## 4 Fretting Wear Analysis Considering Internal Contact Pressure
### Result and Discussion  
As described in the Archard formula, the wear depth \(h_{w}\) is related to the wear coefficient \(k\), contact pressure \(P\), total cycle count \(\Delta N\), and sliding distance \(s\). Integrating Equation (23), the cumulative wear depth can be expressed as follows:  
\[h_{w}=k{\sum_{N_{l}}{P\cdot s\cdot\Delta N}} \tag{24}\]  
Under the established conditions of contact pressure and sliding distance, the interface wear depth is directly proportional to the cycle count. In the analysis, this equation can be employed to exert numerical control on the calculation of wear depth. To gain a deeper understanding of the morphological evolution of the worn surface, Figures 21(a) and 21(b), respectively, show the development of the wear interface and the variation in wear depth at different cycle counts. In order to maintain stable contact at the interface, the FEA does not account for the periodic variation in positive pressure during wire bending. Instead, positive pressure is controlled at 50 N, 100 N, and 200 N, facilitating the calculation of the wear condition at a specific contact point on the wire. It is not difficult \begin{table}
\begin{tabular}{c c c c} \hline \hline Wire radius \(R\) & mm & 3.5 \\ \hline Young modulus \(E\) & GPa & 210 \\ \hline Poisson’s ratio \(v\) & & 0.3 \\ \hline Positive pressure \(F\) & N & 50 & 100 & 200 \\ \hline Wear coefficient \(k\) & MPa\({}^{-1}\) & & \(8.2\times 10^{-8}\) \\ \hline Wire displacement \(s\) & \(\mu\)m & 100 \\ \hline Cycle number \(N_{l}\) & & & 100–700 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fetting wear coefficient. Figure 21: Finite element model for contact wear between two wires. to discern that wear depth and cycle count exhibit a positive correlation, and the overall wear rate demonstrates a progressively slowing trend. This phenomenon is attributed to the increasing contact area as wear develops, leading to a gradual \"averaging\" of wear effects at the contact interface. Consequently, the development of wear depth at various points gradually decelerates, as shown in Figure 22c,d. Figure 22c shows the variation in interface contact pressure with an applied load of 100 N as the cycle count increases. The peak contact pressure occurs at the center of the contact width. As the contact width increases during the wear process, the peak contact pressure decreases continuously, with the reduction becoming progressively smaller. It is expected that with the development of wear, the contact pressure distribution across the interface will become more uniform, resulting in minimal further changes to the numerical values. According to the Archard theory, under constant wear coefficient \(k\), cycle count \(\Delta N\), and sliding distance \(s\), the development of wear is primarily related to contact pressure. Therefore, the reduction in contact pressure disparity at different positions is the main reason for the \"averaging\" of interface wear. Figure 22d shows the variation in contact width with an increasing cycle count under loads of 50 N, 100 N, and 200 N. Under these three loads, the contact width gradually increases with an increasing cycle count, aligning with the results calculated using [PERSON]'s formula. The contact width experiences rapid growth in the initial few hundred sliding cycles, while the growth trend notably slows down after 500 cycles, consistent with the earlier analysis of contact area. The wear progression rate is primarily determined by contact pressure. Therefore, regions farther from the contact center exhibit slower wear progression and a smaller increase in contact width. The impact of load on the growth of contact width is significant. A higher load results in a larger contact width under the same cycle count. However, it is  
Figure 22: Finite element wear analysis: (**a**) evolution of contact surface profile; (**b**) evolution of wear depth; (**c**) evolution of contact pressure; (**d**) evolution of contact width. important to note that such positive correlation is not unlimited.