Rough Surface Contact

This paper studies the contact of general rough curved surfaces having nearly identical geometries, assuming the contact at each differential area obeys the model proposed by Greenwood and Williamson. In order to account for the most general gross geometry, principles of differential geometry of surface are applied. This method while requires more rigorous mathematical manipulations, the fact that it preserves the original surface geometries thus makes the modeling procedure much more intuitive. For subsequent use, differential geometry of axis-symmetric surface is considered instead of general surface (although this “general case” can be done as well) in Chapter 3.1. The final formulas for contact area, load, and frictional torque are derived in Chapter 3.2.


INTRODUCTION
The need of understanding rough surfaces contact has long been recognized.One primary focuses of the early studies is to predict real contact area as it varies with load.Since a rough surface is known to include layers of micro-asperities, the real area of contact can be extremely small comparing to the apparent area observed by our eyes and is very difficult to measure.This problem has been addressed and resolved for the first time by Archard, Greenwood and Williamson using novel fractal and statistical models to mathematically describe the microscopic surface structure.Their works have been the basis for various subsequent studies on contact mechanics, describing the surface geometry (asperities distribution, geometry) and material behavior (elastic, plastic flow) (Yastrebov et al. 2014).Recently, a deterministic approach to model rough surface contact has grown rapidly with the advance of computational capability, providing further insights to the study of contact mechanics.
The fact that only nominally flat rough surfaces case is focused has limited the scope of this model.One reason for this shortage was given by Greenwood and Trip, as generally the curvatures difference of curved surfaces creates a cluster effect which makes asperities interaction becomes significant.Thus an intensive analysis similar to those performed in the nominally flat rough surfaces contact is not frequently performed in the case of rough curved surfaces contact.Rather, the latter in only loosely studied through the inspection of axial contact between two rough curved surfaces having constant curvatures, by replacing them with a nominally flat rough surface and a smooth curved surface having anequivalentcurvature (Johnson 1985).This method although gives a quick approximation of pressure distribution, it does not allow one to account for: • More general analysis, such as the contact is non-axial or the surfaces have varying curvatures ___________________________________ *Corresponding Author: balzahab@kettering.edu • More detailed analysis, such as the true distribution of contact pressure which is important to the calculation of cumulative frictional torque in rotating parts.
This paper studies the contact of general rough curved surfaces having nearly identical geometries, assuming that the contact at each differential area obeys the model proposed by Greenwood and Williamson (GW model for short).In order to account for the most general geometry, principles of differential geometry of surface are applied.This method requires more rigorous mathematical manipulations, as it preserves the original surface geometries (i.e.not require the original system to be replaced by any equivalent system) makes the modeling procedure much more intuitive.For subsequent use, differential geometry of axis-symmetric surface is considered instead of general surface (although this "general case" can be done as well) in Chapter 3.1.The final formulas for contact area, load, and frictional torque are derived.
One direct application of this study is the analysis of roughness-dependent frictional torque occuring rotating parts, whose geometry is often axis-symmetric.For flat surfaces contact (i.e. two flat surfaces slide across each other), effect of friction is generally quantified with the calculation of frictional force value.Similarly, for curved surface contact (e.g. in journal bearing), the value of frictional torque is frequently required.Unlike the former situation, where surface roughness does not affect the frictional force if one uses the Coulomb's friction model (since the total reaction force at the points of asperity contact always equal to the load), surface roughness changes the distribution of contact pressure across the curved surfaces (even when Coulomb's model holds), thus frictional torque value would be different.Furthermore, the frictional torque will not vary linearly with the load like when one models contacting surfaces smooth, rather it will also be dependent on the roughness.This topic is clarified through two specific examples.Lastly, additional analysis on the load -contact area and frictional torque -load relationship is presented.

MODELING ROUGH SURFACE CONTACT:
Greenwood and Williamson (1966) proposed a method to mathematically model the stochastic nature of surface's microscopic structure by using a probabilistic approach, which introduce the concept of "asperity" and consider their height to be normally distributed over the entire rough surface.In practice, such statement is valid for most high-end engineering surfaces (i.e.homogeneous, isotropic surface), yet not quite so for other lower-end ones (Bhushan 2001).For the latter situation, Kotwal and Bhushan (1996) have developed an analytical method to generate probability density functions of non-Gaussian distributions, but will not be considered in this study.One key assumption in the work of Greenwood and Williamson is that each individual contact does not affect the deformation of its neighbors and the asperity is spherical with curvature 0 ℜ at its peak (Figure 1).This conveniently allows the each individual contact to be modeled independently by implementing the Hertzian theory.As mentioned previously, this method limits the contact to only be between nominally flat rough surfaces, so that the above assumption holds.Consider two rough surfaces in contact which could be replaced by a system of two other surfaces with equivalent asperities' curvature and RMS roughness parameter.The first surface is perfectly smooth and is located at some distance 0 l from the reference line 0 h .
The second surface is considered rough with the asperities' height z varies randomly around the reference line which is described by the Gaussian distribution (Skewness = 0 and Kurtosis = 3) (Figure 2): where q R is the RMS roughness parameter.
If the total number of asperities on this surface is 0 N , the number of asperities having height in the interval [ , ] and thus the total number of asperities in contact is Furthermore, according to the Hertzian contact theory, when a elastic sphere is indented to depth 0 z l − (from now refer as the indentation depth) in an elastic half-space: The contact area is and the required force is with * E is the equivalent modulus of elasticity and can be found using , , , E E ν ν are the moduli of elasticity and Poisson's ratios of the two bodies) (Figure 3).
Noting that if a thin layer of coating is present, according to Liu et al (2005) a different equivalent modulus of elasticity should be used.Very recently, the exact solutions for these integrals have been found in Jackson and Green (2011): , then: where:

NEARLY-IDENTICAL ROUGH CURVED SURFACES CONTACT:
Since the two curved surfaces considered in this study are nearly-identical, contact at each infinitesimal area could be treated as the contact between two nominally flat rough surface, which can then be integrated to describe the overall contact behavior between gross geometries.This approach requires the number of asperities and the indentation depth at each differential surface contact to be found.Furthermore, the magnitude, direction and location of application of contact pressure (i.e.define a bound vector) at each individual asperity as well as corresponding differential surface area should also be stated.What is known is the applied load, geometry and material properties of considered surfaces, and therefore any expression should be written in terms of these given parameters.

Differential geometry of surfaces: i) Vector formalism of line in 3D space:
It is very convenient to express a general bound vector in 3D space using vector formulation.A bound vector is completely defined if its initial point, magnitude and direction are specified.In this problem, three quantities need to be expressed vectorially are the asperity's direction, the reaction force and the friction force.Consider a straight line L is defined by two parameters ( , ) P x l (Figure 4).An arbitrary point Q on the line has position vector Q x is given by Q p x x λ = + l where λ is an arbitrary scalar.
We are also interested in he point where an asperity comes into contact: a point I is the intersection of two line L1 ( , ) x = l has position vector I x given by: ( , ) p p ξ ξ = .In orthonormal coordinates the surface of revolution can always be expressed as (Gray 1997): The unit normal vector n of the surface is: For a surface of revolution, it is natural to pick 1 2 , r ξ ξ φ = = (Figure 6)., from Eq. ( 5) any point on this surface has the position vector: with ( ) r r ϕ = and ( ) From Eq. ( 6) and Eq. ( 7) , the corresponding unit normal vector n and the corresponding area of a differential surface element dA is: , the (apparent) area of this surface given by Anton (1999) is:   ( , , ) j j j ℜ = (Figure 7).
n r φ is normal vector of the smooth surface S2 at point 2 I (resolved in 2 ℜ ).From Eq. ( 9) and Eq. ( 10): n , 2 n represent the height, the direction of an individual asperity and the direction of reaction force respectively (Figure 8).I can then be derived from Eq. ( 4) and Eq. ( 8): ( , ) r φ φ into Eq.( 13), we could express 2 n in terms of 1 r and 1 φ .The indentation depth of an individual asperity can be found as: 9).
In a differential area, the asperities can be assumed to be unidirectional (i.e.having the same direction vector).If the asperities density is 0 ℵ , The number of asperities in that differential area is . Since the asperities' height is normally distributed described by the Gaussian distribution, the number of asperities on a differential area that height in the interval [ ] In terms of contact pressure distribution: In terms of contact pressure distribution: Furthermore, the position vector of a single point of contact 1 I and the moment arm (Figure 10) respectively are: Finally, we attain the expressions for the number of asperities, real contact area, reaction force components and frictional torque in terms of an individual asperity, a differential surface area and the entire contacting surfaces:

Real contact area
Single asperity

Reaction force (x -component)
Single asperity

Reaction force (y -component)
Single asperity

Reaction force (z -component)
Single asperity

Frictional torque
Single asperity Eq. ( 20) shows the number c N of asperities that are in contact.Eq. ( 21) can be solved using Eq. ( 3) that approximates the real contact area.Eq. ( 22), ( 23), (24) yield the components of the cumulative reaction force which according to Newton's third law have to be equal to the components of the applied load.Finally, Eq. ( 25) gives the expression for the cumulative frictional torque.Consider two concentric spherical surfaces (Figure 12) created by the revolution of: Substitute Eq. ( 1), ( 26), (32), (34) into Eq.(25), we attain the frictional torque expression: where: I can be solved using Eq. ( 2))

ANALYSIS:
However, the eccentricity ( e ) is hard to measure without special instruments.Thus it is more convenient to consider the frictional torque -load relationship.By eliminating e using Eq.
, then from Eq.(24) and Eq. ( 25 Eq. ( 53) is exactly the formula for the frictional torque -applied load relationship if the contacting surfaces are modeled as smooth that is obtained in several studies such as the one from Grégory (2014).
Considering the example in Section 4.2 where no additional geometric assumption is made.First, the consistency between theories of contact mechanics should be taken into account.In the early days, Hertz set the foundation of contact mechanics by analytically predicting the compressive force required to indent a smooth sphere into a infinite smooth half-space, which was then broadened to account for other shapes.According to Sneddon 1965, in the case of parallel-axis cylinders contact, the applied force as function of indentation depth Thus Eq. ( 48) should be identical to Eq. (54) as 0 = q R since both describe contact in the "smooth" case.Taking the limit of Eq. (48): Noting that the Gaussian distribution function becomes the Dirac delta function as 0 → q R , which has a special property: )) ( sin ) (cos( )) cos( ( The analytical solution for this definite integral could be found using Wolfram Alpha: Next, using Eq. ( 2), (3) we could analytically evaluate the first integrations in Eq. ( 47), (48), (49).Based on these expressions, a MATLAB script is written to numerically evaluate Eq. ( 47), (48) and (49) at increments of ε and q R .The integration command "integral" uses global adaptive quadrature method "integral" with absolute error tolerance of 1e-10 (Shampine, 2008).Graphs 1 to Graph 6 show the relationships between contact area, applied force, frictional torque with eccentricity and roughness.Graph 7 indicates that even though the contact areaapplied load relationship might be linear at a particular roughness, the contact area increases faster with applied load as roughness goes up.We even observe this behavior more clearly in the case of frictional torque -applied load relationship.

Figure 1 .
Figure 1.GW model of a single asperity

Figure 3 .
Figure 3.Contact between a elastic sphere and elastic half-space ⋅ and ( ) K ⋅ are the modified Bessel function of the first & second kind respectively ( ) erfc ⋅ is the complimentary error function ( ) Γ ⋅ is the gamma function

Figure 6 .
Figure 6.A general surface of revolution

11 ) 3 . 2 . 1 O
Calculation of contact area, contact pressure and frictional torque: Consider a surface of revolution S1 with vertex is at the origin of frame of revolution S2 with vertex 2 O is at the origin of frame

1 2 3 ( 2 O
, , ) i i i can always be chosen such that the position vector of with respect to 1 ℜ is the eccentricity vector surface is axis-symmetric and the asperities are randomly distributed, the eccentricity vector is assumed to be parallel to the load a known vector.

Figure 7 . 3 Consider a differential area at point 1 I
Figure 7. Contacting surfaces of revolution P is any point on the line 1 2 I I .Physically, δ , 1

Figure 8 .
Figure 8. Representation of several surface parameters

Figure 9 .
Figure 9. Contact of a single asperity

Figure 10 .
Figure 10.Representation of moment arm

Figure 11 .
Figure 11.Schematic of a concentric spherical annulus

(F
22) to Eq. (25), we can find: and µ are the applied load and the friction coefficient q R is RMS roughness parameter i G 's are the surfaces geometric parameters frictional torque is linearly dependent on the applied load value.values become closer if more assumptions about the geometries are made.For example, in Section IV.1, if we assume

F
and ( ) ⋅ E are the incomplete elliptic integrals of first and second kind Since it is required that

Graph 1 .
Contact area as function of eccentricity and roughness Graph 2. Contact area as function of eccentricity at various roughness Graph 3. Applied force as function of eccentricity and roughness Graph 4. Applied force as function of eccentricity at various roughness Graph 5. Frictional torque as function of eccentricity and roughness Graph 6. Frictional torque as function of eccentricity at various roughness Graph 7. Contact area as function of applied force at various roughness Graph 8.Frictional torque as function of applied force at various roughness

Table 1 .
Expressions for the number of asperities, real contact area, reaction force components and frictional torque