## 2 Notch Root Approximations in the FKM Guideline Nonlinear  
Two different notch root approximations are available in the calculation algorithm of the FKM Guideline nonlinear:  
* Notch root approximation according to [PERSON] [27] in the variant according to [28]. * Notch root approximation according to [PERSON] and [PERSON] [30; 31; 32]. Both notch root approximations are used in the FKM Guideline nonlinear in such a way that they establish the relationship between a linear-elastically determined stress and the local elastic-plastic stresses and strains. The elastic-plastic stress state is thus estimated. This relationship between the linear-elastic stress and elastic-plastic stress is called the load-notch-strain curve of the component. This is used as a template with discrete values for the respective stresses in the Rainflow HCM algorithm [33] to simulate the path of the local stresses and strains depending on the path of the linear-elastic stress (load sequence)  
Figure 1: Estimation of local elastic-plastic stresses and strains from linear-elastic stresses using a notch root approximation (schematically). and to identify closed hysteresis loops. The HCM algorithm takes into account the effects of [PERSON]'s behavior during load reversal [34] and material memory [1; 33]. The FKM Guideline nonlinear uses the approach according to Ramberg and Osgood to describe the elastic-plastic material behavior [35]. \[\varepsilon_{\text{el,pl}}=\frac{\sigma_{\text{el,pl}}}{\text{E}}+\left(\frac{ \sigma_{\text{el,pl}}}{\text{K}^{\prime}}\right)^{\frac{1}{\eta}} \tag{3}\]  
Here, E is Young's modulus, K' is the cyclic hardening coefficient, and n' is the cyclic hardening exponent. The determination of these parameters can be carried out either experimentally or by using estimation methods. For the evaluation of strain-controlled tests, the FKM Guideline nonlinear suggests a procedure based on [36] and [37]. In addition, a method developed by [PERSON] et al. [38; 39] for the estimation of the cyclic properties using tensile strength is provided for use. This enables an analytical fatigue assessment to be carried out without the need for expensive experiments. Using the cyclic parameters K' and n', it is possible to calculate a cyclic yield strength R'p0,2 as in Equation (4). R'p0,2 will later be of importance to estimate a maximum load. \[\text{R}^{\prime}_{\text{p0.2}}=0.002^{n^{\prime}}\cdot\text{K}^{\prime} \tag{4}\]  
For the material behavior on a hysteresis branch, the doubled curve applies with regard to stress and strain because of [PERSON]'s behavior, illustrated in Figure 2:  
\[\Delta\varepsilon_{\text{el,pl}}=\frac{\Delta\sigma_{\text{el,pl}}}{\text{E}} +2\cdot\left(\frac{\Delta\sigma_{\text{el,pl}}}{2\cdot\text{K}^{\prime}} \right)^{\frac{1}{\eta}} \tag{5}\]  
Due to this different behavior at the initial load and on load reversal, two load-not-strain curves have to be determined. This can be performed before applying the HCM algorithm. In the FKM Guideline nonlinear, a total of 100 values with equally spaced intervals between the respective elasticity-theoretical values are specified for the discretization of the load-not-strain curve for the initial load. According to Equation (5), the number of values for the hysteresis branch must be twice as large and amount to 200. The procedure is shown schematically for only eight values in Figure 3.  
Figure 2: Schematic illustration of [PERSON]’s behavior (**a**) and the shape of the stress-strain curve due to [PERSON]’s behavior (**b**). The load-notch-strain curves of the component for the initial load and for the hysteresis branch are related. It is possible to derive the initial load-notch-strain curve from the load-notch-strain curve of the hysteresis branch.