## 3 Results  
The major mode of deformation induced by the masticatory force F (Figures 3a and 7) was bending, leading to compressive- and tensile-loaded zones of the NiTi shaft. The surfaces at the SMA shaft where compressive stresses were dominant were named oral according to their location in the oral cavity (right patient side: O\({}_{1}\); left patient side: O\({}_{2}\)), while the opposite sides were associated with tensile stresses and named buccal (right patient side: B\({}_{1}\); left patient side: B\({}_{2}\)) (Figure 7). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{\(g\) Function (Fitting) Parameters} \\ \hline
**Param./[Unit]** & \(g_{a}\) [-] & \(g_{b}\) [-] & \(g_{c}\) [-] & \(g_{d}\) [-] & \(g_{e}\) [-] & \(g_{f}\) [-] & \(g_{o}\) [MPa] \\ \hline Values & \(-2\) & \(2\) & \(0\) & \(2.75\) & \(0\) & \(3\) & \(300\) \\ \hline
**Param./[unit]** & \(M_{s}\) [\({}^{\circ}\)C] & \(M_{f}\) [\({}^{\circ}\)C] & \(A_{s}\) [\({}^{\circ}\)C] & \(A_{f}\) [\({}^{\circ}\)C] & \(C_{M}\) [MPa/\({}^{\circ}\)C] & \(C_{A}\) [MPa/\({}^{\circ}\)C] \\ \hline Values & \(-63\) & \(-65\) & \(-13\) & \(-10\) & \(5.45\) & \(6.25\) \\ \hline
**Param./[unit]** & \(E_{M}\) [**GPa**] & \(E_{A}\) [**GPa**] & \(v_{M}\) [-] & \(v_{A}\) [-] & \(e_{M}^{T}\) [-] & \\ \hline Values & \(19\) & \(61\) & \(0.33\) & \(0.33\) & \(0.049\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters of the SMA constitutive model, for more details, refer to supplementary material in ref [30]. Figure 6: The uniaxial mechanical response obtained by fitting parameters of the FEM model to the experimental results recorded by the manufacturer of the prototype NiTi attachment system (ADMEDES GmbH, Pforzheimer, Germany), and considering the assumed tension-compression asymmetry [32,33]. Force application in rising magnitude resulted in two subsequent stages of deformation. In the first stage, it caused a maximum vertical displacement of 5 mm at the posterior end of the prosthesis until mechanical contact with the binding sheet (Figure 8). In this stage, the dominant deformation observed was the bending of the SMA shafts. In the course of contact, at the second stage, the continuation of the loading induced compressive stresses on the SMA shafts. The surface distribution of the axial stresses and martensite volume fractions at maximum load for all studied geometries (parallel and non-parallel implant fixation for prototype abutment design, as well as the non-parallel fixation for the optimized/alternative designs) are given in Figure 9. Throughout this section, the term axial denotes the z direction of the reference cartesian system (Figure 3). Here, the SMA material is used for the male part, shaft, and base parts. Figure 9(a1-d1) show the axial stress at surfaces that are dominantly under compressive stresses (Oral Surfaces) and denoted by O\({}_{1}\) and O\({}_{2}\), while in Figure 9(a2-d2,a3-d3) their corresponding axial strain and volume fraction of detwinned martensite are depicted, respectively. In contrast, Figure 9(a4-d4) exhibit the axial stress distribution at Buccal Surfaces (denoted by B\({}_{1}\) and B\({}_{2}\)), and their associated strains and martensite volume fractions are represented in Figure 9(a5-d5) and Figure 9(a6-d6), respectively. In Figure 10 the corresponding stress-strain responses at representative points (maximum martensite fraction points) at Oral and Buccal surfaces from Figure 9a-d are represented. For comparative purposes, Figure 11 represents the case study in which the material for the SMA parts was replaced by titanium (in the non-parallel fixation case study, see Figure 4). Figure 8: Vertical prosthesis displacement at the posterior border was limited to 5 mm by simulating a binding sheet. Figure 7: (a) Iso view of the FE model showing the SMA shaft (purple) with bending-induced zones (**a**) \(O_{1}\) and \(O_{2}\) dominated by compressive stresses (at _Oral Surface_), (**b**) \(B_{1}\) and \(B_{2}\) dominated by tensile stresses (at _Buccal Surface_).