**1. Introduction**

The introduction of nickel–titanium alloy (NiTi) for the manufacturing of root canal instruments entailed a grea<sup>t</sup> revolution in the field of endodontics, as the consequent endodontic files decreased the incidence of iatrogenic complications [1,2]. However, despite the continuous mechanical and chemical improvements made by manufacturers, the failure of endodontic files during root canal treatments remains a concern for clinicians [3], as the incidence of their fracture still ranges from 0.09% to 5% [4,5].

The fracture of rotary instruments occurs mainly due to two different mechanisms, usually referred to as torsion overload and flexural fatigue [6,7]. On one hand, the torsion overload failure mechanism corresponds to a static failure that typically occurs when the tip of the endodontic file becomes blocked in the root canal whilst the instrument continues rotating [8]. In static failure, the file fails because the stress value reaches the elastic limit of

**Citation:** Roda-Casanova, V.; Pérez-González, A.; Zubizarreta-Macho, A.; Faus-Matoses, V. Influence of Cross-Section and Pitch on the Mechanical Response of NiTi Endodontic Files under Bending and Torsional Conditions—A Finite Element Analysis. *J. Clin. Med.* **2022**, *11*, 2642. https://doi.org/10.3390/ jcm11092642

Academic Editors: Massimo Amato, Giuseppe Pantaleo and Alfredo Iandolo

Received: 18 April 2022 Accepted: 5 May 2022 Published: 8 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the material, such that the file undergoes permanent deformation and finally fractures. On the other hand, flexural fatigue is a failure mechanism produced mainly by the alternating compressive and tensile stresses and strains that appear in any point of a file rotating inside a curved root canal [8,9]. This type of fatigue failure results in a sudden fracture of the file after a certain number of rotations, even if the stress levels are far below the elastic limit of the material, due to the nucleation and progression of small cracks in some stressed sections of the file. Thus, bending and torsion are essential conditions to evaluate the mechanical behavior of endodontic instruments [10]. The unexpected failure of NiTi endodontic files may condition the outcome of the root canal treatment by blocking the advancement of disinfecting agents beyond the fractured instrument [11–13], which may lead to subsequent pulp necrosis and the formation of periapical lesions [14], or decrease the success rate of root canal treatment of teeth with periapical pathology [15]. In addition, extraction of the fractured NiTi endodontic rotary file from the root canal system requires root dentin removal to provide access to the fractured instruments [16]. This causes a loss of dentin tissue, which can negatively affect the structural integrity of the tooth [17]. Furthermore, it can lead to root perforation and increase the risk of vertical root fracture, especially in the apical third [16]. For these reasons, a better understanding of the independent and combined effects of the different parameters that affect these failure mechanisms is desirable, and additional research must be addressed to this end.

Several works have been conducted to analyze the influence of both the NiTi alloy [18] and the geometrical parameters on the torsional and bending resistance of endodontic instruments. Both the chemical composition and crystalline structure of the NiTi alloy have been studied, and it has been shown that they highly influence the strength of the endodontic file [19]. In particular, endodontic rotary systems with a higher concentration of the martensitic phase and manufactured using electropolishing, ion implantation, cryogenic treatment, and heat treatments improve the mechanical behavior of NiTi endodontic rotary files, increasing their cyclic fatigue resistance. The geometric parameters of the endodontic files have also been reported to influence the instrument's performance, including the taper and apical diameter [20], cross-section design [21,22], flute length, helix angle, and pitch [23]. The influence of these variables has been analyzed using static and dynamic custom-made cyclic fatigue testing devices, which have not been submitted to a standardization normative, and do not allow for independently assessing the influence of each geometric parameter associated with flexural fatigue or torsional overload. There are other standardized testing devices, such as those described in ISO 3630-1:2008 [24], which allow for the independent assessment of both torsional and bending phenomena, although their capability to reproduce the actual operating conditions of endodontic files has not ye<sup>t</sup> been verified.

Computer simulation has proven to be an interesting tool for studying the failure of endodontic rotary files. In the simplest cases, analytical methods can be used for such a purpose, which are usually based on the small strain theory of elasticity. In this line, Zhang et al. [25] have analyzed the mechanical behavior of NiTi endodontic files under torsional and bending loads. Tsao et al. [26] have developed analytical models to study the flexibility of NiTi instruments subjected to bending loads. These analytical models have the advantage of being fast and easy to implement, but their capabilities to consider non-linear behaviors (i.e., material non-linearity) or complex loading scenarios are limited. These limitations can be overcome by using numerical methods such as the finite element method.

The ability of the finite element method to reproduce the results obtained from experimental tests using endodontic rotary files has been proven in several works [7,10,27–29], whose main conclusions have been summarized in a recent bibliographical review [30]. This review concluded that the finite element method is a reliable tool for evaluating the behavior of NiTi rotary instruments, and has the advantage of reducing instrument development time and costs. Another important advantage of the finite element method is that it also allows us to assess aspects of the mechanical behavior of the instruments, such as the stress distribution, which are difficult to obtain in laboratory tests [10]. The finite element

method has been previously used to analyze the influence of cross-section design and pitch on the stiffness and stress distribution under bending and torsional conditions [10,31–37]. Appendix B collects detailed information about these previous studies, including their main conclusions and limitations. Some of these studies have used proprietary file models, such as ProTaper, ProFile, Mtwo, and others, which hampers the independent evaluation of parameters such as cross-section geometry, cross-section area, or pitch [10,32–35]. Other studies have used theoretical file models to avoid this problem, but with some limitations; for example, in [36], the authors analyzed four different cross-sections and three pitch values under torsion, but did not provide detailed information about the material model for the shape memory alloy (SMA) of the files or about the quality of the finite element mesh. In another study, Versluis et al. [33] analyzed the effects of pitch and cross-section geometry on flexural stiffness and stresses using a representative SMA material model. However, the boundary conditions were specified differently to those in ISO 3630-1:2008 [24] and the bending applied was low, leading to maximum von Mises stresses below the initial stress for transformation from austenite to martensite, and, thus, the effect of the super-elasticity of the files was not analyzed; furthermore, torsion behavior was not included in the study. In [37], the effect of cross-section geometry and pitch on the 'screw-in' tendency of the files was analyzed, but a linear material model was used for the file. A more recent study investigated different geometric options for the sides of a triangle-shaped cross-section (straight, convex, and concave), as well as the use of files with combinations of these geometries along the file [31], but the pitch effect was not analyzed.

Some of these finite element models are limited in their accuracy, in terms of representing the correct geometry and boundary conditions of the endodontic files, or use simplified material models that are incapable of representing their actual mechanical response under load. In this study, we address all of these partial limitations of previous studies by undertaking a comprehensive analysis of the effects of pitch and cross-section using an accurate finite element model that allows us to simulate the testing conditions of the ISO3630 Standard to the best extent possible. The method used to obtain the parametric geometrical representation of the endodontic instrument and the corresponding finite element mesh has been proposed in our previous work [38]. The use of an accurate numerical model in these tests can foster improvements in new generations of more resistant and flexible endodontic files, reducing the need for expensive and time-consuming experiments in the early design stages. From a clinical perspective, these improvements are expected to reduce the risk of failure of endodontic instruments, thus preventing clinical complications.

The aim of this study was to analyze and compare the effects of the cross-section and the pitch on the mechanical response (in terms of strength and stiffness) of NiTi endodontic files under bending and torsional conditions, similar to those indicated in the ISO 3630 Standard [24], using the finite element method. The study was conducted using a set of eight different endodontic rotary files whose geometries were obtained from combinations of two cross-sections (triangular and square) and four pitches (1 mm, 2 mm, 4 mm, and 8 mm). Under these conditions, the following individual objectives were pursued: (i) to develop a finite element model which reproduces the experimental tests conducted in the ISO 3630 Standard; (ii) to conduct a bending analysis of the selected endodontic rotary files, in order to predict the stiffness and strength of the files under static and cyclic loading conditions; and (iii) to conduct a torsional analysis of the selected endodontic rotary files, in order to predict the stiffness and the strength of the files under static loading conditions.

#### **2. Materials and Methods**

For this study, different endodontic instruments were analyzed using numerical simulation with finite elements. Figure 1 shows the geometries of the eight endodontic files considered. The different geometries were obtained by varying the cross-section (square and triangular) and the pitch (*pz* = {1 mm, 2 mm, 4 mm, 8 mm}) of the files. All of them had a total length of *Ltotal* = 25 mm, the length of their active part was *La* = 16 mm, and their tip and shaft diameters were *da* = 0.25 mm and *dsh* = 1.20 mm, respectively. The taper of the endodontic files was 6%.

**Figure 1.** Geometries of the analyzed endodontic files: endodontic files with square cross-section (**a**); endodontic files with triangular cross-section (**b**); normalized square cross-section (**c**); and normalized triangular cross-section (**d**).

The material for all the files was considered to be NiTi, which exhibits a super-elastic stress–strain curve, as shown in Figure 2. Here, *EA* and *EM* represent the Young's moduli of austenite and martensite, respectively. The beginning and end of the loading phase transformation are denoted by *σSL* and *σEL* , respectively, whereas the beginning and the end of the unloading transformation phase are denoted by *σSU* and *<sup>σ</sup>EU*. Finally, *εL* represents the uni-axial transformation strain, and *σEME* indicates the end of the martensitic elastic regime.

 **Figure 2.** Sample stress–strain curve for NiTi material.

#### *2.1. Devices for Experimental Bending and Torsion Analysis*

Endodontic files are usually tested in terms of bending and torsional loads, and the typical standardized procedure for these tests has been described in the ISO 3630 Standard [24], as summarized in Figure 3. For the torsion analysis (Figure 3a), the last 3 mm at the tip of the endodontic file are inserted inside a clamping jaw. After checking that the endodontic file is properly fixed and aligned with the axis of rotation, the top of the file is rigidly connected to the torsion device. This torsion device is increasingly rotated at angle *θ<sup>z</sup>*, and the torsional moment *Mz* is measured using a torquemeter attached to the clamping jaw. The test ends with the failure of the endodontic file. At this point, the maximum rotated angle *θ<sup>z</sup>*,*max* and maximum torsional moment *Mz*,*max* are registered.

**Figure 3.** Devices used for torsion (**a**) and bending (**b**) analyses.

In a similar way, in the bending analysis (Figure 3b), the last 3 mm at the tip of the endodontic file are inserted inside a clamping jaw. After checking that the endodontic file is properly fixed and aligned with the axis of rotation, the bending device is positioned until it contacts the endodontic file. Then, the bending device is increasingly rotated at angle *θ<sup>x</sup>*, and the bending moment *Mx* is measured using a torquemeter attached to the clamping jaw. The test ends with the failure of the endodontic file. At this point, the maximum rotated angle *θ<sup>x</sup>*,*max* and maximum bending moment *Mx*,*max* are registered.

#### *2.2. Definition of the Finite Element Model for the NiTi Endodontic File*

Figure 4 shows an example of the finite element model created for the endodontic file simulation experiments, as described in Section 2.1. Here, only the portion of the endodontic file subjected to stresses and strains was considered in the analysis (i.e., the part of the endodontic file inserted into the clamping jaw was not included in the finite element model). The geometry of the endodontic file was generated and then discretized into quadratic finite element tetrahedrons following the meshing procedure developed in our previous work [38]. Using this procedure, the finite element mesh of a endodontic file was automatically built from its geometrical parameters (*dsh*, *da*, *La*, *Ltotal*, and *pz*, as shown in Figure 1) and the average element size.

**Figure 4.** Definition of the finite element model.

To select the average element size, a mesh sensitivity study was conducted in our previous work [38] for a finite element model of an endodontic file with similar geometry, element type, boundary, and loading conditions, as described in Figure 4. In this study, the variations in the maximum element energy error and energy norm error with respect to the

average element size were observed, and it was concluded that an average element size equal to 0.1 mm provided a good compromise between accuracy and computational cost. For these reasons, this average element size was used to perform this study, resulting in a finite element model with 89,295 nodes and 58,749 elements.

The super-elastic behavior of the NiTi alloy used to manufacture the endodontic files was modeled using the material model developed by Auricchio [39]. The material properties that characterize this material model were extracted from [10], and are shown in Table 1.

**Table 1.** Material properties to characterize the super-elastic behavior of NiTi alloy. Reprinted/adapted with permission from Ref. [10]. 2014, Elsevier.


The surface at the fixed end of the endodontic file was defined as a rigid surface (denoted as rigid surface A in Figure 4). This rigid surface was rigidly connected to reference node A, which was used to introduce the boundary conditions for the finite element model. To simulate the effect of the clamping jaw over the endodontic file, all of the degrees of freedom of reference node A were restricted. At the other side of the file, the top surface was also defined as a rigid surface (denoted as rigid surface B in Figure 4). This rigid surface was rigidly connected to reference node B, which was used to define the loading conditions of the model. Two different loading conditions were considered in the analyses, one for the bending analysis and the other for the torsional analysis:


The finite element model was solved through transient analysis using the large displacements formulation, which was conducted using the ABAQUS software. Hence, material and geometric non-linearities were considered in the study. In each one of these analyses, the rotation at reference node B (*θx* for bending analysis and *θz* for torsional analysis) and the reaction moment at reference node A (*Mx* for bending analysis and *Mz* for torsional analysis) were registered for each analysis frame. The maximum von Mises stress and the maximum principal strain were also retrieved for each analysis frame, using the

method indicated in Appendix A.1, in order to minimize possible numerical singularities in the model. Finally, the bending fatigue life was estimated following the method described in Appendix A.2, based on the Coffin–Manson relation, considering the material properties indicated in Table 2.

**Table 2.** Material properties used to characterize the fatigue behavior of NiTi alloy [28,40].

