## 2 Methodology
### Analysis of Input Values for Local Fatigue Design Methods
#### 2.1.1 Requirements for N-SIF Method  
A generalized notch stress intensity factor-based fatigue analysis, as shown in Equation (3), requires the following key inputs for each point of the defect contour:  
* Notch opening angle \(2\alpha\),
* Notch radius \(\rho\),
* Principal stress scalar at the evaluated defect contour edge point, as well as path values to calculate \(K_{I}^{V}\) (in the case of a sharp notch assessment) and/or \(K_{\rho,I}^{V}\) (for the blunt notch case). In addition, the equivalent quadrilateral cell size \(d\) of the attached elements for each defect node must be known to apply the peak stress method (Equation (2)). As noted in the introduction, the peak stress method is strictly applicable only for notch opening angles up to \(2\alpha\leq 135^{\circ}\). However, defects with a circular or elliptical shape possess arc-length sections with opening angles up to \(180^{\circ}\), or even higher (e.g., in the case of changing from a concave to a convex local shape of the surrounding base material). For the application of the N-SIF method to irregularly shaped defects, convex opening angle values are, therefore, limited to the boundary case as a straight line. Thus, the referred peak stress database in [40] must be extended towards larger opening angles. To check the applicability of the peak stress method for sharp notches, a numerical study using a double V-notched plate under tension was conducted. The scripted parametric study was based on linear-elastic analysis using Ansys(r) Build 21.1, invoking the gRPC-interface [68], wherein the notch stress intensity factor was based on the boundary value \(K_{I}^{V}\), evaluating the stress field \(\sigma_{\theta(\theta=0)}(r)\) along the bisector path. The quadratic element type PLANE183 was utilized, and key options were set according to whether plane strain or plane stress analysis was considered. The peak stress value was based on linear elements and, so, the PLANE182 element type was assigned. The peak stress results are directly comparable to the PLANE42 elements used in [40]. The conducted analysis used the same plate geometry with a thickness of \(W=50\) mm, but we extended the parametric range of the notch opening angle up to \(0^{\circ}\leq 2\)\(\alpha\leq 165^{\circ}\) in 12 steps, notch depth range \(0.3\) mm \(\leq a\leq 10\) mm in 98 steps, and the local notch tip mesh size to \(0.1\) mm \(\leq d\leq 2.9\) mm in 15 steps. A limit value of \(a/d>2.5\) was considered in this study, thus reducing the number of parametric data points to 945 for each load angle. In total, about \(11,340\) peak stress method simulation runs were performed (Figure 3). It was confirmed that the non-dimensional ratio \(K_{FE,I}^{*}=1.38\) was well-applicable for opening angles up to \(135^{\circ}\), if nodal stress results are evaluated. As the observed deviations were more distinctive at notch opening angles \(2\alpha\geq 165^{\circ}\), we concluded a dependency of the new correction factor \(K_{FE,I,mod}^{*}\) on the second eigenvalue \(\lambda_{2}\), as manifested in Equation (8), as this value was non-singular for values above \(103^{\circ}\). In addition, it was found that, for large opening angles, the number of elements at the notch tip is reduced by automatic free-meshing algorithms; thus, the number of elements \(\#d\) at the notch tip, from notch edge to bisector line, must also be addressed. The known value \(q=(2\pi-2\alpha)/\pi\) characterizes the opening angle as a non-dimensional parameter. \[ \begin{array}{ll}K^{*}_{FE,I,mod}=K_{IE,I}/_{\text{max}(1,\lambda_{2})},&\#d=1 \\ K^{*}_{FE,I,mod}=K_{FE,I}\text{${}^{0.9q}/_{\lambda_{2}}${}^{2/3}},&\#d>1\\ \end{array}  \tag{8}\]  
The value of \(K^{*}_{FE,I}\) is almost unchanged for opening angles up to \(2\alpha=120^{\circ}\), and the new non-dimensional correction factor \(K^{*}_{FE,I,mod}\) takes effect for larger opening angles only, as shown in Figure 3b. The conducted numerical study involves only Mode I tension loading.