World Tribology Congress 2017 Beijing, China, September 17 – 22, 2017
Evolution of wear and roughness in mixed lubrication regime N D Chakladar1) *, L Gao2), R M Hall1) and R Hewson2) 1)
2)
School of Mechanical Engineering, University of Leeds, Woodhouse Lane, LS2 9JT, UK Department of Aeronautics, Imperial College London, South Kensington Campus, SW7 2AZ, UK * Corresponding author:
[email protected]
1. Introduction Wear is an important predictor in the performance of mixed lubricated bearing surfaces at the areas of direct solid contact. In addition, the time of computation of wear, accompanied with the update of the evolved surface geometry is one of the challenges due to deterministic definition of roughness on a micro-scale. Researchers since 1970s, proposed solutions to mixed lubrication problems [1], assuming mostly Gaussian surfaces. Patir and Cheng [2] conceptualized a flow factor approach to consider the influence of roughness at the asperity scale on the fluid flow and adapted the Reynolds Equation. During the same time, Greenwood and Williamson [3] developed the asperity contact models based on the Hertzian contact of probabilistically described asperities. The current approach builds on that concept and characterizes the wear of a loaded and lubricated pin-on-plate system, assuming a rough non-wearing pin, being run-in by a rough wearing plate. The evolution of load partition ratio indicates an increase in load sharing by the fluid as wear occurs, accompanied by a reduction in the contact area ratio. The evolution of the cumulative distribution function (CDF) of the wearing surface reflects gradual smoothening of the asperities. The model is numerically efficient for mixed lubricated problems, given the CDFs of two real rough surfaces. 2. Mixed lubrication modelling In the mixed regime, while most of the contacting surfaces are separated by a lubricating film, the asperities still come in contact, sharing only a part of the total load with the remainder borne by the fluid. In order to solve the mixed lubrication problem, the fluid flow parameters are calculated using the flow factor approach of Reynolds equation and the contact parameters using the Greenwood-Williamson (GW) asperity contact model. The lubrication problem is reduced to one-dimension by assuming the depth of the pin and the plate to be infinite (Figure 1). The roughness and asperities of the pin and the plate are described by their CDFs. As the plate runs-in the pin, the roughness of the plate is worn by the roughness of the loaded pin, and the process is continued until steady wear is attained. Each simulation cycle represents the sliding distance of the plate relative to the pin and at each cycle the sum of the fluid and solid load is iteratively balanced with the applied load, adjusting the gap between the pin and the plate.
Figure 1. Schematic of a pin-on-plate system 2.1 Evaluation of contact load The asperity contact force, F and the real contact area, Ar between the pin and the plate are evaluated using the GW model (Eq. 1). The symbols and variables are detailed in Nomenclature (Section 5). The PDF of the overlap of asperities is the combined PDF of the asperities of the pin and the plate. 1 max( d e ) 3 4 F nA Eeq eq 2 (d e ) 2 PDF (d e )d (d e ) 3 min( d e ) (1) Ar nA eq
max( d e )
(d e ) PDF (d e )d (d e )
min( d e )
1 12 1 2 2 Eeq E2 E1 1
1
1
1
eq 1 2 2.2 Evaluation of wear The mechanical wear is calculated from the contact force using the Archard’s wear equation (Eq. 2). Wvol and Wdepth are the asperity volumetric wear and the asperity wear depth of the plate. The magnitude of wear depth is then used to update the asperity heights, and the asperity wear is extrapolated to update the roughness.
Wvol k w Fs Wdepth
Wvol Ar
(2)
2.3 Evaluation of fluid load
fluid
At the level of fluid flow, the film pressure is obtained by solving the flow factor adapted Reynolds equation (Eqn. 3) using a finite difference scheme. Figure 2 illustrates the local film thickness between two rough surfaces (in this case, the pin and the plate) based on which the local gap-dependent flow factors are computed. Slip boundary condition is considered with a minimum slip length of 10-9 m. The pressure and shear flow factors (ϕp, ϕs) are determined from the PDF of the local film thickness which is the combined PDF of the roughness of the pin and the plate. Once, the flow factors are calculated, they are incorporated into Eqn. 3 to determine the fluid pressure. The obtained fluid pressure is then integrated across the true fluid area (i.e. the total contact area when subtracted from the nominal bearing area) to find out the fluid load.
s1 s 2
1 dp hs1 V2 V1 s 2 2 dx h
max( hlocal )
hlocal h min( hlocal )
PDF (hlocal )d hlocal
h h min( hlocal ) local
PDF (hlocal )d hlocal
max( hlocal )
(6)
2.5 Non-dimensionalization The Reynolds and shear stress equations are non-dimensionalized to improve numerical convergence of the system (Eqns. 7). H , H slip
h, hslip c
;X
x l
c2 2 2 P p ; Rq Rq1 Rq 2 12 lV entr z1 , z2 , z Z1 , Z 2 , Z Rq
(7)
2.6 Algorithm of the code:
Figure 2. Local and nominal film thickness 3 d h p dp d (s h ) Ventr dx 12 dx dx
p s
(3)
3
max( hlocal )
hlocal PDF (hlocal )d (hlocal ) h min( hlocal )
max( hlocal )
hlocal h min( hlocal )
PDF (hlocal )d (hlocal )
2.4 Evaluation of friction: To determine the total friction of the system, the total shear load due to the fluid and the solid is divided by the applied normal load (Eqn. 4). The shear load due to solid contact, Fsolidshear, is assumed to follow the Amonton’s law of friction with the asperity contact load, F (Eqn. 5).
Fsolidshear Ffluidshear Fapplied
Fsolidshear s Fsolid
(4) (5)
Eqns. 6 compute the flow factor adapted fluid shear stress. The shear load, Ffluidshear, due to the fluid is obtained by integrating the fluid shear stress, τfluid over the true fluid area.
The code is implemented in Matlab R2016a and solved for up to 5 million simulation cycles (MCs). The algorithm of the code is briefed below. a. Discretize the pin and the plate into equidistant grids. b. Assign a CDF of roughness and asperities to each grid of the pin and the plate. c. Initialize simulation cycle, cycle = 1. d. Initialize iteration, iter = 1, to balance the applied load, Fapplied. e. Determine the asperity contact forces, F , the contact area, Ar and the total solid load, Fsolid. f. Calculate the local flow factors, Φp, Φs. g. Solve non-dimensional Reynolds PDE for the fluid pressure, P and determine the fluid force, Ffluid. h. Check if
F
solid
Ffluid Fapplied Fapplied
iter = iter + 1; hiter 1 hiter
F
104
solid
Ffluid Fapplied
Fapplied (here, α is an appropriate factor which varies the separation between the pin and the plate until the calculated load is balanced); Go To Step e; Else Go To Step i. i. Compute wear of asperity, Eqn. (2) and update asperity and roughness information of the plate. j. Calculate shear stress flow factors, Φs1, Φs2. k. Calculate solid and fluid shear load and total friction coefficient, μ. l. If end of simulation cycle is reached, End; Else cycle = cycle + 1, Go To Step d. 3. Results and discussion In this study, a distribution of roughness of a silicon 2
nitride coated surface is used for the pin and the plate. The material and geometrical properties of the pin-on-plate system is listed in Table 1. Simulation results are presented to demonstrate the evolution of load partition ratio and contact area ratio, the evolution of wear, and the evolution of CDF of the plate asperities. Table 1: Data for input parameters Parameters Width of pin (l mm) Velocity of plate (V2 mm/s) Maximum clearance (c μm) Dynamic viscosity (η kg/mms) Elastic modulus (E1, E2 GPa) Poisson’s ratio (ν1, ν2) Asperity radius (β1, β2 μm) Wear coefficient (kw mm3/Nm) Applied load (F N/mm)
wear at about 2MC. The unit of wear volume is in mm3/mm as the scale of the problem is reduced to one-dimension. Figure 4(b) illustrates the evolution of plate asperity CDF with wear, indicating a gradual smoothing of asperities.
Values 10 1 5 9×10-7 280 0.3 10 10-9 1 (a)
3.1 Load partition and contact area ratio: Figure 3(a, b) presents the evolution of load partition ratio (i.e. the sharing of the total load by the fluid and the solid) and the contact area ratio (i.e. the total contact area divided by the nominal bearing area) with wear simulation cycles. At the beginning of wear, 75% of the applied load was borne by the asperities and the rest by the fluid. In course of wear after 5MC, the load share due to the solid contact reduces to 15%, whereas, the share of the load by the fluid increases from an initial 25% up to a steady value of 85%. As the fluid load share increases, the probability of solid contact reduces, which is indicated in the Figure 3(b) with a reduction of contact area ratio from a value of 7×10-5 at the beginning of wear to a value of 2.5×10-5.
(b) Figure 4. Evolution of (a) wear, and (b) CDF 3.3 Evolution of friction:
(a)
(b) Figure 3. Evolution of (a) load partition ratio, and (b) Contact area ratio 3.2 Evolution of asperity wear and CDF: The cumulative wear volume is plotted across 5MCs (Figure 4(a)) which indicates the steady transition of
A friction study is carried out to identify the regimes of lubrication of the total system. Wear, being closely related to friction, a corresponding wear volume plot is compared, which has not been shown for brevity of the abstract. A range of entrainment velocities were chosen 0.001 mm/s to 1000 mm/s and represented in terms of the Stribeck number. Figure 5 shows the evolution of friction in linear scale and Figure 5(b) in logarithmic scale in order to stretch the boundary lubrication regime. All the three regimes are marked in the figure – the boundary lubrication regime (‘1’), the mixed regime (‘2’) and the full film regime (‘3’). As the entrainment velocity is increased (i.e. from a Stribeck number 2.75 to 180), the friction coefficient drops down by 96% from a solid friction coefficient value of 0.16 to 0.005, accompanied with a wear reduction from 6.0×10-7 mm3/mm to zero at the onset of full film. A further increase of Stribeck number (in the full film regime), shows the total friction is now dependent only on the viscous shear of the fluid, and indicates a little increase to a value of 0.02 (an increase of 3%) 3
Subscripts (1, 2): Pin, Plate; ND: Non-dimensional Symbol Definition Bearing dimension (mm), ND x, X l Width of the pin (mm) Fluid pressure (MPa), ND p, P h, H
hlocal , H local hslip , H slip (a)
A wear model is proposed to capture the evolution of friction and update the geometry of a lubricated real rough surface. The roughness of the surfaces are statistically described by their CDFs. The model characterizes a pin-on-plate system, assuming a non-wearing fixed pin being rubbed by a moving and wearing plate. The scale of the problem is reduced to one-dimension, considering infinite depth of the pin and the plate. The mechanics of contact between the asperities are calculated based on the Greenwood -Williamson contact model and the level of fluid flow is computed through the Patir-Cheng flow factors approach. Mechanical wear is calculated using Archard’s law. The simulations are run up to 5MC until the steady wear is attained. Results on the evolution of load sharing due to wear reported a decrease of asperity contact load with a corresponding increase in fluid load. Due to an increase in the fluid load, the probability of overlap of asperities reduced which in turn decreased the contact area, as observed from the evolution of the contact area ratio. The wear of asperity CDF indicated a gradual smoothing of asperities and the cumulative wear volume was observed to attain a steady wear around 2MCs. A friction study identified all the three regimes of lubrication while varying the entrainment velocities and how load is transferred in the mixed regime. With an increase of Stribeck number from a value of 2.75 to 180 (in the mixed regime), the friction coefficient dropped from a dry friction value of 0.16 to 0.005, whereas beyond a value of 180 there was an increase in the friction due to the viscous shear of the fluid in the full film regime. The model is efficient in solving wear of mixed lubricated surfaces, given their CDFs. 5. Nomenclature
Minimum slip length (mm), ND
de
Overlap of asperities (mm)
A, Ar
Nominal bearing area, Real contact area (mm2) Maximum clearance (mm) Roughness heights (mm), ND
c z, z1, z2 , Z , Z1, Z 2
(b) Figure 5. Evolution of friction in (a) linear and (b) logarithmic scale 4. Conclusion
Nominal film thickness (mm), ND Local film thickness (mm), ND
Ventr , V1 , V2
Entrainment velocity, Velocities of Pin and Plate (mm/s)
PDF , CDF
Probability density function (mm-1), Cumulative distribution function Pressure flow factor, ND
p , p s , s s1 , s1 , s 2 , s 2
Shear flow factor, ND Shear stress flow factors, ND
E1 , E2 , Eeq
Elastic moduli (MPa), equivalent
1 , 2 1 , 2 , eq
Poisson’s ratios
n
Asperity density (mm-2) Dynamic viscosity of the fluid (kg/mms) Dimensional wear coefficient (mm3/Nm) Solid force (N), Fluid force (N), Applied force (N)
kw Fsolid , Ffluid , Fapplied
s Rq1 , Rq 2 , Rq s ,
Asperity radii (mm), equivalent
Sliding distance (mm) Roughness RMS of pin and plate (mm), composite RMS (mm) Solid friction coefficient, total friction coefficient
6. Acknowledgements This research is funded by the European Union’s Seventh Framework Programme (FP7/2007-2013) under the grant agreement no. GA-310477 (for ‘LifeLongJoints’). 7. References [1] [2]
[3]
Christensen, H., “A Theory of mixed lubrication”, Proc. Inst. Mech. Engineers, 186, 1972, 421-430 Patir, N., Cheng, H.S., “An Average flow model for determining effects of three dimensional roughness”, Trans. ASME, 100, 1978, 12-18. Greenwood, J.A. Williamson, J.B.H., “Contact of nominally flat surfaces”, Proc. Royal Soc. London. Series A, 1966, 300-319. 4
