## 3 Issues concerning the Implementation of Notch Root Approximations
### Derivatives of the Functions for Use in [PERSON]'s Method  
To apply [PERSON]'s method, not only the functions for the root finding problem, i.e., Equations (10), (14)-(16), (20) and (21), are required but also their derivatives need to be known. In this section only the derivatives for the hysteresis branch formulations of the root finding problems will be discussed, the full list of derivatives are given in Appendix B. In the case of the extended notch root approximation according to [PERSON], the derivative of Equation (10) is relatively easy to determine analytically. It is given in Equation (22). \[\frac{\mathrm{df}}{\mathrm{d}\Delta\sigma_{\mathrm{el,pl}}}=\mathrm{f}^{\prime }\Big{(}\Delta\sigma_{\mathrm{el,pl}}\Big{)}=\frac{1}{\mathrm{E}}+\frac{ \Big{(}\frac{\Delta\sigma_{\mathrm{el,pl}}}{2\cdot\mathrm{K}^{\prime}}\Big{)} ^{1/n^{\prime}-1}}{\mathrm{K}^{\prime}\cdot\mathrm{n}^{\prime}}+\frac{\mathrm{ K}_{\mathrm{p}}\cdot\sigma_{\mathrm{el}}\cdot\Big{(}2\cdot\Big{(}\frac{\Delta \sigma_{\mathrm{el}}}{2\cdot\mathrm{K}^{\prime}\cdot\mathrm{K}_{\mathrm{p}}} \Big{)}^{1/n^{\prime}}+\frac{\Delta\sigma_{\mathrm{el}}}{\mathrm{E}\cdot \mathrm{K}_{\mathrm{p}}}\Big{)}}{\Delta\sigma_{\mathrm{el,pl}}^{2}} \tag{22}\]  
The derivative for the alternative approach, Equation (20), described in the Section 3.1, is given in Equation (23). \[\frac{\mathrm{df}}{\mathrm{d}\Delta\sigma_{\mathrm{el,pl}}}=\mathrm{f}^{\prime }(\Delta\sigma_{\mathrm{el,pl}})=\frac{\frac{\Delta\sigma_{\mathrm{el,pl}}}{ \mathrm{E}}+2\cdot\Big{(}\frac{\Delta\sigma_{\mathrm{el,pl}}}{2\cdot\mathrm{K }^{\prime}}\Big{)}^{1/n^{\prime}}}{\mathrm{K}_{\mathrm{p}}\cdot\Delta\sigma_{ \mathrm{el}}\cdot\Big{(}2\cdot\Big{(}\frac{\Delta\sigma_{\mathrm{el}}}{2 \cdot\mathrm{K}^{\prime}\cdot\mathrm{K}_{\mathrm{p}}}\Big{)}^{1/n^{\prime}}+ \frac{\Delta\sigma_{\mathrm{el}}}{\mathrm{E}\cdot\mathrm{K}_{\mathrm{p}}} \Big{)}}+\frac{\Delta\sigma_{\mathrm{el,pl}}\cdot\Bigg{(}\frac{1}{\mathrm{E}}+ \frac{\Big{(}\frac{\Delta\sigma_{\mathrm{el,pl}}}{2\cdot\mathrm{K}^{\prime} \cdot\mathrm{K}_{\mathrm{p}}}\Big{)}^{1/n^{\prime}-1}}{\mathrm{K}^{\prime}\cdot \mathrm{n}^{\prime}}\Bigg{)}}{\mathrm{K}_{\mathrm{p}}\cdot\Delta\sigma_{\mathrm{el }}\cdot\Big{(}2\cdot\Big{(}\frac{\Delta\sigma_{\mathrm{el}}}{2\cdot\mathrm{K}^{ \prime}\cdot\mathrm{K}_{\mathrm{p}}}\Big{)}^{1/n^{\prime}}+\frac{\Delta\sigma _{\mathrm{el}}}{\mathrm{E}\cdot\mathrm{K}_{\mathrm{p}}}\Big{)}} \tag{23}\]  
The notch root approximation according to [PERSON] and [PERSON] has clearly more extensive expressions in the root finding problem, namely, Equations (14)-(16) and (21). In principle, the derivatives can also be determined analytically. However, practically, the analytical derivative can only be calculated using a computer algebra system (CAS). The derivative of Equation (14) is shown in Equation (24). In this paper, the following criteria are investigated in more detail:  
1. Termination after 10 fixed steps, as suggested in [25]. 2. Termination after falling below a function value \(\mathrm{f}(\sigma_{\mathrm{n}})<\epsilon_{\mathrm{V}}\). 1. using f according to Equations (10) and (14) (i.e., finding the root of the difference of \(\epsilon_{\mathrm{el,pl,RO}}-\epsilon_{\mathrm{el,pl,NRA}}\)) and \(\epsilon_{\mathrm{Vd}}=10^{-7}\). 2. using f according to Equations (20) and (21) (i.e., finding the root of the quotient of \(\epsilon_{\mathrm{el,pl,RO}}/\epsilon_{\mathrm{el,pl,NRA}}-1\)) and \(\epsilon_{\mathrm{Vq}}=10^{-3}\). Variants 2a and 2b aim to reach a defined accuracy for the solution, while no prediction can be made about the required number of iterations. On the other hand, the number of iterations is fixed for variant 1, but no control of the calculation accuracy is provided. The concrete threshold levels for termination criteria 2a and 2b were determined based on the following considerations:  
* A small remaining strain deviation in criterion 2a has a high influence, especially at low stresses.