Calculation of EHL contacts in mixed lubrication regime

Tribological Research and Design for Engineering Systems D. Dowson et al. (Editors) 9 2003 Elsevier B.V. All rights reserved.

537

Calculation of EHL Contacts in Mixed Lubrication Regime A. C. Redlich a, D. Bartel b and L. Deters b a Robert Bosch GmbH, FV/FLM, P.O. Box 10 60 50, D-70049 Stuttgart, Germany, [email protected]

b

Machine Elements and Tribology, Otto-von-Guericke-University Magdeburg Universitaetsplatz 2, D-39106 Magdeburg, Germany

A calculation model is presented for machine elements operating under mixed lubrication regime. The following conditions are assumed: elastohydrodynamic point contact; stationary rough surface to study pure sliding conditions; non-Newtonian Ree-Eyring fluid model; linear-elastic, ideal-plastic material behaviour and isothermal conditions. Under these constraints some detailed investigations within the mixed lubrication regime are made.

1. I N T R O D U C T I O N For solving tribological problems, simulation tools gain an increasing importance. Particularly the calculation methods for elasto-hydrodynamic contacts were developed further during the last years, so that film thickness-, pressure- and temperature distribution as well as elastic deformations of the surfaces can be calculated successfully. The aim of this study is to broaden this knowledge regarding surface topographies in mixed lubrication regime. It can be seen as a continuation of[1]. An overview of the theoretical models in the mixed lubrication regime can be found at Nakahara [2] or Chang [3]. Starting from stochastic approaches like Makino et al. [4] more and more deterministic studies were developed, e.g. Jiang et al. [5] or Hu and Zhu [6]. Unfortunately the deterministic approach has to cope with some numerical problems, especially with the treatment of the boundary condition between fluid film and solid contact regions. Jiang et al. use a continuous pressure boundary condition whereas Hu and Zhu switch off single terms in the Reynolds equation depending on the local film thickness. In this work a new approach is used to cope with the fictitious solid contact spots. This is demonstrated with the help of a single bump in sliding contact.

This simple surface feature enables insight to some important effects within mixed lubrication regime. After these results three different real surface topographies are compared with each other. Finally the friction calculation will be explained and two Stribeck curves are presented. 2. RESULTS W I T H SINGLE BUMP The basic concept of the EHL model is described in [1], i.e. the non-Newtonian Ree-Eyring fluid model according to Ai et al. [7] and the implementation of the 2D Reynolds-, elasticity- and load equation according to Venner [8]. The lubricant properties are the viscosity-pressure relationship as proposed by Mihailidis et al. [9] and the densitypressure relationship as proposed by Dowson and Higginson. The non-dimensional parameters in this work can be seen in table 1. Table 1 Nondimensional parameters W U G L M

3.188.10 .5 2.16.10 l l 4918 12.61 1892

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2.1. Single asperity (bump) in sliding contact A single bump is studied under pure sliding conditions. The sliding direction is always along the x-axis. Equation (1) describes the geometry of the bump with an amplitude of hbump = 500 nm and a wavelength of 2 = 10 #m.

d(x,y)=hb,mp.lO

t ,t= ).cos 2 n -

2

(1)

Fig. la and lb show the bump in the center of an unloaded ball surface.

The elastic deformations as well as the pressure distribution under the given conditions of table 1 are shown in the following graphs of fig. 2 and 3. In the center of the Hertzian contact one can see that high pressure gradients upstream of an asperity lead to unrealistic high pressure values up to 20 GPa. This leads to strong local elastic deformations ahead of the bump. The reason for this is the Reynolds equation which opens the gap between the two surfaces by a local pressure increase. But in this case

there is still a negative film thickness at the tip of the asperity (fig. 3, left side). This is just a matter of grid refinement. As soon as the asperity is too sharp for a certain grid (in this work a distance of 638 nm could be achieved between the grid points) the pressure gradients cannot be solved properly and a negative film thickness occurs. These are so called "fictitious solid contact spots". Unfortunately, this results has large consequences since it is not possible to make any statements about the number or area of solid contacts on the basis of a film thickness criterion. Also solid friction cannot be calculated with the help of the film thickness distribution. Due to this fact

2.2. Dry contact solution As it will be explained in section 2.3 the mixed lubrication regime needs the solution of the dry contact problem. (Meaning of symbols see [8])

o

o

- ~ AK~m~

o

<,,

= :(,L

+

(2)

539

o f (ni.j )""

~,.,
(3)

~li.j > 0

The system of equations (2) is derived from the EHL problem without terms of Poiseuille and Couette flow. It can be solved in the same way as the EHL solution. The constraints in equation (3) lead to a pressure build up at spots where film thickness is lower than zero whereas all other grid points do not experience any pressure changes. Therefore the pressure of grid points with both solid contact pressure and film thickness above zero must be reduced by a suitable non-linear function.

2.3. Additional pressure Pnum The basis of the new approach is, that the geometrical boundary condition must be valid, i.e. two surfaces cannot penetrate each other. Since the numerical solution of the Reynolds equation on "coarse" grids is not able to fulfil this condition in case of real surface topographies, a new pressure Pnum is introduced at fictitious solid contact spots to maintain this requirement.

[GPa]

2OIPressure p

"

Pressuie Pnum[GPai

20

15

15

'i

Two remarks must be mentioned: The pressure Pnum may not be coupled directly with the hydrodynamic pressure p, but must be calculated by the dry contact solution (section 2.2) and secondly the pressure Pnum is not a solid contact pressure, since the Reynolds equation is still valid and therefore no real solid contact spots can be identified. With this approach, the results of fig. 3 change as follows. At the fictitious solid contact spot in the center of the Hertzian contact a sharp pressure spike Pnum is calculated (fig. 4) which leads to a film thickness equal to zero (fig. 5, left). Additionally, the pressure drop downstream of the bump does not reach zero as before (fig. 5, right). From the numerical point of view, these local pressure gradients cause instabilities during solution process and therefore preventing convergence.

2.4. Linear-elastic, ideal-plastic material behaviour The next step is to cope with the unrealistic high pressure values above material strength. Due to these high pressure values, the material will deform plastically. For this reason a linear-elastic, idealplastic material behaviour was introduced. This is done by the implementation of a new variable hplast into the elasticity equation (4). The value of hplast has to be calculated iteratively: + h(x, y) = /Zoo+ r(x, y)+ ~x~ 2R.

.....y2 +

2Ry

(4)

0 L

-100

-50

0

50

x [lam]

100

-1()0

-50

0

x [~tm]

50

100

Fig. 4. Pressure p and Pnum in the line of symmetry. 500,i ..... I

'

,ootln l

,

20 [GPa]

oot/ /

15 ,o

I \L / / loo~ 0L~. "100

-50

0

x [~.m]

50

100

-100

-50

0

x [lam]

Fig. 5. Film thickness and pressure p + Pnum.

50

100

2 _~=_~**~/( x p(x',y') dx, dy,+hp,.,t(x,y,. :r E' _ x')2 + (y _ y') ~

In this work the plastic pressure limit Pplast is set to 4 Gpa and above this value plastic deformations occur. With the new variable hplast the simulation results of the example change as follows: In figures 6 and 7 one can see that the pressure upstream of the bump is limited by the plastic pressure limit. Therefore the elastic deformations upstream of the bump are less than before. The plastic deformation of the bump with an amplitude of 500nm is hplast = 350 nm. The remaining 150 nm of the bump are deformed elastically and form a micro EHL contact as one can see on the left side of figure 7.

540

solve the elastic deformation equation.) For that reason, solution time increases dramatically, especially in combination with the additional iterations to solve for hplast and Pnum. 3. RESULTS W I T H REAL SURFACE TOPOGRAPHIES In this section three different real surface topographies will be discussed with the help of the new mixed lubrication approach. All surfaces are measured with an atomic force microscope (AFM). The measured values have a distance of 784.3 nm and has to be transferred to the finest simulation grid by linear interpolation.

3.1. Lapped surface topography

Fig. 7. Film thickness and pressure distribution with linear-elastic, ideal-plastic material behaviour (Pplast "- 4 GPa) in the line of symmetry.

The roughness parameters of the lapped surface topography (see fig. 8) are C L A = 0 . 1 2 # m and RMS = 0.15/~m, respectively.

2.5. Some remarks The question if the plastic material behaviour would have been sufficient to solve the problem with the fictitious solid contact spots should be clarified: 1) Since the pressure drops down at the fictitious solid contact spots, there will be no plastic deformation at these spots themselves. High pressure values can only be found directly upstream of the bump. Hence, the implementation of the linearelastic, ideal-plastic material behaviour without Pnum leads to an increasing plastic deformation upstream of the fictitious solid contact spot during the iterations and not at the contact itself. So no useful solution can be obtained. 2) Concerning the FAS-algorithm it was not possible to get a converged solution with micro asperity or real surface topographies. The local events in the vicinity of surface asperity seem to limit the multigrid algorithm which needs a smooth residual distribution during solution process. (Of course, this does not effect the multi-integration algorithm to

Fig. 9. Film thickness and pressure distribution for dry contact situation. The smooth line in the pressure distribution (right side of fig. 9) represents the ideal-smooth surface

541

area is 45% and the plastically deformed contact area is 37%, i.e. 82% of the real contact area are deformed plastically and 18% elastically.

with the same conditions. In case of the real surface topography, one can see that almost every asperity in contact reaches the plastic pressure limit of 4 GPa. Based on the Hertzian contact area, the real contact 2000

100

[nm 80

1500

60 1000 40 500

0

20

-100

-50

0 x [lam]

50

100

0

' -100

' -50

' 0 x [l.tm]

50

0

100

J

,

0

200

,

,

,

400 600 800 Film thickness h [rim]

1000

Fig. 11. Film thickness, pressure distribution and slice plane graph for u2 = 10.0 rn/s. 2000

4

'

,

~,

1

100

f

[GPa]

[nm

80

1500 i 1000

t

60

% 40

I

500

_J -100

-50

0 X [lam]

50

100

-100

I -50

0 x [lam]

100

50

[,p

00

200

400 600 800 Film thickness h [nm] .

1000

_

Fig. 12. Film thickness, pressure distribution and slice plane graph for u2 = 1.0 rn/s. 20001 ........ [nm] 1500

1000

100 i

80

%

60 40

500

20

-100

-50

0 X [l.tm]

50

100

-100

-50

0 x [pm]

0

50

100

o

'

'

2o0

4oo

6;o

8o0

1000

Film thickness h [nm]

Fig. 13. Film thickness, pressure distribution and slice plane graph for u2 = 0.1 m/s. A new graph describing the material and fluid film or solid contact portion within the Hertzian contact area will be presented as follows: The slice plane graph is similar to the well known Abbott curve. Here, slice planes parallel to the x-axis through the film thickness distribution on the left

side of fig. 10 result in material and fluid film areas (or air in case of the dry contact situation). These areas are then divided by the Hertzian contact area and depicted as fluid- and material portions on the right side of fig. 10. The aim of this graph is to

542

enable an easy view into the contact and to decide whether it is good lubricated or not. 2000

9

Fluid 10ovortion f . . . .

9

80[ /

1500

60

1000 500 -100

-50

0 x [t.tm]

50

100

The roughness parameter of the polished surface topography (see fig. 14) are CLA =0.06 #m and RMS = 0.08 #m, respectively.

Material portion

V

20

i

9...............

3.2. Polished surface topography

~

Solid contact portion

200 400 600 800 1000 Film thickness h [nm]

Fig. 10. Film thickness and slice plane graph for dry contact situation. The next step is to investigate pure sliding conditions for three different velocities: u2 = 10.0, 1.0 and 0.1 m/s. Again the smooth lines in the film thickness and pressure distribution correspond to the smooth surface case. Fig. 11 shows the simulation results for u2 = 10.0 rn/s. At this sliding speed, both surfaces are separated completely by a fluid film as one can see in the slice plane graph because there is no material portion at film thickness zero. The pressure distribution in the line of symmetry shows a pressure wedge due to the asperity at about x = 25 #m. Downstream of this point pressure does not recover any more. In case of sliding velocity u2 = 1.0 m/s (fig. 12), the pressure wedge moves upstream due to the microasperity at about x = - 2 0 #m. But in this case pressure builds up again until the next smallest gap. The pressure spikes at the end of the Hertzian contact are due to the additional pressure Pnum to satisfy boundary condition that surfaces may not penetrate with each other. In case of the lowest sliding velocity of u2 = 0.1 rn/s (fig. 13), the overall fluid film between the two surfaces is the least. The slice plane graph shows that the material portion in the range of small film thickness has increased significantly and of course the fluid film portion has decreased. The pressure wedges still move upstream and many local pressure spikes occur in the center of the Hertzian contact.

Fig 15. Film thickness and pressure distribution for dry contact situation with polished surface. Because the polished surface is much smoother than the lapped one, the pressure distribution is more similar to the solution of the smooth surface. Only in the center of the Hertzian contact the pressure reaches the plastic pressure limit of 4 GPa. The right side of fig. 16 shows that in this case the real contact area is 80% of the nominal contact area and only 6% of the nominal contact area is plastically deformed. Fluid

2000

9

9

I00 portion

"

80~=

1500[ 1000I~

I\

4 0 ~ ~olid contact portion

5001 O'

Material portion

% 60 [ ~ ~ '

-100

20 '

-5'0

~ . A /~ 0 50

x [lam]

100

0

0

200

400

600

800 1000

Film thickness h [nm]

Fig 16. Film thickness and slice plane graph for dry contact situation with polished surface.

543

above the curves) are considerably smaller for the polished surface. The pressure distribution for the highest sliding velocity of u2 = 10.0 m/s (fig. 17) is similar to the solution of the smooth surface. With decreasing sliding velocity, some smooth pressure wedges form but do not reach the plastic pressure limit, so no micro asperity will be deformed plastically.

Again, the next step is to investigate pure sliding conditions for three sliding velocities (fig. 17-19). It is interesting that in all cases the slice plane graphs show no material portion at zero film thickness, i.e. also for the lowest sliding velocity it still exists an almost complete fluid film between both surfaces. Compared to the lapped, surface the run of the slice plane curves are quite different because the fluid film portions or fluid volumes, respectively (i.e. area

[Gi:,,,j

2000 [nm

.

.

.

.

.

.

.

.

.

100,6080 . . . . . .

1500

,'

%

1000

40 500

20 -100

-50

0 X [laml

50

100

0

-100

-50

0 x [om]

50

0 0

100

200

400 600 800 ~lmthicknessh[nm]

1000

400 600 800 Film thickness h [nm]

1000

Fig. 17. Film thickness, pressure distribution and slice plane graph for u2 = 10.0 rn/s. 20001 . . . . . . . . . . [nm]

lO0

8O

1500 %

1000

6O 4O

500

2O -100

-50

0 x [l.tm]

50

100

-100

-50

0 x [,um]

50

0

100

0

200

Fig. 18. Film thickness, pressure distribution and slice plane graph for u2 = 1.0 m/s. 20001 - .. [nm]

.

4

1500

3

1000

2

lO0

"'

[GPa]

8O % 6O 4O

500 0

^.A ................

-100

-50

0 x [~m]

2o 50

100

0

-~oo

-5o

o x [tam]

~o

0

100

Fig. 19. Film thickness, pressure distribution and slice plane graph for

U2 --

200

0.1 m/s.

400 600 800 Film thickness h [nm]

1000

544

3.3. Transversal and longitudinal ground surface topographies In this section a comparison between a transversal and a longitudinal ground surface is made, i.e. the same surface topography is studied with different sliding directions (fig. 20 and 21). The

comparison indicates two main differences. First the slice plane graph of the longitudinal ground surface runs more steep than the transversal one. This means that the fluid volume inside the Hertzian contact region of the longitudinal ground surface is smaller than the fluid volume of the transversal ground surface. The third diagram shows the

Fig. 21. Film thickness, slice plane graph and plastic deformations for longitudinal ground surface.

plastically deformed micro asperity of both surfaces. In case of the transversal ground surface, only a few asperity at the outlet of the EHL contact experience a small amount of plastic deformations. On the other hand, one can find severe plastic deformations in case of the longitudinal ground surface due to the ridges along the sliding direction. 4. FRICTION This section describes the calculation of the friction force within the mixed lubrication regime. This is fluid friction due to shear stresses within the

film and solid contact friction due to interaction between micro asperity.

4.1. Fluid friction Fluid friction is calculated from the midplane shear stress rm of a Ree-Eyring fluid. This is already described in [1]. Shear stress rm for the line of symmetry of the Hertzian contact is depicted in fig. 22. The pressure p has the most influence on shear stress due to the exponential effect on viscosity. As a result of the introduction of the plastic pressure limit Pplast the effect on viscosity is

545

if one neglects the influence of film thickness and pressure gradient at these spots.

in fact limited. Therefore the maximum shear stress is also limited to a certain value which is more or less linearly dependent on the Eyring shear stress ~,

400

400

400

[MPa]

[MPa]

[MPa] 300

3O0

300

200

200

200

100

100 0

-100

-50

0 x [pm]

50

100

'

-1 O0

.ILN,

l

-50

0 x [lam]

100

50

i

-50

-100

i

,

.w

0 x [pm]

v Ihlt_Jvb.

50

1

Fig. 22. Shear stress rm with ~ = 1.9, 3.8 and 7.6 MPa, lapped surface, u2 = 0.1 m/s. The friction curves of a ball on disc sliding wear test rig were evaluated (fig. 23).

4.2. Solid friction

The solid friction is calculated by an energetic approach which is described in [1] and in more depth by Bartel [10]. The components of the solid friction are deformation and adhesion. The first one can be calculated from the elastic-, plastic deformations of the macro and micro geometry. To calculate the adhesion force, some assumptions have to be taken into account at the moment: Adhesive bonding can only occur if plastic deformation of the material takes place and secondly if local temperatures exceed a critical value which lead to a desorption of the fluid film. Since the simulation program is not able to determine local temperatures yet, the second assumption cannot be tested. But it seems reasonable that the plastically deformed micro asperity also experience local temperature peaks. Since it is quite sophisticated to get reliable values for the shear strength in adhesive bonding, in this work one sixth of the universal hardness or plastic pressure limit of 100Cr6 (SAE 52100), respectively, is used: "t',,a =

HU 6

=

Pplast

6

N = 6 6 7 ~ mm ~

(5)

In this work the reference oil FVA Nr. 3 was studied. Again the value for the Eyring shear stress is not known. Hence, the following estimation with the help of two experimental results was done.

0,14 ...................................................................... 0,1

J

-Jtl. ,j..... l~,ull ......

l 0,14

/.t.l

o,o6

io,o6

0,04

j 0,04

j

[ 0,020

0,02

~

.............................................................................................

0,1

i

o',~

oi,

o,,

o,,

,

o

,i

o',~

Time t

oi,

o',~

o,,

Time t

Fig. 23. Friction coefficient in sliding wear test rig for lapped and polished discs. Conditions of table 1. Ball and discs of 100Cr6 (SAE 52100). One can see that the lapped disc starts with a friction coefficient of/z = 0.12 whereas the polished surface starts at # = 0.08. These measurements were compared with simulation results for various Eyring shear stresses. It was found that an Eyring shear stress of about 6.8 MPa yields reasonable agreement between experiment and simulation.

4.3. Calculation of a Stribeck curve

The next step is to calculate Stribeck curves from boundary to hydrodynamic lubrication. With the help of the aforementioned assumptions as well as material- and fluid parameters, it is possible to calculate friction coefficients over sliding velocity. Fig. 24 shows the results for the lapped and fig. 25 for the polished surface topography.

546

Unfortunately, the available test rig is not capable to measure friction coefficient over the complete velocity range. Therefore it was not possible to verify all calculated points of the curve at this time. Nevertheless, one can see that in case of the lapped surface, the friction coefficient starts to increase after the hydrodynamic and mixed friction region is passed. However, in case of the polished surface, the friction coefficient is further decreasing. The reason is that no additional plastically deformed asperity will be added with decreasing velocity because of the smoothness of the polished surface. 0,20 ..................................... j I I ,..., 0,16 ~ _ I "" ~""----....~

-A- solid friction coefficient l ..................... ]-/ 1000 I -O--fluid film friction [ /t | ,--, ~ t o t a l friction coefficient ]/ | 800 E . -*-central film thickness J / " |

._~ 0,12

600

.tO

$

r ,3r

'- 0,08 ._o

400

.o_ --=. E ~_

""

200

"E

0,04

o

0,00 0,001

0,01

0,1

1

0 100

10

sliding velocity u2 [m/s]

Fig. 24. Calculated Stribeck curve, ball against lapped disc (100Cr6/100Cr6). 0,20 ] .~. 0,16

1000 ,-,

--/Y-solid friction coefficient --0-fluid film friction ~_~.tcCtnatlr~ilftiil~ c~ ~ _ e~nt

....

[~

800

.=.

~_~ 0,12

600

~

0,08

400

7

r

t-

.to .--

,._

g

""

-

0,04

200

0,00 0,001

0

E o

0,01

0,1

1

10

9

100

sliding velocity u2 [m/s]

Fig. 25. Calculated Stribeck curve, ball against polished disc (100Cr6/100Cr6). Since it is expected that the friction coefficient in the boundary regime is also above the hydrodynamic friction coefficient, it shows that the assumptions which are made concerning the friction calculation are not sufficient yet. It is supposed to be the temperature calculation which must be introduced to

explain the increasing solid friction coefficient in that region. Probably the molecular thin film will be destroyed under these conditions and adhesive forces will increase. The central film thickness is calculated according to Hu and Zhu [6]. This is 2/3 of the Hertzian diameter is taken into account for averaging. 5. CONCLUSION The aim of the study is to investigate real rough surface topographies under pure sliding and mixed lubrication conditions. The new approach was introduced by means of an EHL problem with a single bump as surface asperity in the center of the Hertzian contact. This example offers detailed insight to some important mechanisms: On the one hand "fictitious solid contact spots" at asperity tips and on the other hand huge local pressure gradients upstream of the bumps which cause plastic deformations. The introduction of the variables Pnum and hplast enable the solution of problems with microasperity within the EHL contact. With this algorithm three different real surface topographies were studied. Results for film thickness and pressure distribution were obtained. Further on new slice plane graphs were introduced to offer a quick and easy view into the contact. For example, one can see that the supply with lubricant on a lapped surface is quite different to a polished surface. The lapped surface shows a higher material portion than the polished surface but on the other hand in total the lapped surface has a bigger fluid volume within the contact. Secondly, a comparison between a transversal and a longitudinal ground surface was made. It could be shown that the fluid volume of the transversal ground surface is bigger and the plastic deformations are less than in case of the longitudinal ground surface. The last section describes the friction calculation with fluid and solid friction. The new model enables the calculation of Stribeck curves over several orders of magnitude. But it has proven that the assumptions concerning solid friction calculation are not sufficient yet. Since it could be shown that solid contact spots cannot be calculated by Reynolds equation but rather must be identified by a

b4'/

temperature criterion. This must be done in future work.

6. 7.

REFERENCES

1.

2. 3. 4. 5.

A. C. Redlich, D. Bartel, H. Schorr and L. Deters, Proceedings 26th Leeds-Lyon Symposium on Tribology, pp. 85-93, 2000 T. Nakahara, Japanese Journal of Tribology, Vol. 39, Nr. 3, pp. 335-349, 1994 L. Chang, Wear, Vol. 184, pp. 155-160, 1995 T. Makino, S. Morohoshi and K. Saki, Proc. of the 25th Leeds-Lyon Symposium, 1998 X. Jiang, D. Y. Hua, H. S. Cheng, X. Ai and S. C. Lee, ASME Journal of Tribology, Vol. 121, pp. 481-491, 1999

8. 9.

10.

Y.-Z. Hu and D. Zhu, ASME Journal of Tribology, Vol. 122, pp. 1-9, 2000 X. Ai, H. S. Cheng and L. Zheng, ASME Journal of Tribology, Vol. 115, pp. 102-110, 1993 C.H. Venner, PhD Thesis, Twente University, 1991 A. Mihailidis, J. Retzepis, C. Salpistis and K. Panajiotidis, Wear, Vol. 232, pp. 213-220, 1999 D. Bartel, PhD Thesis, Aachen; Shaker-Verlag, 2001