## 3 Functionality of Key Modules
### Local Fatigue Assessment
#### 3.4.3 Elastic Strain Energy Density
Figure 25: ESED evaluation of coarse-meshed circular hole-in-plate (mesh ‘circle_lin_R1 mm_v1’) by scalar superposition: (**a**) Contour plots for load angle of \(\varphi=0,\ 45,\ 90,\ 135^{\circ}\); and (**b**) ESED plots along defect contour. The cell-based computation of the stress \(\overline{\overline{\sigma}}(\varphi)\) and strain tensor \(\overline{\overline{\overline{\epsilon}}}(\varphi)\) involves the calculation of the elastic strain energy density tensor \(\Delta\overline{\overline{W}}(\varphi)\). The maximum principal value is obtained by eigenvalue decomposition of the matrix. \[\Delta\overline{\overline{W}}(\varphi)=\frac{\overline{\overline{\sigma}}( \varphi)\cdot\overline{\overline{\epsilon}}(\varphi)}{2} \tag{13}\]  
This additional calculation is done within the transformation step of the Abaqus(r) output odd-database to load angle referencing mat-files by extending the'readcae_result.oml' script for the superposition method. Other program modules are not affected. To visualize this cell-based computation of the elastic strain energy density within post-processing, additional output files can be generated by querying the element results in HyperView(r). Figure 26 depicts the tensor-based superposition results for the circular hole-in-plate model, as introduced in Figure 7a. As the tensors are computed per element, nodal averaging is not set to be active in these sub-figures. Each sub-figure depicts the maximum principal values of the element results. As the elastic strain energy density fatigue assessment is applicable for both methods, the result distribution was examined for the three basic geometries (circular hole, elliptical hole, and elongated defect). The load angle step was set to \(\Delta\varphi=10^{\circ}\) for the elastic strain energy density simulation of the three defect geometries. The averaged ESED value was computed for each control volume \(\Omega\), as defined by the corresponding point of the defect contour. Assessing the points of the defect contour as a sharp notch implies that each control volume originates at the corresponding defect point. Figure 27 summarizes the obtained ESED charts for the sharp notch case. For the circular hole-in-plate model, the mean values of the elastic strain energy density distribution in the sharp notch case were \(\Delta\overline{W}_{sharp,M2,CIR}=1.256\times 10^{-5}\) mJ/mm\({}^{3}\) and \(\Delta\overline{\overline{W}}_{sharp,M3,CIR}=1.156\times 10^{-5}\) mJ/mm\({}^{3}\). The evaluated standard deviations were negligible for this geometry, for both simulation methods. The elliptical hole-in-plate model led to mean values of \(\Delta\overline{W}_{sharp,M2,ELL}=2.012\cdot 10^{-5}\) mJ/mm\({}^{3}\) and \(\Delta\overline{W}_{sharp,M3,ELL}=1.769\times 10^{-5}\) mJ/mm\({}^{3}\), with standard deviation values of \(\Delta\tilde{W}_{sharp,M2,ELL}=0.972\times 10^{-5}\) mJ/mm\({}^{3}\) and \(\Delta\tilde{W}_{sharp,M3,ELL}=0.872\times 10^{-5}\) mJ/mm\({}^{3}\). As the ellipse had a varying curvature, the ESED changed along the defect circumference for each load vector, accordingly. Figure 26: ESED evaluation of coarse-meshed circular hole-in-plate (mesh ‘circle_lin_R1 mm_v1’) by tensor superposition: (**a**) Stress contour plots; (**b**) Strain contour plots; and (**c**) Elastic strain energy density contour plots for load angle of \(\varphi=0,\ 45,\ 90,\ 135^{\circ}\). The elongated defect exhibited the highest elastic strain energy density results, with mean values of \(\Delta\overline{W}_{sharp,MB.DEF}=3.481\times 10^{-5}\) mJ/mm\({}^{3}\) and \(\Delta\overline{W}_{sharp,MB.DEF}=2.917\times 10^{-5}\) mJ/mm\({}^{3}\), while the standard deviations were \(\Delta\hat{W}_{sharp,MB.DEF}=1.604\times 10^{-5}\) mJ/mm\({}^{3}\) and \(\Delta\hat{W}_{sharp,MB.DEF}=1.424\times 10^{-5}\) mJ/mm\({}^{3}\). The local fatigue strength relations of the three geometries, in terms of ESED, matched the observed stress-based TCD results.