## 3 Results
The statistics for the maximum reached torsion torques \(T_{\text{max}}\) and twisting fracture angle \(\theta_{\text{max}}\) for each type of instrument are obtained from Table 1 and are presented in Figure 6. Insignificant differences can be remarked between the three study groups both in the case of _maximum torque_--for which Endostar values are insignificantly increased (Kruskal-Wallis test, \(p\) = 0.284), and in the case of _maximum twist_--for which Protaper values are insignificantly increased (Kruskal-Wallis test, \(p\) = 0.108). This allows for ranking the mechanical behavior of the different available instruments in case of blocking their tip. The higher torque values \(T_{\text{max}}\) for the Endostar instruments, at a smaller values of the twisting angle \(\theta_{\text{max}}\) shows a good mechanical strength of these instruments, but also a brittle behavior. On the contrary, the Protaper instruments have a ductile character at fracture (as shown by the higher values of \(\theta_{\text{max}}\)) but lower values of the mechanical strength (as shown by the smaller values of \(T_{\text{max}}\)). The Reciproc instruments are in-between the two other types, with satisfactory (but not outstanding) values regarding both investigated parameters. To observe the sections aspect and thus to be able to calculate the shear stresses, microscopic investigations of the fracture sections for each type of instrument were performed. They are presented in Figure 7 for an example from each group, with the remark that images obtained for all instruments in each group are similar. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multicolumn{2}{c}{**Group Endostar**} & \multicolumn{2}{c}{**Group Reciproc**} & \multicolumn{2}{c}{**Group Protaper**} \\ \cline{2-7}  & \(T_{\text{max}}\) (\(\text{N}\cdot\text{cm  
In these microscopic images the aspect of the structure in the fracture section of each instrument can be seen, as well as the differences in size and shape of each instrument. The crystalline shiny aspect areas confirm fragile fracture zones, while the voids and the waves-type marks of the surface indicate ductile fracture caused by overloads. In addition, on every sample, traces of inter-crystalline slide with circular orientations characteristic to fracture by torsion can be observed. Regarding the capacity of the cross-section to withstand torsion, the optimal shape belongs to Endostar instruments, as they have the most circular-like shape from the three investigated groups. The shape of the cross-section of Protaper instruments is the most detrimental regarding torsion, as it is almost rectangular. These aspects are further detailed in Figure 8, where the presentation starts from the initial shape of the three types of instruments (i.e., Endostar, Reciproc, and Protaper) in the first row, continues with their lateral aspect after fracture in the second row, and is finalized with the geometrical approximation of the fracture sections in the third row (all images being obtained with the optical microscope set at a 25\(\times\) magnification). The failure angle that can be observed in Figure 8(a2,b2,c2) is not specific either to fragile fracture (as the section is not oblique on the longitudinal axis) nor to ductile fracture (as the section is not perpendicular on the longitudinal axis, either) for none of the three instruments. Therefore, the studied endodontic instruments can be considered similar from the ductile-fragile point of view. The maximum tangential stresses \(\tau_{\text{max}}\) were determined using the values of the maximum torque \(T_{\text{max}}\) (Table 1 and Figure 6a), as well as the approximate shape and dimensions obtained for every section of each instrument. The latter are presented in the third row of Figure 8, while their values were provided in Figure 4b,c.