# Quasi-Static Sliding Wear Analysis of 3D Rough Surface Considering Changes in the Point of Contact
## 3 Result and Discussion
### Comparison between Three Wear Models
After 1000 cycles, the wear depth at the peak of the sphere was 68.3 \(\upmu\)m in the truncation model and 65.9 \(\upmu\)m in the fixed model, and the truncation model's prediction is 3.6% greater. In the fixed model, the surface profile itself is used for calculating the pressure, but in the truncated model, it is a cause of error to assume that the tip of the worn asperity is a sphere. Since the point of the flat area is used for fitting in the step of assuming the worn asperity as a sphere, the profile of the fitted asperity becomes a circle inscribed with the real profile of the worn asperity. Therefore, comparing the volume of the fitted sphere with the volume of the actual asperity with worn tips, the former is smaller. This results in the radius being estimated to be small in the analysis, and the contact area is smaller than the real one. Therefore, in the truncation model, the maximum pressure is greater than in the fixed model, which results in greater wear depth. On the other hand, in the quasi-static model, since the gap between the two surfaces (corresponding surface) changes as the upper sliding surface moves, it is difficult to represent the wear analysis result with only the equivalent surface. Figure 4c shows the wear shapes of body 1 and body 2, and the sum of the wear depths of body 1 and body 2 is up to 42.0 \(\upmu\)m. Body 2 wears over a wide area because the contact position changes. Figure 5 shows the results of the wear analysis by applying a truncation model, a fixed model, and a quasi-static model to an actual rough surface with numerous asperities. As described in Section 2.1, a part of the total contact area of the two objects is set as the analysis area, which is a square with a side length of \(L=0.889\) mm, and this analysis area is subjected to a vertical load of \(1N\). The number of nodes is \(n_{x}\times n_{y}=257\times 257\). Under the conditions in which an upper moving body with a length of \(L_{M}=3.6\) mm slides with a sliding distance of \(S=1.8\) mm per cycle, the wear analysis was performed for 200 cycles on two objects with a Gaussian surface, with the same material as the elastic modulus of \(E_{1}=E_{2}=200\) GPa, [PERSON]'s ratio of \(v_{1}=v_{2}=0.3\), and wear coefficient of \(k=5\times 10^{-5}\). Figure 5a-c show the 2D profile change in the gap between the surface pairs analyzed by the truncation model, the fixed model, and the quasi-static model. In the truncation model, the worn parts in each asperity all have the same height and the wear progresses faster than in the other two models since the wear depth is predicted to be large. The analysis model with the slowest wear progress was the quasi-static model, and although the wear depth was small, more peaks were worn than with the fixed model. In the quasi-static model, the  
Figure 4: Wear profile for sliding wear of a single asperity with an equivalent elastic modulus of \(E^{\prime}=110\) GPa, Poisson’s ratio \(v=0.3\), and an equivalent radius of \(R^{\prime}=300\) mm under a normal load of 410 kN: (**a**) Truncation model; (**b**) deterministic fixed model; (**c**) deterministic quasi-static model. gap continues to change as the upper moving surface slides, and as a result, more asperities were worn than in the fixed model. Moreover, even if the asperity is in contact at any step, it does not mean that it is in contact throughout one cycle, so the pressure is distributed over several asperities. On the other hand, in the fixed model, since the point of contact does not change and it is considered to be in continuous contact at that location during the cycle, wear occurs intensively at that point. Figure 6 shows the change in RMS roughness of the Gaussian surface from that in Figure 5 during 400 cycles. RMS roughness decreases as the peak is lowered by wear, and the decrease in RMS roughness is the largest in the truncation model. Analysis using the truncation model was performed for 210 cycles, and the reason will be explained later.