# 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  
Before performing the wear analysis on the rough surface using the truncation model and the deterministic model introduced in Section 2.3, we compared the results on the sliding wear of a plane and one spherical asperity for a basic analysis between the models. The analysis results of the following three models were compared: The truncation model, the deterministic fixed model, and the deterministic quasi-static model. The deterministic model includes a fixed model that does not consider the change in the asperity contact point, such as the assumption of the truncation model, and a quasi-static model that considers the change in the asperity contact point. The truncation model and the deterministic fixed model differ only in the pressure calculation method. The deterministic model obtains the pressure at each node based on a deterministic method. DC-FFT was used as a numerical analysis technique for the non-periodic contact problem because CC-FT, which was used for the rough surface contact analysis, cannot be used in the case of a sphere contact. More details on this can be found in [42]. The truncation model is the simplest model that calculates the contact pressure based on the GW model and does not consider the change in the asperity contact point. In the truncation model, assuming that the tip of each asperity is spherical, the contact load \(F\) and contact area \(A\) were calculated according to the Hertzian contact theory with the following equation [43]. \[F=\left\{ \begin{array}{cc}\frac{4}{S}F^{\prime}R^{\prime 0.5}\omega^{ 1.5}&\omega<\omega_{c}\\ \pi R^{\prime}\omega\big{(}2-\frac{\omega_{c}}{\omega}\big{)}k_{ep}H&\omega> \omega_{c}\\ \pi R^{\prime}\omega\big{(}2-\frac{\omega_{c}}{\omega}\big{)}&\omega>\omega_{c }\end{array} \right. \tag{9}\]  
Here, \(E^{\prime}=1/\left(\left(1-\
u_{1}^{2}\right)/E_{1}+\left(1-\
u_{2}^{2}\right) /E_{2}\right)\) is the equivalent Young's modulus, \(R^{\prime}\) is the equivalent radius of asperity, \(\omega\) is the amount of interference between the rigid plane-surface, \(k_{ep}\) is the elastic-plastic constant, and if \(\omega\) is greater than \(\omega_{c}=\left(\frac{\pi k_{ep}H}{2E^{\prime}}\right)^{2}R^{\prime}\), the asperity is considered to be plastically deformed [43]. The radius of the asperity is determined by its geometry, and as the radius increases as the wear progresses, the radius must be newly estimated every cycle. After the radius of asperity based on the 2D profile in the \(x\) direction and \(y\) direction is calculated, the equivalent radius \(R^{\prime}=\sqrt{R^{\prime}_{x}R^{\prime}_{y}}\) is obtained from the two radii. To calculate \(R^{\prime}_{x}\), \(R^{\prime}_{y}\), a circle is fitted using the two intersection points of the rigid plane-surface profile and the highest point of the asperity. In the truncation model, since the asperities are cut horizontally, a flat area is quickly formed at the tip of the asperities. For the worn asperities, the midpoint of the flat area is set as the peak point. Figure 4 shows the wear process for 1000 cycles for a sphere with an equivalent elastic modulus of \(E^{\prime}=110\) GPa, [PERSON]'s ratio of \(\
u=0.3\), and an equivalent radius of  
\(R^{\prime}=300\) mm slides on a rigid plane under a normal load of 410 kN. The wear coefficient is \(k=5\times 10^{-5}\), and the sliding distance per cycle is \(S=1\) mm. As a result of contact analysis on the unworn surface, the maximum pressure was calculated as 2.2 GPa in all three models, which was consistent with the Hertz contact theory. In the truncation model and the fixed model, which do not consider the change in contact point, wear continues to occur at the same point. Since the gap between the two surfaces (equivalent surface) does not change during one cycle, the wear depth of the two models was compared based on the equivalent surface.