Biology 代写:Determine The Number Of Asperity Peaks

## 1.1 Models for real-contact-area calculations

One of the first real-contact-area models was proposed by Archard in 1957 [3]. He suggested that the asperities should be modelled on different scales. An asperity should have upon it a collection of smaller asperities, each of which supports a collection of even smaller asperities. For a rough, flat surface in contact with a smooth, rigid flat, he discovered that if multiple scale roughness is assumed, the correlation between the real contact area and the load is linear. In 1966 Greenwood and Williamson presented their contact model (GW model) to describe the conditions between a rough surface and an ideally flat, rigid plane, by considering the statistical nature of the surfaces [6]. The deformation theory behind the model is based on Hertz contact theory [12]. This model assumes that a rough surface consists of asperity peaks that have ideal spherical tips of the same radius. The asperity-peak height distribution (a statistical function), however, has to be defined in advance; Greenwood and Williamson assumed a Gaussian distribution of the asperity-peak heights as well as an exponential distribution. For the exponential distribution, the real contact area is directly proportional to the load and almost the same conclusion was drawn for the Gaussian distribution [6].

Several modifications to their model were proposed later [13-16]. Greenwood and Trip expanded the model in 1970 to the contact of two rough surfaces and concluded that the contact between two rough surfaces is not significantly different from the contact between a rough surface and a flat plane, although some modifications to the calculations are needed [13]. In 1975 Bush et al. upgraded the GW contact model (i.e., the BGT model). They still considered a statistical distribution of the asperity-peak heights, but assumed that the asperities are elliptical in shape [15]. In 2006 Greenwood additionally simplified the BGT contact model but obtained very similar results [14].

The above-mentioned real-contact-area models [6, 14-16] are considered to be ”statistical” and they require the asperity-peak properties (the asperity-peak density, the average asperity-peak radii, etc.) as an input parameter for the calculation. These properties are calculated from surface profiles with the use of statistical analyses of the surface profiles or topography. However, such analyses can never replicate the exact behaviour of the surface asperities; instead they just provide an average ”picture” of the surface, on which we base many assumptions and thus develop uncertainties and errors.

In 1971 Nayak introduced a random process model of rough surfaces [17]. He suggested that the power spectral density of an isotropic, Gaussian surface contains all the data required for a calculation of the real contact area. With the help of spectral moments, which can be calculated from the surface profiles, the density and radii of the surface asperity peaks can be estimated [9, 18, 19]. The main problem with this model is that the spectral moments are calculated from the measured surface profiles. Namely, as several authors reported [4, 9, 10, 19], the parameters calculated from the surface profiles or the surface topographies depend greatly on the surface-measuring technique, the instrument, its resolution and the use of filters. As a result, the asperity-peak properties and thus the real-contact-area calculations can vary significantly depending on the measuring instrumentation, the procedures and the post analyses.

On the other hand, the problem of surface-parameter dependence on the measuring instrument and the measuring procedures can be eliminated by the use of a fractal analysis, where the surface roughness becomes scale-independent and thus provides surface-roughness information regardless of the resolution and length scale. Such a model was presented by Majumdar and Bhushan [20, 21], and several other authors [22, 23]. However, these concepts were not adopted for deterministic contact models.

In recent years, with advancing computational power, numerical models with computer simulations are often used to calculate real contact areas [4, 24-27]. With these ”deterministic” models, the statistical functions for the asperity peaks on the surface are replaced with simple, but real, measured geometries. In this way the calculation does not depend on the statistical characterization and the typical ”averaging” of the surfaces. Nevertheless, these models still require input data about the real surface, which needs to be measured with surface-measuring instruments in 2D or 3D, and the level to which the measured surface data are considered as (relevant) micro-asperities must also be determined by using certain arbitrary criteria.

## 1.2 Asperity-peak identification for deterministic contact models

When using deterministic, real-contact-area models instead of, e.g., statistical models, each asperity on the surface can be identified and its height and radius can be calculated. However, the key issue is, how do we identify the ”relevant” asperities on the surface that actually influence the deformation, temperature and load-carrying properties? Namely, if an ”irrelevant” asperity is mistakenly identified, it becomes equal to a relevant one and the contact morphology changes significantly. So it is very important that only relevant asperity peaks are identified on the surface for a realistic evaluation. A few criteria on how to identify the relevant asperities on the surface exist in the literature [7-9]. These are described in more detail below.

## The 3-point peak criterion (3PP criterion)

In 1984 Greenwood suggested using a 3-point peak (3PP) criterion on a 2D surface profile [8]. An asperity peak is defined as a point that is higher than its two closest neighbours, as schematically shown in Figure 2. However, only asperity peaks above a profile mean line were taken into consideration. This is due to the fact that contacts between two rough surfaces are expected to occur on the highest asperity peaks, certainly above the profile mean line [6]. The valleys and the peaks above and below the profile mean line will thus have no effect on the real contact area, at least when the loads are in a realistic engineering range.

Figure 2: The 3-point peak (3PP) criterion with the presented height differences Î”z on a 2D profile.

The 3-point peak on a 2D profile must therefore satisfy the following criteria:

ziÂ >Â zi-1, zi+1,

with the additional condition

ziÂ >Â m.

The asperity-peak radius Î² can be calculated as the radius of a circumcircle through the peak point and its two closest neighbours, shown in Figure 2. Each individual asperity peak i, found according to the 3PP criterion, is thus characterized with an asperity-peak height zi and a radius Î²i.

Obviously, in these analyses, discrete points are used, which are separated from each other by a certain distance Î”x (see Figure 2). This is a distance corresponding to the profile measuring length L divided by the number of acquired discrete (x,z) data points in the profile, typically defined by the software of the measuring machine.

## The 5-point peak criterion (5PP criterion)

A 5-point peak is defined as a point that is higher than its four closest neighbour points (Figure 3b). Basically, it is the same as a 3PP, just that each asperity peak must have two lower neighbour points on each side. This method was rarely used in the literature [7] and its effects on an asperity-peak determination and consequently on the contact properties are thus even less clear than for the 3PP criterion.

## The 7-point peak criterion (7PP criterion)

A 7-point peak is defined as a point that is higher than its six closest neighbour points (Figure 3c). It is a variation of 3PP and 5PP, but with even more restrictive criteria. No such asperity-peak definition was found in the literature, but we introduce it to obtain a trend of the effect of the criterion that is restricted by the number of neighbouring points.

Figure 3 shows the difference between the 3PP, 5PP and 7PP criteria. It is clear that with an increasing number of neighbouring points for the asperity-peak definition, the asperity peaks become wider. These three criteria (3PP, 5PP and 7PP) basically determine the width of an asperity peak at the root of a so-identified asperity peak.

Figure 3: a) 3PP, b) 5PP and c) 7PP criteria on a 2D profile.

Biology 代写:Determine The Number Of Asperity Peaks