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Deformation of Rough Surfaces in Point EHL Contacts

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Abstract

The film thickness profile obtained by the optical interferometry technique merely represents a separation between contact surfaces and does not give the surface shape of contact surfaces. It has been pointed out that the actual shape of the surface roughness in rolling and/or sliding EHL contacts must be evaluated by separately obtaining the surface shape of contact bodies. The degree of the amplitude reduction in actual roughness depends on the mechanical properties of both contacting surfaces. That is, the deformation of roughness and the pressure at which roughness locates are controlled by the interrelationship between the rough surface and the mating surface caused by rolling and/or sliding motion. The method of finding surface profiles of both bodies separately and also the pressure distribution from the interferogram obtained by the optical interferometry technique is proposed in “Appendix”.

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Abbreviations

a :

Radius of Hertzian contact circle, (m)

A a,b :

Height of ridge or depth of groove on surfaces a and b, (m)

B a,b :

Base half-width of ridge or groove on surfaces a and b, (m)

E a, E b :

Elastic moduli of solids, (Pa)

Eʹ:

Equivalent elastic modulus, (Pa) Eʹ =  2{(1 − νa2)/Εa + (1 − νb2)/Εb }−1

G :

Material parameter, (–) G = α E′

h :

Film thickness s, (m)

h a, h b :

Surface shapes of bodies a and b, (m)

h 00 :

Rigid film thickness s, (m)

h ini :

Initial central impact gap, (m)

H :

Dimensionless film thickness, (–) H = h/R

ISA, ISB :

Original shapes of bodies a and b, (m)

L a,b :

Wavelength of ridge or groove on surfaces a and b, (m)

m :

Mass of moving body, (kg)

N :

Number of time step, (–)

p :

Film pressure, (Pa)

p H :

Maximum Hertzian pressure, (Pa)

P :

Dimensionless film pressure, (–) P = p/pH

R :

Reduced radius of contact bodies, [m] R = \({R_{\text{a}}}{R_{\text{b}}}/({R_{\text{a}}}+{R_{\text{b}}})\)

R a, R b :

Radii of contact bodies, (m)

t :

Time, (s)

t 0 :

Time required to wmax, (s)

T :

Temperature, (K)

T 0 :

Ambient temperature, (K)

u, v :

Flow velocities in x and y directions, (m/s)

u a, u b :

Surface velocities of solids in x-direction, (m/s)

u e :

Entrainment velocity, (m/s) ue = (ua + ub)/2

U :

Velocity parameter, (–) U = ue/(E′R)

w A :

Applied load, (N)

w max :

Maximum load, (N)

W :

Load parameter, (–) W = wA/(E′R2)

x, y :

Coordinates, (m)

x in, x out :

Domain boundaries in x-direction, (m)

X :

Dimensionless coordinate, (–) X = x/a

Y :

Dimensionless coordinate, (–) Y = y/a

y in, y out :

Domain boundaries in y-direction, (m)

α :

Viscosity-pressure coefficient, (Pa−1)

β :

Viscosity-temperature coefficient, (K−1)

δ a, δ b :

Roughness on surfaces, (m)

Δ :

Total elastic deformation, (m)

ν a, ν b :

Poisson’s ratios of solids, (–)

η :

Viscosity of lubricant, (Pa s)

η 0 :

Ambient viscosity of lubricant, (Pa s)

ρ :

Density of lubricant, (kg/m3)

ρ 0 :

Ambient density of lubricant, (kg/m3)

τ x, τy :

Shear stresses along x- and y-directions, (Pa)

τ 0 :

Eyring stress, (Pa)

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Acknowledgements

The authors would like to express their thanks to the financial support from Czech Science Foundation 15-24091S.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Brno University of Technology, Brno, Czech Republic

    Ivan Krupka & Martin Hartl

  2. School of Engineering, Kyushu Institute of Technology, Kitakyushu, Japan

    Kenji Matsuda, Hiroshi Nishikawa & Motohiro Kaneta

  3. School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao, People’s Republic of China

    Jing Wang, Feng Guo & Peiran Yang

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  1. Ivan Krupka
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  2. Martin Hartl
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  3. Kenji Matsuda
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  4. Hiroshi Nishikawa
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  6. Feng Guo
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  8. Motohiro Kaneta
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Correspondence to Motohiro Kaneta.

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Appendix

Appendix

It has been revealed that the effect of the surface roughness on the EHL film thickness and pressure distributions can be understood by knowing the surface shape of the contact bodies separately. The optical interferometry technique seems to be the best experimental method to measure the film thickness distribution. Therefore, it is extremely important to develop a method to obtain the surface shape of contact bodies and pressure distribution from the optical interferometric observations. The method is explained below.

Divide an area wider than the Hertzian contact area into small rectangular areas of arbitrary sizes in consideration of the change in film thickness. The deflection \(\Delta (x,y)\) of a general point \((x,\,y)\) on the surface due to a uniform pressure \(p(x^{\prime},y^{\prime})\) acting on a rectangular area 2a × 2b at \((x^{\prime},\,y^{\prime})\) is given as follows [41]:

$$\begin{aligned} \Delta (x,y) & =p(x^{\prime},y^{\prime})\frac{2}{{\pi E^{\prime}}}\int_{{ - a}}^{a} {\int_{{ - b}}^{b} {\frac{{{\text{d}}{x_1}{\text{d}}{y_1}}}{{\sqrt {{{(x - x^{\prime} - {x_1})}^2}+{{(y - y^{\prime} - {y_1})}^2}} }}} } \\ & =\frac{2}{{\pi E^{\prime}}}\left[ \begin{gathered} (x - x^{\prime}+a)\ln \left\{ {\frac{{(y - y^{\prime}+b)+\sqrt {{{(y - y^{\prime}+b)}^2}+{{(x - x^{\prime}+a)}^2}} }}{{(y - y^{\prime} - b)+\sqrt {{{(y - y^{\prime} - b)}^2}+{{(x - x^{\prime}+a)}^2}} }}} \right\} \hfill \\ +(y - y^{\prime}+b)\ln \left\{ {\frac{{(x - x^{\prime}+a)+\sqrt {{{(y - y^{\prime}+b)}^2}+{{(x - x^{\prime}+a)}^2}} }}{{(x - x^{\prime} - a)+\sqrt {{{(y - y^{\prime}+b)}^2}+{{(x - x^{\prime} - a)}^2}} }}} \right\} \hfill \\ +(x - x^{\prime} - a)\ln \left\{ {\frac{{(y - y^{\prime} - b)+\sqrt {{{(y - y^{\prime} - b)}^2}+{{(x - x^{\prime} - a)}^2}} }}{{(y - y^{\prime}+b)+\sqrt {{{(y - y^{\prime}+b)}^2}+{{(x - x^{\prime} - a)}^2}} }}} \right\} \hfill \\ +(y - y^{\prime} - b)\ln \left\{ {\frac{{(x - x^{\prime} - a)+\sqrt {{{(y - y^{\prime} - b)}^2}+{{(x - x^{\prime} - a)}^2}} }}{{(x - x^{\prime}+a)+\sqrt {{{(y - y^{\prime} - b)}^2}+{{(x - x^{\prime}+a)}^2}} }}} \right\} \hfill \\ \end{gathered} \right]p(x^{\prime},y^{\prime}) \\ & ={I_{xy,\,x^{\prime}\,y^{\prime}}}p(x^{\prime},y^{\prime}) \\ \end{aligned} $$
(17)

where \((x^{\prime},y^{\prime})\) is the coordinates of the center of the rectangular area 2a × 2b.

The total deflection \(\Delta ({x_i},{y_j})\), i.e., \({\Delta _{i,j}}\), at an arbitrary point \(({x_i},{y_j})\) due to the pressure distribution \(p({x_k},{y_l})\), i.e., \({p_{k,l}}\), defined by discrete elements, can be found as follows [41]:

$${\Delta _{i,j}}=\sum\limits_{{l=1}}^{N} {\sum\limits_{{k=1}}^{M} {{I_{ij,kl}}{p_{kl}}} } $$
(18)

where l = 1, N, k = 1, M, and M × N is the total number of grid points of the rectangular elements.

Equation (1) leads

$$\Delta ({x_i},{y_j})=h({x_i},{y_j}) - {h_{00}} - (ISB({x_i},{y_j}) - ISA({x_i},{y_j}))$$
(19)

The deflection \(\Delta ({x_0},{y_0})\) at a reference point \(({x_0},{y_0})\) can be obtained by

$$\Delta ({x_0},{y_0})=h({x_0},{y_0}) - {h_{00}} - (ISB({x_0},{y_0}) - ISA({x_0},{y_0}))$$
(20)

\({h_{00}}\) can be eliminated by subtracting \(\Delta ({x_0},{y_0})\) from \(\Delta (x,y)\). If the center of the contact is adopted as the reference point, \(\Delta ({x_0},{y_0})=\Delta (0,0)\). So the difference in deflection between the reference point (0, 0) and the target point (xi, yj), i.e., \(\Delta ({x_i},{y_j}) - \Delta (0,0)\), becomes

$$\sum\limits_{{l=1}}^{N} {\sum\limits_{{k=1}}^{M} {\left( {{I_{ij,kl}} - {I_{00,kl}}} \right){p_{kl}}} } ={h_{i,j}} - (IS{B_{i,j}} - IS{A_{i,j}})+(IS{B_{0,0}} - IS{A_{0,0}})$$
(21)

The applied load \({w_A}\) should be related to the following equation

$${w_A}=4\sum\limits_{{l=1}}^{N} {\sum\limits_{{k=1}}^{M} {{a_k}{b_l}{p_{kl}}} } $$
(22)

The total deflection \(\Delta ({x_i},{y_j})\) can be obtained by inserting the pressure distribution obtained by Eqs. (21) and (22) into Eq. (18). Then, one can find h00 in Eq. (19), and as a result, surface profiles can be obtained from Eqs. (5a) and (5b) or (6a) and (6b).

The same method has already been used by the authors [42] for a contact between smooth surfaces without describing detailed analytical method.

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Krupka, I., Hartl, M., Matsuda, K. et al. Deformation of Rough Surfaces in Point EHL Contacts. Tribol Lett 67, 33 (2019). https://doi.org/10.1007/s11249-019-1145-z

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