# Quasi-Static Sliding Wear Analysis of 3D Rough Surface Considering Changes in the Point of Contact
## 2 Theoretical Background
### Wear Model  
This section describes the wear analysis method using the truncation model and the wear analysis method using the deterministic model considering the change in the asperities contact points. First, as inferred from the name, the truncation model is a method of analyzing wear by cutting the asperities in contact with a rigid plane, and the analysis proceeds with the following procedure: (1) The contact between two rough surfaces is replaced by the contact between the rough surface and the rigid plane. (2) After detecting the contact asperities by the displacement of the rigid body and the surface profile, the contact pressure of the asperity is calculated using the Greenwood-Willamson statistical method based on the Hertzian contact theory. The method of calculating the contact pressure of the asperity is described in detail in Section 3.1. (3) The wear depth is obtained using the Archard wear model. (4) The average wear depth for the entire surface is calculated, and the height of the nodes in contact is replaced with the value of maximum surface height minus average wear depth. The truncation model is one of the methods currently widely used in wear analysis due to its convenience. However, there is a disadvantage in that the initially formed contact points of the asperities do not change and the wear at the arbitrary location is overestimated.  
The quasi-static model supplements the shortcomings of the truncation model, considering the change in the point of contact and the pressure distribution of the asperity as the surface slides. This makes it possible to perform the wear analysis on each of the moving and stationary surfaces separately, which means that the wear analysis is possible when the physical properties and wear resistance of the two surfaces are different. The wear depth at each node in contact is calculated with the modified Archard wear model, and the equation is:  
\[h_{w_{i}}(x,y)=\frac{k_{i}p(x,y)s_{i}}{H_{i}}\ (i=1,2) \tag{8}\]  
where \(h_{w}\) is the wear depth, \(k\) is the wear coefficient, \(p\) is the contact pressure, \(s\) is the sliding distance, \(H\) is the hardness, \(i=1\) is the upper moving surface, and \(i=2\) is the lower stationary surface. According to the Archard wear equation, since wear depth is highly dependent on contact pressure, the statistical contact analysis is replaced with a deterministic method to calculate the contact pressure between rough surfaces more accurately. The surface profile is updated by subtracting (removing) the wear depth of each node obtained through Equation (8) from the previous surface height. By averaging the pressure distributions calculated in each step, the average pressure for 1 cycle at each node is obtained and then used for the wear analysis. It should be noted that since the lower stationary surface is not in constant contact, the sliding distance is smaller than that of the upper moving surface, resulting in a relatively small amount of wear depth on the lower stationary surface. When the upper moving surface slides by the sliding distance \(S\), the lower stationary surface slides by the length of the upper moving body \(L_{M}\). Based on this, the wear depth is calculated by varying the sliding distance between the two surfaces. This wear analysis model was also used in this literature [21] to study the influence of the surface topography, load, and slide-to-roll ratio on wear, and the analysis results were verified with experiments.