## 2 Notch Root Approximations in the FKM Guideline Nonlinear
### Extended Version of Neuber's Rule  
The original version of [PERSON]'s rule [27] postulates, assuming elastic net-section behavior, the equality of the products of stress and strain for linear-elastic and elastic-plastic material behavior, Equation (7):  
\[\sigma_{\text{el,pl}}\ \cdot\ \varepsilon_{\text{el,pl}}=\sigma_{\text{el}}\ \cdot\ \varepsilon_{\text{el}} \tag{7}\]  
As shown by [PERSON] et al. [28; 30; 31; 32] but also noted by [PERSON] and [PERSON] [42], this equation must be modified to account for high loads. To take into account the increasing net-section plasticity, [PERSON] and [PERSON] proposed a modification of the Neuber equation from Equation (7):  
\[\sigma_{\text{el,pl}}\cdot\ \varepsilon_{\text{el,pl}}=\sigma_{\text{el}}\cdot \varepsilon_{\text{el}}^{*}\cdot\text{K}_{\text{p}} \tag{8}\]  
Figure 3: Creation of a discretized load-notch-strain-curve via notch root approximations [40].  
The use of \(\varepsilon_{\text{el}}^{*}\cdot\text{K}_{\text{p}}\) modifies the linear-elastic material behavior so that on the one hand, the real material behavior and on the other hand, the increasing net-section plasticity are taken into account [28; 40]. The net section plasticity is modelled by \(\varepsilon_{\text{el}}^{*}\) using fictitious nominal stresses, obtained by means of the linear-elastic local stress \(\sigma_{\text{el}}\) and the limit load factor \(\text{K}_{\text{p}}\), and inserting these into the material law. In this case the Ramberg and Osgood Equation (3) approach is used.  
\[\varepsilon_{\text{el}}^{*}=\frac{\sigma_{\text{el}}/\text{K}_{\text{p}}}{ \text{E}}+\left(\frac{\sigma_{\text{el}}/\text{K}_{\text{p}}}{\text{K}^{\prime }}\right)^{1/n^{\prime}} \tag{9}\]  
As mentioned above the load-notch-strain curve for initial load can be derived from the load-notch-strain curve of the hysteresis branch. Therefore, in the following only the procedure and the root finding equations for the hysteresis branch are explained. The detailed derivation, which also includes the formulas for the initial load curve, is shown in Appendix A. Equation (8) applied to the hysteresis branch can be transformed, as described above, into the root finding problem specified in the FKM Guideline nonlinear Equation (10).  
\[\text{f}(\Delta\sigma_{\text{el,pl}})=0= \frac{\Delta\sigma_{\text{el,pl}}}{\text{E}}+2\,\cdot\,\left( \frac{\Delta\sigma_{\text{el,pl}}}{2\cdot\text{K}^{\prime}}\right)^{1/n^{ \prime}}-\,\left(\frac{\Delta\sigma_{\text{el}}}{\Delta\sigma_{\text{el,pl}} }\cdot\text{K}_{\text{p}}\cdot\left(\frac{\Delta\sigma_{\text{el}}/\text{K}_ {\text{p}}}{\text{E}}+2\cdot\left(\frac{\Delta\sigma_{\text{el}}/\text{K}_{ \text{p}}}{2\cdot\text{K}^{\prime}}\right)^{1/n^{\prime}}\right)\right) \tag{10}\]  
Equation (10) has to be solved for the 200 discrete required load steps of the load-notch-strain curves, respectively. The FKM Guideline nonlinear specifies [PERSON]'s method for this purpose. (Other solution methods would also be possible, but since this paper focuses on the FKM Guideline nonlinear, only [PERSON]'s method will be investigated, as specified).  
[PERSON]'s method is a numerical solution method in which, starting from an initial value \(\text{x}_{0}\), the following iteration (iteration step n) is executed up to a termination criterion:  
\[\text{x}_{n+1}=\text{x}_{n}-\frac{\text{f}(\text{x}_{n})}{\text{f}^{\prime}( \text{x}_{n})} \tag{11}\]  
The objective is to find the root of the function f(x).  
In the FKM Guideline nonlinear [25], the start values of iterations \(\Delta\sigma_{0}\) are defined to be the corresponding elasticity-theoretical solutions \(\Delta\sigma_{\text{el}}\). The iteration is terminated after a fixed number of ten steps.  
To apply [PERSON]'s method, Equation (11), the user must first determine the derivatives f\({}^{\prime}\) of the function f from Equation (10).  
With the value for the stresses \(\Delta\sigma_{\text{el,pl}}\) determined in this way, the corresponding strain value can be calculated using Equation (5).