# Numerical-Experimental Analysis to Optimize the Geometry of a Prosthetic Abtment
## 2 Preliminary Theoretical Considerations  
The correct geometric preparation of the abutment must meet specific geometric and morphological requirements to ensure anchor stability and prevent destabilization from the forces generated during the chewing process. In evaluating the phenomena of destabilization of the prosthetic product, it is necessary to take into account the extent of external forces, their point of application, the position of the fulcrum, and the location and the extent of resistance areas on the abutment surface. \begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline
**Abutment Marginal Design Profile** & **Name** & **Differences** \\ \hline \hline \multirow{4}{*}{Knife-Edge} & Features a very thin, sharp margin that tapers to a point. It is often used for esthetic restorations but may provide less resistance to fracture. \\ \cline{2-3}  & Feather-Edge & Similar to the knife edge, this design has a slight bevel that creates a thin margin. It provides a transition between the tooth structure and the restoration, often used in esthetic areas. \\ \hline \hline \multirow{4}{*}{Chamfer} & Characterized by a rounded or beveled edge, this design offers a smooth transition and is less prone to fracture. It is commonly used for crowns and provides good retention \\ \cline{2-3}  & Shoulder & Features a flat, horizontal margin that provides a strong, well-defined edge. This design is ideal for porcelain or metal–ceramic restorations, offering excellent support and esthetics \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Abutment marginal designs. In particular, in cases where a cusp is made at the _interface of the abutment_, the resistance areas of a prosthetic abutment are divided into axial and cusp areas, depending on whether the resistance is opposed by the sidewall or by the internal cusp of the abutment. For the localization of the axial and cusp resistance areas, the reference parameters are the base width, the height and taper of the abutment, and the length of the cusps at the interface. The basic assumptions of the theories proposed in the scientific literature are the following [27]:  
* Perfect adherence between the internal surface of the anchor and the abutment surface along the entire interface;
* Abutment and prosthetic structure materials considered as perfectly rigid;
* Destabilizing force directed parallel to the occlusal plane;
* Plane stress state. Figure 1 shows the geometric references of the prosthetic abutment. In the cross-section, the following parameters are indicated: the upper vertices (\(V_{c1}\) and \(V_{c2}\)) and the intersection of the cusp walls (\(C\)), the height of the abutment \(h\), and the cusp height \(h_{c}\). \(\gamma\) is the slope of the cusp walls. The semicircle, having as center \(O_{1}\) and diameter the base \(b\) of the abutment (\(F_{1}F_{2}\)), intersects the axial surface at point \(T_{1}\). \(F_{1}\) represents the fulcrum around which the crown rotates with a radius \(r_{a}\) (left rotation arc as highlighted as a dashed line). The plane parallel to the base and passing through \(T_{1}\) intersects the abutment, determining a trace on the axial surface called the axial tangent line. This line divides the axial surface into two portions, an upper and a lower one. The upper portion, opposing the rotation and, therefore, the dislodgement of the crown, constitutes the axial resistance area; the lower one provides no resistance. The semicircle centered in the midpoint \(O_{2}\) of the axial wall intersects the cusp wall in \(T_{2}\). In this way, the cusp tangent line is identified. The cusp resistance area is the portion between the trace \(T_{2}\) of the cusp tangent line and \(V_{c2}\). The arm of the cusp resisting moment is \(r_{c}\) (even this rotation arc is highlighted as a dashed line). The cusps must be appropriately shaped on the abutment in order to increase the overall resistance area and, therefore, the stability of the coupling [12].