# Quasi-Static Sliding Wear Analysis of 3D Rough Surface Considering Changes in the Point of Contact
## 2 Theoretical Background  
### Problem Description and Hypotheses  
Figure 1 shows the periodic sliding motion between two objects with apparently flat contact surfaces. In this case, it is assumed that the sliding speed is constant and the load acts in the direction perpendicular to the contact surface. In a sliding motion such as that in Figure 1, the contact surface of the upper moving body with length \(L_{M}\) in the sliding direction is not changed, whereas the contact surface of the lower stationary body with length \(L_{S}\) in the sliding direction varies over time. The contact surfaces are called the upper moving surface and the lower stationary surface. Since a very dense grid is required to represent the rough surface asperities, the analysis domain must be carefully set. Depending on the object, the area of the contact surface varies from micro-scale to metric-scale. The larger the contact surface area, the longer the analysis time and the higher the computer cost. On the other hand, reducing the number of nodes increases the sampling interval, making it impossible to accurately represent rough surfaces. That is, the analysis domain should be selected as an area that minimizes the computational burden while having a grid size that sufficiently represents the roughness of the surface including numerous asperities. Therefore, the analysis domain is limited to a very small area at the level where the periodicity of the rough surface is held among the total contact area. In order to select such an appropriate analysis domain, it is necessary to establish a standard. Parameters representing rough surfaces include statistical parameters for height distribution and autocorrelation function that provides information about spatial distribution. The autocorrelation function (ACF) represents the spatial correlation of the rough surface height and is expressed as the following equation [33]. \[R(\Delta x,\ \Delta y)=\mathbb{E}\{z(x,y)z(x+\Delta x,y+\Delta y)\} \tag{1}\]  
where \(\mathbb{E}\) is the expectation operator and \(z\) is the rough surface height. Equation (1) means that the autocorrelation function indicates the degree of correlation between the height at an arbitrary point and the height at a point at a distance of \(\Delta x\) in the \(x\) direction and \(\Delta y\) in the \(y\) direction. It is known that most autocorrelation functions of engineering surfaces have the form of exponential functions [34]. An important parameter for the autocorrelation function is the autocorrelation length \(\beta\), which is mainly defined as the length to the point at which the value of the autocorrelation function is 0.1 [33]. For a numerical generation of rough surfaces, it is assumed that the autocorrelation function has a value of zero after a certain length, where this length is the truncation length [35]. In a past study on rough surface generation, the truncation length was arbitrarily determined, but in [PERSON]'s work, it was reported that the truncation length should be 6 or more times the value of \(\beta\) to obtain an accurate surface without losing statistical information [35]. Therefore, since the minimum surface length for given statistical and spatial information is equal to the truncation length, in this study, we set the truncation length to be more than 10 times \(\beta\) and the length of the analysis domain \(L\) to be twice the truncation length to ensure a sufficient surface grid. The most significant feature of the analysis method we used in this study was that, in the model, we took into account that the point of contact changes according to the relative motion of the asperities. This phenomenon was implemented by the direct movement of the data. As shown in Figure 2, in this process, the data of the upper moving surface are stacked on top of the data of the lower stationary surface, and the data of the upper moving surface are shifted by \(\delta_{\rm s}\) in the sliding direction to consider the movement of asperities.