**3. Results**

Figure 8 shows the strain results for endodontic rotary files P2 and P3 when they are rotating inside the most curved root canal RC9. Figure 8a shows the maximum principal strain plot for the case of study P2/RC9 in the instant of the rotation step of the analysis where the bending strain at the critical node reaches its maximum value. Figure 8b shows the minimum principal strain plot, for the same case of study, in the instant of the rotation step of the analysis where the bending strain at the critical node reaches its minimum value. The figure also shows the location of the critical node in both instants of time, as well as the critical plane for such a node. Figure 8c,d show the equivalent results for case P3/RC9 in which the file has a different pitch. The highest strains, both in tension and compression, are located at the edge in the surface of the file.

**Figure 8.** Principal strains in two different analysis frames for cases of study P2/RC9 and P3/RC9.

The evolution of the bending strain and the maximum and minimum principal strains during an entire rotation of the file are shown in Figure 9 for P2/RC9 and P3/RC9, reflecting a similar pattern for both files, but with a slightly higher strain range for the file with pitch 3 mm. The phase shift is due to a different orientation of the edges of the file in the critical plane. The points marked with a star correspond to the frames of maximum and minimum bending strain values, shown in Figure 8.

**Figure 9.** Strain history at critical nodes of cases of study P2/RC9 (**a**) and P3/RC9 (**b**).

The strain ranges obtained for all the studied cases are summarised in Figure 10. In these plots, the horizontal axis indicates the angle of curvature of the root canal and the vertical axis indicates the radius of curvature. The black dots indicate the combinations of radius and angle of curvature that have been studied, and the isolines are interpolated from these results.

**Figure 10.** Maximum bending strain range as a function of the geometry of the root canal. (**a**) P2 and (**b**) P3.

Figure 11 shows the expected life for each file as a function of the radius and the angle of curvature of the root canal, calculated as indicated in Section 2.2, also with interpolated isolines.

The results in Figures 10 and 11 indicate that root canals with higher curvatures (higher angles of curvature or/and smaller radii of curvature) force the files to a higher strain range and reduce their fatigue life. The effect of the angle of curvature is less significant for smaller radii of curvature, as indicated by the lower inclination of the isolines in the bottom of the figures. The file with pitch 3 mm (P3) showed higher strain and shorter fatigue life as compared with that of pitch 2 mm (P2), but this effect is slight, especially for less curved canals.

**Figure 11.** Expected life in number of fatigue cycles *Nf* as a function of the geometry of the root canal. (**a**) P2 and (**b**) P3.

Finally, Figure 12 summarizes the effect of the degree of insertion of the endodontic rotary file inside the root canal for the case of study P2/RC9. Here, the degree of insertion of the rotary file inside the root canal is expressed as the percentage of the active part that is inserted. Figure 12a shows the evolution of the bending strain at the critical node during a entire revolution of the file for different degrees of insertion of the file inside the root canal (70%, 85% and 100%).

**Figure 12.** Effect of the degree of insertion of the endodontic file within the root canal. (**a**) Bending strain history at critical nodes and (**b**) bending strain range as a function of the degree of insertion.

Figure 12b shows the evolution of the bending strain range with the degree of insertion of the endodontic rotary file inside the root canal. Here, additional data points are considered for a better observation of the evolution of this magnitude. This figure shows that the bending strain range is reduced as the degree of insertion of the endodontic rotary file inside the root canal is decreased.

## **4. Discussion**

This study analysed numerically the effect of the root canal geometry on the bending fatigue life of NiTi rotary instruments. Some previous studies have shown the predictive capacity of this simulated approach based on the use of FE models to predict the location of the file fracture [7,8]. However, a recent review highlighted some limitations introduced in previous FE studies in this field, especially in the representation of the boundary conditions, the accuracy of the mesh to represent the real file geometry or the lack of consideration of friction in the FE analysis [21]. Our investigation solved these main limitations, with a very accurate mesh of quadratic elements for the file, and undertook a transient non-linear simulation of the introduction of the file inside the root canal and its rotation inside the canal, including contact and friction simulation. We also used the Coffin–Manson relation

to estimate the expected life for the file working on a representative set of root canal geometries, proposing an adequate methodology for translating FE results to clinically relevant variables, which can be useful for manufacturers of NiTi rotary instruments.

Our results confirm that more curved canals are prone to reduce fatigue life, which was also previously observed in several in vitro studies [5,8,14]. We also found that the radius of curvature of the canal has a higher effect than the angle between the shaft and the apical portion of the file, especially for the lower radii of curvature. Changing the radius of curvature from 15 mm to 5 mm multiplies the strain amplitude by a factor of close to three for curvature angles of 30◦ and by a factor close to 2.5 for curvature angles of 60◦. Due to the logarithmic relationship between strain and NCF, this effect is higher in the expected life, which is reduced by a factor close to ten for 30◦ of curvature angle and by a factor six for 60◦. This result is in agreemen<sup>t</sup> with the experimental study by Chi et al. [5], although their setup was different because the curved part of the file reached the apical end for all conditions.

In most of the experimental studies, NCF is represented against a theoretical bending strain value obtained from geometrical approximations of the radius of curvature of the file and its diameter at the failure section using Equation (3) [14,16,17]:

$$
\varepsilon\_d = \frac{d}{2R} \tag{3}
$$

where *d* is the file diameter, and *R* is the approximate radius of curvature of the file. Thus, the higher effect of the radius of curvature is expected from a theoretical point of view, because the ideal strain amplitude in bending is defined by Equation (3), depending only on the diameter of the file and the radius of curvature. However, the angle between the entrance and the apical portion of the root canal and also the clearance between the file and the canal walls affect the actual deformation of the file, which can be different to that defined by the geometrical approximation indicated by Equation (3), as observed in Figure 8. This would explain the effect observed for the angle of curvature.

The difference in strain amplitude and fatigue life for the files with pitch 2 mm and 3 mm is limited, with a slightly higher fatigue life for P2 and more pronounced for less curved root canals. Ha et al. [30] also observed lower stresses for closer pitch when analysing the screw-in tendency of NiTi rotary files with a transient FE model.

In Figure 12, we also studied the effect that the degree of insertion of the rotary file has on bending strain range. We observed that the bending strain range at the critical node decreases with a reduction in the degree of insertion. According to the Coffin–Manson relation, this increases the fatigue life of the rotary file. This reduction in the bending strain range could be explained because the diameter *d* of the rotary file at the curved part of the root canal is smaller with partial insertion and, according to Equation (3), this results in a reduction in the bending strain. Additionally, the greater clearance between the rotary file and root canal implies that the effective curvature radius of the file is higher than that of the root canal. Again, according to Equation (3), this implies an additional contribution to the reduction in bending strain range.

In this study, we based the analysis of file behaviour on the strain values observed, as recommended for LCF with significant strain values and fatigue life below some thousands of cycles [25]. We also proposed a method for calculating the strain amplitude during the cyclic rotation of the file, based on the use of a critical plane defining the direction of the maximum and minimum principal strains in the critical points. The results shown in Figure 9 reveal that the method proposed here for defining the critical plane is correct, because when the bending strain reaches its maximum and minimum points, its values coincide with the maximum and minimum principal strains, respectively.

This study has some limitations. We only considered a geometry of the file section, with a constant taper and pitch through its length. We also limited the analysis to root canals for which its geometrical axis can be represented by a planar curve. Moreover, despite our work trying to be as representative as possible of the clinical situation of the

file rotating inside the root canal, some possible improvements remain and are commented in the following paragraphs. They can be observed as challenges for future studies trying to improve the accuracy of the model.

The material model used in this study is non-linear and included the phase transformation plateaus in the stress–strain curve and the different Young's modulus of martensite and austenite. However, heat dissipated due to the hysteresis loop in the loading–unloading curve, and heat generated by sliding friction between the file and the canal can also induce phase transformations from martensite to austenite due to the shape memory effect of the material, adding a complex thermal–mechanical coupling that is not considered here. On the other hand, we have made estimations of the fatigue life based on the Coffin–Manson equation, but the fatigue ductility coefficient and fatigue ductility exponent were taken from the literature. We have not considered the effect of the phase transformations between austenite and martensite on fatigue response and, hence, on these parameters. Previous studies have shown that higher strains did not necessarily imply less cycles to failure, because a higher fraction of martensite results in a better response under cyclic loads [16]. To our knowledge, there is no clear approach for the moment to include this effect in the FE simulations.

In this study, we considered a transient simulation, but this simulation does not include the dynamical effects, which are dependent on the rotational speed of the file. Moreover, our model included normal and shear contact between the file and root canal walls, but these walls are simplified as rigid elements. Considering the dentine properties for the walls would be necessary for estimating the risk of canal transportation or ledging.

We have proposed a method for dealing with the multiaxial strain state in order to predict fatigue life, but the fatigue phenomenon in shape memory alloys is still under investigation [32], and there is no clearly established fatigue criterion for analysing multiaxial fatigue in those materials [7,32].

In addition, since the first introduction of NiTi rotary instruments in the last decade of the twentieth century, several changes have been introduced to new families of files in terms of composition, manufacturing methods and thermomechanical treatments, which is not always publicly known, affecting greatly the percentage of martensite or austenite present in the file in clinical use and also the fatigue life of instruments [15,33].
