# 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  
Figure 4 shows the residual stress field in the surface layer induced by setting the thermal mismatch. A uniform residual stress in the surface layer, which is equi-biaxial in the plane parallel to the contact surface, is obtained. The dimensionless critical yield load of the surface layer/substrate system for \(E_{s}=E_{l}=200\) GPa and \(Y_{s}=Y_{l}=600\) MPa pressed by a rigid hemisphere for dimensionless residual stresses change from \(-1\) to \(1\) and dimensionless thicknesses of 0.001, 0.008, and 0.064 are presented in Figure 5. The locations of the yield inception of the points labeled 1-24 in Figure 5 were checked and will be discussed later. For \(t/R=0.001\), as shown in Figure 5a, as the dimensionless residual stress changed from \(-1\) to \(1\), the dimensionless critical yield load first increased then plateaued, and then decreased. The maximum dimensionless critical yield load (the plateau value) equals \(1\)  
Figure 4: Residual stress field induced in surface layer by thermal mismatch. Figure 5: Dimensionless critical load versus dimensionless residual stress and dimensionless thickness of 0.001 (**a**), 0.008 (**b**), and 0.064 (**c**) for surface layer/substrate system pressed by a rigid hemisphere (\(E_{s}=E_{l}=200\) GPa, \(Y_{s}=Y_{l}=600\) MPa). which is equal to the contact system of zero residual stress (without residual stress). For excessive residual stress, whether compressive or tensile, the critical yield loads are less than the contact system of zero residual stress--that is, the contact system is weakened. The curve is asymmetric to the zero residual stress; the range is larger for the systems possessing compressive residual stress to obtain the maximum critical yield load than those possessing tensile residual stress. For \(t/R=0.008\), Figure 5b, a very similar ternary form rule is presented; as the dimensionless residual stress changed from \(-1\) to \(1\), the dimensionless critical yield load increases first, then levels to a plateau value, and then decreases. The maximum dimensionless critical yield load (plateau value of \(2.09\)) increases when moderate compressive residual stress is applied, which means that moderate compressive stress can promote the plasticity onset threshold to enhance the load-carrying capability. Tensile residual or excessive compressive stress leads to a decrease in the critical yield load, and the layer/substrate system is weakened. For \(t/R=0.064\), Figure 5c, as the dimensionless residual stress changes from \(-1\) to \(1\), the dimensionless critical yield load first increases, reaches a peak value, and then decreases. The maximum dimensionless critical yield load is \(4.1\). Moderate compressive residual stress strengthens the system, whereas tensile or excessive compressive residual stress weakens it. Figure 6 shows the variation in the critical yield load with both the residual stress and surface layer thickness for \(E_{s}=E_{l}=200\) GPa and \(Y_{s}=Y_{l}=600\) MPa. For all the surface layer thicknesses, as the dimensionless residual stress varies from \(-1\) to \(1\), the critical yield load increases first to a peak value or is leveled to a plateau value, and subsequently decreases. The curves are asymmetric to the zero residual stress. Generally, a moderate compressive residual stress in the surface layer results in higher critical yield loads than that without residual stress or with tensile residual stress. The dimensionless critical yield load of the system with dimensionless thicknesses of \(t/R=0.0005\) and \(0.002\) is similar to that of \(t/R=0.001\), as shown in Figure 6. When the thickness of the surface layer is small (\(t/R=0.0005\), \(0.001\), and \(0.002\)), the maximum yield load (the plateau value) is equal to that of the zero residual stress; in other words, from the viewpoint of an increasing critical yield load, the contact system cannot be strengthened by applying residual stress in the surface layer.