## 5 Results and Discussion
When \(\sigma_{R}/Y\leq 0\), the curve of the normalized hardness difference and residual stress was a downward parabola. When \(\sigma_{R}/Y>0\), the relationship between the normalized hardness difference and residual stress was a downward straight line. Figure 8: Relationship between normalized hardness difference and residual stress. Figure 7: Relationship between normalized depth of indentation and normalized mean contact pressure. The influence of compressive residual stress on hardness was not as significant as that of tensile residual stress. The yield strength of Superalloy GH4169 was about 550 Mpa. When the residual stress exceeded the yield strength of GH4169, the material began to harden and deform, thus increasing the hardness of the sample. Therefore, due to the limitation of applicable conditions, Equation (9) was further modified as follows:  
\[\sigma_{R}=\left\{ \begin{array}{l}-\frac{Y}{2B}\times\left(C+\sqrt{C^{2}-4B \left(\frac{E^{*}}{Y}\right)^{-0.5}\times\left(\frac{H}{Y}-\frac{H_{0}}{Y} \right)}\right),-550\;\mathrm{MPa}\leq\sigma_{R}\leq 0\\ \left(\frac{E^{*}}{Y}\right)^{-0.5}\times D(H-H_{0}),0<\sigma_{R}\leq 550\; \mathrm{MPa}\end{array} \right. \tag{10}\]  
Tensile residual stresses exist mainly on the machined surface, while compressive residual stresses are found mainly on the machined subsurface [31; 32]. The higher the cutting amount, the larger the surface tensile stress [33]. Due to the needs of this funded project, this study mainly focused on the side milling research of Superalloy GH4169. In this study, a side milling experiment of Superalloy GH4169 was carried out, so the residual stresses of the machined surface were all tensile residual stresses. During the experiments, dry cutting was adopted in this study. The milling method was down milling due to the consideration of milling surface quality. Since the influence of tool wear on residual stress was not considered in this research, a new tool was used for each milling. By fitting the simulation and test data, the residual stress prediction formula of this study was fitted as follows:  
\[\sigma_{R}=51.2(H-H_{0}),\;0<\sigma_{R}\leq 550\;\mathrm{MPa} \tag{11}\]  
Table 3 lists the hardness test values, measured residual stress values and predicted residual stress values. It can be observed that when the hardness is 443 HV, the percentage difference between the experimental and numerical results is 28.5%. When the hardness is 445 HV, the percentage difference between the experimental and numerical results is 8.1%. When the hardness is 448 HV, the percentage difference between the experimental and numerical results is 15.8%. It can be found that the closer the hardness of the material is to 445 HV, the more accurate the prediction accuracy is. The average percentage difference between the simulation and test results is 17.5%. Figure 9 shows the forecast value and the measured value of the residual stress of the fitted prediction formula. It can be found that the prediction model is more accurate in predicting tensile stress based on hardness, and the prediction accuracy of the prediction model is about 82.5%. Although the method proposed by [PERSON] et al. [24] has the advantage of measuring the component size of residual stress compared with the method proposed in this study, it needs to consider the load-displacement curve in the state without residual stress, it is difficult to accurately measure the height of accumulation and it consumes too much time with related work. [PERSON] proposed a method for estimating the magnitude and direction of non-equal biaxial residual stress components based on indentation [34]. The proposed method utilizes the force penetration response provided by the improved Berkovich tip to improve the prediction accuracy, but it also requires longer testing time and costs more money than the method in this study.