## Appendix B Derivatives of the Root Finding Equations  
In this section the analytical derivatives of the different root finding equations, derived in the Appendix A are listed. In the case of the extended notch root approximation according to Neuber, the derivatives of Equation (A5) (initial load curve) and Equation (A6) (hysteresis branch) are given in Equations (A20) and (A21). \[\text{f}^{\prime}\!\left(\sigma_{\text{el,pl}}\right)=\frac{1}{\text{E}}+\frac {\left(\frac{\sigma_{\text{el,pl}}}{\text{K}^{\prime}}\right)^{1/n^{\prime}-1 }}{\text{K}^{\prime}\cdot n^{\prime}}+\frac{\text{K}_{\text{p}}\cdot\sigma_{ \text{el}}\cdot\left  
\[ \begin{array}{ll}\frac{\mathrm{df}}{\mathrm{d}\sigma_{\mathrm{el,pl}}}=\mathrm{f}^{ \prime}\Big{(}\sigma_{\mathrm{el,pl}}\Big{)}=&\frac{1}{\mathrm{E}}+\frac{\left( \frac{\sigma_{\mathrm{el,pl}}^{\mathrm{d}}}{\mathrm{K}^{\prime}}\right)^{1/n^{ \prime}-1}}{\mathrm{K}^{\prime}\cdot n^{\prime}}-\frac{1}{\sigma_{\mathrm{el}} }\cdot\left(\mathrm{K}_{\mathrm{p}}\cdot\epsilon_{\mathrm{el}}^{\ast}\cdot \left(\frac{8\sigma_{\mathrm{el}}^{\mathrm{d}}\ln\left(\frac{1}{\mathrm{cos}( \mathrm{u})}\right)\cdot\left(\mathrm{K}_{\mathrm{p}}-1\right)^{2}}{\sigma_{ \mathrm{el,pl}}^{2}\cdot n^{2}\cdot\left(\frac{\sigma_{\mathrm{el}}}{\sigma_{ \mathrm{el,pl}}^{2}}-1\right)^{2}}-\frac{\sigma_{\mathrm{el}}}{\sigma_{ \mathrm{el,pl}}}+1\right)\right)\\ &-\frac{\mathrm{K}_{\mathrm{p}}\cdot\epsilon_{\mathrm{el}}^{\ast}\cdot\sigma_ {\mathrm{el,pl}}}{\sigma_{\mathrm{el}}}\\ &\cdot\left(\frac{\sigma_{\mathrm{el}}}{\sigma_{\mathrm{el,pl}}^{\ast}}-\frac{ 16\sigma_{\mathrm{el}}^{2}\cdot\ln\left(\frac{1}{\mathrm{cos}(\mathrm{u})} \right)\cdot\left(\mathrm{K}_{\mathrm{p}}-1\right)^{2}}{\sigma_{\mathrm{el,pl} }^{3}\cdot n^{2}\cdot\left(\frac{\sigma_{\mathrm{el}}}{\sigma_{\mathrm{el,pl}} ^{\ast}}-1\right)^{2}}+\frac{16\sigma_{\mathrm{el}}^{3}\cdot\ln\left(\frac{1}{ \mathrm{cos}(\mathrm{u})}\right)\cdot\left(\mathrm{K}_{\mathrm{p}}-1\right)^{2 }}{\sigma_{\mathrm{el,pl}}^{4}\cdot n^{2}\cdot\left(\frac{\sigma_{\mathrm{el}} }{\sigma_{\mathrm{el,pl}}^{\ast}}-1\right)^{3}}-\frac{4\sigma_{\mathrm{el}}^{ 3}\cdot\sin(\mathrm{u})\cdot\left(\mathrm{K}_{\mathrm{p}}-1\right)}{\sigma_{ \mathrm{el,pl}}^{4}\cdot n^{2}\cdot\left(\frac{\sigma_{\mathrm{el}}}{\sigma_{ \mathrm{el,pl}}^{\ast}}-1\right)^{2}}\right)\\ \end{array}  \tag{100}\]  
\[ \begin{array}{ll}\frac{\mathrm{df}}{\mathrm{d}\delta\sigma_{\mathrm{el,pl}}}= \mathrm{f}^{\prime}(\Delta\sigma_{\mathrm{el,pl}})=&\frac{1}{\mathrm{E}}+ \frac{\left(\frac{\Delta\sigma_{\mathrm{el,pl}}}{\mathrm{K}^{\prime}}\right)^{1 /n^{\prime}-1}}{\mathrm{K}\cdot n^{\prime}}-\frac{1}{\Delta\sigma_{\mathrm{el}} }\left(\mathrm{K}_{\mathrm{p}}\cdot\Delta\epsilon_{\mathrm{el}}^{\ast}\cdot \left(\frac{8\alpha_{\mathrm{el}}^{\ast}\ln\left(\frac{1}{\mathrm{cos}( \mathrm{u})}\right)\cdot\left(\mathrm{K}_{\mathrm{p}}-1\right)^{2}}{\Delta \sigma_{\mathrm{el,pl}}^{2}\cdot n^{2}\cdot\left(\frac{\sigma_{\mathrm{el}}}{ \sigma_{\mathrm{el,pl}}^{\ast}}-1\right)^{2}}-\frac{\Delta\sigma_{\mathrm{el}} }{\Delta\sigma_{\mathrm{el,pl}}}+1\right)\right)\\ &\cdot\left(\frac{\Delta\sigma_{\mathrm{el}}}{\Delta\sigma_{\mathrm{el}}}- \frac{16\alpha\sigma_{\mathrm{el}}^{2}\cdot\ln\left(\frac{1}{\mathrm{cos}( \mathrm{u})}\right)\cdot\left(\mathrm{K}_{\mathrm{p}}-1\right)^{2}}{\Delta \sigma_{\mathrm{el,pl}}^{4}\cdot n^{2}\cdot\left(\frac{\sigma_{\mathrm{el}}}{ \sigma_{\mathrm{el,pl}}^{\ast}}-1\right)^{3}}-\frac{4\alpha_{\mathrm{el}}^{3} \cdot\sin(\Delta\mathrm{u})\cdot\left(\mathrm{K}_{\mathrm{p}}-1\right)}{\Delta \sigma_{\mathrm{el,pl}}^{4}\cdot n^{2}\cdot\left(\frac{\sigma_{\mathrm{el}}}{ \sigma_{\mathrm{el,pl}}^{\ast}}-1\right)^{3}}\right)\\ \end{array}  \tag{101}\]  
In case of the notch root approximation according to [PERSON] and [PERSON] the derivative of the quotient formulation of the root finding problem Equation (100) (initial load curve) is shown in Equation (101) and of Equation (101) (hysteresis branch curve and quotient formulation) in Equation (102).