# Effect of Residual Stress in Surface Layer on Plastic Yield Inception
## 3 Results
### Effect of Residual Stress on the Locations of Yield Inception  
The locations of yield inception are quite important in avoiding the failure of the contact system because these failures are usually triggered by crack initiation and propagation and are significantly influenced by the location of the plasticity initiation [30; 31]. Thus, the evolution rule of the location of the yield inception and its association with the corresponding critical yield load is significant. Table 3 presents the contour plots of the von Mises stress during the yield inception for the point marked in Figure 5. In Table 3, because we mainly focused on the location of the yield inception, the specific value of von Mises equivalent stress was not given. Red represents a greater von Mises stress, whereas blue represents smaller von Mises stress in the contour plots. Typical locations of the yield inception and variation rule are summarized in Figure 8, as the points labeled a-i are shown. The evolution rule in the locations of the yield inception as the residual stress varied from compressive to tensile for different surface layer thicknesses is also summarized in Figure 8.  
Figure 7: Dimensionless critical load versus dimensionless residual stress and dimensionless thickness for surface layer/substrate system pressed by a rigid hemisphere. (**a**) \(E_{s}=200\) GPa, \(E_{l}=400\) GPa, \(Y_{s}=600\) MPa, \(Y_{l}=952\) MPa. (**b**) \(E_{s}=200\) GPa, \(E_{l}=400\) GPa, \(Y_{s}=600\) MPa, \(Y_{l}=1200\) MPa. (**c**) \(E_{s}=100\) GPa, \(E_{l}=300\) GPa, \(Y_{s}=200\) MPa, and \(Y_{l}=600\) MPa. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & & \\ \hline
**Point (\(t\)/\(R\) = 0.001)** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Dimensionless Residual stress & \(-0.915\) & \(-0.863\) & \(-0.815\) & \(-0.343\) & \(0.200\) & \(0.257\) & \(0.428\) & \(0.771\) \\ Contour plot & & & & & & & & \\ Location of yield inception & a & a & b & b & b & c & c & d \\ \hline
**Point (\(t\)/\(R\) = 0.008)** & **9** & **10** & **11** & **12** & **13** & **14** & **15** & **16** \\ \hline Dimensionless Residual stress & \(-0.857\) & \(-0.785\) & \(-0.743\) & \(-0.515\) & \(-0.315\) & \(-0.228\) & \(0.115\) & \(0.572\) \\ Contour plot & & & & & & & & \\ Location of yield inception & e & e & f & f & f & g & g & g \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Contour plots of von Mises stress at yield inception of cases with different residual stresses and different surface layer thicknesses (\(E_{s}=E_{l}=200\) GPa, \(Y_{s}=Y_{l}=600\) MPa, number 1–24 are the labels of the points in Figure 5, and letters a–i denote the locations of the yield inception summarized in Figure 8). The contour plots of the von Mises stress during the yield inception of the points labeled 1-8 in Figure 5a for \(t/R=0.001\) are presented in Table 3a. The evolution rule of plastic inception location on change of the residual stress is summarized in Figure 8a. When the points are on the left part of Figure 5a, where the residual stress is highly compressive and the dimensionless critical yield load increases with the increase in the dimensionless residual stress, the yield inception occurs at location a in Figure 8a, which is a circular region in the surface layer near the surface. When the points are on the central part (plateau) of Figure 5a, where the residual stress is moderately compressive or slightly tensile and the dimensionless critical yield load remains unchanged with the dimensionless residual stress, the yield inception occurs at location b in Figure 8a, at the axis of symmetry in the substrate. When the points are on the right side of Figure 5a, where the residual stress is tensile and the dimensionless critical yield load decreases with the increase in the dimensionless residual stress, the yield inception occurs at location c in Figure 8a, at the axis of the symmetry and surface layer side of the surface layer/substrate interface.