## 2 Methodology
The main disadvantage of this concept is that the mesh in the vicinity of the defect is not preserved between different studies. This requires additional post-processing effort, as the evaluation nodes in the surrounding defect area are not identical for each load vector study. Thus, it is not easily applicable for stress path-dependent fatigue assessments, such as in the case of the TCD or ESED methods, as the evaluation procedure must be repeated for each single load case. For further reference, in ([PERSON], 2016), an irregularly shaped planar defect was evaluated through variation of the loading direction in the linear-elastic finite element analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notch Tip Radius** & \(\sigma_{\theta}\)(\(a_{+}\)/2) 1  & \(\sigma_{\theta}\)(\(a_{+}\)/2) 2  \\ \hline \hline \(\rho=0\) mm & 2.493 MPa & 2.476 MPa \\ \(\rho=1\) mm & 2.428 MPa & 2.438 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 2: TCD results of a basic study on double-notched V-plate (\(W=10\) mm, \(a=1\) mm, \(2a=90^{\circ}\)) in plane strain formulation. Second, the given finite element model can be split up into an inner region, which is kept unchanged between different load vector studies, and an outer region, which rotates in accordance with the given load vector angle. This approach is denoted as Method 2. For each load case, tension loading along the \(x\)- and \(y\)-axes is applied, where the load vector coordinate system rotates with the outer region. Both regions are glued together, and the post-processing focuses on the inner region with unchanged mesh entities. This method necessitates an increased number of finite element simulations, but the computing time for each linear-elastic simulation run is rather low and the post-processing effort is minimized due to the defect mesh patterns remaining unchanged. Moreover, each simulation run provides results for two load angles, as two load steps with perpendicular load vectors are solved at once. Figure 2b depicts the schematic principle of this sub-case approach. Third, the transformation equations for a mechanical stress tensor can be utilized to derive the stresses based on three principle basic load cases for each evaluated load angle \(\varphi\). Although the number of linear-elastic simulation runs is minimized (three, in the case of planar studies), the post-processing effort is increased. This principle is hereafter denoted as Method 3 and is schematically depicted in Figure 2c. With varying load angle \(\varphi\) in the plane system, the resulting stress tensor is obtained by superposition of the three basic load cases; namely, unified tension loading in the \(x\)-axis, tension in the \(y\)-axis, and shear in the \(xy\)-plane. The stress transformation equations to transform a plane coordinate system \(\{x,y\}\) by an angle \(\varphi\) into the rotated system \(\{\ \xi,\ \eta\}\) have been previously detailed in mechanical handbooks [67]. As only uniaxial loading under varying load angle \(\varphi\) is considered in this study, Equation (6) details the scale factors for tension load in the \(x\)-axis \(f_{XX}\), tension load in the \(y\)-axis \(f_{YY}\), and the shear load \(f_{XY}\). For example, if the uniaxial load vector is inclined by 45 degrees, all three load scale factors equate to one half. The transformation equations are applied to load angles up to 180 degrees in the uniaxial load study. \[ \begin{array}{c}f_{XX}(\varphi)=0.5+0.5\ \cos(-2\ \varphi)\\ f_{YY}(\varphi)=0.5-0.5\ \cos(-2\ \varphi)\\ f_{XY}(\varphi)=-0.5\ \sin(-2\ \varphi)\\ \end{array}  \tag{6}\]  
Figure 2: Principle methods of solution to obtain local stress and elastic strain energy density results for irregularly shaped defects under uniaxial loading with varying load vectors: (**a**) Individual load case study; (**b**) Sub-case approach; and (**c**) Transformation and superposition of basic load steps. The stress tensor \(\overline{\overline{\sigma}}(\varphi)\) is obtained by linear superposition of the weighted tensors for each basic load step.