## 3 Results
From these values one may see that the Endostar instruments can be considered circular in their cross-section (Figure 8(a3)), therefore the maximum tangential stresses can be calculated with the well-known equation:  
\[\tau_{max}=T_{max}\cdot d/2I_{p}, \tag{1}\]  
where \(I_{p}=\pi\cdot d^{4}/32\) is the polar moment of inertia of the cross-section, \(d\) is the radius of the cross-section (Figure 4b,c), and \(T_{max}\) is the measured torsional torque at failure (Figure 6a). The cross-sections of instruments Reciproc and Protaper can be considered rectangular (Figure 8(b3,c3)), therefore the equation of the tangential stress is:  
\[\tau_{max}=T_{max}/W_{t}, \tag{2}\]  
Figure 7: Fractography of the endodontic instruments: (**a**) Endostar E3, (**b**) Reciproc R 25, and (**c**) Protaper-Next X2 (i.e., an example from each of the three groups)-study carried on with optical microscopy with a 50\(\times\) magnification. where \(W_{t}=k\cdot d_{1}\cdot d_{2}^{2}\) is the torsion modulus of strength, while \(k=f(d_{1}/d_{2})\) is the parameter that takes into consideration the dimensional ratio of the section; the minimum and maximum dimensions \(d_{1}\) and \(d_{2}\) are measured from the microscopy images in Figure 8(b3,c3) for Groups Reciproc and Protaper, respectively (as provided in Figure 4b,c). The calculated values of the maximum stress are presented in Table 2, considering also the standard deviations of the measured values. They are shown in parallel with the values of the maximum twist angle \(\theta_{\text{max}}\), corresponding to those presented in Figure 6b. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Group Endostar**} & \multicolumn{2}{c}{**Group Reciproc**} & \multicolumn{2}{c}{**Group Protaper**} \\ \cline{2-7}  & **Mean** & \begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & **Mean** & \
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & **Mean** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} \\ \hline \(\tau_{\text{max}}\) (MPa) & 1306.8 & 301.9 & 1160.0 & 473.0 & 1050.6 & 287.4 \\ \(\theta_{\text{max}}\) (\({}^{\circ}\)) & 161.2 & 60.2 & 154.2 & 51.0 & 204.2 & 82.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of the maximum shear stresses \(\tau_{\text{max}}\) and of the twist angle \(\theta_{\text{max}}\) at fracture. Figure 8: (**1**) Initial longitudinal profile, (**2**) lateral aspect of the fracture site, and (**3**) geometrical approximation of the fracture sections for an example of instruments from each study group: (**a**) Endostar, (**b**) Reciproc, and (**c**) Protaper. All images were obtained with optical microscopy with a 25\(\times\) magnification. Statistical processing was performed by SPSSv17. For numeric variables average values, standard deviations, minimum and maximum values were calculated. The comparisons between the numerical series were performed with the non-parametric Kruskal-Wallis test for more than two series and with the Mann-Whitney test for comparisons between two sets of values with no Gaussian distribution. Insignificant differences were remarked between the three study groups both for \(\tau_{\text{max}}\)--for which the Endostar values are insignificantly increased (Kruskal-Wallis test, \(p=0.207\)), and for \(\theta_{\text{max}}\)--for which the Protaper values are insignificantly increased (Kruskal-Wallis test, \(p=0.287\)). One may see that the values of both shear stress and twist angle in Table 2 have large standard deviations, because of the different fracture points: closer to 2 mm or to 3 mm from the tip of the instrument. For the former fracture length, the values of both torque and twist angle are smaller, while for the latter length they are higher--as pointed out regarding the individual values presented in Table 1 and Figure 5. This is due to the instrument shape and conicity. This imposes a separate analysis for the two lengths of the tip in the root canal.