# Effect of Residual Stress in Surface Layer on Plastic Yield Inception
## 4 Discussion  
### Discussion on Effect of Residual Stress on the Critical Yield Load  
It was observed in the simulation that intermediate compressive residual stress can strengthen the contact system, whereas tensile stress weakens the contact system. To explain this, the stress components were analyzed. Figure 9 shows examples of contact stress field for cases without residual stress, and with tensile, compressive, and excessive compressive residual stress in the surface layer. The parameters adopted in these examples are as follows: \(E_{s}=200\) GPa, \(E_{l}=400\) GPa, \(Y_{s}=600\) MPa, \(Y_{l}=952\) MPa, \(t/R=0.006\), and \(P=0.05\) N. The induced tensile residual stress in the surface layer is 274 MPa, the compressive residual stress is \(-274\) MPa, and the excessive compressive residual stress is \(-754\) MPa. As can be observed in Figure 9, the existence and variation of the residual stress in the surface layer only has a slight influence on the von Mises equivalent stress in the substrate but significantly changes the von Mises equivalent stress field in the surface layer. Thus, it may affect the critical load and locations of the plastic yield inception and the performance of the contact surface. The von Mises equivalent stress \(\sigma_{v}\) can be expressed as [37]:  
\[\sigma_{v}=\frac{1}{\sqrt{2}}\Big{[}(\sigma_{rr}-\sigma_{\theta\theta})^{2}+( \sigma_{\theta\theta}-\sigma_{zz})^{2}+(\sigma_{zz}-\sigma_{rr})^{2}+6(\tau_{ r\theta}^{2}+\tau_{\theta z}^{2}+\tau_{zr}^{2})\Big{]}^{1/2} \tag{2}\]  
where \(\sigma_{rr}\) is the radial stress, \(\sigma_{\theta\theta}\) is the circumferential stress, \(\sigma_{zz}\) is the stress parallel to the axis of symmetry, and, \(\tau_{r\theta}\), \(\tau_{\theta z}\), and \(\tau_{zr}\) are the shear stresses. Due to the rotational symmetry, \(\tau_{r\theta}\) and \(\tau_{\theta z}\) are equal to zero. It can be observed in Figure 9 that the compressive residual stress causes \(\sigma_{rr}\) and \(\sigma_{\theta\theta}\) to be more compressive (the value is more negative) and the tensile residual stress causes \(\sigma_{rr}\) and \(\sigma_{\theta\theta}\) to be more tensile (the value is more positive). The influence of the residual stress on \(\sigma_{zz}\) and \(\tau_{zr}\) is small. From Equation (2) and the above discussion, it can be inferred that the von Mises equivalent stress \(\sigma_{v}\) depends on the difference between, \(\sigma_{rr}\), \(\sigma_{\theta\theta}\), and \(\sigma_{zz}\). The stress contour plots in Figure 9 give a full view of the stress field, however, the value of the stresses are difficult to compare. Since the yield begins either at the axis of symmetry or near the surface according to the results of locations of yield inception, stresses at the axis of symmetry and the surface are extracted and presented in Figure 10 for a more detailed analysis. Contact stresses at axis of symmetry and surface without residual stress, with tensile residual stress, with compressive residual stress, and with excessive compressive residual stress are shown Figure 10a-d. The left part of the figure shows stress at the axis of symmetry, and the right part shows stress at the surface. Without residual stress (Figure 10a), \(\sigma_{zz}\) is more compressive (negative) than \(\sigma_{rr}\), and \(\sigma_{\theta\theta}\), at the axis of symmetry and most parts of the surface except near the contact edge. The maximum differences between, \(\sigma_{rr}\), \(\sigma_{\theta\theta}\), and \(\sigma_{zz}\) are located on the axis, as well as the maximum von Mises equivalent stress. With tensile residual stress (Figure 10b), \(\sigma_{rr}\) and \(\sigma_{\theta\theta}\) become more tensile (or less compressive), and the differences between \(\sigma_{rr}\), \(\sigma_{\theta\theta}\), and \(\sigma_{zz}\) are enlarged at the axis. The maximum von Mises equivalent stress is still located at the axis and is increased. Therefore, tensile residual stress makes the contact system easier to yield.