## 2 Materials and Methods
### Fracture Mechanical Investigations  
Considering Equation (3), the control radius \(R_{c}\) necessary for application of the linear-elastic SED concept depends on the long-crack growth threshold stress intensity factor range \(\Delta K_{th,lc}\). Hence, fracture mechanical investigations in terms of crack propagation experiments were conducted to evaluate \(\Delta K_{th,lc}\) for the investigated stress ratios of \(R=-1,0\) and \(0.5\). Single-edge-notched bending samples (SENB) were utilized, as depicted in Figure 6.  
After milling of the specimens from bulk material, the initial notch featuring a width of \(0.35\,\mathrm{mm}\) and a notch depth of \(4\,\mathrm{mm}\) was manufactured by wire-cut electrical discharge machining, followed by polishing of the sides of the sample geometry. The polishing procedure assisted the measurement of the initial notch geometry by microscopy. In order to achieve a defined pre-crack necessary for subsequent crack propagation experiments, the specimens were razor-blade polished utilizing diamond polishing suspension. The evaluation of \(\Delta K_{th,lc}\) was conducted according to the rising load amplitude crack growth test procedure recommended by [PERSON] and [PERSON] [48]. This procedure applies compressive pre-cracking [49; 50] of respective specimens, which was carried out on a SincoTe\({}^{\text{\textregistered}}\) Power Swing MOT100 kN (SincoTe Test Systems GmbH, Clausthal-Zellerfeld, Germany) resonance test rig at a stress ratio of \(R=20\). Subsequently, crack propagation experiments were conducted under constant amplitude loading at room temperature, using a RUMUL\({}^{\text{\textregistered}}\) Cracktrronic (Russenberger Pri\(\ddot{\text{u}}\)rmaschinen AG, Neuhausen am Rheinfall, Switzerland) resonance test rig. Utilizing a constant stepping of the applied load, the stress intensity factor range \(\Delta K\) was increased until long-crack propagation occurred. Crack propagation was measured by the direct current potential drop method, incorporating compensation for crack length fluctuations due to ambient temperature changes [51]. Figure 7 depicts crack propagation curves of three representative experiments for the investigated stress ratios.  
Figure 6: SENB geometry utilized for crack propagation experiments.  
Figure 7: Crack propagation curves for investigated stress ratios.  
Regarding Figure 7, the influence of \(R\) on the propagation behaviour of fatigue cracks can be perceived clearly. As the value of \(R\) increases, contact of the crack flanks during cyclic loading is restricted [52]. Hence, the build-up of crack closure mechanisms, such as plasticity-induced or roughness-induced crack closure, is less pronounced as the crack transitions from short- to long-crack growth, resulting in lower values of \(\Delta K_{th,lc}\)[53; 54; 55].  
The experimental results were evaluated for the investigated stress ratios, where \(\Delta K_{th,lc}\) was estimated by a least-mean-squares regression of the NASGRO Equation [54]. The determined values of \(\Delta K_{th,lc}\), as well as the number of specimens tested, are listed in Table 3.  
As the effect of crack closure mechanisms is less pronounced at higher values of \(R\), the variation of derived threshold values is also reduced, which is given in Table 3 by comparing the min-max range of obtained stress intensity factors. Although the experimental data shows only small scatter indices, the arithmetic mean value of \(\Delta K_{th,lc}\) for each investigated stress ratio was utilized for calculations reported in the following sections.