# Effect of Residual Stress in Surface Layer on Plastic Yield Inception
## 4 Discussion
### Effect of Surface Layer Thickness  
As shown in Figures 5 and 6 and Table 3, the thickness of the surface layer, together with the residual stress, co-determines the critical yield load and yield inception location of the contact system. The von Mises equivalent stress at the axis of symmetry and the surface under contact with different surface layer thicknesses and different residual stresses are presented in Figure 11. The blue line in Figure 11 gives the boundary between substrate and surface layer and the yield strength value. The result for \(t/R=0.001\) is presented in Figure 11a. For the residual stress of 0 MPa, under \(P=0.081\) N, the von Mises equivalent stress in the substrate has reached the yield strength (600 MPa, indicates by the arrow A), while the surface layer has not (952 MPa). Changing the residual stress from 0 MPa to \(-514\) MPa, under \(P=0.081\) N, the von Mises equivalent stress in the substrate remains unchanged and has reached the yield strength. Although the von Mises equivalent stress in the surface layer changes, it has not reached the yield strength and has no effect on the yield position and the critical yield load. That  
Figure 10: Contact stresses at the axis of symmetry and the surface with different surface layer. The parameters are \(E_{s}=200\) GPa, \(E_{l}=400\) GPa, \(Y_{s}=600\) MPa, \(Y_{l}=952\) MPa, \(t/R=0.006\), and \(P=0.05\) N. (**a**) Without residual stress. (**b**) With tensile residual stress. (**c**) With compressive residual stress. (**d**) With excessive compressive residual stress. is to say, a change of residual stress does not affect the critical yield load. That is the reason for the critical yield load plateau to come out. Corresponding yield position is at the axis of symmetry in the substrate, as indicated by the arrow A in Figure 11a. It can be inferred that when possessing excessive residual stress, whether tensile or compressive, the von Mises equivalent stress in the surface layer will increase, and the yield inception will happen in the surface layer. According to the discussion in Section 4.1, tensile residual stress causes the von Mises equivalent stress at the axis of symmetry to rise, while excessive compressive residual stress causes the von Mises equivalent stress at the contact edge near surface to rise. As a result, yield occurs at the axis of symmetry under tensile residual stress, and at the contact edge near surface under excessive compressive residual stress. From the above discussion, the variation rule for locations of yield inception in Figure 8a can be understood. The result for \(t/R=0.008\) is presented in Figure 11b. For residual stress of \(-171\) MPa and \(-514\) MPa, under \(P=0.180\) N, the yield begins in the substrate (600 MPa, indicated by the arrow B). The residual stress change does not cause von Mises equivalent stress in substrate to change. It also corresponds to a critical load plateau. Compared to case \(t/R=0.001\), the difference is that the yield happens on the axis of symmetry, and the substrate side of the surface layer/substrate interface, as indicated by the arrow B in Figure 11b. Combined with the above discussion of contact with tensile residual stress and excessive compressive residual stress, the variation for locations of yield inception in Figure 8b can be understood. The result for \(t/R=0.064\) is presented in Figure 11c. It can be observed that both changes to the residual stress (\(-343\) MPa and \(-576\) MPa) and load (\(0.256\) N and \(0.418\) N) lead to an apparent von Mises equivalent stress change in the surface layer, while only having a minor effect on the stress of surface layer. In addition, the von Mises equivalent stress in the substrate is very small. This is due to the fact that the substrate is far away from the contact zone. Therefore, the yield only occurs in the surface layer. The variation of residual stress in the surface layer will inevitably lead to the changes in the contact stress field of the surface layer, thereby changing the critical load.