## 3 Functionality of Key Modules
### Local Fatigue Assessment
#### 3.4.3 Elastic Strain Energy Density  
The ESED is computed as the accumulated elastic strain energy divided by the control volume. It can be computed considering the sharp or blunt notch case, which affects the size and location of the control volume \(\Omega\), as depicted for the V-notch in Figure 1. To prove the validity of the constructed simulation model, computation of the elastic strain energy density was conducted for the circular hole-in-plate model (see Figure 7a). The standard value of \(R_{c}=0.19\) mm and plane strain formulation were used for the simulation. The computed blunt notch ESED had a maximum value of \(\Delta\overline{W}=1.126\cdot 10^{-5}\) mJ/mm\({}^{3}\), which can be directly compared to the blunt notch ESED reference simulation value of \(\Delta\overline{W}_{ref}=1.1282\cdot 10^{-5}\) mJ/mm\({}^{3}\), as given in Table 1 for the bilateral blunt V-notched plate. The simulation result for the circular hole-in-plate model was about 15% lower than the reference ESED for the double V-notched plate. Figure 22 shows the stress and displacement contour plots of the V-notched plate and circular hole-in-plate model. Table 8 provides details about the number of cells within the control volume, as well as the average mesh-seed. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Evaluation of \({}^{*}\)** & **Control Volume \(\Omega\)** & **\# Cells in \(\Omega\)** & **Cell Edge Length** & **ESED \(\Delta W\)** \\ \hline sharp notch & 0.058 mm\({}^{3}\) & 30 & \textasci{-}45 μm & \(1.374\times 10^{-5}\) mJ/mm\({}^{3}\) \\ blunt notch & 0.183 mm\({}^{3}\) & 88 & \textasci{-}45 μm & \(1.126\times 10^{-5}\) mJ/mm\({}^{3}\) \\ \hline \hline \end{tabular} \({}^{*}\)The size and origin of the control sphere are set as sharp notch or blunt notch (see Figure 1).  
\end{table} 
Table 8: Evaluated elastic strain energy density by manual partitioning of the control volume of circular hole-in-plate model with \(\rho=1\) mm (mesh ‘circle_fine_quad_R1 mm_v1_M2_rot0_single’). This difference in elastic strain energy density between the V-notched plate and the circular hole-in-plate was not negligible. As a fine-meshed circular hole-in-plate model with 800 points on the defect loop was used, the corresponding number of elements in the control volume was quite high, as detailed in Table 8. With reference to the results in Table 4, the control volume was evaluated correctly for the hole-in-a-plate model. The evaluated notch stress results in Figure 22a also matched, so, the observed difference was considered to be related to the strain field in the vicinity of the notch. The distinction in ESED value of the V-notched plate, compared to the circular defect, was based on differences in the strain field, which can be visualized by comparing the notch tip displacement in Figure 22b. The circular hole-in-plate exhibited a decreased stiffness in the notch vicinity, with the evaluated change in local displacement at the notch tip being up to 16%. This relates roughly to the strain field as well, which affects the elastic strain energy within the control volume. The evaluated ESED values in Table 8 for the circular hole-in-plate model were, therefore, valid; as the observed difference was caused by a changing local strain field depending on the macro-geometry. Figure 23 depicts the ESED evaluation for the fine-meshed circular hole-in-plate model, including some plots of the elastic strain energy density course. The maximum peak of the ESED result in the sharp notch case was obtained as \(\Delta W_{max,sharp}=1.368\cdot 10^{-5}\) mJ/mm\({}^{3}\) while, for the blunt case, it was \(\Delta W_{max,blunt}=1.126\cdot 10^{-5}\) mJ/mm\({}^{3}\). For an exceedance limit of 97.5%, the calculated elastic strain energy density mean values of the fine-meshed hole-in-plate model were \(\Delta\overline{W}_{sharp}=1.332\cdot 10^{-5}\) mJ/mm\({}^{3}\) and \(\Delta\overline{W}_{blunt}=1.062\cdot 10^{-5}\) mJ/mm\({}^{3}\), accordingly.