## 2 Theoretical Analysis
### Modeling for Stress Intensity Factor  
In the existing literature, [PERSON] et al. [16; 17] suggested that the maximum local stress \(\sigma_{local,\max}\) at the machined surface was considered a good indicator for evaluating fatigue life. They defined the stress \(\sigma_{local,\max}\) as the following Equation (20):  
\[\sigma_{local,\max}=\sigma_{applied,\max}+\sigma_{RS,surface} \tag{20}\]  
where \(\sigma_{applied,\max}\) represents the maximum stress that is applied on a specimen during fatigue tests. \(\sigma_{RS,surface}\) is the residual stress measured in the axial direction at the machined surface. They proposed a new law that described the relationship between the maximum local stress \(\sigma_{applied,\max}\) and fatigue life \(N_{f}\):  
\[\sigma_{local,\max}=1.535\times 10^{8}\times N_{f}{}^{-1.105}+362.2 \tag{21}\]  
However, the residual shear stress is not taken into account to evaluate the effect of residual stress on fatigue life in Equations (20) and (21). As analyzed in Section 1, residual shear stress is produced during machining, and it should be considered when determining the equivalent stress. When the stress \(\sigma_{applied}\) is applied in the axial direction of a machined specimen during fatigue tests, the equivalent stress can be calculated following the von Mises equation:  
\[\sigma_{v}=\frac{1}{\sqrt{2}}\sqrt{\left(\sigma_{1}-\sigma_{2}\right)^{2}+ \left(\sigma_{2}-\sigma_{3}\right)^{2}+\left(\sigma_{3}-\sigma_{1}\right)^{2}} \tag{22}\]where \(\sigma_{v}\) is the equivalent stress. \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are the principal stresses which can be determined by Equation (23) for a plain stress condition:  
\[\sigma_{1,2}=\frac{\left(\sigma_{xx}+\sigma_{applied}\right)+\sigma_{yy}}{2} \pm\sqrt{\left(\frac{\left(\sigma_{xx}+\sigma_{applied}\right)-\sigma_{yy}}{2} \right)^{2}+\tau_{xy}^{2}} \tag{23}\]  
Once the equivalent stress is calculated by the Equation (22), the stress intensity factor (SIF) can be determined according to the feature of the specimen. Figure 3 exhibits the schematic diagram of the fracture surfaces for surface failure. The region with small facets is generally regarded as a rough area where the crack is prone to initiate. Hence, the region with small facets is defined as the initiation area and the crack shape is assumed to be semi-elliptical. For the crack initiation area, the value of the SIF range, \(\Delta K_{ini}\), determines whether the crack propagates or not. Consequently, the value of \(\Delta K_{ini}\) can be considered as the threshold for crack propagation \(\Delta K_{th}\)[21]. For a cylinder specimen, the value of the SIF range \(\Delta K_{ini}\) under tension-tension fatigue test can be given [21]:  
\[\Delta K_{ini}=\Delta K_{ht}=n\cdot\Delta\sigma\cdot\sqrt{n\cdot\sqrt{area}} \tag{24}\]  
where \(n\) is the correction factor which is related to the geometry of the crack. For a crack initiated at the surface, \(n=0.65\)[21]. \(\sqrt{area}\) is the equivalent size of the projected area which includes facets. \(\Delta\sigma\) is the applied stress range and can be expressed by Equation (25):  
\[\Delta\sigma=\sigma_{applied,\max}-\sigma_{applied,\min} \tag{25}\]  
Due to the presence of residual stress, the effective stress \(\Delta\sigma_{\mathit{eff}}\) can be obtained according to Equation (19) [16; 17]. \[\Delta\sigma_{eff}=\left(\sigma_{applied,\max}+\sigma_{RS,surface}\right)- \left(\sigma_{applied,\min}+\sigma_{RS,surface}\right)=\Delta\sigma \tag{26}\]  
Based on the description of Equations (25) and (26), it is interesting to find that Equation (26) is equal to Equation (25). It is indicated that the presence of residual stress has no influence on the effective stress that acted on the specimen. It can be deduced that fatigue life will not be changed when residual stress is introduced by machining. Thus, Equation (20), proposed by [PERSON] et al. [16; 17], may not be suitable to describe the effect of residual stress on the SIF range.