**5. Conclusions**

In this study, we simulated the mechanical response of NiTi rotary endodontic files with different cross-sections and pitches using an accurate finite element model under bending and torsion according to the conditions of the ISO 3630 Standard.

From the results obtained, we can conclude that, with equivalent shaft diameter and taper, endodontic files with a square-shaped cross-section have more than double the stiffness of those with a triangular-shaped cross-section under both bending and torsion. The effect of the pitch on stiffness was less significant, but the use of a pitch lower than 3 mm made the files more flexible for bending and stiffer for torsion when using a triangular cross-section, with beneficial effects seen in clinical use. The phase transformation from

austenite to martensite led to a significant decrease in file stiffness both in bending and torsion, which was noticeable in the moment versus deformation curve. When the files were deformed under bending or torsion up to failure, a higher angle of rotation was possible before failure for the triangular section, especially in torsion and, for small pitches, in bending. A higher fatigue life can be expected in clinical use with the triangular-shaped cross-section than for the square cross-section under equivalent file deformations, especially with small pitch values. These results sugges<sup>t</sup> a clinical recommendation for the use of files with triangular-shaped cross-sections and small pitch, in order to minimize ledging and maximize fatigue life.

Under the conditions of the ISO 3630 standard, the orientation of the bending plane with respect to the cross-section of the file had a significant effect on the stiffness and the strength of the file. This effect should be taken into account when designing, reporting, and interpreting similar test results.

Further works on this topic could be focused on studying the mechanical response of endodontic instruments with variable parameters (e.g., in terms of pitch and crosssection) throughout their active length. The bending fatigue life of the endodontic files in cases where the loading conditions do not represent a fully reversed fatigue phenomenon (e.g., adaptive or reciprocating motions) also deserves attention in future investigations.

**Author Contributions:** Conceptualization, A.P.-G. and V.R.-C.; methodology, A.P.-G. and V.R.-C.; software, V.R.-C.; investigation, A.P.-G. and V.R.-C.; resources, A.Z.-M. and V.F.-M.; writing—original draft preparation, A.P.-G.; writing—review and editing, V.R.-C.; supervision, A.Z.-M. and V.F.-M.; project administration, A.Z.-M. and V.F.-M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Post-Processing of the Finite Element Analysis Results**

*Appendix A.1. Assessment of the Maximum von Mises Stress and Maximum Principal Strain Values in the Endodontic File*

Due to the nature of the finite element method, stress and strain singularities may appear in the vicinity of those regions of the model where boundary conditions are applied or in those areas nearby geometric stress increases. These singularities imply that unrealistically large values of stress–strain are obtained as a consequence of the numerical treatment used to derive these magnitudes from the nodal displacement results. There are many researchers who have claimed that the stress–strain results at singularity points cannot be considered when evaluating the strength of endodontic files [41,42].

Several strategies can be found in the literature to address this issue. Zmudzki [ ˙ 43] proposed to exclude the stress results at these points, instead extrapolating the extreme value from the stress values in the surrounding nodes. A different approach has been used by Baek [36], who determined the maximum stress level as the mean value of the top 1% von Mises equivalent stress values in the finite element model. In this work, the maximum von Mises stress *σmax* at a given analysis frame is defined as the maximum stress level that is reached by a certain amount *λ* of the total volume of the file (*Vtot*). To determine this magnitude, the steps below were followed:

• Let *i* ∈ [1, *ne*] refer to each of the *ne* tetrahedral finite elements in the model, and *j* ∈ [1, 4] refer to the integration points in each tetrahedral element. The von Mises stress at a given element and integration point is denoted as *<sup>σ</sup>ij*, and the volume associated with each integration point is denoted as *Vij* = *Vi*/4 (where *Vi* is the volume of element *i*).

• The von Mises stress *<sup>σ</sup>ij* and the volume *Vij* at each integration point of the model are retrieved and stored in an array Σ with *ni* = 4 · *ne* rows. Each row *m* in Σ contains the von Mises stress and the volume associated with a given integration point, with the shape

$$
\Sigma[m] = [\sigma\_{ij}, \mathcal{V}\_{ij}].\tag{A1}
$$

• The rows in Σ are rearranged in such a way that the von Mises stresses are sorted in descending order. Then, the algorithm shown in Figure A1 is applied to determine the maximum von Mises stress in the analysis frame.

**Figure A1.** Algorithm to search for the maximum von Mises *<sup>σ</sup>ij* stress in the analysis frame after the array Σ is created.

In this work, the magnitude of *λ* is set arbitrarily to 0.1‰, which has been shown to be a good value to avoid stress singularities while maintaining the actual stress level of the file. The same strategy was applied to determine the maximum principal strain in each analysis frame.

#### *Appendix A.2. Determination of Bending Fatigue Life of the NiTi Endodontic Files*

When the endodontic files are continuously rotated inside the root canal, they are typically subjected to a purely reversed fatigue phenomenon in which, for each rotation of the file, the bending strain alternates between nearly equal positive and negative peak values following a sinusoidal function [44]. The difference between these peak values is called the bending strain range, which is denoted by Δ*ε*. Several studies [45–47] have demonstrated that the bending strain range and the number of cycles to failure (NCF) are correlated, and this correlation can be adequately represented by the Coffin–Manson relation:

$$\frac{\Delta\varepsilon}{2} = \varepsilon\_F^{\prime} \cdot \mathcal{N}\_f^{\varepsilon} + \frac{\sigma\_F^{\prime}}{E} \cdot \mathcal{N}\_{f^{\prime}}^{b} \tag{A2}$$

where *Nf* is equivalent to the NCF, *εF* is the fatigue ductility coefficient, *σF* is the fatigue strength coefficient, *c* is the fatigue ductility exponent, and *b* is the fatigue strength exponent.

Two issues arise when applying the Coffin–Manson relation to predict the NCF of the endodontic files from the strain results obtained from the proposed finite element model:


According to Roda-Casanova et al. [44], and in order to convert the multi-axial strain state into uni-axial strain, the bending strain range Δ*εi* at node *i* of the finite element model can be successfully approximated by:

$$
\Delta \varepsilon\_i = \max\_{j=1...n\_f} \left( \varepsilon\_{ij}^{\max} \right) - \min\_{j=1...n\_f} \left( \varepsilon\_{ij}^{\min} \right),
\tag{A3}
$$

where *εmax ij* and *εmin ij* are the maximum and the minimum principal strains that take place at node *i* at time frame *j* of the transient analysis, respectively. Considering that the endodontic file is continuously rotating inside the root canal, it is fair to assume that the maximum and minimum principal strains that take place at node *i* have the same modulus and different sign. Under this assumption, Equation (A3) can be simplified to:

$$
\Delta \varepsilon\_{\bar{i}} = 2 \cdot \max\_{\bar{j} = 1...n\_f} \left( \varepsilon\_{\bar{i}\bar{j}}^{\max} \right). \tag{A4}
$$

Thus, by determining the maximum magnitude of the maximum principal strain in the finite element model and calculating the strain range Δ*εi* at such a node using Equation (A4), the NCF for a given specimen can be predicted through Equation (A2). The material parameters considered for the application of the Coffin–Manson relation are reflected in Table 2.

#### **Appendix B. Literature Review**




**Table A1.** *Cont.*
