## 3 Results  
The remaining lengths of the instruments and the extreme values of the dimensions of their transversal sections were measured after fracture using the same type of digital caliper and micrometer (as for the initial values), respectively (Figure 4). By comparing the total length of each type of instruments (\(L_{1}\)) with their length after fracture (\(L_{2}\)), it can be observed that the broken element has a length of 2 to 3 mm (with a higher average, of 3.24 mm for the Protaper instruments). Therefore, they correspond to the lower third of the instrument. This confirms the clinical experience of instruments failure inside the root canals or in simulated root canals from polycarbonate training blocks [26]. In addition, it can be observed from Figure 4a that the fracture length of all three types of instruments included in the study are comparable, with dimensional variations of around 0.5 mm. Regarding the minimum (\(d_{1}\)) and the maximum dimension (\(d_{2}\)) of the cross sections, the statistical results concerning all considered instruments in the three study groups are presented in Figure 4b,c. In the following, these data are correlated with microscopic images to determine the geometrical properties of the sections after fracture. After torsional loading, the diagrams of the torsion torque versus the twisting angle for each instrument of each group were drawn. This type of approach is necessary for this  
Figure 4: The measured dimensions of the instruments of each study group after fracture: (**a**) length \(L_{2}\) (mm); (**b**) minimum \(d_{1}\) (mm) and (**c**) maximum \(d_{2}\) (mm) dimension of the fracture section. study to highlight the different behavior of instruments inside each group, because of the variations in the fracture lengths. The diagrams present the behavior of each instruments in the elastic zone, but also the maximum moment of torsion recorded at fracture point (Figure 5). The recorded values of these maximum torsional torques and twist angles, both corresponding to the fracture points (also for each of the fifteen instruments) are presented in Table 1.  
The variability of the measurement results for a certain type of instrument are caused by the different gripping points in relation to the specific taper of the instruments. Thus, for gripping tip lengths closer to 2 mm, fractures occur at smaller torque values and twisting angles as compared to gripping lengths closer to 3 mm. Thus, clinical situations corresponding to a range of gripping lengths are actually simulated, because this is what real-life situations refer to. Regarding the limits of this considered gripping lengths, the lower one corresponds to the situation in which the blockage on the root canal occurs in the first 2 mm of the instrument, while the higher one to the situation in which the first 3 mm of the active part of the instrument get stuck into the root canal. Figure 5: The diagrams of the torsion torque \(T\) (N \(\cdot\) cm) versus the twisting angle \(\theta\) (\(\lx@overaccentset{{\circ}}{\sim}\)) determined for the samples of (**a**) Group Endostar, **(b**) Group Reciproc, and (**c**) Group Protaper. From Table 1, in the case of Group Endostar, the values of the maximum torsion torque \(T_{\text{max}}\) are superior, i.e., closer to \(2\text{ N}\cdot\text{cm}\). In comparison, the maximum reached torque of the Group Protaper instruments is \(1\text{ N}\cdot\text{cm}\). The angles \(\theta_{\text{max}}\) at which the fracture occurs range from 150 to 200\({}^{\circ}\); they indicate the energy required to fracture the instruments by torsional load.