edge contact effect on thermal elastohydrodynamic lubrication of finite contact lines

Edge contact effect on thermal elastohydrodynamic lubricationof finite contact lines

Morteza Najjari, Raynald Guilbault n

Department of Mechanical Engineering, École de technologie supérieure, 1100 Notre-Dame Street West, Montréal, Québec, Canada H3C 1K3

a r t i c l e i n f o

Article history:Received 10 May 2013Received in revised form4 November 2013Accepted 7 November 2013Available online 21 November 2013

Keywords:Thermal EHLNon-Newtonian lubricationFinite line contactEdge contact

a b s t r a c t

Minimum lubricant film thickness and maximum pressure every so often appear close to roller ends. Thisstudy combines the Boussinesq–Cerruti half-space equations with a free boundary correction procedurefor precise modeling of edge contact conditions. The thermal EHL model developed associates thisrepresentation to a standard finite difference of the energy equation, and to a modified finite differenceexpansion of the Couette term of the Reynolds equation. To complete the model, the Carreau expressiondescribes the shear-thinning response of the lubricant. The investigation includes different roller profilecorrections. The results show that a large radius crowning modification combined with a rounding of thecorners constitutes the most effective profile adjustment.

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1. Introduction

An elastohydrodynamic lubrication regime (EHL) developswhen high pressures (compared to Young modulus of the bodies)generate significant surface deformations, impacting the lubricantfilm shape. Usually, in real applications such as gears, cams androller bearings, contact lines are of a finite length, a conditionwhich leads to the edge contact problem, with the most severecase arising between contacting surfaces of different lengths. Toreduce the damaging effect of finite contact lines, mechanicaldesigners commonly round off or axially profile the surfaces closeto the body ends. The well-known studies published by Gohar,Cameron, Wymer and Bahadoran [1–4] are among the very few tohave investigated the finite line contact problem. Their experi-mental investigations analyzed the effects of roller geometry onEHL lubricant film shape and thickness. For instance, the opticalfringe obtained by Wymer and Cameron [3] demonstrated that thefilm thickness thins down near the roller ends. Even though recentreviews [5–7] show that analyses of infinite line contact andpoint contact problems have been well developed over the pastdecades, the early finite line contact numerical simulations (Mos-tofi and Gohar [8], Kuroda and Arai [9], Xu et al. [10] and Parkand Kim [11]) were limited to light or moderate loads. More recentpublications have examined the influence of assembly precisionand surface modifications. For example, Kushwaha et al. [12]

investigated the influence of alignment on the film shape betweenrollers and raceways; Chen et al. [13] studied the effect ofcrowning and logarithmic roller end profiles, and Liu and Yang[14], and Sun and Chen [15] analyzed the thermal EHL of finite linecontact under heavy loads with the multigrid approach developedby Lubrecht [16]. Recently, Zhu et al. [17] presented a mixed EHLinvestigation including realistic geometries and surface roughnesseffects on finite line contact modeling. Xue et al. [18] carried outexperimental EHL studies of finite rollers with logarithmic endprofiles under heavy loads, and found that the film at the rollerends may be thinner than the outlet film at the mid-lengthposition.

While increasing the treatment sophistication of the Reynoldsequation, the advent of the multigrid approach facilitated numer-ical investigations of high pressure EHL problems, and conse-quently, the description of the lubricant behavior in finite linecontact conditions. However, in addition to the oil flow perturba-tions accounted for in the Reynolds equation, the free surfaces atthe ends of a contact line also strongly affect the deformation ofthe loaded surfaces. Nevertheless, to the author's knowledge, thepotential localized solid–fluid interactions have never been thor-oughly described.

Under dry contact conditions, the free boundaries have asignificant influence on the contact stresses and deformation[19]. For example, the finite length roller-half space contactcondition is well known to generate high stress concentration atthe roller limits. Conversely, the plane stress condition at the freeboundaries of coincident end rollers permits small axial expan-sions, and consequently, local contact pressure reductions, whichmay be approximated by Eq. (1) [19].

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n Corresponding author. Tel.: þ1 514 396 8862; fax: þ1 514 396 8530.E-mail addresses: [email protected] (M. Najjari),

[email protected] (R. Guilbault).

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General contact solutions are often based on the classicalelastic half-space theory (Boussinesq–Cerruti), which establishesthe relation between the surface tractions and displacements.However, because of the underlying half-space assumption, whenemployed without any correction, the relation produces incorrectpressure increases near free edges. Over four decades ago, Hetényi[20,21] proposed a correction process involving a shear stresselimination from mirrored pressure distributions, in combinationwith an iterative treatment for normal stress correction. Recently,Guilbault [22] introduced a correction factor (Eq. (2)) whichmultiplies the mirrored pressures to simultaneously correct theshear and normal stress influence on displacements, therebyguaranteeing significantly lower calculation times as comparedto a complete Hetényi process.

p′0 � ð1�ν2Þp0 ð1Þ

ψ ¼ 1:29� 11�ν

ð0:08�0:5νÞ ð2Þ

This paper presents a detailed numerical investigation of thepotential influence of solid-fluid interactions on the pressure, filmthickness and temperature distributions at the ends of finitecontact lines. The study develops a model combining accuracyand high solution speed for low- to extreme-pressure EHL pro-blems. In this model, the general non-Hertzian contact solutionpresented by Hartnett [23] to calculate surface deformations andpressure distributions is completed with the Hetényi shear stresselimination process and the Guilbault correction factor for therelief of the normal stress effect on the free boundaries. While

a standard finite difference formulation ensures an energy equa-tion solution, a simple algorithm based on a modified forwardfinite difference iterative method, presented by Cioc [24], resolvesthe Reynolds equation for the thermal EHL part of the globalsolution. The section following the model preparation comparesthe numerical results to experimental measurements published byWymer and Cameron [3]. In the third section, the free boundarycorrection contribution is analyzed in two steps: first using onlythe mirrored pressures for shear correction, and thereafter inte-grating Guilbault correction factor for a complete correction. Thelast section investigates the influence of common roller profileaxial modifications.

2. Model preparation and governing equations

2.1. Contact problem

The general dry contact problem resolution procedure is welldescribed and validated in Ref. [22]. The EHL model developed inthe present paper uses the same algorithm: the solution domain isdivided into constant pressure cells, and the flexibility matrixwritten for the resulting mesh. The pressure cells are mirroredwith respect to the free boundaries, and their influence isintegrated into the flexibility matrix to eliminate the free bound-ary artificial shear stress. To remove the remaining normal stressinfluence, each mirror cell contribution is multiplied by Guilbault'sfactor prior to its integration into the flexibility matrix. This lastoperation completely releases the boundaries.

Nomenclature

α pressure-viscosity coef. (Gpa�1)β density-temperature coef. (K�1)γ viscosity-temperature coef. (K�1)ψ Guilbault's correction factorρ density (kg/m3)ν coefficient Poisson's ratioη viscosity at given P, T and shear rate (Pa s)η0,1 shear-independent viscosity (Pa s)ρa,b density of solids a and b (kg/m3)τL limiting shear stress (Pa)Λ limiting shear-pressure coefficienta half width of Hertzian contact area (m)cp specific heat of the fluid (J/(kg K))cpa specific heat of body a (J/(kg K))cpb specific heat of body b (J/(kg K))fi,j,k,l flexibility matrixh0 minimum film thickness (m)k thermal cond. of fluid (W/(m K))ka,b thermal cond. of bodies a and b (W/(m K))

n slope in the lubricant shear-thinning zonep fluid pressure (Pa)p0 maximum dry pressure (Pa)p'0 maximum plain stress dry pressure (Pa)ua,b velocities of surfaces a and b (m/s)ue rolling speed (m/s)w total load (N)yd distance from the roller end (m)z0 dimensionless viscosity-pressure indexE′ equivalent modulus (GPa)G dimensionless material parameterGf lubricant modulus (Pa)L roller length (m)Rx Equivalent radius in rolling direction (m)Ry crown radius (in y direction) (m)S0 dimensionless slope of viscosity–temperature

relationshipTbulk bulk temperature (K)T0 ambient temperature (K)U dimensionless speed parameterW dimensionless load parameter

Fig. 1. Coordinate system.

M. Najjari, R. Guilbault / Tribology International 71 (2014) 50–61 51

2.2. Thermal EHL problem

Assuming that the lubricant flow is parallel to the x-axis andthat the roller length coincides with the y-axis (see Fig.1), andconsidering the following hypotheses, the Reynolds equation isgiven by Eq. (3).

� The film thickness and fluid density are time-independent� The surface speed in the y direction is negligible

∂∂x

ρh3

η

∂p∂x

!þ ∂∂y

ρh3

η

∂p∂y

!¼ 12ue

∂ðρhÞ∂x

ð3Þ

where ue is the entraining speed (x-axis), and the pressureboundary conditions are: pressure equal to zero at the body limitsas well as at positions far before the inlet and after the outlet (p¼0at y¼7 L/2, xmin and xend) and pressure gradient equal to zeroafter the outlet p¼ ∂p

∂x ¼ 0 at xend.

2.3. Lubricant modeling

2.3.1. Viscosity–pressure–temperature relationshipThe shear-independent viscosity is often expressed by one of

the following two variants of the Roelands equation (Eqs. (4a,b)).Eq. (4b) is however considered more accurate, but nevertheless,Sadeghi and Lee Refs. [25–27] employed Eq.(4a). For the modelvalidation, Section 3.2 repeats the tests published in Ref [25–27],and compares the temperatures and friction coefficients obtainedwith published values. Therefore, for consistency, during thiscomparison, the present model also uses Eq. (4a). For all othercalculations of this study, Eq. 4b is preferred.

η1ðp; TÞ ¼ η0expfð lnðη0Þþ9:67Þð�1þð1þ5:1� 10�9pÞz0 ÞþγðT0�TÞgð4aÞ

η1ðp; TÞ ¼ η0exp ð lnðη0Þþ9:67Þ T�138T0�138

� ��S0ð1þ5:097� 10�9pÞz0 �1

!( )

ð4bÞwhere z0 is related to α as follows:

z0 ¼α

5:1� 10�9ð lnðη0Þþ9:67Þð5Þ

2.3.2. Rheological modelThe influence of severe shear conditions leading to a shear-

thinning response of the lubricant is also well-established [28],and is often described by the Carreau expression. On the otherhand, the limiting shear stress is nearly proportional to thepressure, and influenced by the temperature. Eq. (6) establishesthe limiting shear stress. The value of Λ oscillates around 0.04–0.08. Incorporating Eq. (6) into the Carreau relation leads to asimple and accurate rheological model [28].

τL ¼ Λp ð6Þ

η¼ min η1 1þ η1Gf

_χ

� �2" #ðn�1Þ=ð2Þ

;τL_χ

24

35 ð7Þ

2.3.3. Density–pressure–temperature relationshipThe equation proposed by Dowson and Higginson [29] for a

compressible fluid formulates the density–pressure–temperaturerelationship:

ρðp; TÞ ¼ ρ0 1þ 0:6� 10�9p

1þ1:7� 10�9p

" #ð1�βðT�T0ÞÞ ð8Þ

2.4. Energy equation

Neglecting the heat conduction along x and y directions, theenergy equation within the lubricant film is written as:

k∂2T∂z2

¼ ρcpu∂T∂x

�βTu∂p∂x

�η∂u∂z

� �2

ð9Þ

where

u¼ � 12η

∂p∂x

zðh�zÞþuah�zh

� �þub

zh

ð10Þ

The heat repartition between the loaded solids is calculated duringthe solution of the energy equation, which is written for solidbodies as follows (Eq. (11)):

k∂2T∂z2

¼ ρcpu∂T∂x

ð11Þ

The solution process ensures that the temperature inside thesolids at a depth greater than 3a is equal to the bulk temperature(Tbulk). Moreover, if the bulk temperatures of bodies a and b areassumed to be equal to the ambient temperature (T0), theboundary conditions are: Tajza ¼ �3a ¼ T0,Tb

��zb ¼ 3a ¼ T0 and

ka∂T∂za

��za ¼ 0

¼ k∂T∂z

��z ¼ 0

; kb∂T∂zb

��zb ¼ 0

¼ k∂T∂z

��z ¼ h

2.5. Film thickness

The EHL film thickness is classically formulated as given byEq. (12 a).

hðx; yÞ ¼ h0þgðx; yÞþ 2πE′

Z ZΩ

pðx′; y′Þdx′ dy′ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx�x′Þ2þðy�y′Þ2

q ð12aÞ

where h0 corresponds to the initial body separation, g(x,y) repre-sents the axial profile of the roller, and finally, the integral term isthe elastic deformation of the contact surfaces, established herewith the model of Ref. [22]. Therefore, when adopting thediscretization process of the contact model [22], the solutiondomain is discretized into constant pressure rectangular cells.The film shape expression then becomes

hi;j ¼ h0þgi;jþ2πE′ ∑

nx

k ¼ 1∑ny

l ¼ 1f i;j;k;lpk;l ð12bÞ

where fi,j,k,l is known as the deformation matrix [23].

2.6. Load

The load balance equation (Eq. (13)) ensures the load equili-brium over the solution domain. This equation is expressed asZ Z

Ωpðx; yÞdx dy¼w ð13Þ

2.7. Numerical thermal EHL model

2.7.1. Modified forward iterative methodTo determine the pressure distribution within the lubricant

film, the Reynolds equation (Eq. (3)) must simultaneously besolved with the equations for film thickness (Eq. (12b)), lubricantproperties (Eqs. (7 and 8)) and load balance (Eq. (13)). The solutionof the energy equation (Eqs. (9 and 11)) gives the temperaturewithin the fluid and the solid bodies. Therefore, since the viscosity,the density, the pressure and the temperature are nonlinearlyinterconnected elements, the solution of the equation system isextremely sensitive, and may rapidly become unstable whensubmitted to high loads. Consequently, special approaches, such

M. Najjari, R. Guilbault / Tribology International 71 (2014) 50–6152

as the combination of the line relaxation scheme and the multigridmethod, are required to overcome instability problems. The for-ward iterative method is probably the simplest approach used inthe past to solve this nonlinear problem. Unfortunately, in pre-sence of heavy loads resulting in pressures greater than 1 GPa, themethod was rapidly proven to be unstable. However, recently Cioc[24] observed that the instability was related to the Couette termfinite difference formulation, and thus suggested a modifiedapproach capable of overcoming the solution instability for pres-sures higher than 1 GPa; the modified method separates thepressures of finite difference mesh points i, i�1 and iþ1 betweenthe two consecutive iterations k�1 and k. The Couette term is thuswritten as in Eq. (14)

∂ðρhÞ∂x

��ki;j¼ ðρhÞki;j� ρhð Þki�1;j

Δx

¼ 1Δx

ρiðf i;j;i�1;jpki�1;jþ f i;j;i;jp

ki;jþ f i;j;iþ1;jp

kiþ1;jÞ

�ρi�1ðf i�1;j;i�1;jpki�1;jþ f i�1;j;i;jp

ki;jþ f i�1;j;iþ1;jp

kiþ1;jÞ

þρiðhk�1i;j � f i;j;i�1;jp

k�1i�1;j� f i;j;i;jp

k�1i;j � f i;j;iþ1;jp

k�1iþ1;jÞ

�ρi�1ðhk�1i�1;j� f i�1;j;i�1;jp

k�1i�1;j� f i�1;j;i;jp

k�1i;j � f i�1;j;iþ1;jp

k�1iþ1;jÞ

26666664

37777775

ð14ÞFig. 2 presents the flowchart of the modified iterative method

developed in Ref. [24].

3. Model validation

The model validation involves a comparison along the two axes:the film thicknesses in the rolling direction (x-axis), at the mid-length section (y¼0), and along the initial contact line (y-axis) arecompared to those published by Wymer and Cameron [3]. Thegeometry of the roller and the lubricant properties corresponding tothe experiments of Ref. [3] are given in Table 1. The roller profile

approximation is identical to that of Ref. [30]. Since the preciseinformation was not available in the study, Sun and Chen assumedthat the location of the first measured point in the Wymer andCameron's experiments was located at yd¼0.2 mm. However, thelateral film constriction they calculated did not precisely coincidewith the experimental results. Their assumption is thereforeadjusted in this study to yd¼0.1 mm. Fig. 3 draws the resultingaxial modification of the roller profile. As mentioned in Ref. [3], ShellHVI 650 mineral oil was used during the experiments. The lubricantproperties are obtained from Ref. [31,32].

After validation with experimental measurements, the tem-perature distributions and friction coefficients obtained from themodel were compared to numerical values published by Sadeghiand Sui [25], Hsu and Lee [26,27] and Guilbault [28]. The roller andthe lubricant properties definition of Ref. [25] are given in Table 2.

For the following results, the label “No-Correction” refers tocalculations made without accounting for the mirrored pressure orGuilbault's correction factor. The designation “Mirror Correction”refers to results obtained after the integration of the mirrored

Fig. 2. Flowchart for model solution.

Table 1Roller and lubricant properties [3,31,32].

Roller

Radius R 4.1 mmLength L 13.7 mmTotal cone angle 7.91Young modulus E 206 GPaPoisson ratio ν 0.3Density ρ 7850 kg/m3

Thermal conduct k 46 W/(m K)Specific heat c 470 J/(kg K)

GlassYoung modulus E 75 GPaPoisson ratio ν 0.22Density ρ 2500 kg/m3

Thermal conduct k 0.78 W/(m K)Specific heat c 840 J/(kg K)

LubricantAmbient temperature T0 313 KViscosity at 303 K 0.900 Pa sViscosity at 393 K 0.015 Pa sVisc.-Press. coef. Α at 303 K 30.2 GPa�1

Visc.-Press. coef. Α at 393 K 16.4 Gpa�1

Density ρ at 313 K 888 kg/m3


Thermal conduct k 0.125 W/(m K)Specific heat c 2000 J/(kg K)Modulus G at 303 K 0.1þ3.0�P GPaSlope factora n at 313 K 0.570Slope factor n at 373 K 0.993

a The values were taken from Ref. [28].

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0 0.1 0.2 0.3 0.4 0.5

Z (m

m)

Distance from the roller end (mm)

Curve fit

Experiment

Fig. 3. Roller profiling.


pressure cell contribution into the flexibility matrix. Finally, thelabel “Complete Correction” identifies the values established fromcalculations incorporating both the mirrored pressure cell con-tribution and Guilbault's factor into the flexibility matrix.

3.1. Film thickness comparison

The central film thickness in the rolling direction is presentedin Fig. 4. The graph in Fig. 4 includes three dimensionless speeds.The results were established with a complete relief of the freeboundaries at the ends of the roller (Hetényi and Guilbaultcorrections). The calculated results show high correspondencewith the experiments. Fig. 5 shows the film thickness in the axialdirection. The calculations again include three dimensionlessspeeds. The graph also compares the results obtained with acomplete relief of the free boundaries to the no-correction condi-tions. The film thickness distributions are in close agreement; theconstriction location is well predicted by the model regardless ofthe condition of the boundaries at the ends of the roller. In fact,because of the precision of the experimental measurements, andthe axial profile modification of the roller, which reduces the endinfluence, it is difficult to describe the contribution of the freeboundaries to the simulation results with a high level of certainty.Nevertheless, the curves indicate that because of the rigidityreduction associated with the complete correction, the axialconstriction shape and position are affected by the free boundaryrepresentation. Therefore, since the film thickness is significantlylower at the roller ends than at the central outlet position, anaccurate depiction of the boundary behavior is very important.

3.2. Temperature and coefficient of friction comparison

The simulations integrate one dimensionless load W(¼1.3�10�4) and four dimensionless speeds U(1.8�10�11,3.6�10�11, 5.5�10�11 and 7.3�10�11). The material parameteris G¼3500. Table 3 compares the maximum mid-film and averagetemperature rise, and the dimensionless minimum film thicknessevaluated at the central position of the roller (y¼0) to thereference values. Table 3 indicates that compared to the result ofRef. [25], the maximum difference for the maximum mid-filmtemperature increase is 4 1C (within 8% margin of error). On theother hand, considering the maximum average temperature rise,compared to Ref. [27] the difference is less than 2 1C. Finally, themaximum deviation of the minimum film thickness is 8.3% (caseof U¼1.8�10�11 and 30% slip). All simulations integrated acomplete relief of the free boundaries.

Figs. 6 and 7 compare the coefficients of friction (loadW¼1.3�10�4 and material parameter G¼3500) at different roll-ing speeds to those published in Refs. [25, 27]. The curvesdemonstrate a very good agreement.

4. Profiled roller

Cylindrical bodies are often designed with axial crown profilesin order to compensate for misalignment errors. The ends are alsooften rounded or chamfered to eliminate stress concentrationsresulting from edge contact. Logarithmic profiles representanother alternative to smooth contact pressure distributions. Inthis section, five roller profile sample cases are examined andcompared to unprofiled roller responses (crowned, rounded cor-ners, crowned with rounded corners, chamfered corners andlogarithmic profile). The general properties of the roller andlubricant are as shown in Table 4. For all simulations, the loadand speed are 8000 N (corresponding to a maximum Hertzpressure of 1.25 GPa) and 7.5 m/s, with the slip ratio equal to20%. The computational domain stretches from xmin¼�3a toxend¼1.5a in the x direction, while half of the roller length isconsidered in the y direction. Along the z-axis, the solutiondomain includes the film thickness and extends to a depth of 3ainto both solids.

Table 2Roller and lubricant properties, from Ref. [25].

Roller

Equivalent radius R 20 mmYoung modulus E 200 GPaPoisson ratio ν 0.3Density ρ 7850 kg/m3

Thermal conduct k 47 W/(m K)Specific heat c 460 J/(kg K)

LubricantAmbient temperature T0 313 KViscosity at T0 0.04 Pa.sVisc.–Press. coef. α 15.9 GPa�1

Visc.–Temp. coef. γ 0.042 K�1

Density ρ 846 kg/m3

Density–Temp. coef. β 6.4�10�4 K�1


0

10

20

30

40

50

60

70

80

90

-6 -4 -2 0 2 4 6

Film

Thi

ckne

ss (i

n/10

6 )

x (in/103)

U=5.4e-11 (Experiment)U=5.4e-11 (Model-No Correction)U=5.4e-11 (Model-Complete Correction)U=29e-11 (Experiment)U=29e-11 (Model-No Correction)U=29e-11 (Model-Complete Correction)U=82e-11 (Experiment)U=82e-11 (Model-No Correction)U=82e-11 (Model-Complete Correction)

Fig. 4. Central film thickness in rolling direction.

0

10

20

30

40

50

60

70

80

1 3 5 7 9 11

Film

thic

knes

s (in

/106 )

Distance from roller end (in/103)

U=5.4e-11 (Experiment)U=5.4e-11 (Model-No Correction)U=5.4e-11 (Model-Complete Correction)U=29e-11 (Experiment)U=29e-11 (Model-No Correction)U=29e-11 (Model-Complete Correction)U=82e-11 (Experiment)U=82e-11 (Model-No Correction)U=82e-11 (Model-Complete Correction)

Fig. 5. Film thickness in the axial direction.


The model convergence is illustrated in Table 5 for the coin-cident end straight roller case. Nx and Ny correspond to thenumber of divisions along the x- and y-axes, respectively. The leftcolumn of the table illustrates the influence of Nx, and the rightcolumn shows the influence of Ny. Although the pressure variation

is less than 0.3% for the different mesh sizes, the central filmthickness convergence trend demonstrates the more significantinfluence of the mesh size. The value variations indicate that thesolution stabilizes between the 120�80 and 120�100 meshes.Therefore, comparing all film thicknesses to the 120�100 solutionleads to the precision levels shown in the table. Table 5 alsoincludes the calculation times. Thus, when accounting for the timeaspect, the 120�80 mesh assuring a precision of 99.8% appears tobe the optimal option, while the 100�40 mesh with a precisionhigher than 88% represents an efficient trade-off. All of the resultspresented in this section were obtained with the mesh size120�80. Along the z-axis, the film thickness and solid bodieswere described by 30 and 20 nodes, respectively.

4.1. Crowned roller

The axial crowning modification is defined by a crowningradius Ry. Therefore, when the roller radius is Rx, the profile

Table 3Temperature comparison for W¼1.3�10�4.

U Slip (%) Maximum mid-film temperature increase (1C) Maximum average temperature increase (1C) Dimensionless minimum film thickness

Sadeghi [25] Hsu–Lee [26] Guilbault [28] Model Lee–Hsu [27] Guilbault [28] Model Sadeghi [25] Hsu–Lee [26] Model

1.8 �10�11 0 0.91 – 0.43 0.53 – 0.43 0.49 0.0555 – 0.060010 10.64 – 14.66 14.71 – 12.64 12.52 0.0551 – 0.058520 31.61 – 31.55 32.54 – 27.17 27.8 0.0526 – 0.056430 46.57 – 44.55 46.09 – 38.35 39.47 0.0502 – 0.0544

3.6 �10�11 0 2.76 – 1.59 3.28 – 1.59 2.63 0.0891 – 0.092410 26.64 27.28 28.16 28.42 21.31 22.94 22.51 0.0867 0.0871 0.089320 53.46 51.17 51.02 52.48 40.00 41.38 41.61 0.0831 0.0834 0.086030 71.12 67.19 66.90 69.03 53.09 54.27 54.83 0.0803 0.0808 0.0832

5.5 �10�11 0 6.15 – 3.34 7.00 – 3.34 5.34 0.1145 – 0.115510 40.37 39.54 40.43 40.30 29.76 32.15 30.89 0.1118 0.1078 0.111320 69.95 67.50 66.66 67.83 50.09 52.67 52.01 0.1077 0.1035 0.107330 89.10 85.50 84.11 86.09 64.46 66.45 66.08 0.1037 0.1001 0.1039

7.3 �10�11 0 9.97 – 5.27 10.43 – 5.27 7.83 0.1351 – 0.135810 51.33 51.18 50.26 50.46 38.77 39.59 37.77 0.1305 0.1201 0.130420 82.95 79.70 78.45 80.42 59.76 61.29 60.16 0.126 0.1156 0.126030 102.67 99.18 96.78 99.73 73.28 75.58 74.67 0.1226 0.1119 0.1226

0

0.01

0.02

0.03

0.04

0.05

0 0.05 0.1 0.15 0.2 0.25

Coe

ffici

ent o

f fric

tion

Slip

Sadeghi (5.5e-11) Model (5.5e-11)Sadeghi (7.3e-11) Model (7.3e-11)

Fig. 6. Friction coefficient for W¼1.3�10�4, G¼3500.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2

Coe

ffici

ent o

f fric

tion

Slip

Hsu-Lee (3.6e-11) Model (3.6e-11)Hsu-Lee (5.5e-11) Model (5.5e-11)Hsu-Lee (7.3e-11) Model (7.3e-11)

Fig. 7. Friction coefficient for W¼1.3�10�4, G¼3500.

Table 4Roller and lubricant properties, from Ref. [28].

Roller

Radius R 17.5 mmLength L 10.0 mmYoung modulus E 200 GPaPoisson ratio ν 0.3Density ρ 7850 kg/m3


LubricantAmbient temperature T0 313 KViscosity at 313 K 0.19580 Pa.sViscosity at 373 K 0.01664 Pa.sVisc.-Press. coef. Α at 313 K 20.2031 GPa�1

Visc.-Press. coef. Α at 373 K 14.8490 GPa�1



Thermal conduct k 0.14 W/(m K)Specific heat c 1880 J/(kg K)Modulus G at 313 K 7.0 MPaModulus G at 373 K 0.9 MPaSlope factor n at 313 K 0.570Slope factor n at 373 K 0.993


modification g(x,y) is given by

gðx; yÞ ¼ Rx�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiδ2�x2

pð15Þ

where δ¼ Rx� Ry�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRy

2�y2q� �

For this section, the crowning radius is Ry¼560mm. Fig. 8(a)illustrates the modification. Because of the symmetry, only half of thedomain is considered in the computations. This case is similar to thepoint contact problem, since the film thickness forms the horse-shoeshape. Fig. 8 shows the axial film thickness, pressure and temperature

at x¼0. The results presented in Fig. 8 were obtained with nocorrection, a mirror correction and a complete correction of the freeboundaries of the roller.

Fig. 8(b) indicates that the correction effect is localized near theroller end. At Y/L¼0.456, the graph reveals that the 0.317 pressureestimated with no correction is reduced to close to zero as soon asthe mirror or the complete correction are integrated into thecalculations. At the first non-zero pressure point established withthe complete correction procedure (Y/L¼0.443), compared to theno-correction case, the mirror correction induces a pressure reduc-tion of 11.5%, while the complete correction reduces the pressure by

Table 5Mesh convergence.

Mesh refinement along x Mesh refinement along y

Nx�Ny pcenter (MPa) hcenter (μm) hcenter Precision (%) CalculationTime (min)

Nx�Ny pcenter (MPa) hcenter (μm) hcenter Precision (%) CalculationTime (min)

50�40 1280.1 1.084 32.7 0.82 120�60 1279.8 0.652 99.4 28.0260�40 1279.8 0.981 48.6 1.72 120�80 1279.7 0.649 99.8 59.0870�40 1278.1 0.902 60.8 2.07 120�100 1279.8 0.648 100 137.980�40 1280.9 0.833 71.5 2.6390�40 1281.2 0.769 81.3 3.47

100�40 1280.8 0.723 88.4 5.87110�40 1281.1 0.675 95.8 7.52120�40 1280.9 0.658 98.5 8.68

0.060.080.10.120.140.160.180.20.220.240.260.280.30.320.340.360.38

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0.2 0.25 0.3 0.35 0.4 0.45 0.5

Dim

ensi

onle

ss F

ilm T

hick

ness

Dim

ensi

onle

ss P

ress

ure

and

Tem

pera

ture

Y/L

P/Ph - No correctionP/Ph - Mirror correctionP/Ph - Complete correctionT/T0 - No correctionT/T0 - Mirror correctionT/T0 - Complete correctionh*Rx/a^2 - No correctionh*Rx/a^2 - Mirror correctionh*Rx/a^2 - Complete correction

Fig. 8. Correction effect for crowned roller.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49

Dim

ensi

onle

ss F

ilm T

hick

ness

Dim

ensi

onle

ss P

ress

ure

and

Tem

pera

ture

Y/L


Fig. 9. Correction effect for roller with rounded corners.


21.1%. The maximum film thickness increases are 51.7% and 69.0% atthe roller end (Y/L¼0.494) for the mirror and complete correction,respectively. The effect of the correction procedure on the tempera-ture is less significant, and limited to 1.2% for the mirror correctionand 2.3% for the complete correction at Y/L¼0.443.

4.2. Roller with rounded corners

For this case, a rounding radius of 30 mm is added at a distance of1 mm from the roller end. Fig. 9(a) illustrates the modified roller. Thecurves in Fig. 9(b) demonstrate that the lateral pressure spike is notsharp for this case. In addition, the graph shows that the 0.452pressure estimated at Y/L¼0.468 with no correction is reduced to0.183 and to zero with the mirror correction and complete correctionprocedures, respectively. Furthermore, at the first non-zero pressurepoint established with the complete correction procedure (Y/L¼0.456), the pressure is reduced by 11.2% with the mirror correc-tion and by 16.0% with the complete correction, when compared tothe no-correction calculations. The film thickness at the roller end (Y/L¼0.494) is increased by 33.3% with the mirror correction and by47.6% with the complete correction. Finally, at Y/L¼0.456 thetemperature values present a 0.8% and a 1.7% reduction for themirror correction and the complete correction, respectively.

4.3. Crowned roller with rounded corners

For this roller shape (Fig. 10(a)), a crowned profile (Ry¼1500mm)is combined with a rounded corner (rounding radius of 30 mm at a

distance of 1 mm from the roller end). Fig. 10 presents the pressure,the temperature and film thickness evaluations along the axialdirection. For the modeled shape, the no-correction conditionresulted in a 0.287 pressure at Y/L¼0.456, while when includingthe mirror or complete corrections the pressures obtained reduce tozero. At the first non-zero pressure point found with the completecorrection procedure (Y/L¼0.443), the mirror and complete correc-tions produce pressure reductions of 10.6% and 15.6%, respectively,when compared with the no-correction case. At the same location,the corresponding temperature reductions are 1.2% and 1.7%. Finally,at the roller ends (at Y/L¼0.494), the film thickness increases by15.1% and 21.4% when incorporating the mirror and completecorrections, respectively.

4.4. Roller with chamfered corners

This roller profile modification involves a 51 chamfering appliedat a 0.5 mm distance from the roller end (Fig. 11(a)). Fig. 11 presentsthe pressure, temperature and film thickness curves. Because of theprofile shape, the figure shows no considerable difference betweenthe mirror and complete correction results. Compared to the resultsof the uncorrected condition at Y/L¼0.443, the pressure spike isreduced by 11.3%, while the temperature decreases by 2.2%, and thefilm thickness increases by 7.7%. More importantly, for the uncor-rected free boundary condition, the results predict a completecollapse of the lubricant film close to the chamfer beginning(Y/L¼0.456). Nevertheless, as evaluated with the correction pro-cesses, the bodies remain separated.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0.3 0.35 0.4 0.45 0.5D

imen

sion

less

Film

Thi

ckne

ss

Dim

ensi

onle

ss P

ress

ure

and

Tem

pera

ture

Y/L


Fig. 10. Correction effect for crowned roller with rounded corners.

0

0.5

1

1.5

2

2.5

3

3.5

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49

Dim

ensi

onle

ss fi

lm T

hick

ness

Dim

ensi

onle

ss P

ress

ure

and

Tem

pera

ture

Y/L


Fig. 11. Correction effect for roller with chamfered corners.


4.5. Logarithmic profile

This section analyzes the Lundberg profile, well known for itssmoothing of pressure distributions. Although the pressure andtemperature distributions are more uniform, as compared tothe previous cases, a lateral film constriction remains visible.The correction process has a significant effect on the results.As shown in Fig. 12, the estimated pressure value near the roller

extremity obtained without any free boundary relief is 0.293(at Y/L¼0.494), while the mirror and complete correction pro-cesses predict a null pressure at this location. At the firstnon-zero pressure point found with the complete correctionprocedure (Y/L¼0.481), the mirror correction generates a pres-sure reduction of 22.0%, when compared with the no-correctioncase. This reduction increases to 32.0% with the complete correc-tion. At the same point, the maximum temperature reduction is

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49

Dim

ensi

onle

ss F

ilm T

hick

ness

Dim

ensi

onle

ss P

ress

ure

and

Tem

pera

ture

Y/L


Fig. 12. Correction effect for roller with logarithmic profile.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1

2.3

2.5

0.4 0.42 0.44 0.46 0.48 0.5

Dim

ensi

onle

ss F

ilm T

hick

ness

s

Dim

ensi

onle

ss P

ress

ure

and

Tem

pera

ture

Y/L


Fig. 13. Correction effect for unprofiled coincident roller ends.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

0.4 0.42 0.44 0.46 0.48 0.5

Dim

ensi

onle

ss F

ilm T

hick

ness

Dim

ensi

onle

ss P

ress

ure

and

Tem

pera

ture

Y/L


Fig. 14. Correction effect for unprofiled non-coincident roller ends.


2.4% for the mirror correction and 4.1% for the complete correc-tion. Moreover, the film constriction predicted at the rollerboundary for the no-correction case is moved toward the center

when the free boundaries are not artificially restrained. Thecompletely corrected minimum film thickness is 9.3 times higherthan the uncorrected value.

Fig. 15. Contours of film thickness, pressure and temperature (a) crowned roller, (b) roller with rounded corners, (c) crowned roller with rounded corners, (d) roller withchamfered corners, (e) logarithmic profile, (f) unprofiled roller with coincident ends, (g) unprofiled roller with non-coincident ends.


4.6. Non-profiled roller

The free edge effect of non-profiled rollers is evaluated followingtwo conditions: 1- coincident ends, and 2- non-coincident ends.These two conditions are common in cam and gear applications.

4.6.1. Coincident endWhen the boundaries of two contacting bodies coincide, the free

surface expansion causes a pressure reduction at the extremities.Fig. 13 shows the pressure, temperature and film thickness dis-tributions. The curves clearly indicate that the pressure drop at theboundary (Y/L¼0.494) is predicted only when the correctionprocess is integrated into the simulations; the uncorrected bound-ary pressure is 2.6 times the pressure calculated with the mirrorcorrection, and 4.1 times that of the complete correction. At thesame point, the temperature evaluated with the no-correctioncondition is more than 1.2 times the temperatures evaluated withthe mirror or complete correction procedures. In addition, over the0.400rY/Lr0.494 range, the average film thickness increases by1.9% and 4.4% with the mirror and complete correction, respectively.

4.6.2. Non-coincident endThis condition arises when the mating surface extends beyond

the roller boundaries. In this case, the pressure domain remainsunchanged. However, to simulate the influence of the longercylinder, the flexibility matrix only integrates the mirrored pres-sure cells and Guilbault's correction factor for the shorter roller.Fig. 14 presents the results. As with the previous case, theminimum film thickness occurs at the roller end (Y/L¼0.494).The partial mirror correction reduces the edge pressure by 37.6%,while the complete correction produces a 48.4% decrease. At thesame position, the temperature calculated with the mirror andcomplete corrections are 92.9% and 90.5% of the non-correctedcondition. Over the 0.400rY/Lr0.494 range, the average filmthickness increases by 1.1% and 1.5% with the mirror and completecorrection, respectively.

Fig. 15 compares the seven cases studied; the film thickness(H), pressure (P) and temperature (T) contour plots obtained witha complete relief of the free boundaries of the roller are juxtaposedto show the influence of the profile modification. In addition,Table 6 gives the film thickness, pressure and temperature valuesevaluated along the contact line at the mid-length position and atthe constriction location.

The values of Table 6 clearly demonstrate that profiling theroller reduces the pressure and increases the film thickness at theconstriction location. On the other hand, the modification hasinverse consequences at the mid-length position. Therefore, it isrational to assume that an optimal profile modification would leadto a pressure ratio between the pressure at the constriction andthe pressure at the mid-length position (Pcons./PM�L) being close toa unitary value. Hence, Table 6 shows that the crowned roller with

rounded ends and the logarithmic profile are the most efficientoptions among the cases studied. Table 6 also indicates that thetemperature ratio follows a behavior similar to the pressureresponse. Additionally, the film thickness ratio reveals that thefilm at the constriction position is thinner with a logarithmicmodification than with a roller with a crowned rounded end.Therefore, even though the study did not involve any profilingoptimization, the results suggest that a large radius crowningmodification combined with rounded ends probably representsthe most advantageous profile alteration.

In addition to the optimal mechanical response of the mod-ification, since the pressure values at some points largely exceededthe 1 GPa limit of the original finite difference formulation of theReynolds equation, Table 6 also demonstrates the numericalstability of the modified method promoted in this study.

5. Conclusions

The influence of the edge on pressure distribution has beenwell documented for dry contacts. Conversely, the lubricationproblem involving finite contact lines has undergone much lessinvestigation. Moreover, the high pressure condition presents aparticularly challenging problem, since resolving the Reynoldsequation may rapidly lead to unstable computations.

This paper has presented an efficient solution approach to theproblem of thermal lubricated edge contacts submitted to highpressures. The model includes a standard finite difference solutionof the energy equations, combined with a modified finite differ-ence treatment of the Reynolds equation; to eliminate computa-tion instability, the finite difference expansion of the Couette termof the Reynolds equation is distributed over successive iterationsduring the solution process. The modified iterative method resultsin a very stable, precise and simple calculation approach. Finally, anon-Hertzian contact representation completes the model, whilethe solution procedure integrates the non-Newtonian response ofthe lubricant by means of the Carreau expression.

The general non-Hertzian contact representation is based onthe Boussinesq and Cerruti solution, complemented by a correc-tion procedure to eliminate the shear and normal internal stressesartificially generated on the traction-free surfaces defining anyfinite contact line. The final contact simulation offers fast calcula-tions and precision, and ensures a particularly efficient descriptionof edge contact conditions.

The first sections of the paper validate and evidence theaccuracy of the evaluation made with the complete thermalelastohydrodynamic model, through a comparison with experi-mental measurements and numerical results obtained from theliterature.

The third part of the paper investigates the consequences ofedge contact on the film thickness, pressure and temperaturedistributions. The study demonstrates the importance of an

Table 6Film thickness, pressure and temperature at mid-length position and constriction location along contact line.

H (10�6 m) P (GPa) T (1C)

Mid-length Constric. Hcons./HM�L Mid-Length Constric Pcons./PM�L Mid-length Constric. Tcons./TM�L

Crowned 0.6643 0.9211 1.39 1.6239 0.4165 0.26 112 62.9 0.56Rounded 0.6648 0.6031 0.91 1.2777 1.6302 1.28 103.2 112.1 1.09Crowned with rounded 0.6607 0.6362 0.96 1.4212 1.3634 0.96 107 106.2 0.99Chamfered 0.6594 0.3952 0.60 1.2668 2.7863 2.20 102.9 147.5 1.43Logarithmic 0.6428 0.5893 0.92 1.3121 1.1004 0.84 104.1 97.7 0.94Unprofiled-coincident 0.649 0.0921 0.14 1.2797 1.2959 1.01 103.2 103.7 1.00Unprofiled-noncoincident 0.6374 0.0989 0.16 1.2358 1.4969 1.21 102 115.7 1.13


accurate representation of the free boundaries. For example, theHartnett non-Hertzian contact model applied without any freeboundary relief predicted a null film thickness at the contact edgeof the chamfered roller. Conversely, both Hetényi's partial mirrorcorrection and complete procedures indicated that while present-ing a substantial constriction, the film thickness separates thesurfaces along the complete contact line. Likewise, for the straightrollers with non-coincident ends, the pressure distributions revealthe significant consequence of an inadequate treatment of the freeboundaries; at the constriction position, the partial mirror correc-tion reduces the pressure evaluation by 37.6%, while the completerelief showed a reduction of 48.4% as compared to the non-corrected conditions. Similarly, with coincident ends, the non-corrected representation leads to pressure overestimations of 160%and 310%, as compared to partial mirror correction and completecorrection, respectively. The obvious influence of the free bound-ary calls for precise modeling. Therefore, since the combination ofthe correction factor of ref. [22] with the Hetényi shear stresscorrection does not increase the calculation times, and offersaccurate estimations, the procedure is recommended for reliabledescriptions of elastohydrodynamic conditions of finite contact lines.

The last part of the study utilizes the proposed thermal modelto investigate the influence of axial profiling. The analysis includesseven common roller profile forms and contact conditions. Insummary, the simulations show that chamfering the ends gen-erates high-pressure concentration, leading to a complete collapseof the film thickness close to the chamfer beginning. On the otherhand, assuming that an optimal profiling should produce constantmaximum pressure along a contact line, the well-known logarith-mic modification and a crowning profiling combined with arounding of the corner were shown to offer the best pressuredistributions. In addition, the crowned with rounded cornerprofile also ensures a more uniform film thickness along thecontact line. Hence, the simulations presented suggest that a largeradius crowning modification combined with a rounding of thecorners probably represents the most effective profile adjustment.

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