Tribology Online, 12, 1 (2017) 1-7. ISSN 1881-2198

DOI 10.2474/trol.12.1

Copyright © 2017 Japanese Society of Tribologists 1

Article 

Prediction of Shallow Indentation Effects in a Rolling-Sliding EHL Contact

Based on Amplitude Attenuation Theory

Petr Šperka*, Ivan Křupka and Martin Hartl Faculty of Mechanical Engineering, Brno University of Technology

Technicka 2896/2, 616 69 Brno, Czech Republic *Corresponding author: sperka@fme.vutbr.cz

( Manuscript received 31 May 2016; accepted 04 December 2016; published 15 January 2017 )

In this paper amplitude attenuation theory is used to predict film thickness variations caused by shallow indentation inside a rolling-sliding elastohydrodynamic (EHL) contact. The model providing a complete analytical description of roughness passage through the contact is considered. The estimation model is compared with film thickness measurement obtained by optical interferometry. The in-contact effects for several shapes of indentation were simulated. It is clearly shown that optimal indentation profile should have smooth shape with large diameter and small depth. The results were generally explained on theoretical curves which bring a clear insight into the transient phenomenon and can be used as a rapid engineering tool. Keywords: micro-elastohydrodynamic lubrication, surface roughness, amplitude attenuation theory, film thickness

1. Introduction

Basic requirement for lubrication of concentrated contacts in machine elements is to separation surfaces by a coherent film to maintain reliable operation with low wear and friction. Every engineering surface has some roughness which is connected with film thickness and pressure variations. The roughness could have significant effect on damaging processes such as wear and fatigue. In lubricated contacts, surface roughness is deformed by a lubricant meanwhile it influences the lubrication process not only by its deformed shape. Therefore, a rapid prediction of surface pattern related effects in lubricated contacts represent essential demand of high importance.

An indentation is one of the surface features that can be present on the bearing surface. It can be produced purposefully as surface treatment called surface texturing or accidentally by contamination particles. According to operating conditions, lubrication regime and particular application an indentation on contact surfaces can have beneficial or negative effect.

The first experimental observation of indentations influence on film thickness in an EHL contact was done by Wedeven and Cusano [1,2]. Film variations around the indentation and implications to pressure distribution under rolling and sliding conditions has been shown and discussed. Further insight into processes involved in lubrication of surfaces with indentations has been provided by Kaneta et al [3,4]. A film reduction in

vicinity of indentation was explained in the moments when dent enters, exits or takes place on the edge of central zone under both orientations of sliding and pure rolling. Mourier et al. [5] used numerical simulation and film thickness measurement to study the effect of dent geometry and operating conditions. Their conclusion was that deep dents lead to film decrease or collapse while shallow indentations produce significant film increase. The importance of shallow dent geometry and conditions of faster surface with indentations was highlighted by other study [6], further extended to mixed lubrication regime [7,8] and the case of motion start-up and reversal [9]. The effect of dent bottom shape was studied in [10] by numerical simulations. Mourier et al. [11] has shown the importance of maintaining high pressure and lubricant viscosity inside dent to allow large outflow from the dent providing explanation for beneficial function of shallow indentations. Finally, further effect of dent geometry was studied by mixed EHL numerical model [12].

In summary, an increase of film thickness during transient phenomenon of dent passage through an EHL contact was reported for shallow indentations. This film increase has to be connected with deformation of indentation and the scale of film increase is therefore related with the magnitude of deformation. The scale of deformation depends on pressure drop inside an indentation. When pressure is maintained, high lubricant viscosity allowing viscous forces to drag lubricant out of the dent and cause significant dent

Petr Šperka, Ivan Křupka and Martin Hartl

Japanese Society of Tribologists (http://www.tribology.jp/) Tribology Online, Vol. 12, No. 1 (2017) / 2

deformation. This leads to large lubricant outflow promoting film increase helping separate surfaces in mixed regime and decrease damage of surfaces.

Other authors studied aspects of friction [13,14], effects on pressure [15,16] and contact fatigue [17-27]. During recent years several review papers dealing with surface texturing have been published. They provide more detail and widespread overview of current state of knowledge [28-30].

In this paper general amplitude attenuation theory is used to predict film thickness variations caused by indentation. The model providing a complete analytical description of roughness passage through the contact is considered and its prediction compared with film thickness measurement. It is shown that effects induced by shallow indentation with various shapes could be rapidly assessed and optimal geometry found.

2. Prediction scheme

The film thickness and pressure variations connected to low-amplitude roughness passage through an EHL contact can be predicted by amplitude attenuation theory [31-33]. Under rolling-sliding conditions it is necessary to consider two components model [34], i. e. a roughness deformation (RD) and a complementary effect (CE). This theory considers harmonic surface as an essential roughness feature defined by the amplitude and wavelength and describes how this harmonic wave is changing inside the contact. Together with Fourier transform it enables to predict effects of real (complex) roughness shapes.

The part of the model used in this study was published by Hooke [35-37] and other part by Lubrecht and Venner [29]. Inside an EHL contact a harmonic wave of amplitude and phase expressed by complex number Ai is change to Ad. This modification can be described for a rolling-sliding by amplitude ratio Ad/Ai

1

1d

i

A iCQ

A iQ iCQ

, (1)

where Q is parameter including operating conditions and shear properties of the fluid under contact conditions, C holds fluid compressibility effect and i is imaginary unit. For Eyring shear thinning model [35,36] these parameters can be expressed as

2

2 2

6sign( ) eQ u

E'h

(2)

2

hE'C

B

(3)

where e is Eyring shear stress a parameter of non-Newtonian (shear thinning) model, λ is wavelength in x direction, h is steady-state central film thickness and B is bulk modulus of a lubricant. The dependency of amplitude ratio on wavelength defined by Eqs. (1)-(3) is plotted in Fig. 1 and labeled as RD curve. Recently, this prediction scheme was compared with detailed experiments and good agreement was found

[38]. Under rolling-sliding conditions, it is also necessary

to consider a second component called complementary effect. This part of film variations is generated in the inlet as a roughness enters into the contact. These variations are moving across a contact by mean speed. This study assumes that the amplitude and the wavelength of CE do not change inside contact during passage. The amplitude ratio of CE is based on modified

Lubrecht and Venner parameter for rolling-sliding conditions [32,39]. It is combined with so called short wavelength limitation (or decay rate) introduced by Hooke [35]

20

exp( 2 )

1 0.15 0.015ch b

A

(4)

M u

b L v

(5)

where β introduces the short wavelength limitation. It was shown by experiments [40] that the CE does not contain short wavelengths. The definition of β comes from Hooke [35]

3

2 1v v

Fu u

(6)

F

Q

(7)

The final attenuation ratios that describes CE is plotted by red dashed line in Fig. 1 labeled as CE curve·SW limitation. The individual contributions of CE curve and short wavelength (SW) limitation can be seen in Fig. 1 too.

The calculation steps to obtain prediction are illustrated in Fig. 2. Initial surface topography is analyzed by fast Fourier transform. Obtained spectrum is modified by RD amplitude ratios given by Eq. (1) and separately the second copy of the spectrum is modified by CE amplitude ratios given by Eq. (4). After both modified spectra are processed by inverse Fourier transform. The RD data set already holds prediction of an in-contact profile while the CE data has to be scaled

Fig. 1 Amplitude attenuation curves under rolling-sliding conditions

Prediction of Shallow Indentation Effects in a Rolling-Sliding EHL Contact Based on Amplitude Attenuation Theory

Japanese Society of Tribologists (http://www.tribology.jp/) Tribology Online, Vol. 12, No. 1 (2017) / 3

by factor u/v. It is because the initial roughness feature of wavelength λ generate complementary effect with wavelength λc = λu/v. Finally, the complementary component is shifted by distance (u-v)t against roughness deformation component to reflect different speed of both components. After, both components are summed up and the complete in-contact effect is obtained. By this method a complex roughness profiles can be used for the prediction.

Computational time for calculation of the prediction on standard desktop PC for 512 × 512 pixels topography data is <10 s. Compared to full transient solution of the problem based on multilevel multi-integration approach with 257 × 257 domain is in the order of 104 s. The amplitude attenuation theory was originally introduced for roughness features smaller or the same height as the mean film thickness. Breaking

this limit can lead to the error which tends to overestimate both minimum film thickness and pressure peaks. Present authors have shown deviations obtained by direct comparison of real rough surface measurement of film thickness and prediction based on amplitude attenuation theory [41]. Deviations were observed at ratio of surface height to film thickness equal or greater than 4. For these reasons, the method is considered as a rapid method to estimate film thickness and pressure. Moreover, thanks to analytical definition it enables to observe trends and gain more detailed understanding of the phenomena.

3. Experimental methods

Film thickness measurement was done on a ball-on-disk optical tribometer using an interferometric principle. This device consists of a microscope and an optical test rig where an elastohydrodynamic film is formed between a steel ball and a flat glass disk coated by a thin chromium layer on the bottom side. Interferograms are recorded by a hi-speed digital camera. Thin film colorimetric interferometry was used for the film thickness evaluation. This technique uses robust calibration procedure based on monochromatic and chromatic interferogram of a static contact. The method is described elsewhere [42,43]. It enables evaluation of film thickness in a range of 0 to 800 nm with resolution of 1 nm or better. Experiments were conducted with a paraffinic mineral base oil SR600 of 0.24 Pa·s viscosity, the pressure-viscosity coefficient of 24 GPa−1, the Eyring shear stress of 5 MPa and bulk modulus of 6 GPa; all at ambient temperature.

4. Results and discussion

Figure 3 presents a measurement with surface indentation of 0.2 μm depth and 30 μm diameter compared with prediction given by the current scheme and measured initial topography. Operational parameters were entrainment speed u = 0.04 m/s, slide/roll ratio (SRR) +1 (smooth disc is faster) and maximum Hertzian pressure 0.6 GPa. The Moes dimensionless parameters are M = 372 and L = 5.6.

Fig. 2 The scheme of prediction steps including fast Fourier transforms (FFT)

Fig. 3 Film thickness measurements with surface indentation compared with prediction and initial topography

Petr Šperka, Ivan Křupka and Martin Hartl

Japanese Society of Tribologists (http://www.tribology.jp/) Tribology Online, Vol. 12, No. 1 (2017) / 4

These conditions give steady-state central film thickness of 125 nm. The indentation was produced mechanically by Rockwell indenter and the topography was measured by optical profilometer. In Fig. 3 film variations that correspond to CE can be clearly seen in the interferogram on the right side from the indentation. The prediction plotted in Fig. 3 agrees to measurement rather well but some minor differences can be seen. The first is a significant difference ahead of the indentation where the prediction overestimates deformation of indentation and predicts lower film thickness. The second is in the CE part (the part of x > 170 μm) the prediction give slightly larger amplitude than the measurement shown. Nevertheless, these differences are small, so, it can be concluded that the rapid estimation model used in this study can capture an essential physics of low-amplitude roughness effects in a point EHL contact.

Figure 4 presents simulations of film thickness variations for several indentation shapes including rectangular, rounded bottom, triangular and two profiles with smooth transitions. This simulation was made for the same conditions of 125 nm central film thickness to that of measurement in Fig. 3. In Fig. 4, there are three columns showing initial shape, in-contact deformed shape (RD component) and complete in-contact effect. It is visible that a deformed shape heavily depends on the initial geometry. The first three cases (Fig. 4(a-c)) contain sharp edges, therefore, these configurations have higher spikes in deformed profile than the last two (Fig. 4(d,e)). Sharp edges persist in the in-contact profile. It is because sharp edges contain short wavelengths that are not deformed, see RD amplitude ratios of short wavelengths in Fig. 1. The RD curve says that long wavelengths are deformed much more than short wavelengths.

As to complete effects a local film thickness increase at front of the indentation is the biggest for rectangular shape. It is due to the largest amplitudes relative to other cases of mid-size wavelengths included in Fourier spectra of initial surface profile of rectangular indentation.

The simulation of the effect of different indentation size is shown in Fig. 5. The shape from Fig. 4(d) and four indentation diameters were studied. It clearly shows how deformation depends on the indentation diameter. Oppositely, the CE film variations are growing with increasing diameter. Note that the case with diameter of 70 μm has the highest average film thickness and also quite evenly distributed film thickness across a large area. From these results it can be concluded that large diameter and smooth indentations lead to better film thickness distribution than small diameter and sharp indentations.

The current model is valid for low-amplitude roughness. Usually it is assumed that roughness height is comparable to mean film thickness. Under these conditions the behavior does not depend on roughness

amplitude. In the current case the ratio of roughness height to mean film thickness was approximately 1.6. The higher value of this ratio will lead to overestimated prediction of both deformed roughness and complementary effect. It means that the largest deformation can be observed when surface height is comparable with mean film thickness and therefore the shallow indentation produces the highest film increase.

The situations in simulations can be generally explained by plotting relative amplitude spectra of initial profiles together with amplitude attenuation curves for both components. This is done for indentation profile from Fig. 4(e) and it is shown in Fig. 6. The gray dash dot line depicts amplitude spectrum of indentation from Fig. 4(e). It contains mainly wavelengths longer than 20 μm. The amplitudes of these harmonic components will be substantially deformed according

Fig. 4 Simulations of various indentation shapes and its effects on in-contact film thickness variations

Fig. 5 Simulations of different size of indentation on in-contact film thickness variations

Prediction of Shallow Indentation Effects in a Rolling-Sliding EHL Contact Based on Amplitude Attenuation Theory

Japanese Society of Tribologists (http://www.tribology.jp/) Tribology Online, Vol. 12, No. 1 (2017) / 5

RD curve and besides a significant CE film variations given by CE curve will be generated. From this plot it is evident that by designing indentations with large amplitudes in spectra close to peak of CE curve will lead to large CE film variations and with low amplitudes at wavelengths smaller than the transition wavelengths of RD curve will give in-contact deformed profiles without sharp edges. Therefore, this method provides clear quantitative insight into mechanism involved in lubrication effects of shallow indentations. It can be used to rapidly design texture with optimal deformation (pressure variations) and lubricant outflow (CE part) for maximum overall film enhancement. This represents a difference of present approach to other studies.

5. Conclusions

In this study, it was shown that in-contact effects of shallow indentations under rolling-sliding conditions can be predicted by amplitude attenuation theory. Prediction based on this model was compared with experimental interferometric measurements where rather minor differences were found. The in-contact effects for several shapes of indentation were simulated. It was demonstrated that optimal indentation profile should to have smooth shape with large diameter and small depth. The results were generally explained on theoretical curves.

This theory brings a clear insight into the influence of surface topography on an EHL in-contact effects and can be used as a rapid engineering tool.

Acknowledgements

The research leading to these results has received funding from the Ministry of Education, Youth and Sports under the National Sustainability Program I (Project LO1202).

Nomenclature

Ad deformed amplitude Ai initial amplitude Ad/Ai amplitude ratio of the roughness deformation,

Eq. (1)

b radius of the Hertz contact circle B bulk modulus of lubricant C compressibility parameter E' reduced elastic modulus, 2/[(1 − μ1

2)/E1 + (1 − μ2

2)/E2] F parameter of short-wavelength limitation Fp load

G material parameter, αE' h steady-state film thickness hc amplitude of complementary effect hc/Ai amplitude ratio of the complementary effect, Eq.

(4) i imaginary unit M Moes parameter, G(2U)0.25 L Moes parameter,W(2U)−0.75 Q dimensionless wavelength parameter, Eq. (2) r1x, r2x radii of curvature of surface 1 and 2 in

x-direction R reduced radius of curvature Rx reduced radius of curvature in x-direction,

r1xr2x/(r1x + r2x) SRR slide to roll ratio, 2(w − v)/(w + v)

u mean speed, (v + w)/2 U speed parameter, 0u/E'Rx

v speed of rough surface (ball) w speed of smooth surface (disc) W load parameter, Fp/E'Rx

2

α pressure-viscosity coefficient of lubricant β parameter of short-wavelength limitation η0 viscosity of lubricant at ambient pressure λ wavelength of initial roughness in x-direction λc wavelength of the complementary effect in

x-direction dimensionless wavelength parameter (λ/b)

(M/L)0.5 modified dimensionless wavelength parameter,

Eq. (5)

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