table 8.1. The agreement of the traction coefficient calculated for the midplane with the measured friction is excellent. Comparing the error in the calculated friction using a full simulation of the pressure distribution in Fig. 13.10 with the assumption of Hertz pressure in Fig. 8.2 shows the importance of a full numerical simulation for the prediction of EHL friction. However, the simple calculation, based upon the Hertz pressure distribution, has for many years provided needed guidance in the application of rheological data to EHL traction which would not have been possible otherwise. The average shear stresses on the upper and lower surfaces are not equal as shown in Fig. 13.10. The pressure gradient in the inlet imposes a frictional penalty for the generation of the film. In experiments, the friction force is measured on one of the two surfaces. When friction curves are experimentally generated, it is customary to “zero out” the friction force under pure rolling and this is why the experimental friction coefficient is zero at Σ 5 0. For this low pressure (pH 5 0.528 GPa), the initial traction gradient is small compared with the gradient for elastic creep of the ball and disc so that the slide-to-roll ratio need not be corrected for roller compliance. The local shear stress did not exceed 0:05p except near the pressure spike so that the stress limit was unnecessary. These insights into the mechanism of friction generation, especially the importance of the accurate film thickness prediction, were hidden from classical EHL because of the lack of a description of real viscosity.

13.3.2 The thermal solution Although the solution of the thermal EHL problem is more difficult, solutions of the classical type were presented as early as 1980 [9,21] and, interestingly, the subject liquid was the mineral oil, LVI 260. See Chapter 6, Correlations for the Temperature and Pressure and Composition Dependence of Low-Shear Viscosity, for low-shear viscosity correlations. A general characteristic of these thermal EHL solutions was the extreme understatement of the temperature dependence of viscosity [22,23], which was apparently required to generate reasonable solutions when applying the second rheological assumption of classical EHL. The classical EHL approach to the thermal problem has assumed viscosity correlations which have limited the temperatureviscosity coefficient, β, to unreasonably

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345

low values, typically no more than 0.1K21, whereas for pressure approaching the glass transition, β may reach 0.4K21 [24]. Temperature affects the response of the liquid film to shear primarily through the temperature dependence of viscosity. Therefore any understanding of thermal effects in EHL can only come with accurate temperature dependence of the liquid properties, especially viscosity. The quantitative EHL approach to the thermal problem first appeared in 2008 [25]. In this early work, Hertz pressure was low at 0.47 GPa. Two years later [26], a complete solution was obtained for a low viscosity mineral oil, Shell T9, to pH 5 1.35 GPa. The volumetric heat capacity, ρcp , and thermal conductivity, k, are both strongly dependent on pressure. These thermal properties appear in the energy equation which must be solved simultaneously with the Reynolds and elasticity equations. This equation must be solved for both the liquid film and the solid bodies. It is written here for the liquid. "



2 # @T @T @2 T T @ρ @p @p @u 2 @v 1v 5k 2 2 u 1v 1η ρcp u 1 @x @y @z ρ @T @x @y @z @z (13.10)

The second term on the right-hand side describes heat of compression and the third describes viscous dissipation in shear. Both volumetric heat capacity and thermal conductivity increase with increasing pressure [27], the conductivity approximately doubling as pressure increases to 1 GPa while the effect on heat capacity is less. The volumetric heat capacity increases slightly with temperature at all pressures in the liquid state; however, it decreases substantially at the glass transition [27] which will often invalidate such rules-of-thumb when applied to EHL. At the glass transition, the volumetric heat capacity changes from a large, liquid-like, to a smaller, solid-like value. The thermal conductivity decreases slowly with temperature at ambient pressure; however, the pressure dependence increases with temperature so that at some pressure in the range of 200500 MPa it is independent of temperature. Some space should be devoted here to methods of measuring and modeling the temperature and pressure dependences of volumetric heat capacity, ρcp , and thermal conductivity, k, in liquids under high pressure. Today the transient hot-wire technique [28] has displaced nearly all others. A thin metal wire is immersed in the liquid at the test temperature and pressure. A metal is chosen that has a known temperature dependence

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of the electrical resistance. A pulse of electrical current is sent through this wire while the voltage is recorded as a function of time. From the known temperature dependence of the electrical resistance, the temperature rise, ΔT ðt Þ, is obtained as a function of time. For large values of the Fourier number, ΔT ðt Þ 5 θ0 1 θ0 lnðt Þ where heating of the wire begins at t 5 0. There is theory [29] which permits the calculation of k from θ0 and the calculation of ρcp from θ0 once k is known. Such measurements have proven to be more difficult than measurements of viscosity or of relative volume since the hot wire is extremely fragile and the oil is so viscous at high pressure that the wire cannot resist being deformed by the flow of it. Fortunately there is an alternative to the measurement of the thermal properties for each liquid at high pressure. The EHL film temperature is not as sensitive to heat capacity as it is to conductivity. Therefore the volumetric heat capacity may be approximated by a constant value representative of the average pressure, perhaps ρcp 5 2 MJ K21m23, without great error. Friction calculations are, however, quite sensitive to the thermal conductivity of both the liquid and of the solids [26,30]. Thermodynamic scaling has been applied to the temperature and pressure dependences of volumetric heat capacity and thermal conductivity [26,31] of liquid lubricants; however, a simple density scaling rule may be applied to the isothermal pressure dependence of the thermal conductivity [32]. @ ln k

T 5g @ ln ρ

(13.11)

Here g is a material specific constant which is between 2.79 and 3.08 for very low molecular weight organic liquids [32]. Using this rule, it should be possible to estimate the conductivity from
g
2g ρ V k 5 k0 5 k0 (13.12) ρ0 V0 Here the subscript, zero, denotes ambient pressure. For the low viscosity mineral oil, T9, g 5 3.3 [26], for the aircraft turbine oil, L23699, g 5 3.0 [33], and for the pure hydrocarbon, squalane, g 5 3.2 [31]. For the general case, it should be sufficiently accurate to model the thermal conductivity as k 5 k0 ðV =V0 Þ23 . The equation of state (Chapter 4: Compressibility and the Equation of State) will provide values of V =V0 as a function of pressure at each temperature and the determination of k0 as a function of temperature is a routine measurement performed on commercial instruments.

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347

Figure 13.11 Comparisons of measured with calculated traction coefficient of a very low viscosity turbine oil extending into the thermal regime. The thermophysical properties, rheological and thermal, all came from primary measurements. For the dashed curve, the sinh-law (8.2) was substituted for the CarreauYasuda equation.

There should be little loss of accuracy in a friction prediction from treating k0 as temperature independent; however the treatment of k0 as pressure independent results in large errors [26]. The thermal properties were correlated with temperature and pressure with thermodynamic scaling relations for a low viscosity mineral oil in a full numerical prediction of friction extending well into the thermal regime by Habchi and coworkers [26]. A finite element “full system” approach was used. An example of the prediction compared with measurements is shown in Fig. 13.11 where the agreement is excellent. Thermal properties were measured by the transient hot-wire method described above and the pressure and temperature-dependent low-shear viscosity was developed from high pressure falling cylinder viscometer measurements (Section 5.2.4) and correlated with AshurstHoover thermodynamic scaling (Section 6.7). The shear dependence of viscosity was determined from pressurized thin-film Couette viscometry (Section 7.3.1) and correlated with the CarreauYasuda equation (Section 8.6). The Murnaghan equation of state (Section 4.3) was employed to fit the data from a metal bellows piezometer (Section 4.2.2). The only property that was not obtained from a primary measurement was the limiting stress

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High Pressure Rheology for Quantitative Elastohydrodynamics

coefficient, Λ 5 0.083, that was found from a traction measurement at very high pressure, pH 5 2.44 GPa. The quantitative full numerical simulations of the thermal EHL behavior of a mineral oil, described above, can be used to investigate the necessity of reducing the temperature dependence of viscosity [22,23] when the sinh-law, Eq. (8.2) is employed to describe the shear dependence. In Fig. 13.11, this viscosity function was substituted for the CarreauYasuda equation. A value of the Eyring stress, τ E , is needed to implement this substitution and a value of 5 MPa yields a similar response in the transition to shear thinning seen in the measured viscosity. The result of this new calculation, kindly provided by Professor Habchi of Lebanese American University, is the dashed curve in Fig. 13.11. Here, as is customary, the limit to the shear stress has not been implemented with the sinh-law; however, the stress limit merely makes the two calculated curves nearly identical for low side-to-roll ratio, Σ , 0:2. The two viscosity functions predict substantially different friction response in the thermal regime, the sinh-law appearing to be overly sensitive to temperature. This fault may be mitigated by the usual strategy of reducing the temperature dependence of the low-shear viscosity at high pressure, but such behavior is not representative of real liquids. Quantitative full numerical simulations may also be used to investigate the procedure of assigning the value of the limiting stress coefficient to the friction coefficient along a plateau in the traction curve. This requires much care as the friction coefficient may appear to be independent of rate (flat on a friction curve) while a significant portion of the contact remains controlled by thermally softened shear-thinning. Measurements of traction versus slide-roll ratio for squalane in a sapphire/steel contact at 40 C with pH 5 1.2 GPa were kindly provided by Philippe Vergne of INSA Lyon in Fig. 13.12. The friction coefficient is reasonably independent of Σ from 0.3 to 0.5 at τ=p 5 0.060. However, the value of Λ for this pure hydrocarbon is 0.075 [31]. Quantitative full numerical simulations of the thermal EHL friction were provided by Professor Habchi of Lebanese American University and are plotted in Fig. 13.12 for two values of the power-law exponent, the correct (Table 8.1) value, n 5 0.46, and an overstated value, n 5 0.65. The agreement for the actual value of n is excellent with τ=p 5 0.060 along the “plateau.” For the overstated exponent, there remains a plateau, however, τ=p 5 0.065. Clearly the plateau in the friction does not mean that the entire contact is stress-limited in this case. Without knowledge of

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349

Figure 13.12 Measurements of traction versus slide-roll ratio for squalane provided by INSA Lyon compared to full simulations provided by Lebanese American University for two values of n, the value obtained from a viscometer, 0.46 from Table 8.1, and an overstated value.

the pressure dependence of viscosity it would be difficult to make any assessment of the conditions within the film.

13.4 REGIMES OF FRICTION The explanation for the different regimes of friction as seen in a traction curve and the prediction of the transition from regime one to another have been subjects of discussion for many years. The classical approach is exemplified by the often cited work of Evans and Johnson [34]. There, visual inspection of experimental traction data, that is to say, observing the shape of a traction curve, was used to specify the rheological response of the film. This approach has lacked the use of measurable properties of the liquid to delineate the various regimes and thus is not predictive. The conditions at the contact center are critical to the commencement of a new regime as sliding speed is increased. See Fig. 13.4 where the start

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High Pressure Rheology for Quantitative Elastohydrodynamics

of the logarithmic portion of the traction curve begins at 100 reciprocal seconds which is the shear rate at which the shear stress is shown to reach the limiting value at the contact center in Fig. 13.3. Therefore, in the following, the delimiting of each regime will be based upon dimensionless numbers with properties evaluated at the pressure of the contact center, pH . The dimensionless groups are listed below. 1. The Weissenberg number (Section 2.3.1) is, in terms of shear rate, _ Wi 5 λγ_ 5 μγ=G. However, an estimation of the shear rate in the contact at very low slide-to-roll ratio, Σ, is bedeviled by roller compliance. Therefore the Weissenberg number, in terms of shear rate, for the film at the contact center should be evaluated using Eq. (13.9) for the slide-to-roll ratio at the contact as  μðp 5 pH Þu 3τ Wi 5 (13.13) Σ2 Ghc 2Gs The second term in brackets (order 1023) will sometimes be negligible. 2. The NahmeGriffith number (Section 7.4) for Couette shear at the contact center is Na 5

β ðp 5 pH Þμðp 5 pH ÞðuΣ Þ2 kðp 5 pH Þ

(13.14)

Properties are evaluated at the contact center. 3. A new dimensionless number, the limiting stress number, was introduced for the delimiting of regimes by Habchi et al. [35].  τ ðp 5 pH Þ ηðp 5 pH Þu 3τ Li 5 (13.15) Σ2 5 τ L ðp 5 pH Þ ΛpH hc 2Gs Properties are evaluated at the contact center. 4. A fourth dimensionless number, the thermoviscous number, Ti, is required because the thermal regime at high shear rate is strongly influenced by the shear dependence of viscosity. Ti 5

NaUWi Li

(13.16)

Of course the correction for roller compliance will be negligible here and for Na as well.

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These definitions were applied to the properties of the low viscosity mineral oil, Shell T9, employed in the previous section, in many friction calculations to observe the transition from one regime to another [35]. The resulting demarcation of regimes follows. The traction curve can be divided into four regimes: linear; nonlinear; plateau, where the friction is at first glance independent of sliding speed; and thermoviscous, where friction decreases with sliding speed. It must be emphasized that all four regimes generally do not exist in a single friction curve. 1. The linear regime of EHL friction could be attributed to three possible effects that occur for very low slide-to-roll ratio. a. The response is the short time elastic response of the liquid with shear modulus, GN . b. The response is that of the liquid with Newtonian viscous shear. c. The response is that of the solid rollers with shear modulus, Gs . See Eq. (13.8). The first, the elastic behavior of the liquid, can be quickly eliminated since the limiting high frequency shear modulus, GN , is of the order of 1 GPa at EHL pressure (Section 8.3.1) and this combined with the very thin films means that elastic response is limited to extremely small shear displacements. Such small displacements are obscured by the elastic creep of the rollers. 2. The nonlinear regime of EHL friction could be attributed to two possible effects. a. The viscosity has become shear dependent at the contact center. b. The stress limit has been reached at the contact center. The nonlinear regime begins at Wi . 1 or Li . 1. In this regime, the traction is not so sensitive to the particular nature of the shear dependence [36] as it is in the thermoviscous regime. 3. The plateau regime does not begin at Li 5 1, but at the sliding speed at which the stress controlled region has occupied nearly all of the contact area. Since this condition depends also on the shear-dependent viscosity, all that is certain is that the plateau regime begins at Lic1. 4. It is not well-appreciated that it is the thermoviscous regime that is most sensitive to the shear dependence of viscosity as can be seen in Fig. 13.11. Both the temperature dependence (through Na) and the shear dependence (through Wi) are important here and both must be modeled correctly. For the mineral oil, the thermoviscous regime begins when Ti . 100. Again, all four regimes generally do not exist in a single friction curve.

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