## Appendix B  
In this section, the determination of the parameters \(\text{a}_{i}\), \(\text{b}_{i}\), \(\text{c}_{i}\), and \(\text{d}_{i}\) for the spline interpolation in Section 3.4.3 is described. Boundary conditions are defined for the individual sections of the function: to avoid jumps occurring in the load-notch-strain curve, the strain ranges at the beginning of each section must match those at the end of the previous section, and the strain ranges at the end of each section must match those at the beginning of the following section; see Equations (10) and (11). These conditions correspond to the boundary conditions of interpolation with a polygonal chain; see Section 3.4.1. In addition, Equation (12) guarantees a kink-free transition at the support points, and Equation (13) generates a two-times continuously differentiable function:  
\[\Delta\varepsilon_{\text{S},i}(\Delta\sigma_{\text{el,FEA},i})=\Delta \varepsilon_{\text{FEA},i}\] with i \[=1,\ldots,n_{\text{H}} \tag{20}\]  
\[\Delta\varepsilon_{\text{S},i}(\Delta\sigma_{\text{el,FEA},i+1})=\Delta \varepsilon_{\text{FEA},i+1}\] with i \[=1,\ldots,n_{\text{H}} \tag{21}\]  
\[\Delta\varepsilon_{\text{S},i}^{\prime}(\Delta\sigma_{\text{el,FEA},i+1})= \Delta\varepsilon_{\text{S},i+1}^{\prime}(\Delta\sigma_{\text{el,FEA},i+1})\] with i \[=1,\ldots,n_{\text{H}} \tag{22}\]  
\[\Delta\varepsilon_{\text{S},i}^{\prime\prime}(\Delta\sigma_{\text{el,FEA},i+1})= \Delta\varepsilon_{\text{S},i+1}^{\prime\prime}(\Delta\sigma_{\text{el,FEA},i+1})\] with i \[=1,\ldots,n_{\text{H}} \tag{23}\]In order to determine the parameters \(\text{a}_{i}\), \(\text{b}_{i}\), \(\text{c}_{i}\), and \(\text{d}_{i}\) with a linear system of equations, two further boundary conditions must be defined. At the left edge of the load-notch-strain curve, the modulus of elasticity E is specified as the gradient; at the right edge, the load-notch-strain curve should end with a point of inflection. This results in:  
\[\Delta\epsilon_{\text{S,1}}^{\prime}(\Delta\sigma_{\text{e,l,FEA,l}}=0)=\frac{ 1}{\text{E}}\] (A16)  
\[\Delta\epsilon_{\text{S,n}}^{\prime\prime}\big{(}\Delta\sigma_{\text{e,l,FEA,n }_{\text{H}}+1}\big{)}=0\] (A17)  
With the linear system of equations in Equation (A18), the vector \(\mathbf{b}\) containing the parameters \(\text{b}_{i}\) is calculated:  
\[=1,\ldots,n_{\text{H}} \tag{20}\]0 (A18)  
The matrix \(\mathbf{A}\) and the vectors \(\mathbf{b}\) and \(\mathbf{g}\) are defined as follows:  
\[=1,\ldots,n_{\text{H}} \tag{20}\]1 (A19)  
\[\mathbf{A}=\left[ \begin{array}{cccc}1&0.5&&&\\ \text{h}_{1}&2\ (\text{h}_{1}+\text{h}_{2})&\text{h}_{2}&&\\ &&\ddots&&\\ &&&\text{h}_{\text{n}_{\text{H}}-1}&2\ (\text{h}_{\text{n}_{\text{H}}-1}+\text{h}_{ \text{n}_{\text{H}}})&\text{h}_{\text{n}_{\text{H}}}\\ &&&1\end{array} \right]\] (A20)  
\[\mathbf{b}=\left[ \begin{array}{cccc}\text{b}_{1}&\\ \text{b}_{2}&\\ \vdots&\\ \text{b}_{\text{n}_{\text{H}}}&\\ 0&\end{array} \right]\] (A21)  
\[\mathbf{g}=\left[ \begin{array}{cccc}\text{g}_{1}&\\ \text{g}_{2}&\\ \vdots&\\ \text{g}_{\text{n}_{\text{H}}}&\\ 0&\end{array} \right]\] (A22)  
\[\mathbf{g}_{1}=\frac{3}{2}\bigg{(}\frac{\Delta\epsilon_{2}-\Delta\epsilon_{1} }{\text{h}_{1}}-\frac{1}{\text{E}\cdot\text{h}_{1}}\bigg{)}\] (A23)  
\[\mathbf{g}_{i}=3\ \bigg{(}\frac{\Delta\epsilon_{i+1}-\Delta\epsilon_{i}}{ \text{h}_{i}}-\frac{\Delta\epsilon_{i}-\Delta\epsilon_{i-1}}{\text{h}_{i-1}} \bigg{)}\quad\text{with i}=2,\ldots,\text{n}_{\text{H}}\] (A24)  
The remaining parameters \(\text{a}_{i}\), \(\text{c}_{i}\), and \(\text{d}_{i}\) are calculated with Equations (A25)-(A27):  
\[\text{a}_{i}=\frac{\text{b}_{i+1}-\text{b}_{i}}{3\ \text{h}_{i}}\qquad\qquad \qquad\text{with i}=1,\ldots,\text{n}_{\text{H}}\] (A25)  
\[\text{c}_{i}=\frac{\text{b}_{i+1}-\text{b}_{i}}{\text{h}_{i}}-\frac{2\ \text{b}_{i}+\text{b}_{i+1}}{3}\ \text{h}_{i}\qquad\text{with i}=1,\ldots,\text{n}_{\text{H}}\] (A26)  
\[\text{d}_{i}=\Delta\epsilon_{i}\qquad\qquad\qquad\text{with i}=1,\ldots,\text{n }_{\text{H}}\] (A27)