# Rough Surface Contact Modelling—A Review

## Abstract

**:**

## 1. Introduction

^{2/3}, where W is the applied load. However, in practice, a rough surface has a distribution of asperity heights, and a more detailed analysis by Greenwood and Williamson [17] of rough, non-adhering surfaces as a collection of asperities of varying height, with spherical tips, deforming elastically, found a linear relationship between the real contact area and load. This initial analysis by Greenwood and Williamson has since been refined and extended by many other researchers [18,19,20,21,22,23,24,25,26,27]. The analysis of Greenwood and Williamson [17] was substantially aided by earlier useful insights from Archard and others [28,29,30,31,32,33]. In particular, in reference [30], Archard showed that if there is one level of “protuberances”, of radius R

_{2}, superimposed on a spherical tip of radius R

_{1}(with R

_{2}<< R

_{1}), then elastic deformation leads to the real area of contact being proportional to W

^{8/9}(and that if a further level of “protuberances” are added, of radius R

_{3}, with R

_{3}<< R

_{2}, then the real contact area is proportional to W

^{26/27}). Archard’s work [30] showed that the real area of contact can be almost proportional to the load even when there is elastic contact only, for multi-asperity contact. It should also be mentioned that Archard’s concept of protuberances on protuberances on protuberances clearly has a link to fractals, where multiple length scales are important, but at the time of Archard’s work, fractals were not yet known about, as they were only introduced in the 1970s [34]. The consensus of these various studies is that, in the elastic contact case, the real contact area is approximately proportional to the applied load, and therefore Amontons’ first law of friction follows. Persson [8] has reported that the real contact area is also approximately proportional to the applied load even when there is adhesive interaction between surfaces.

## 2. The Friction and Load Carrying Capacity of Lubricated Rough-Surface Contact

_{TOTAL}, this can be split between the load carried by the asperities W

_{asp}and the load carried by the fluid film W

_{fluid}as:

_{TOTAL}, of the contact can be written as

_{asp}is the friction coefficient of the asperities (i.e., when λ = 0) and μ

_{fluid}is the friction coefficient of the fluid (i.e., when λ > 3). The friction coefficient, f, of the contact can then be written as f = F

_{TOTAL}/W

_{TOTAL}, which is equal to

_{asp}/W

_{TOTAL}, which can be regarded as the proportion of mixed/boundary lubrication. Clearly, it is expected that X = 1 when λ = 0 and X is approximately zero when λ > 3. It is of great interest, for the modelling of mixed/boundary friction, to know how X varies with λ, and it is also of experimental interest to measure this variation.

^{−2}.

## 3. The Statistics of Rough Surfaces

_{o}, m

_{2}, and m

_{4}, where m

_{o}is related to the RMS roughness of the surface (σ), m

_{2}is related to the RMS surface gradient, and m

_{4}is related to the RMS radius of the curvature of the rough peaks.

_{o}(the RMS roughness), m

_{2}(the RMS surface slope) and m

_{4}(the RMS of the radius of the curvature of the asperities). As the size of the measuring stylus decreases, the curvature of the asperities, and their slope, increase (i.e., the asperity radius of the curvature decreases) although the RMS roughness stays relatively constant [57].

_{o}is a constant. If β = 0, then C(β) = σ

^{2}as required.

^{2}for a wide range of surfaces spanning length scales differing by eight orders of magnitude. The factor β

_{o}that appears in the equations above can be regarded as the length scale of the measurement stylus, and so the values of m

_{o}, m

_{2}and m

_{4}will vary depending on the value of β

_{o}. It could be argued that in calculating the moments of the spectral density, rather than integrating between 0 and ∞, one should instead integrate between 2π/L

_{1}and 2π/L

_{2}, where L

_{1}is the size of the sample (typically a few mm) and L

_{2}is the size of the stylus (typically a few μm). It should be pointed out that the higher moments of the spectral density function do not exist for the spectral density function of Equation (9) (if the integral limits are 0 and ∞). The relationships between m

_{o}, m

_{2}, m

_{4}and the spectral density function are

## 4. Rough-Surface contact Models

#### 4.1. Bowden and Tabor’s Model

^{2}). If a critical shear stress, τ

_{s}, was needed to shear the asperity junctions, then the frictional force would be (τ

_{s}W/H), and so friction is proportional to normal load. A useful retrospective look-back at this work was published by Tabor in 1996 [64].

#### 4.2. Archard’s Work

_{1}, contacting a smooth surface under a load W, there will be a single circular area, A

_{1}, of radius b, where

_{1}) was covered in smaller asperities, the radii of which were R

_{2}(where R

_{2}<< R

_{1}), which were evenly distributed over the larger asperity with a number m per unit area. The annulus load above was assumed to be unchanged, with the load δW supported by q “protuberances” (with q = 2πrmδr) and that the load w

_{2}on each one being given by

_{2}of each of these contacts is

_{2}is the constant of proportionality appropriate to a sphere of radius R

_{2}, and the area of contact within the annulus is qa

_{2}, such that the total contact area, A

_{2}, is

^{8/9}, whilst the area of a single asperity is proportional to W

^{2/3}. Archard [30] showed in the same paper, using a similar analysis to that above, that the addition of a third level of asperities (with radius R

_{3}, where R

_{3}<< R

_{2}<< R

_{1}) results in a total contact area that is proportional to W

^{26/27}.

#### 4.3. The Greenwood–Williamson and Greenwood–Tripp Models

^{1/2}, and the Hertzian load, P, is P = 4E*R

^{1/2}δ

^{3/2}/3 (where E* is the effective combined elastic modulus of the surfaces). Therefore, the total (real) contact area, A

_{r}, is given by:

_{r}) is directly proportional to the load (P), and in addition, the number of contact points (N) is also directly proportional to the load. Greenwood and Williamson [17] pointed out that an exponential distribution of summit heights is a good approximation to a Gaussian distribution for the important range from z = σ to z = 3σ.

_{n}(λ) for the case where 2n is an integer (which applies to the interesting cases of n = 3/2 and n = 5/2):

#### 4.4. Bush, Gibson, Thomas Model

_{o}, m

_{2}and m

_{4}in their work, and found a linear relationship between the load and the real contact area. They also found that the real contact area, A

_{r}, was related to λ by

#### 4.5. Models Due to Persson

_{o}is a constant. Persson [75] commented that earlier theories, based on elastic contact, predicted that p(u)~u

^{−a}exp(−bu

^{2}), where a = 1 for the Bush, Gibson and Thomas model [19], and a = 5/2 for the Greenwood and Williamson model [17]. Persson [75] also stated that the functional variation of pressure with the separation of Equation (34) was in better agreement with experimental data than the variation of pressure with separation from the Greenwood and Williamson [17] or Bush, Gibson and Thomas [19] models. It should be mentioned that Persson’s models [22,23,75] are not thought to be widely used by the average tribologist for the calculation of mixed/boundary friction, as they are generally seen as being difficult to apply, and no simple equations are available for the prediction of the variation of the real contact area and load carried as a function of surface separation (such a relationship is most likely the reason why the Greenwood and Williamson [17] and Greenwood and Tripp [21] models are still so widely used).

#### 4.6. Recent Work on Rough-Surface Contact Models

## 5. The Impact of Lubricant Additives

_{2}, graphite, glycerol mono-oleate, and oleyl amide are well-known examples.

## 6. Experimental Data on Rough-Surface Friction

_{o}was the friction coefficient measured at λ = 0, and f

_{EHD}was the friction coefficient measured at large values of λ, then the value of X was defined to be (f − f

_{EHD})/(f

_{o}− f

_{EHD}), where f was the measured friction coefficient at a particular value of λ. Figure 1 shows measured data from reference [126]. It was pointed out that if the correct value of surface roughness was used to calculate λ, then all of the lubricants would be found to fit onto a universal curve. It should be pointed out that although the RMS roughness of the MTM ball and disk were only about 4 nm each, when lubricants containing ZDDP anti-wear additives were used, relatively thick tribo-films were formed (between 100–200 nm), and the RMS surface roughness at the top of the tribo-films was significantly higher, at around 40 nm. λ was estimated by calculating the elastohydrodynamic oil film thickness in the contact (the radius of the curvature of the ball, the elastic moduli, the loads, speeds, temperatures were known, and the viscosity and pressure–viscosity coefficient of the lubricant were calculated).

## 7. Application of Rough-Surface Models to the Prediction of Mixed/Boundary Friction

## 8. Discussion

## 9. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Graph showing the measured value of X (the proportion of mixed/boundary friction) versus λ for two ZDDP-containing lubricants, as measured in the Mini-Traction Machine (the specific operating conditions are reported in [121]).

λ | F_{0}(λ) | F_{1/2}(λ) | F_{1}(λ) | F_{3/2}(λ) | F_{2}(λ) | F_{5/2}(λ) |
---|---|---|---|---|---|---|

0.0 | 0.50000 | 0.41109 | 0.39894 | 0.43002 | 0.50000 | 0.61664 |

0.5 | 0.30854 | 0.22534 | 0.19780 | 0.19520 | 0.20964 | 0.24024 |

1.0 | 0.15865 | 0.10415 | 0.08332 | 0.07567 | 0.07534 | 0.08056 |

1.5 | 0.06681 | 0.03988 | 0.02931 | 0.02464 | 0.02285 | 0.02286 |

2.0 | 0.02275 | 0.01248 | 0.00849 | 0.00665 | 0.00577 | 0.00542 |

2.5 | 0.00621 | 0.00316 | 0.00200 | 0.00147 | 0.00120 | 0.00106 |

3.0 | 0.00135 | 0.00064 | 0.00038 | 0.00026 | 0.00020 | 0.00017 |

3.5 | 0.00023 | 0.00010 | 0.00006 | 0.00004 | 0.00003 | 0.00002 |

4.0 | 0.00003 | 0.00001 | 0.00001 | 0.00000 | 0.00000 | 0.00000 |

X | ||||||
---|---|---|---|---|---|---|

λ | $X=\frac{1}{{\left(1+{\lambda}^{k}\right)}^{a}}$ k = 3/2, a ≈ 4/3 | exp(−λ) | Linear fit (Equation (38)) | Olver and Spikes [44] (Equation (36)) | Greenwood-Tripp [21] | Bush, Gibson, Thomas [19] |

3 | 0.0879 | 0.0498 | 0 | 0.0625 | 0.00028 | 0.0027 |

2 | 0.167 | 0.135 | 0.333 | 0.111 | 0.0088 | 0.0455 |

1 | 0.397 | 0.368 | 0.667 | 0.25 | 0.131 | 0.317 |

0.5 | 0.668 | 0.607 | 0.833 | 0.444 | 0.390 | 0.617 |

0.2 | 0.892 | 0.819 | 0.933 | 0.694 | 0.698 | 0.895 |

0.1 | 0.959 | 0.905 | 0.967 | 0.826 | 0.838 | 0.920 |

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Rough Surface Contact Modelling—A Review. *Lubricants* **2022**, *10*, 98.
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Rough Surface Contact Modelling—A Review. *Lubricants*. 2022; 10(5):98.
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2022. "Rough Surface Contact Modelling—A Review" *Lubricants* 10, no. 5: 98.
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