The Film Thickness in Piezoviscous Regime; Film Thickness Regimes Transition Criteria 1

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1 The Film Thickness in Piezoviscous Regime; Film Thickness Regimes Transition Criteria by Harry van Leeuwen Eindhoven University of Technology As published in the Journal of Tribology, Trans. ASME, Vol.06, No., July 984, pp Johnson [4] showed in his magisterial 970 paper that lubrication regimes play a decisive roll in the determination of the film thickness in a counterformal contact under full film lubrication conditions. The numerical and pictorial definition of the transition areas between these regimes deserves careful attention. The author has elaborated this topic for elliptical contacts with arbitrary ellipticity ratio. In addition, he has derived a film thickness equation for the so-called piezoviscous rigid (PVR) lubrication regime. The author deserves congratulations for this welcome addition to the literature. The following is a rather long commentary on the author's ideas, and some occasional queries on the consequences.. Lubrication regimes in isothermal full film lubrication In the discusser's opinion, the paper is slightly ambiguous concerning the definition of lubrication regimes. To make this clear, a closer look at counterformal line contacts will suffice. In his lucidly written paper Johnson [4- defines four lubrication regimes, e.g. IVR, PVR, EHD, and IVE (in the author's notation). The IVR regime exists where film thickness values are not more than about 5% beyond the Martin value. The PVR regime is entered when viscosity effects yield higher film thickness values, in the mean time elasticity effects staying unimportant. This is the area studied by Blok [A]. According to Johnson [4], the EHD regime is entered if theories for rigid bodies predict real pressures with values larger than infinity, or when elastic effects outperform viscous effects: Hooke [A] considers the farmer situation as PVR, because the film thickness behaves in the same way as in the region studied by Blok [A], and the latter regime as EHD lubrication. In the IVE regime elastic effects dominate viscous effects, resulting in the absence of the pressure viscosity coefficient a. The IVE/EHD transition has been studied in detail by Hooke [A]. Because line contacts have received much more attention in the past than elliptical contacts, and film thickness results can be compared so well in a qualitative way, it is worthwhile to draw some results from the existing literature and to compare them with the author's findings, in particular for the PVR regime.. The PVR lubrication regime Although Blok [A] does not provide a film thickness formula for the PVR regime at line contacts, it can be deduced from his Fig. that, for a fluid having an exponential pressure-viscosity relationship, h g 0/9 (A) where Johnson's [4] notation has been used. This equation has the same properties as the author's equation (0), viz. an approximation which merges in the IVR solution for low g or low A values. Discussion on ASME Paper No. 8-Lub-7 by L. Houpert: The film Thickness in Piezoviscous-Rigid Regime; Film Thickness Lubrication Regimes Transition Criteria", Journal of Tribology, Trans. ASME, Vol.06, No., July 984, pp Capital letter and number combination between square brackets refer to additional references at the end of the discussion.

2 The Film Thickness Transition Criteria discussion by Harry van Leeuwen page If the maximum isoviscous pressure P o max equals (GU) -, or A =, and therefore the maximum real pressure p approaches infinite values, it follows from the definition of g that necessarily / h crit 665 g crit (A) Consequently, at infinite real pressures p ([A]): h crit. (Aa) g 7.70 (Ab) crit In the absence of elasticity, it is impossible to attain higher values of the film thickness than. times the Martin value. Johnson [4] represented the limiting value g crit = 7.70 in his map by locating the PVR-EHD transition at this level. It is only possible to arrive at values beyond g = 7.70 by allowing elasticity effects. Johnson called this area the "extrapolated Blok" region, because equation (A) is a good fit here. Equation (Al) does not provide an adequate description of the film thickness here. At very high g values the Grubin region is entered ([4]). This means that, for the line contact case, the author's parameter A approaches the value for a finite value of A, or, for a finite value of the load parameter W: g crit corresponds to A crit.5. Fig. seems to be in contradistinction with this observance: if A approaches, W PVR becomes infinite. In addition, PVR film thickness formula (0) allows for an increase with the viscosity parameter A without ever meeting any limit due to elasticity effects. However, every film thickness formula is an approximation and has a limited domain of validity. What are the limits of the proposed expression? Because line contact solutions cannot be obtained from elliptical contact results, e.g. by just stating that k, Johnson [4] derived a separate set of non-dimensional parameters for these two different contact situations. Hence, equations () and () look a bit peculiar. How can viscous or elastic influences affect the IVR solution? And how should equation () be used, while A cannot be written as an explicit function of g v according to equation (4), so C cannot be expressed in g v? It seems that the criterion g el is not applicable.. PVR or EHD Regime? Greenwood [A] observed the remarkable property of the extrapolated Blok solution (A) that it blends so well with solutions of the Grubin type, for low g l values. For low g values this solution reads / h min 784 g (A4) Now, it can be appreciated that the author remarks that his predictions based on equation (0) show only 0% inaccuracy when compared to EHD film thickness results. This can be seen by considering A, which approaches the value in this situation. It follows from equations (4) and (5) that hence C A / H PVR A / H IVR (A5) (A6) By substituting equations (4) and (5) in (A6) it is found that H PVR.4k k ( GU ) (A7) This value is obtained by multiplying the central film thickness expression by 0.80

3 The Film Thickness Transition Criteria discussion by Harry van Leeuwen page Consequently, for line contacts, with k,, and, for elliptical contacts, h PVR.65 g (A8) Hˆ.4k PVR g v k (A9) + Since equation (A8) is very close to equations (A) and (A4), it can be concluded that equation (A7) is a reliable guide for film thickness predictions in the extrapolated Blok region, at least for large k values. Moreover, equation (A9) is a very good approximation of equation (0) in this region, that is at large values of A, g or g v. Typically, A > 4, which is in accordance with the aforementioned A crit =.5. On the other hand, equation (0) blends into equation (A9) at high g v values, placing confidence in the predictions of the film thickness at high g v values. Thus it is clear that equation (0) is not far from reality, at the least for large k values. Evidently, elastic deformations must exist because g > On the contrary, they are small compared with viscous effects, because the film thickness behaves in almost the same manner as if only viscous effects with g = g crit were present. Since the film thickness in the Grubin region is determined by the entry zone, the shape of the inlet zone will not be greatly affected by surface deformation. The bulk of the contact retains its cylindrical shape. This is especially true for relatively rigid surfaces, where a Grubin approach applies. Hence Hooke [A] considers this situation as PVR, just as the author does, because the deformation is small compared with the film thickness. In consequence, Hooke gives preference to a criterion g -/ g 0.5 over g = 7.70 for a transition PVR/EHD. The former criterion originates from Greenwood [A4], who studied deformation effects on the film thickness of a line contact with a piezoviscous lubricant. 4. PVR film thickness formulae In the discusser's opinion, both Hamrock and Dowson [], and Dowson, Dunn, and Taylor [] were aware of the similarity in solutions for the extrapolated Blok and the Grubin area. Hence, they derived expressions for the PVR case where the maximum isoviscous pressure equals /α, yielding formulae that will never merge into IVR solutions for low g v (or A) values, but will constitute a reasonable estimate of the film thickness beyond the critical value of g v! Hence, a comparison with PVR film thickness values from [] or [] is only fair at high g v values, corresponding to, say, A > 4. Under these conditions the line contact results in Fig. 6b and the elliptical contact results in Fig. 7 show only small deviations, except for the Hamrock and Dowson formula () at low ellipticity ratio. But this fact has already been diagnosed by Dowson et al in [], resulting in expression (4), which shows excellent agreement with the author's results in Fig PVR/EHD transition criteria The parameter Φ describes the position in the intermediate region between IVR and EHD behavior of a line contact. Markho and Clegg's work [] is an elaboration of the area investigated by Dowson and Higginson [A5]. This area is marked in Fig. A, which is based on Johnson's [4] Fig.. This region has been named after Weber and Saalfeld [A6], who were the first to study it. Both viscous and elastic effects exist, but their influence is relatively small. This is exactly the area between the PVR asymptote and the line marked H EHD /H IVR =.96 in the author's Fig. 5. It is easy to see this by considering that Fig. 5 is a generalization of an earlier map for line contacts from Moes [A7]. Unfortunately, it is impossible to visualize the PVR region of Fig. A in Fig. D6 of

4 The Film Thickness Transition Criteria discussion by Harry van Leeuwen page 4 [A7], or in Fig. 5, due to the choice of the particular representation in the latter figures: this parameter choice results in coinciding PVR and Weber and Saalfeld regimes. As can be concluded from Fig. A, the regime described by Markho and Clegg [l], viz. the Weber and Saalfeld region, does not coincide with the PVR region. This remains so, even when the definition of the PVR regime is extended according to Hooke [A]. Hence, the criterion Φ = also contains some elasticity influences, and therefore is not a correct criterion to describe PVR/EHD transitions. Granting, Φ = describes a transition, that is a transition beyond which elastic and viscous effects are both very strong, but the point is that a PVR/EHD transition should describe a treshold beyond which elastic effects cannot be ignored and under which viscous effects are important. It just happens that Φ = is close to g crit = 7.70 in the line contact case. Actually, it corresponds to g.9. In consequence, the Φ = criterion may yield results which are not too bad, but the physical base is not a sound one. It is now possible to compare the author's physical criterion, developed in equation (9), with criteria of Hooke [A] and Greenwood [A4]. From equation (A8) it can be found that C = hpvr / hivr = 0.74 g (A0) For line contacts, with d=, k yielding ( ) ( ) p Hertz = π g g α (A) max ( αpmax ) ( αp ) max PVR Hertz = 8.64 g g 0.74 g (A) A physical definition of the PVR/EHD transition as proposed by the author, viz. {(αp max ) PVR /(αp max ) Hertz } =, results in 0.57g g e = (A) 0.74g which is also represented in Fig. A. For large values of g equaiton (A) can be approximated by or g e 0. 4 g (A4a) ( ) 4 g g Q π (A4b) according to Greenwood [A4]. It can be concluded that the author's physical criterion for the PVR/EHD transition in line contacts is a bit too conservative, but has the same properties as the criterion developed by Greenwood [A4], which puts confidence in this criterion. 6. Questions For ease of survey, the main questions are: () Is it correct to make a qualitative comparison between line contacts and elliptical contacts? () Does the new PVR film thickness formula (0) show limitations, and if so, how do they look like? () What is the author's opinion on the accuracy of the PVR formulae proposed by Hamrock and Dowson [, and by Dowson, Dunn, and Taylor []? (4) How should equation (0) be applied to l ine contacts?

5 The Film Thickness Transition Criteria discussion by Harry van Leeuwen page 5 (5) Does the author agree that a PVR/EHD transition criterion should be based on transitions around the "extrapolated Blok", or in the Grubin area, rather than on transitions in the Weber and Saalfeld regime? In addition, I would like to express my appreciation for the paper, which has many ideas in it, worth some more consideration. ADDITIONAL REFERENCES [A] Blok, H., "Discussion of a Paper by Ewen McEwen", Journal of the Institute of Petroleum, Vol. 8, No. 44, Aug. 95, pp [A] Hooke, C.J.,"The elastohydrodynamic lubrication of heavily loaded contacts", Journal of Mechanical Engineering Science, Vol. 9, No. 4, 977, pp [A] Greenwood,J.A.,"Presentation of elastohydrodynamic film-thickness results", Journal of Mechanical Engineering Science, Vol., No., 969, pp.8-. [A4] Greenwood, J.A., "An extension of the Grubin theory of elastohydrodynamic lubrication", Journal of Physics D: Applied Physics, Vol. 5, 97, pp [A5] Dowson, D., and Higginson,G.R., Elastohydrodynamic Lubrication, SI Edition, Pergamon Press, Oxford, 977, p.0. [A6] Weber, C., and Saalfeld, K., "Schmierfilm bei Walzen mit Verformung", Zeitschrift für angewandte Mathematik und Mechanik, Vol. 4, No. /, Jan./Feb. 954, pp [A7] Moes, H., "Discussion on Paper R by D. Dowson: Elastohydrodynamic Lubrication: An introduction and a review of theoretical studies", Proc. Instn. Mech. Engrs., , Vo. 80, Pt. B, pp

6 The Film Thickness Transition Criteria discussion by Harry van Leeuwen page 6 Fig.Al: A map of the lubrication regimes in isothermal line contacts after Johnson [4]., showing some transition curves. The area investigated by Dowson and Higginson '[AS]. is marked by crosshatching. = Blok [A]. = Greenwood {A4] o-o-o-o = Hooke [A] = Markho and Clegg [] Φ = resp. - = Houpert physical criterion