## 3 Results and Discussion
### Calculation of the Linear-Elastic Strain Energy Density of Bulk Defects  
Following the definition of the linear-elastic design limit curve, the methodology was extended to assess the fatigue strength of defect-afflicted specimens considered in Section 2.2. Following the considerations of a preceding study [26], a numerical assessment strategy was formulated. As the evaluation of the fatigue behaviour bases on the geometrical appearance of the inherent imperfections, the two recorded radiographs, denoted as X- and Y-view were digitalized for each imperfect large-scale specimen. Due to the discrete nature of the digital images, a smoothing algorithm [61] was applied to derive continuous defect contours. Subsequently, geometrical shape parameters of the digitalized defect structures were calculated in order to characterize the depicted imperfections on the assessed radiographs. Table 5 summarizes the minimum and maximum values of the respective shape parameters derived from the 66 assessed radiographs of the present numerical study. The numerical calculations were conducted by linear-elastic finite element analyses utilizing the software package Abaqus(r) 2021. As the radiographs depict two-dimensional projections of the corresponding imperfection, planar simulation models were utilized,  
Figure 8: Linear-elastic strain energy density-based design limit curve. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Minimum** & **Maximum** \\ \hline Area [mm\({}^{2}\)] & 53.97 & 227.58 \\ Feret Max. [mm] & 11.15 & 61.13 \\ Feret Min. [mm] & 4.80 & 11.64 \\ Aspect ratio [\(-\)] & 1.72 & 7.46 \\ Circularity [\(-\)] & 0.08 & 0.75 \\ Convexity [\(-\)] & 0.30 & 0.99 \\ Elongation [\(-\)] & 2.30 & 24.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Shape parameters of assessed defect projections. in which the respective defect structures were implemented, respectively. Concerning the angular orientation of the defect contour, the rotation angle reflects the one investigated in the experiments described in Section 2.2. Plane stress elements with quadratic shape functions of type CPS6/CPS8 were used. It should be noted, that the size of the finite element models as well as the element sizes were proportional to the absolute dimensions of the corresponding defect in order to provide a simulation methodology that was applicable to arbitrary defect structures. The numerical models were loaded under uniaxial tension, evoking a maximum stress of 1 MPa in the gross-section of the model. Figure 9 depicts the simulation results of selected defect structures. The evaluated stress constitutes the stress concentration factor induced by the respective imperfections. Figure 9: Numerical results based on the radiographs of specimens #10, #16, #20 and #26. As the numerical simulations represent static analyses, the SED evoked by the maximum stress of \(1\,\mathrm{MPa}\) can be interpreted as the cyclic SED induced due to an acting stress range of \(\Delta\sigma_{g}=1\,\mathrm{MPa}\) at a stress ratio of \(R=0\). Due to the linear-elastic material behaviour considered in the numerical analyses, the evaluated strain energy density values can be conveniently scaled to load levels investigated in the corresponding experiments. As the spatial imperfections possess an irregular shape, the planar projections also feature an arbitrary contour, which leads to a wide variability in the observed SED values. In order to derive a characteristic assessment parameter that reflects the fatigue behaviour of the defect-afflicted specimens, a statistical evaluation of the SED distribution along the defect contour was conducted. This procedure includes the definition of a control volume, which moves along the nodes of the respective projection contour. In order to provide a numerically efficient assessment methodology, finite element sub-modelling technique was applied. Figure 10 summarizes the numerical simulation strategy.