A numerical study of fatigue life in non-Newtonian thermal EHL rolling–sliding contacts with spinning

Tribology International 80 (2014) 156–165

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A numerical study of fatigue life in non-Newtonian thermal EHL rolling–sliding contacts with spinning Xiao-Liang Yan, Xiao-Li Wang n, Yu-Yan Zhang School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China

art ic l e i nf o

a b s t r a c t

Article history: Received 6 March 2014 Received in revised form 3 June 2014 Accepted 2 July 2014 Available online 10 July 2014

This paper presents a study on fatigue life in non-Newtonian thermal elastohydrodynamic lubrication (TEHL) point contacts with spinning. A numerical procedure is developed and extended to rolling contact fatigue (RCF) life. The results show that the effect of entraining velocity on the RCF life is closely related to ellipticity. The RCF life first decreases steeply and then gradually with increase in slide–roll ratio. However, the RCF life may increase slightly at a large slide–roll ratio. Spinning is beneficial for reduction of longitudinal friction coefficient; however, even for smooth surface contact, the RCF life can be slightly reduced by spinning. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Thermal EHL Non-Newtonian Fatigue life Spin

1. Introduction In numerous mechanical systems, for instance in the lubricated conjunctions in angular contact with ball bearings and variablespeed traction drive systems, a combination of rolling, sliding and spinning motion is involved. This supplementary kinematic component can create severe conditions and extremely high temperatures may generate at the contact points. Thermal EHL characteristics of pure rolling or rolling together with some sliding in the direction of lubricant entraining in non-conformal contacts have been widely investigated. Some tribological parameters, such as film thicknesses, temperature, friction, and power losses, have received great attention in engineering. Numerical results of nonNewtonian TEHL show that the friction coefficient is found to decrease when the entraining velocity is increased [1–4], and this is consistent with experimental results [4,5]. However, when the entraining velocity is very large, the changing rate of the friction coefficient becomes very small. The increase of the entraining velocity imposes a significant increase of the lubricant temperature. The peak pressure rises with the increase of entraining velocity for point contact [1]. The slide–roll ratio is the most important factor on the traction behavior of EHL contacts, so that the influence of the slide–roll ratio on the friction coefficient has undergone intensive investigation. The results show that when the slide–roll ratio increases, the friction coefficient first increases steeply to its maximum value and then decreases gradually

n

Corresponding author. Tel.: þ 86 13718955298. E-mail address: [email protected] (X.-L. Wang).

http://dx.doi.org/10.1016/j.triboint.2014.07.001 0301-679X/& 2014 Elsevier Ltd. All rights reserved.

for high-speed and heavy-load applications [1,3,4]. However, for low-speed and light-load applications, the friction coefficient first increases rapidly and then gradually with increase in slide–roll ratio [6]. The variation of the friction coefficient versus the slide– roll ratio has close relationship with the entraining velocity and the load [7]. The non-Newtonian TEHL analyses conducted by Liu et al. [1] show that when the slide–roll ratio increases, the pressure spike drops, becomes dull and moves towards the inlet side. Relatively few works on spinning contacts have been published in the field of elastohydrodynamic lubrication. However, in machine elements the spinning motion can be of some tribological significance. Based on Newtonian fluids, Dowson [8] and Yang et al. [9,10] studied the effects of spin. The main conclusion that arose from these works is that film thickness distributions lose their symmetry when spinning is involved. The minimum film thickness also tends to decrease with spinning and appears more influenced than pressure. However, the spin has almost no effect on the pressure distribution and central film thickness. Based on Newtonian TEHL analyses with spin conditions, Ehret et al. [11] pointed out that only a few percent of spin has a tremendous impact on heat dissipation and temperature distribution. They also mentioned that a nonNewtonian analysis was necessary to converge towards more realistic solutions. Recently, a non-Newtonian TEHL analysis conducted by Doki-Thonon et al. [12] shows that not only the minimum film thickness but also the central film thickness can decrease drastically with the increase of spin. This analysis is verified by experiment [13]. Experimental results [14–16] and numerical studies [16,17] have shown that friction decreases when spinning increases. Dormois et al. [18] have given an explanation about

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Nomenclature a b B c cf cs e E´ h h0 H ke kf ks N p P pH q Rx Ry S T T0 T1, T2

radius of Hertzian contact ellipse in the entraining direction (m) radius of Hertzian contact ellipse perpendicular to the entraining direction (m) spin–roll ratio, ωRx/ue stress exponent specific heat of lubricant (J/kg K) specific heat of solid (J/kg K) Weibull slope effective Young's modulus (Pa) film thickness (m) normal approach (m) dimensionless film thickness, hRx/a2 ellipticity, b/a thermal conductivity of lubricant (W/m K) thermal conductivity of solid (W/m K) the number of contact cycles (fatigue life) hydrodynamic pressure (Pa) dimensionless pressure, p/pH maximum Hertzian pressure (Pa) heat flux density (W/m2) effective radius in the x–z plane (m) effective radius in the y–z plane (m) probability of survival temperature (1C) ambient temperature (1C) surface temperatures of solids 1 and 2 (1C)

why the longitudinal friction decreases as spinning increases even if the thermal effects are ignored. Rolling contact fatigue (RCF) life is another major concern in engineering. When inertia and other effects are not considered, the RCF life is closely related to the subsurface stress which is caused by two parts of stress, surface normal stress and surface tangential stress. As entraining velocity increases, a reduced friction coefficient is generally observed due to the reduced viscosity; however, the variation of normal pressure distribution maybe different for different contact geometries. The influence of entraining velocity on RCF life should be carefully investigated. Previous studies indicate that both the normal pressure and the shear stress are affected by the slide roll ratio, but they do not have the same variation. The problem of how the slide roll ratio affects the RCF life remains numerically unexplored. It can be found from previous investigations that longitudinal friction coefficient decreases when spinning increases, and only a few percent of shear stresses perpendicular to velocity is superimposed on contact surface. Contrarily to shear stresses, no major change was revealed in the pressure profile between the cases with and without spinning, and this conclusion was reported for isothermal conditions. However, the question that spin has beneficial or harmful effect on the RCF life of smooth surface TEHL contact remains open. A numerical non-Newtonian TEHL model with spin is developed to investigate the effects of velocity, slide roll ratio and spin on the RCF life under thermal conditions within nonconformal contacts.

Tm T ΔT1, ΔT2 ue U 1 , U2 V w x, y, z X, Y, Z κ υ μ ν ξ ρ ρ0 ρs ω ωe α ηn η0 β βf τ0 τeq τeq

157

mean temperature across the film (1C) dimensionless temperature (T/T0) temperature rises of surfaces 1 and 2 (1C) entraining velocity, ue ¼ (U1 þU2)/2 (m/s) surface velocities of solids 1 and 2 in the x direction (m/s) stress volume (m3) applied load (N) coordinates in the x, y and z directions (m) dimensionless coordinates in the x, y and z directions, X ¼x/a, Y ¼y/a, Z¼z/a diffusivity (m2/s) Poisson's ratio friction coefficient surface deformation (m) slide–roll ratio, 2(U2  U1)/(U1 þ U2) density of lubricant (kg/m3) ambient density of lubricant (kg/m3) density of solid (kg/m3) spin velocity of surface (rad/s) mean spin velocity of surface ωe ¼ (ω1 þω2)/2 (rad/s) viscosity–pressure coefficient of lubricant (G Pa  1) effective viscosity of lubricant (Pa s) ambient viscosity of lubricant (Pa s) viscosity–temperature coefficient of lubricant (K  1) density–temperature coefficient of lubricant (K  1) characteristic shear stress of non-Newtonian fluid (Pa) equivalent stress (Pa) dimensionless equivalent stress, τeq ¼ τeq =pH

under fully flooded lubrication condition is presented. The equations are established assuming the characteristics of the lubricant remain unchanged across the film. Density and viscosity of the lubricant are thus considered to depend on the pressure and the average temperature across the film. Smooth contacting surfaces are assumed.

2.1. Kinematics The contact geometry of the problem is described by the contact of an ellipsoid on a plane as shown in Fig. 1. The spinning contact kinematics for the two surfaces is restricted to the following particular cases in which only one surface (surface 2) possesses spin. Consider a Cartesian coordinate system fixed in

2. Modeling the TEHL contact with spin In the following, the stationary non-Newtonian thermal elastohydrodynamic lubrication model for non-conformal contacts

Fig. 1. Schematic representation of the contact with spinning kinematics between an ellipsoid and a plane.

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relation to the geometry such that the axis OZ is perpendicular to the surface, and the axis OX has the same direction as the translation velocities of both surfaces. The origin is located at the center of the contact area. The surface kinematical formulae are expressed as follows: ( u1 ðx; yÞ ¼ U 1 ; v1 ðx; yÞ ¼ 0 ð1Þ u2 ðx; yÞ ¼ U 2  ω2 y; v2 ðx; yÞ ¼ ω2 x where U1 and U2 are the velocities of surface 1 and surface 2, respectively. ω2 represents the spin velocity of surface 2. 2.2. Reynolds equation and force balance equation Under the conditions of constant density and viscosity across the film, the Reynolds equation of TEHL has the same form for isothermal conditions, ! ! 3 3 ∂ ρh ∂p ∂ ρh ∂p ∂ ∂ ð2Þ þ ¼ 12ðue  ωe yÞ ðρhÞ þ12ωe x ðρhÞ ∂x ηn ∂x ∂y ηn ∂y ∂x ∂y where ue and ωe represent respectively the mean entrainment velocity, and mean spin velocity of the two surfaces. p is the hydrodynamic pressure, ηn is the effective viscosity of the nonNewtonian lubricant, and ρ is the density of lubricant. The Reynolds boundary conditions are used as boundary conditions in the solution of the hydrodynamic lubrication equation. The film thickness equation takes into account the curvature of the solid and the elastic deformation hðx; yÞ ¼ h0 þ

x2 y2 þ þ vðx; yÞ 2Rx 2Ry

ð3Þ

where h0 denotes the lubricant gap in the contact center when no deformation occurs. Rx and Ry are the effective radii of curvature in the x–z and y–z planes, respectively. Surface deformation v can be calculated by the Boussinesq integration, vðx; yÞ ¼

pðX 0 ; Y 0 Þ ∬Ω qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidX 0 dY 0 πE' ðx X 0 Þ2 þ ðy  Y 0 Þ2 2

ð4Þ

where Eʹ is the effective Young's modulus. An effective viscosity ηn has been introduced in Eq. (2) to consider the non-Newtonian properties of lubricant. The effective viscosity ηn of the sinh-law model [19,20] can be defined as   1 1 τ0 τe ð5Þ ¼ sinh n η η τe τ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where τ0 is the characteristic shear stress, τe ¼ τx 2 þ τy 2 , and τx and τy are the shear stresses along the x and y directions respectively. The apparent viscosity η of lubricant depends strongly on the temperature and pressure. A relation proposed by Roelands et al. [21] is used in this study and can be expressed as      T  138  s0 η ¼ η0 exp ðln η0 þ 9:67Þ ð1 þ 5:1  10  9 pÞz0 1 T 0  138 ð6Þ

The load balance equation ensures that the applied load (w) is carried by the fluid pressure w ¼ ∬Ω pðx; yÞdx dy

ð10Þ

2.3. Lubricant energy equations According to Liu et al. [1], the energy equation for a Newtonian lubricant and a sinh-law non-Newtonian lubricant has the same form when replacing the viscosity by the effective viscosity ηn . Under steady state conditions, the energy equation for a Newtonian fluid given by Ehret et al. [11,23] is also suitable for the sinh-law non-Newtonian model. The energy equation can be reduced to the following form: "       2 # ∂T ∂T ∂2 T ∂p ∂p ∂u 2 ∂v ρcf u þv ¼ kf 2 þ β f T u þ v þ ηn þ ∂x ∂y ∂x ∂y ∂z ∂z ∂z ð11Þ where the conduction in the lubricant film plane is neglected as well as convection in the lubricant film thickness. Kim and Sadeghi et al. [23–25] and Kazama et al. [26] have shown that the profile of film temperature across the film was close to the parabolic shape in the high-pressure region. A parabolic temperature profile across the film is introduced, and the three-dimensional equation is thus converted into a twodimensional equation. The temperature profile across the film is expressed as the following form [11,23]: z 2 z

Tðx; y; zÞ ¼ ð3T 1 þ 3T 2  6T m Þ þT 1 ð12Þ þ ð6T m  4T 1  2T 2 Þ h h where T1(x, y) and T2(x, y) represent temperatures of two surfaces. Tm(x, y) is the mean temperature across the film. Substituting Eq. (12) into the energy Eq. (11) and averaging each term of the equation across the film, a simplified form of the energy equation can be obtained [11,25]. 2.4. Surface energy equations As boundary conditions for solving the energy equation, the surface energy equations enable the temperatures of the bounding surfaces to be specified. Based on the full expression for the moving heat source equation, surface temperature equations given by Carslaw and Jaeger [23] can be expressed as ( T 1 ðx; yÞ  T 0  ΔT 1 ðx; yÞ ¼ 0 ð13Þ T 2 ðx; yÞ  T 0  ΔT 2 ðx; yÞ ¼ 0 where T0 is the inlet temperature of the lubricant. The temperature rises ΔT1(x, y) and ΔT2(x, y) have the following expression: 1 ∬Ω q ðx0 ; y0 Þ ΔT i ðx; yÞ ¼ 2πkSi 0 i  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   exp ui ðx  x0 Þ2 þ ðy  y0 Þ2  ðx  x0 Þ =ð2κ i Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  dx0 dy0 ði ¼ 1; 2Þ ðx  x0 Þ2 þ ðy  y0 Þ2 ð14Þ

The dimensionless constants z0 and s0 in Eq. (6) are calculated 0

by z0 ¼ α=½5:1  10

9

ðln η0 þ 9:67Þ

s0 ¼ βðT 0  138Þ=ðln η0 þ 9:67Þ

ð7Þ ð8Þ

The density–pressure–temperature relationship of lubricant is described by a relation proposed by Dowson and Higginson [22] " # 0:6  10  9 p ð9Þ ρ ¼ ρ0 1 þ  βf ðT  T 0 Þ 1 þ1:7  10  9 p

0

In Eq. (14), qi ðx ; y Þ represents the heat flux flowing into the surface i. 2.5. Fatigue life model In order to predict the RCF life, the Zaretsky fatigue model [27] is employed in the present study. Zaretsky dropped the depth factor (first introduced by Lundberg and Palmgren [28]) which is not needed for deterministic calculation [29] and the fatigue limit stress (first introduced by Ioannides and Harris [30]) in his

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proposed model. Fatigue limit stress is not used in the present study, mainly due to the lack of reliable experimental data. However, such a treatment yields a conservative estimation for the fatigue life. The Zaretsky fatigue life model can be expressed as follows: ln

1  N e ∭V ðτeq Þc U e dV S

ð15Þ

where S is the probability of survival, N is the number of contact cycles (fatigue life), e is the Weibull slope, c is the stress exponent, and V is the stressed volume. The relevant constants for typical bearing-type steels are [31] S ¼0.9, c¼9.1, and e¼ 10/9. Equivalent stress τeq is usually defined as von Mises stress because it can be used to predict yield according to the von Mises yield criterion, which has been found to match experimental results for metallic contact better than other yield theories [32]. In the present study, the equivalent stress is selected as von Mises stress.

3. Numerical method In the present study, the Reynolds and energy equations are solved numerically using the finite difference method. The dimensionless pressure, film thickness, and temperature are defined as P ¼p/pH, H¼hRx/a2, and T ¼ T=T 0 respectively. Note pH and a are the maximum Hertzian pressure and radius of Hertzian contact ellipse in the entraining direction respectively. The computation is initiated with the Hertzian pressure, the corresponding solid elastic deformation, and an initial overall temperature (T0). The isothermal EHL solution is obtained by solving the Reynolds equation, the elasticity equation, and the load balance equation simultaneously. The surface deformation is calculated by the discrete convolution and fast Fourier Transform (DC–FFT) method [33]. Pressure and film thickness are served as inputs for the computation of temperature distribution. Viscosity and density of lubricant are updated according to the new temperature distribution and a new isothermal EHL solving process is started. Iterations are repeated until both pressure and temperature obtain convergence. Since the surface temperatures are used as boundary conditions for the energy equation, it is essential that these three equations are relaxed together to avoid any numerical instabilities [23]. At each point, there are just three unknowns and three equations, so the new increments of the lubricant and surface temperatures can be obtained through the same iteration loop. A point relaxation scheme similar to that used by Ehret et al. [11] was adopted for temperature relaxation. The method proposed by Bos [34] is adopted to calculate the influence of coefficients for temperature rise and a discrete convolution and fast Fourier transform (DC–FFT) method is used to accelerate convergence. Obtaining subsurface stress is a preliminary but necessary step for the stress-based fatigue life prediction. In the present study, the subsurface stress is caused by two parts of stress, surface normal stress and surface tangential stress. The DC–FFT method is adopted to calculate the subsurface stress [33,35]. In this paper, relative lives are calculated instead of calculating the absolute lives, that is, the fatigue life results are normalized to the highest smooth surface result, making the choice of the material constant irrelevant. A simplified flowchart for fatigue life prediction is indicated in Fig. 2. The dimensions of the stressed volume are selected as 2.5a r x r1.5a,  1.5br yr1.5b, and 0 rz r2a. The computational grid covering the domain consists of 257  257 equally spaced nodes. The grid node number along the surface downward direction is chosen as 65. The convergence on pressure, load and temperature was reached when the maximal relative difference between two

Fig. 2. Simplified flowchart for the prediction of non-Newtonian TEHL fatigue life.

Fig. 3. Temperature rise results in comparison with those from Ehret et al.

Fig. 4. Midplane (Z¼0.6) temperature results in comparison with those from literature [1].

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consecutive iterations became less than 1  10–5, 1  10  5, and 5  10  5, simultaneously.

4. Verification The validity of solving the Reynolds equation and subsurface stress is illustrated in Reference [36], and here the validation of the algorithm for temperature rise is shown in Fig. 3. The maximum value of temperature given by Ehret et al. [11] is 117 1C, and the present solution is 115.3 1C which shows good agreement with Ehret et al. [11].

Fig. 4 is a comparison of the midplane (Z¼0.6) temperature result with the full solution from the literature [1] for nonNewtonian thermal analyses of point EHL contact. The ambient temperature is T0 ¼ 303 K (30 1C). As seen from the comparisons, the present solution well approximates the full solution of energy equation (the maximum temperature difference does not exceed 8 1C). As can be seen from the above validation results, the numerical solver of the present study is credible.

5. Results and discussions Table 1 Thermophysical and rheological parameters of the lubricant. Ambient viscosity, η0 (Pa s) Viscosity–pressure coefficient, α (GPa  1) Viscosity–temperature coefficient, β (K  1) Ambient density, ρ0 (kg/m3) Density–temperature coefficient, βf (K  1) Specific heat, cf (J/kg K) Thermal conductivity, kf (W/m K)

0.08 22 0.042 870 6.5  10  4 2000 0.14

Table 2 Material and thermal properties of steel. Elastic modulus, E (GPa) Poisson's ratio, υ Density, ρs (kg/m3) Thermal conductivity, ks, (W/kg K) Specific heat, cs (J/kg K)

206 0.3 7850 48 460

In this paper, two different geometries are used, corresponding to the ellipticity ke ¼1.0 and ke ¼6.0. For ellipticity ke ¼ 1.0, the geometric parameters are Rx ¼ Ry ¼20 mm, while for ellipticity ke ¼6.0, the geometric parameters are Rx ¼6.568 mm, Ry ¼ 104.273 mm. Two different loads, w¼ 82.61 N and w ¼278.85 N, are used; the corresponding maximum Hertzian pressures are pH ¼0.8 GPa and pH ¼ 1.2 GPa respectively for both circular contact and elliptical contact. The dimensions of the stressed volume for circular contact are selected as follows:  3a rx r1.5a,  1.5br yr1.5b, and 0 rz r2a. The surface area and depth of elliptical contact are the same as that for circular contact; thus making the stressed volume are the same. In the present study, both contact bodies are assumed to be steel, and the lubricant is assumed to be a sinh-law non-Newtonian fluid. The characteristic shear stress τ0 of lubricant is selected as τ0 ¼18 MPa. The inlet temperature of lubricant is T0 ¼30 1C. All other characteristics parameters for lubricant and solid are summarized in Tables 1 and 2 respectively.

Fig. 5. Pressure distribution for three different entraining velocities (ke ¼ 1.0, ξ ¼0.2, and pH ¼ 0.8 GPa).

Fig. 6. Subsurface stress distribution for three different entraining velocities (ke ¼1.0, ξ¼ 0.2, and pH ¼ 0.8 GPa).

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5.1. Influence of the entraining velocity

Fig. 7. Effects of entraining velocity on the friction coefficient, highest film temperature and fatigue life (ke ¼ 1.0 and pH ¼ 0.8 GPa).

Effects of entraining velocity with different ellipticity ratios, ke ¼1.0 and ke ¼6.0, are investigated. The dimensionless pressure and subsurface stress distributions corresponding to three different entraining velocities in point contacts are presented in Figs. 5 and 6 respectively. The maximum subsurface stresses are also marked in Fig. 6. As can be seen from Figs. 5 and 6, the pressure and subsurface stress distributions are significantly affected by the entraining velocity. As entraining velocity increases, not only the maximum pressure and subsurface stress, but also the regions involved with greater pressures and subsurface stresses are significantly increased for circular contacts. The pressure spike moves towards the inlet side and the maximum subsurface stress moves towards the surface. The effects of the entraining velocity on the minimum film thickness, friction coefficient, the highest film temperature and fatigue life in circular contacts are shown in Fig. 7. The entraining velocity increases from ue ¼0.25 m/s to ue ¼ 5.0 m/s. It can be seen from Fig. 7 that for circular contacts (ke ¼1.0), the film thickness and film temperature increase significantly with the increasing of entraining velocity. The viscosity of lubricant and the friction coefficient decrease considerably due to high temperature. It can also be seen that the effect of the entraining velocity on temperature and friction coefficient is related to slide–roll ratio. For slide–roll ratio ξ ¼0.2, the temperature rise ΔT is 81.3 1C, and the friction coefficient decreases from 0.049 to 0.033. While for ξ¼ 0.8, the corresponding ΔT is 131.9 1C and the friction coefficient decreases from 0.053 to 0.025. Generally speaking, the entraining velocity has more obvious effect on the temperature and friction coefficient for a larger slide–roll ratio. This is due to the fact that when the entrainment velocity is small, there is no much difference of film thickness and temperature for different slide–

Fig. 8. Pressure distribution for three different entraining velocities (ke ¼ 6.0, ξ ¼0.2, and pH ¼ 0.8 GPa).

Fig. 9. Subsurface stress distribution for three different entraining velocities (ke ¼ 6.0, ξ¼ 0.2, and pH ¼ 0.8 GPa).

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roll ratios. In this case, a larger slide–roll ratio will cause greater friction coefficient. However, when the entrainment velocity is large, the generated heat from compression and shear is obvious; a slightly increase in slide–roll ratio will result in a significant rise in temperature. An increase in temperature will cause a significant reduction of lubricant viscosity and friction coefficient. It can be noticed from Fig. 7(b) that the fatigue life decreases considerably with the increase of entraining velocity. The fatigue life decreases 42.95%, 42.72%, and 37.89% for ξ¼ 0.2, ξ¼ 0.5, and ξ¼ 0.8, respectively. The conditions for another ellipticity ratio, ke ¼ 6.0, are presented below. The dimensionless pressure and subsurface stress

Fig. 11. Effects of the slide–roll ratio ξ on the minimum film thickness, highest film temperature, friction coefficient and fatigue life (ke ¼1.0 and pH ¼ 1.2 GPa). Fig. 10. Effects of entraining velocity on the minimum film thickness, friction coefficient, highest film temperature and fatigue life (ke ¼ 6.0, pH ¼ 0.8 GPa) (ke ¼ 6.0, pH ¼0.8 GPa).

distributions corresponding to three different entraining velocities in elliptical contacts are presented in Figs. 8 and 9 respectively. The maximum subsurface stresses are also illustrated in Fig. 9.

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It can be observed from Fig. 8 that with the increase of entraining velocity, the high pressure region becomes narrower, and the pressures near the center of Hertz contact are reduced significantly in elliptical contacts. This is different in the case of point contacts. Fig. 9 shows that not only the maximum subsurface stress but also the zone containing high subsurface stress decreases with the increase of entraining velocity. The variations of the minimum film thickness, friction coefficient, the highest film temperature and fatigue life as functions of the entraining velocity in elliptical contacts (ke ¼6.0) for three different slide–roll ratios are shown in Fig. 10. The entraining velocity increases from ue ¼0.25 m/s to ue ¼5.0 m/s. As can be seen from Figs. 10 and 7 the variation trends of the minimum film thickness, friction coefficient and the highest film temperature are similar for point contacts and elliptical contacts. However, the fatigue life has opposite trends for circular contacts and elliptical contacts. The fatigue life increases monotonically with increasing entraining velocity for elliptical contacts. The fatigue life increases 40.85%, 26.14%, and 26.74% for ξ¼ 0.2, ξ ¼0.5, and ξ¼0.8, respectively. A closer look of the pressure allowed to explain the different variations: the pressure near Hertz contact center and secondary pressure peak increase significantly with the increase of entraining velocity for circular contacts; while for elliptical contacts, the high pressure region moves towards the outlet side with the increase of entraining velocity, thus making the pressure near Hertz contact center decrease obviously. 5.2. Influence of the slide–roll ratio Fig. 11 presents the effects of the slide–roll ratio ξ on the minimum film thickness, the highest film temperature, friction

163

coefficient and relative fatigue life corresponding to three different entraining velocities in circular contacts. It can be seen from Fig. 11(a) that the minimum film thickness decreases with the increase of slide–roll ratio; this is due to a decrease in viscosity of lubricants caused by temperature rise ( see Fig. 11(b)). In Fig. 11(b), with the increase of slide–roll ratio, the temperature increases monotonously. For ue ¼0.1 m/s, the temperature increases from 30 1C to 72.1 1C, while for ue ¼ 2.0 m/s, the temperature experiences greater increase, from 33 1C to 200.5 1C. It can be seen from Fig. 11(c) that the friction coefficient first increases steeply and then almost remains unchanged with increase in ξ for the low-speed condition (ue ¼ 0.1 m/s). However, for ue ¼ 0.5 m/s and ue ¼ 2.0 m/s, the friction coefficient first increases steeply to a maximum value and then gradually decreases. Like changes in the film thickness, the variation of friction coefficient has a close relationship with temperature. For a large entraining velocity, the higher temperature will result in a smaller lubricant viscosity and friction coefficient. It can be noticed from Fig. 11(d) that the fatigue life first decreases steeply and then gradually with increase in slide–roll ratio ξ. However, for relatively large slide–roll ratio, the fatigue life remains almost unchanged with the increase of slide–roll ratio. It can also be seen that the effect of slide–roll ratio on the fatigue life is related to entraining velocity. A more significant impact of the slide–roll ratio on fatigue life is observed for larger entraining velocity. For the case of ue ¼2.0 m/s in Fig. 11(d), a detailed comparison of the midplane pressure corresponding to three different slide– roll ratios (ξ ¼0.0, 0.7, and 1.5) is illustrated in Fig. 12. It can be seen from Fig. 12 that the maximum pressure distributions occur when ξ ¼0.7; then is the case of ξ¼1.5, the pressure distributions for ξ ¼0.0 are obviously less than the other two cases. Therefore, the fatigue life is almost the same for the case of ξ¼ 0.7 and ξ¼ 1.5, while less in the case of ξ ¼0.0.

5.3. Influence of the spin–roll ratio

Fig. 12. Distributions of the dimensionless pressure profile for three different slide–roll ratios (ke ¼ 1.0, pH ¼ 1.2 GPa, and ue ¼ 2.0 m/s).

In many machine elements the spin–roll ratio will be less than 0.5 [8], but in the present study the range of conditions are appropriately extended to a maximum of 1.0. Fig. 13 shows the film temperature distributions for three different spin–roll ratios in elliptical contacts. It can be observed from Fig. 13 that with spinning, the symmetry of the temperature distribution gets lost. The higher the spin–roll ratio, the higher the film temperature and the more skewed the temperature shape. The maximum temperature area is located in the contact area where the spin components contribute to a decrease of the local velocity. This is because the fluxes cannot evacuate the produced heat timely towards solid 2, so the maximum temperature occurs in this area.

Fig. 13. Temperature distributions for three different spin–roll ratios (ke ¼6.0, ξ ¼0.2, ue ¼0.5 m/s, w¼ 278.85 N, and pH ¼ 1.2 GPa).

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Fig. 15. Variations of the relative fatigue life versus the spin–roll ratio (ke ¼ 1.0, ue ¼ 0.5 m/s, and pH ¼ 1.2 GPa).

contacts, but the influence is very weak for point contacts, which is not presented in this paper. Variations of the relative fatigue life with the spin–roll ratio for point contacts are shown in Fig. 15. A similar trend to that of in elliptical contacts can be observed. As the spin–roll ratio increases from 0.0 to 1.0, the fatigue life is reduced by 2.5% (ξ¼ 0.2), 0.8% (ξ¼ 0.5), and 0.4% (ξ ¼0.8). These values are small compared to that in elliptical contacts. It can also be observed from Figs. 14 and 15 that the effects of the spin–roll ratio become more significant with the decrease of slide–roll ratio.

Fig. 14. Variations of the minimum film thickness, highest film temperature, friction coefficient, and relative fatigue life versus the spin–roll ratio (ke ¼ 6.0, ue ¼ 0.5 m/s, and pH ¼1.2 GPa).

Variations of the dimensionless minimum film thickness, the highest film temperature, friction coefficient and the relative fatigue life with the spin–roll ratio in elliptical contacts are shown in Fig. 14. It can be seen from Fig. 14(a) that as spin–roll ratio increases, minimum film thickness becomes thinner, and the highest film temperature becomes larger. Weak dynamic lubrication effects and viscosity decrease due to thermal thinning are the two main factors responsible for decrease of minimum film thickness. It can be observed from Fig. 14(b) that the friction coefficient decreases as spinning increases. The physical interpretation is as follows [18]: with the addition of spin, the longitudinal stresses decrease in a part of the contact area; the integration of these stresses leads to lower longitudinal friction than in a non-spin case. Decrease of viscosity in the lubricant by shear-thinning effect is an additional explanation to the fact that friction decreases with spin. For ξ ¼0.2, the friction coefficient is decreased by about 10%. However, the fatigue life also decreases instead of increasing. As the spin–roll ratio increases from 0.0 to 1.0, the fatigue life is reduced by 3.1% (ξ ¼0.2), 1.2% (ξ ¼0.5), and 0.7% (ξ ¼0.8). This means that even for smooth surface contact, without considering the effects of rough, the spinning also has negative effects on the fatigue life. For the purpose of comparison, the conditions in point contacts (ke ¼ 1.0) are also analyzed. The input parameters are the same as that in elliptical contacts except the geometric parameters. The results show that the influence of the spin–roll ratio on the minimum film thickness, the highest film temperature, and friction coefficient exhibits similar trends to that of in elliptical

6. Conclusions Numerical solution of non-Newtonian thermal EHL problems in non-conformal contacts with spinning motion is extended to RCF life. The sinh-law non-Newtonian rheological model is adopted and smooth contacting surfaces are assumed. The effects of entraining velocity, slide, and spin on RCF life are investigated. The following conclusions can be drawn from the above numerical results. 1) Influence of entraining velocity on the RCF life has a close relationship with ellipticity. The RCF life is significantly reduced with the increase of entraining velocity for point contact (ke ¼1.0). In contrast to this, the RCF life will be significantly increased with the increase of entraining velocity for elliptical contact with large ellipticity (ke ¼ 6.0). This is mainly due to the fact that the pressure distributions have different variations for different ellipticity ratios. 2) The RCF life decreases steeply with an initial increase in slide– roll ratio. A subsequent increase in slide–roll ratio causes a gradual decrease in the RCF life. However, at a large slide–roll ratio, a slight increase in the RCF life maybe observed for smooth surface contact. This is because the viscosity of lubricant is decreased due to shear-heating effect and shearthinning effect, and this can cause decrease in both the surface shear stress and the pressure which are beneficial to improve the fatigue life. The slide–roll ratio has more obvious influence on the RCF life for a larger entraining velocity. 3) The longitudinal friction coefficient is reduced with the presence of spinning; however, even for smooth surface contact, the RCF life can be slightly reduced by spinning. The influence of spin–roll ratio becomes more significant with the decrease of slide–roll ratio.

X.-L. Yan et al. / Tribology International 80 (2014) 156–165

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