**3. Results**

*3.1. Bending Analysis*

Figure 5 shows the von Mises stress plot for the bending analysis of two representative endodontic files with pitch *pz* = 4 mm and analysis angular position given by *ϕz* = 0◦, for the analysis frame in which the maximum von Mises stress in the model reaches the end of the loading transformation phase (*<sup>σ</sup>max* = *σEL* ). Figure 5a shows the von Misses stress plot over an endodontic file with square cross-section and Figure 5b shows the von Misses stress plot over an endodontic file with triangular cross-section. The figure shows that, under these boundary conditions, the highest stresses were located in the apical third of the file.

**Figure 5.** The von Mises stress plots for the bending analysis of endodontic files with *pz* = 4 mm and *ϕ* = 0◦.

Figure 6 shows the relationship between the rotation *θx* and the reaction bending moment *Mx* obtained from the bending analysis of the endodontic files with square (Figure 6a) and triangular (Figure 6b) cross-sections and pitch *pz* = 4 mm. Here, the abscissa axis shows the rotation of the reference node B along the *x*-axis, while the ordinate axis shows the reaction bending moment at reference node A. The figure also shows the points where the maximum von Mises stress in the finite element model reaches the start of the phase transformation, the end of the phase transformation, and the end of the martensitic elastic regime. The curves in the figure exhibit a significant decrease in the slope for a rotation close to 20◦, corresponding to a change in the stiffness of the file, as the transformation from austenite to martensite progresses in part of the file. As the bending response of an endodontic file is dependent on its orientation (given by the angle *ϕz*), different curves were obtained for each cross-section. For clarity, only the lower and upper curves are shown for each case, along with another intermediate representative curve. The figures also show the cross-section orientation at the encastré for each case.

**Figure 6.** Bending moment–rotation relationships for the bending analysis of endodontic files with *pz* = 4 mm: squared cross-section (**a**) and triangular cross-section (**b**).

Figure 7 shows the bending overload failure mechanism evaluation, which occurs when the maximum von Mises stress in the endodontic file reaches the end of the martensitic elastic regime (*<sup>σ</sup>max* = *<sup>σ</sup>EME*). On one hand, Figure 7a shows, for each considered pitch and cross-section, the rotation that needs to be applied at the free end of the endodontic files to reach the end of the martensitic elastic regime in the bending analysis. On the other hand, Figure 7b shows, for each considered pitch and cross-section, the maximum bending moment that can be applied at the free end of the endodontic files before they reach the end of the martensitic elastic regime in the bending analysis. As different angular positions were evaluated for each cross-section and pitch, the results shown are the range between the minimum and maximum obtained values. The bold lines represent the mean value within this range.

**Figure 7.** Bending analysis: effect of the pitch on the maximum rotation (**a**) and maximum applied torque (**b**) when the end of the martensitic elastic regime is reached.

Figure 7a shows that the maximum rotation was, on average, quite similar for triangular and square cross-sections when the pitch value was larger than 3 mm. For these pitch values, it was nearly independent of the pitch, but with a slight tendency to increase with the pitch when using a square cross-section and to decrease when using a triangular cross-section. For pitches below 3 mm, files with triangular cross-sections exhibited larger rotations than files with square cross-sections. From Figure 7b, it can be observed that the moment required to bend the square cross-section to failure was almost twice that for the triangular cross-section. The results shown in Figure 7a,b indicate that square cross-sections

are more sensitive to the orientation of the file (*ϕz*) than triangular cross-sections, as the results exhibited larger variability.

Figure 8 shows the bending stiffness of the endodontic rotary files for the austenite and transformation phases. The stiffness in the austenite phase was calculated as the slope of the bending moment–rotation curve before *σ<sup>S</sup> L* , while that in the transformation phase was calculated as the slope of the bending moment–rotation curve between *σ<sup>E</sup> L* and *<sup>σ</sup>EME*. In general, it was observed that the stiffness of the endodontic files with square cross-sections was larger than that of the files with triangular cross-sections, both in the austenite and transformation phases. Moreover, the sensitivity to the orientation of the files with square cross-sections was larger than that of those with triangular cross-sections, especially in the austenite phase. The effect of the pitch on the stiffness was negligible for pitches larger than 3 mm. With smaller pitches, a reduction in the stiffness was observed, except for the austenite phase with the square cross-section.

**Figure 8.** Bending analysis: bending stiffness of the endodontic rotary files with (**a**) square and (**b**) triangular cross-section.

Finally, Figure 9 shows the evaluation of the expected fatigue life of the endodontic files when cyclically subjected to a purely reversed bending, which produced a rotation of *θx* = 20◦ at the free end of the file. As explained in Appendix A.2, the bending fatigue life depends on the maximum principal strain in the file. Figure 9a shows the maximum principal strain predicted by the finite element model as a function of the pitch, for both square and triangular cross-sections. In both cases, the effect of file orientation with respect to the bending moment was significant, and the effect of the pitch was noted especially for pitches smaller than near 3 mm, for which a decrease in the strain was observed. For the square cross-section, the increase was almost linear; meanwhile, for the triangular crosssection, this increase approximated a logarithmic function. Figure 9b shows the number of cycles that the endodontic files could bear before bending fatigue failure, calculated from the maximum principal strains using the Coffin–Manson relation. It was observed that endodontic files with triangular cross-sections can withstand a larger number of cycles than those with square cross-sections, especially for small pitches.

**Figure 9.** Bending analysis: effect of the pitch on the maximum principal strain (**a**) and the expected number of cycles (**b**) when the rotated angle is *θx* = 20◦.

#### *3.2. Torsional Analysis*

Figure 10 shows the von Mises stress plot for the torsional analysis of the endodontic files with square (Figure 10a) and triangular (Figure 10b) cross-sections and pitch *pz* = 4 mm, for the analysis frames in which the maximum von Mises stress in the model reached the end of the loading transformation phase (*<sup>σ</sup>max* = *σEL* ). As in the case of the bending analysis, the highest stresses were located near the apical part of the file.

**Figure 10.** The von Mises stress plots for the torsional analysis of endodontic files with *pz* = 4 mm.

Figure 11 shows the relationship between the rotation *θz* and the reaction torque *Mz*, obtained from the torsional analysis of the endodontic files with square (Figure 11a) and triangular (Figure 11b) cross-sections. Here, the abscissa axis shows the rotation of reference node B along the *z*-axis, while the ordinate axis shows the reaction torsional moment measured at reference node A. The figure also shows the points where the maximum von Mises stress in the finite element model reaches the start of the phase transformation, the end of the phase transformation, and the end of the martensitic elastic regime.

**Figure 11.** Torque–rotation relationships for the torsional analysis of endodontic files with different axial pitch: squared cross-section (**a**) and triangular cross-section (**b**).

Figure 12a shows, for each considered pitch and cross-section, the maximum rotation that needed to be applied at the free end of the endodontic files so that they reached the end of the martensitic elastic regime in the torsional analysis. The results show that the triangular cross-section was able to bear larger rotations before plastic deformation than the square cross-section. The rotation before failure was nearly independent of the pitch with the square cross-section, whereas it increased with the pitch for the triangular cross-section and pitch values between 1 mm and 4 mm. Figure 12b shows, for each considered pitch and cross-section, the maximum torque that could be applied at the free end of endodontic files before they reached the end of the martensitic elastic regime in the torsional analysis. It was observed that a square cross-section was able to bear almost double the torsional moment of the triangular cross-section. The strength of the files was independent of the pitch for these loading conditions.

**Figure 12.** Torsional analysis: effect of the pitch on the applied torque (**a**) and rotation (**b**) when the end of the martensitic elastic regime is reached.

Finally, Figure 13 shows the torsional stiffness of the endodontic rotary files for the austenite and transformation phases. The stiffness in the austenite phase was calculated as the slope of the torque–rotation curve before *σSL* , while the stiffness in the transformation phase was calculated as the slope of the torque–rotation curve between *σEL* and *<sup>σ</sup>EME*. In general, it was observed that the stiffness of the endodontic files with a square cross-section was larger than that of those with a triangular cross-section, both in the austenite and transformation phases.

**Figure 13.** Torsional analysis: torsion stiffness of the endodontic rotary files with (**a**) square and (**b**) triangular cross-section.

## **4. Discussion**

In this study, we applied an accurate non-linear finite element model to better understand the effects of the cross-section and pitch of NiTi endodontic files on their mechanical response under bending and torsion loads, according to the ISO 3630 Standard. Finite element analysis has been shown to be a good tool for this type of analysis, providing information about the stress distribution and circumventing experimental variability limitations [24]. Previous research using simulation with the same or similar objectives was first thoroughly analyzed, and the main conclusions and limitations of these studies are summarized in Appendix B, as a reference for further research. The importance of this research is supported by fact that the failure of endodontic files during root canal treatments remains a serious concern for clinicians.

The results of this study demonstrated that, for equal file diameter and taper, the crosssection shape, either triangular or square, has a greater effect than the pitch on the flexural and torsional stiffness of the file. The use of a square cross-section more than doubled the stiffness, compared to that of the triangular cross-section, as explained by the greater second moment of the area of the cross-section. The effect of pitch on stiffness was only appreciable for pitches lower than 3 mm, and was more important for triangular than for square cross-sections. When a NiTi file is bent or twisted, according to the conditions of ISO 3630, the super-elastic behavior of the material appears—which is evident from a significant decrease in the stiffness of the file—as a result of the progression of the transformation from the austenite to martensite phase in the most stressed areas of the file (see Figures 6 and 11). Our results indicate that, for a file with a shaft diameter of 1.2 mm and 6% taper, this change in stiffness appears when the rotation of the shank end section, with respect to the tip end section, is approximately 20◦ in bending or 30◦ in torsion. The stiffness of the file decreases by a factor greater than 2 after this transformation point (Figures 8 and 13). The file pitch has the opposite effect on the stiffness for torsion and bending: decreasing the pitch reduces the flexural stiffness, but increases the torsional stiffness. This effect is common for triangular and square cross-sections in the austenite phase, but it is less clear in the transformation phase, where the stiffness is less affected by pitch. This result is in agreemen<sup>t</sup> with those obtained in [33,36] for bending and torsion, respectively. As indicated in [33], pitch reduction could benefit both cutting efficiency, due to the higher torsional stiffness, and better adaptation to the canal shape, due to lower bending stiffness.

The obtained stress distributions (Figures 5 and 10) indicate that, for the boundary conditions imposed by the ISO 3630 Standard, the highest stresses were located near the tip of the file (where it is clamped), both in terms of bending and torsion and for both cross-section shapes. The stresses in the proximal part of the file were negligible when the stress corresponding to the end of the loading transformation phase ( *σmax* = *σ<sup>E</sup> L* ) was reached in the tip of the file. This can be explained by the smaller section at the tip and, in the case of bending, by the higher bending moment in this area.

Static failure under bending was obtained for comparable rotations—close to 40◦ for pitch greater than 3 mm and ranging between 40◦ and 60◦, depending on the pitch—for both triangular and square cross-sections (see Figure 7a). However, the bending moment necessary to reach this bending (and, thus, the reaction in the clamp) was quite different, given the difference in stiffness between the cross-section shapes (Figure 7b). This implies greater reaction forces (close to double) in the root canal with the square cross-section than with the triangular cross-section, for comparable bending deformations. The effect of the pitch on bending strength was only significant for pitches below 3 mm, where a progressive reduction in strain was observed when the pitch decreased (Figure 9a). This allows for bending of the file to a greater deformation before failure for small pitches, with a corresponding higher expected fatigue life for the same bending deformation (Figure 9b). This effect was especially observed for the triangular cross-section and, to a lesser extent, for the square cross-section. The analysis carried out to estimate the fatigue life also showed that, for the same pitch, the triangular cross-section had a higher expected life than the square cross-section, in agreemen<sup>t</sup> with [36], the difference being remarkable for

the smallest pitch analyzed (1 mm), for which the expected life may be more than three times longer.

Our results showed that the orientation of the bending moment, with respect to the cross-section, had a significant effect on the results, changing the results by up to 19.1◦ and 0.97 N · mm for the square cross-section and up to 13.0◦ and 0.27 N · mm for the triangular cross-section. This should be taken into account when designing, reporting, and interpreting experimental bending tests according to ISO 3630.

On the other hand, for torsion, the triangular cross-section files could be rotated to a higher angle before failure than those with a square cross-section, as can be observed from Figure 12. However, due to the difference in stiffness, this failure was reached for a torque less than half that for the square cross-section. The effect of the pitch was opposite to that observed in bending, with a reduction in the pitch leading to a lower strength, as shown by the lower possible rotation before failure, which was also in agreemen<sup>t</sup> with the results in [36].

From a clinical perspective, the results obtained in this study sugges<sup>t</sup> that the use of a triangular-shaped cross-section with small pitch for endodontic files could be better for the safe shaping of curved root canals, as its lower stiffness would produce less reaction forces in the channel, thus reducing the possibility of ledging and canal transportation. At the same time, files with a triangular cross-section and 1 mm pitch could exhibit a fatigue life more than double that of files with higher pitches or with a square cross-section. This is accompanied by a lower rotational stiffness, which could be beneficial for improving cutting efficiency [36]. The use of a smaller pitch can only partially compensate for this lower torsional stiffness of the triangular cross-section.

The results obtained in this simulation study refer to the boundary conditions established for the tests described in ISO 3630; however, it should be noted that the stress distribution within the file in these tests is not always comparable to the clinical situation, as the bending of the file is also constrained by contact with the canal walls, resulting in a different deformation, depending on the root curvature. As shown in [38], in a curved canal, the maximum strain is usually located near the highest curvature of the curved root canal axis and the fatigue life is clearly dependent on the radius of curvature. Under the conditions of ISO 3630, the highest curvature of the deformed file is close to the tip, so the conclusions in this study are especially valid for root canals with the highest curvature located near the apical end.

Finally, this work has certain limitations that deserve to be mentioned. This investigation was conducted through theoretical studies, by means of finite element analyses of endodontic rotary files; as such, no experimental tests were conducted. Regarding the investigated endodontic file geometries, all of them had uniform parameters (pitch and cross-section) throughout their entire length, even though there exist endodontic instruments in which these parameters vary through their active length. Finally, the bending fatigue life of the endodontic instruments was assessed considering a fully reversed fatigue phenomenon corresponding to a continuous rotation motion of the file within the root canal. The study of the bending fatigue under other types of motion (e.g., reciprocating and adaptive motions) is left for future research.
