# Effect of Residual Stress in Surface Layer on Plastic Yield Inception
## 3 Results
### Effect of Residual Stress on the Locations of Yield Inception
As the tensile residual stress continues to increase, the yield inception transfers to location d in Figure 8a, at the axis of symmetry in the surface layer. Generally, the location is transferred from a to b to c t o d (Figure 8a) when the dimensionless residual stress changes from \(-1\) to \(1\). For a relatively thin surface layer (\(t/R=0.0005\), \(0.001\), and \(0.002\)), the rule for the residual stress at the locations of the yield inception is similar. The contour plots of the von Mises stress during the yield inception of the points labeled 9-16 in Figure 5b for \(t/R=0.008\) are presented in Table 3b. The evolution rule of plastic inception location on change of the residual stress is summarized in Figure 8b. When the points are on the left part of Figure 5b, where the dimensionless critical yield load increases with the increase in the dimensionless residual stress, the yield inception occurs at location e in Figure 8b, which is a circular region near the surface center about the axis of symmetry. When the points are on the central part (plateau) of Figure 5b, the onset yield occurs at location f in Figure 8b, at the axis of symmetry, and the substrate side  
Figure 8: Schematic description of the typical locations (labeled a–g) of yield inception and evolution rule as the residual stress varied from compressive to tensile for different surface layer thicknesses. (**a**) \(t/R=0.001\). (**b**) \(t/R=0.008\). (**c**) \(t/R=0.0064\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**(c)**} \\ \hline
**Point (\(t/R=0.064\))** & **17** & **18** & **19** & **20** & **21** & **22** & **23** & **24** \\ \hline Dimensionless Residual stress & \(-0.857\) & \(-0.785\) & \(-0.715\) & \(-0.628\) & \(-0.572\) & \(-0.285\) & \(0.143\) & \(0.570\) \\ Contour plot & & & & & & & & \\ Location of yield inception & h & h & h & h & i & i & i & i \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._of the surface layer/substrate interface. When the points are on the right side of Figure 5b, where the dimensionless critical yield load decreases with the increase in the dimensionless residual stress, the onset yield occurs at location g in Figure 8b, at the axis of symmetry in the surface layer. Generally, the location is transferred from e to f to g (Figure 8b) when the dimensionless residual stress changes from \(-1\) to 1. For a medium-thick surface layer (\(t/R=0.006\), 0.008, 0.01, and 0.012), the rule for the residual stress at the locations of the yield inception is similar. The contour plots of the von Mises stress during the plastic inception of the points labeled 17-24 in Figure 5c for \(t/R=0.064\) are presented in Table 3c. The evolution rule of plastic inception location on change of the residual stress is summarized in Figure 8c. When the points are on the left part of Figure 5c, where the dimensionless critical yield load increases with the increase in the dimensionless residual stress, the onset yield occurs at location h in Figure 8c, which is a circular region near the surface centers on the axis of symmetry. When the points are on the right side of Figure 5c, where the dimensionless critical yield load decreases with the increase in the dimensionless residual stress, the yield inception occurred at location i in Figure 8c, at the axis of symmetry in the surface layer. Generally, the location is transferred from h to i when the dimensionless residual stress changes from \(-1\) to 1. For a relatively thick surface layer, \(t/R=0.016-0.128\), the rule for the residual stress at the locations of yield inception is similar. By associating the location of the yield inception with the critical yield load, it can be observed that when the data points are on the plateau, where the critical yield load remains unchanged with the residual stress, the yield inception occurred in the substrate. When the data points are on the other parts of the curves where the critical yield load changes with the residual stress, yield inception occurs in the surface layer.