Ficha de Observación

6.5

The relationship analysis between friction,

S

and

S

For description of surface topography, the roughness parametersSa andSq do not

capture all the information. Very different surface profiles can show similar or identical values of standard roughness parameters. The opposite is also possible, very similar surfaces may end up having very different roughness parameters. Sa describes

the overall height variations, without any details on waviness and it is not sensitive to small changes in profile height. Compared with Sa,Sq is more sensitive to the

variance of height, while it does not pick up detailed information on the surface roughness.

Skewness and kurtosis are the third and fourth moment of the density function, which are used to characterise asymmetry and the flatness of the surface distribution. For engineer surfaces that have non-Gaussian surface topography, Ssk gives the

skewness on the distribution and is sensitive to occasional deep valleys or high peaks. Zero skewness by definition appears in symmetrical height distributions, while positive skewness describes surfaces with high peaks or filled valleys, and negative skewness describe surfaces with deep scratches or lack of peaks. Sku is defined as

kurtosis which describes the sharpness of the probability density distribution of the height profile. Sku is less than 3 when surfaces have relatively few high peaks and

low valleys, whileSku is more than 3 when the surfaces have relatively high number

of high peaks and low valleys [67,78].

©2012 SAGE Publications, reprinted from Sedlaˇcek, Podgornik and J.Viˇzintin [67]

Figure 6.16: Surfaces with various skewness and kurtosis values

6.5. The relationship analysis between friction, Ssk and Sku

The probability density functionPh defines the probability of locating a point at a

height h. Denote ¯h to be the mean height (the first moment of height). This is not to be confused with Sa, which is the L1-norm of height, i.e. the averaged absolute

height values about the mean. Sa is the first moment only if the heights are strictly

positive. Sq is the second moment of height. Fig. 6.16shows the effect ofSsk and

Sku on the surface roughness [73].

hn= Z ∝ −∝ (h₋¯h)nPhdh (6.12) Rsk= 1 σ3 Z ∝ −∝ (h₋¯h)nPhdh (6.13) Rku= 1 σ4 Z ∝ −∝ (h₋¯h)nPhdh (6.14)

whereσ is theSq of the surface and the standard deviation of Ph. For a Gaussian

distribution, the skewness is zero and the kurtosis is 3. Sedlaˇcek, Podgornik and J.Viˇzintin [67] reported that for dry sliding, positive Sskvalues lead to a greater real

contact area and a large number of peaks in the contact, as well as the tangential and adhesion forces compared to a Gaussian distributed surface. It means that negative Rsk lead to lower friction. NegativeSsk values describe surfaces with deep scratches

or a lack of peaks; the real contact area decreases in this case[152]. For Sku greater

than 3, surfaces is filled with high peaks and low valleys. The real contact area will decrease due to partial contacts. The influence of Sskand Sku on dry sliding friction

can be used as a guideline for designing surface topography with reduced friction. Compared to the Gaussian distribution(Sku = 3, Ssk = 0),surfaces with a

highSku and a positive Ssk should result in a lower static friction coefficient during

dry contact [74]. When Sku increases from 2 to 10, the static friction decreases by

a factor of about 6 [75], mainly due to an increased contact area [76]. Under dry sliding higher values of parameter Sku and more negative values of Ssk led to lower

friction, indicating that deep valleys act as wear particle traps [152]. 6.5.1 The theoretical analysis

According to the GW contact model described in Section4.2.4, if the surface height distribution is known, the real contact area between surfaces can be calculated using the equations given. The probability density function (pdf) of the surface height defines the probability of locating a point at a height hand is denoted by Ph. The

6.5. The relationship analysis between friction, Ssk and Sku

the height distribution, using mn=

Z ∝

−∝

(h₋¯h)nPhdh (6.15)

where ¯his the mean height of h, which is generally removed during data processing and therefore is usually zero. As such, the first moment is zero. The second moment m2 is the variance σ2, which is the square of the standard deviation when ¯h = 0.

The third moment m3 is the skewness, which shows degree of symmetry of the

surface profiles. If the mean is on the left side of the distribution mode, the skewness will be negative with a relatively large numbers of peaks than valleys at a certain height. The skewness of Gaussian distribution surfaces is 0. The fourth moments m4 represents the peakedness(degree of pointedness or bluntness) of the distribution. A surface with low kurtosis has a relatively larger number of peaks than valleys at a certain height. The influence ofSsk and Sku to surface height probability as shown

in Figures6.17 and 6.18[73].

Figure 6.17: The influence of skewness to surface height probability distribution [73] As the skewness and kurtosis are the third and fourth moments of the probability density function, the curve fitting parameters can be determined in terms of the skewness and the kurtosis. Once the probability density function is determined, a contact model can be developed using the GW modelling approach [147]. Based on the classical theory of friction[153], the kinetic friction is proportional to the real area of contact which is higher for smoother surfaces [147].

The Pearson system of frequency curves, based on the methods of moments, provides a family of curves which can be used to generate an equation for a distribution for which the first four moments are known [154], i.e. the probability density function can be generated for a distribution having a known mean, standard deviation, skewness and kurtosis.

6.5. The relationship analysis between friction, Ssk and Sku

Figure 6.18: The influence of kurtosis to surface height probability distribution [73]

In Pearson’s curve fitting, κ is defined as the type of the height probability distribution in Eq. (6.16). Different values ofκdetermine different equations obtained for the probability density functions. The value of κ ranges from _∞ to_−∞, and depending on the range it calculated, the appropriate equation of the density function is obtained. The list of the different types of curves and the range ofκfor which they are applicable is shown [155]. There are three main types of Pearson curves which cover the majority of the cases. These are types I, IV and VI. The parameters for the different types of density functions depend solely on the skewness and kurtosis [75]. κ= Ssk 2_(S ku+ 3)2 4(2Sku−3Ssk2 −6)(4Sku−3Ssk2) (6.16) The value of κ determines the type of the curve. There are three main types of Pearson curves which cover the majority of the cases. According to the contact model described in Section 4.2.5, the probability distribution of the surface height can be determined. The parameters in the curves for the non-Gaussian probability density functions are functions of the standard deviation, skewness and the kurtosis. Once the non-Gaussian probability density functions are obtained in this manner, the calculated results can be substituted into the Greenwood-Williamson model.

Based on the classic GW contact model, the normalised contact area, norm- alised number of contacts and the normalised contact force can be calculated. The following figures show the simulated contact area and the friction results in [73] on one material. In order to better understand how Ssk andSku affect the contact and

the friction, simulation based on our samples was performed compared with the friction coefficient experiment results.