# Effect of Residual Stress in Surface Layer on Plastic Yield Inception
## 3 Results
### Effect of Residual Stress on the Critical Yield Load and Contact Area
The dimensionless critical yield load of the system with dimensionless thicknesses of \(t/R=0.006\), \(0.01\), and \(0.012\) is similar to that of \(t/R=0.008\), as shown in Figure 6. The  
Figure 6: Dimensionless critical load versus dimensionless residual stress and dimensionless thickness for surface layer/substrate system pressed by a rigid hemisphere (\(E_{s}=E_{l}=200\) GPa, \(Y_{s}=Y_{l}=600\) MPa). maximum dimensionless critical yield load (the plateau value) is larger than that of the zero residual stress, and the contact system can be strengthened by applying compressive residual stress in the surface layer. Generally, the maximum critical yield load increases with the surface layer thickness. The plateau corresponding to the maximum critical yield load becomes narrower as the surface layer thickness increases. The dimensionless critical yield load of the system with dimensionless thicknesses \(t/R=0.016\), 0.032, and 0.128 is similar to that of \(t/R=0.064\), as shown in Figure 6. The same peak value of the critical yield load is much larger (4.1 times) than that of the zero-residual-stress case, which appears at an approximately dimensionless residual stress of \(\sigma_{r}/Y_{l}=-0.6\). This value can be used as the design criterion for the residual stress during the fabrication of the surface layer. In the above-mentioned simulations, the elastic moduli of the surface layer and substrate are the same. Sometimes, the elastic modulus is less sensitive to surface treatment, but residual stress originates after surface treatment [32; 33]. However, in most practical cases, the material parameters of the surface layer and substrate are not identical. Simulations are also performed for contact systems with different surface layers and substrate material parameters. The dimensionless critical yield load of the surface layer/substrate system for \(E_{s}=200\) GPa, \(E_{l}=400\) GPa, \(Y_{s}=600\) MPa, and \(Y_{l}=952\) MPa pressed by a rigid hemisphere for dimensionless residual stresses change from \(-1\) to \(1\), and the dimensionless thicknesses of 0.0005-0.128 is presented in Figure 7a. The effect of the residual stress on the dimensionless critical load is basically the same as that in the cases of \(E_{s}=E_{l}=200\) GPa and \(Y_{s}=Y_{l}=600\) MPa. The dimensionless critical yield load (the plateau) is less than 1 for a small surface layer thickness (\(t/R=0.0005-0.004\)). This is because of the \"weakening effect\" caused by the elastic mismatch between the surface layer and substrate [34; 35; 36]. The maximum dimensionless critical yield load for all surface layer thicknesses is 3.9 obtained at a dimensionless critical residual stress of \(-0.6\). The results for \(E_{s}=200\) GPa, \(E_{l}=400\) GPa, \(Y_{s}=600\) MPa, and \(Y_{l}=1200\) MPa are presented in Figure 7b. This figure is almost the same as that for \(E_{s}=200\) GPa, \(E_{s}=400\) GPa, \(Y_{s}=600\) MPa, and \(Y_{l}=952\) MPa, except that the maximum dimensionless critical yield load is 7.6, due to the increase of yield strength of surface layer, also obtained at a dimensionless critical residual stress of approximately \(-0.6\). The results for \(E_{s}=100\) GPa, \(E_{l}=300\) GPa, \(Y_{s}=200\) MPa, and \(Y_{l}=600\) MPa are presented in Figure 7c. The curves have a similar shape to the above cases, and the maximum dimensionless critical yield load is also obtained at a dimensionless critical residual stress of approximately \(-0.6\). Cases of the wide range of material parameter (100 GPa \(\leq\)\(E_{s}\)\(\leq\) 200 GPa, 400 GPa \(\leq\)\(E_{l}\)\(\leq\) 1000 GPa, 400 MPa \(\leq\)\(Y_{s}\)\(\leq\) 600 MPa, 400 MPa \(\leq\)\(Y_{l}\)\(\leq\) 2000 MPa) are calculated, and the value of \(-0.6\) still holds.