**1. Introduction**

The use of nickel-titanium (NiTi) rotary files for shaping root canals has spread in endodontics during the last decades, in detriment of manual preparation with traditional stainless-steel instruments. The superelasticity of NiTi and its lower Young's modulus reduce the risk of canal transportation and ledging in the treatment of curved root canals [1]. The superelasticity of the NiTi refers to the capacity of the material for undergoing large elastic deformations that can be restored after the forces producing the deformation are released. During these large deformations of the superelastic material, a phase transformation is induced within the material from austenite to martensite at a nearly constant stress. Due to this superelastic behaviour, files made of NiTi can adapt easily to strongly curved root canals. Successive modifications introduced during the last two decades in these instruments have allowed improving the quality of the cleaning and shaping, as well as saving time for both clinicians and patients [2–4]. However, the main problem that persists is the fracture of the files inside the root canal [5].

Fracture of rotary instruments occurs mainly by two different mechanisms, usually referred to as torsion overload and flexural fatigue [6,7]. A torsion overload mechanism corresponds to a static failure and occurs when a section of the file is locked within the canal, and the shank continues to rotate. In this static failure, the file fails because the stress value reaches the elastic limit of the material, and the file undergoes permanent deformations and finally it fractures. Flexural fatigue is a failure mechanism produced

**Citation:** Roda-Casanova, V.; Pérez-González, A.; Zubizarreta-Macho, A.; Faus-Matoses,V. Fatigue Analysis of NiTi Rotary Endodontic Files through Finite Element Simulation: Effect of Root Canal Geometry on Fatigue Life. *J. Clin. Med.* **2021**, *10*, 5692. https:// doi.org/10.3390/jcm10235692

Academic Editors: Massimo Amato, Giuseppe Pantaleo and Alfredo Iandolo

Received: 5 November 2021 Accepted: 29 November 2021 Published: 3 December 2021

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mainly by the alternating compressive and tensile stresses and strains that appear in any point of a file rotating inside a curved root canal. This 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. The typical number of cycles to failure (NCF) is between some hundreds to several thousands [8]. This is equivalent to an expected life below some few minutes if a typical speed of rotation of 300 rpm is considered.

There is no definitive conclusion about which is the predominant mechanism of failure in the clinical practice [6]. Satappan et al. [9] indicated a higher prevalence of torsional fracture (55.7%) than flexural fatigue (44.3%). However, Peng et al. [10] and Wei et al. [11] observed the opposite, with a clear preponderance of flexural fatigue. Notwithstanding, flexural fatigue seems to be the main concern for clinicians, because there is no easy method to avoid or anticipate this failure [7], resulting in a common practice of discarding the files after a certain number of uses to prevent it. However, there is no clear rule about the recommended number of uses, mainly due to the variety of factors potentially affecting NCF, such as root canal anatomy, file geometry or the operator's experience, among others [12]. Therefore, a better understanding of the independent and combined effect of the different parameters on the flexural fatigue failure mechanism is desirable and additional research should be addressed to this end.

Experimental and simulated approaches have been used in the literature to analyse the effect of clinical and design parameters on the expected life of NiTi rotary files. Experimental approach has been mainly tackled by using in vitro studies in order to improve reproducibility. In general, those studies make the file rotate inside a curved path, reproducing the root canal geometry and registering NCF [6]. However, the differences among previous studies in the methodology and the setup used to bend the file hamper the comparability of results and limit their clinical relevance [6,12]. Due to this, a call for an international standard on the cyclic fatigue testing of rotary endodontic instruments is recurrent in the literature [6,13]. Despite these difficulties, the results from previous experimental studies on experimental fatigue tests on NiTi wires, or directly on endodontic files, have allowed drawing some conclusions about the fatigue behaviour of NiTi:


These previous experimental studies have shown that, under constant value for other parameters, strain amplitude and NCF for NiTi wires are correlated, and this correlation can be adequately represented by the Coffin–Manson relation [14,16,20].

The simulated approach for analysing flexural fatigue has been mainly undertaken through the use of finite element (FE) models. FE analysis is a mathematical technique that can be used for predicting the state of stress and strain in a body or group of bodies under applied external loads and constraints. It is based on a fine discretisation of the geometry of bodies in a high number of small finite elements. This method allows gaining some insight into the stress and strain distributions inside the file, helping gain a better understanding of the failure mechanism. A recent study made a critical review of the use of this method applied to NiTi endodontic instruments [21] and highlighted some of the main limitations of the analyses performed to date. According to this study, very few studies modelled

cyclic fatigue using FE simulation. The authors cited those of Lee et al. [8], Scattina et al. [7] and Ha et al. [22].

In [8] the authors performed a simulation of four different file models on three root canal geometries with different curvature and compared the results with those obtained from in vitro tests on equivalent systems. In the FE model, the file was rotated inside the simulated FE model of the root canal, and the maximum von Mises stress on the file nodes was analysed. They found that the location of the maximum von Mises stress in the FE model is a good predictor of the fractured section observed experimentally. Additionally, they confirmed a negative correlation between the maximum von Mises stress in the file and the NCF. The authors cited computational problems that forced them to reduce the rotational speed to 240 rpm and to consider a friction coefficient of 0.01 in order to avoid nodal binding. The non-linear behaviour of the material was considered by using data from [23], which did not include the lower plateau in the stress–strain curve characteristic of the phase transformation for the unloading path, which corresponds to a lower stress level than that observed for the loading phase.

Scattina et al. [7] tried to predict NCF using FE simulations. They compared in vitro tests and FE simulations for three file models on a single root canal geometry. The model considered the contact between the file and the root canal, represented with rigid shell elements, and the simulation included the rotation of the file and the analysis of the stress state every 0.2 s during 2 s at a rotation speed of 300 rpm. The authors used a multiaxial random fatigue criterion [24] to predict the NCF based on the stress history. They tuned the material properties with an optimisation procedure to match NCF predictions with experimental results on two of the file models and used these properties to predict NCF for the third model, finding a good agreemen<sup>t</sup> with experiments in both NCF and fracture location. However, the paper neither cited the final material parameters obtained from this optimisation nor the specific parameters considered.

In [22], the authors used FE simulation to develop a new file model intermediate between G-1 and G-2 models (Dentsply Maillefer, Ballaigues, Switzerland), but in this case the FE model did not include a fatigue simulation.

Cheung et al. [25] also performed an FE based fatigue analysis for comparing two different cross section geometries for the file, NiTi and steel based on a fully reversed bending analysis without including the root canal in the model. They applied the Coffin– Manson equation for predicting NCF.

The objective of the present study is to contribute to a better understanding of the effect of root canal geometry on the expected life of NiTi rotary files using FE simulation. To our knowledge, only Lee et al. [8] attempted a similar study, but they only considered three canal geometries with a different curvature, without changing the length of the straight part at the entrance of the root canal. Moreover, they based their analysis on the von Mises stress instead of analysing strain, which is the relevant parameter for predicting the fatigue life for low-cycle fatigue, according to the Coffin–Manson relation [25]. They also used a constitutive material model that did not include the hysteresis cycle formed in the stress– strain cycle due to the different stress levels corresponding to the phase transformation during loading and unloading.

In the present study, we used transient FE simulation for analysing the fatigue behaviour of a NiTi endodontic file with two different pitch values on a greater variety of root canal geometries, with changes in both the angle between the initial part and the apical part of the root and the radius of curvature in the connection between both sections. The model also includes a more comprehensive constitutive model for NiTi material and a very detailed discretisation of the file into quadratic finite elements. It simulates the introduction of the file into the canal and its rotation, including contact and friction. With this model, we calculated the strain range during a cycle for each point of the file. We used the Coffin–Manson relation to predict the expected NCF of the file in each root canal geometry.

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

The present investigation was conducted by using finite element analysis of a set of cases of study in which several combinations of endodontic rotary files and root canal geometries were studied.

Two different geometries of endodontic rotary file were considered, which are denoted as P2 (Figure 1a) and P3 (Figure 1b). Both of them have a convex ProTaper cross section shown in Figure 1c, their total length being *Ltotal* = 25 mm, the length of their active part is *Lap* = 16 mm and the diameter of their shaft and their tip is *dsh* = 1.20 mm and *dap* = 0.25 mm, respectively. The only difference between P2 and P3 resides in their axial pitch: *pz* = 2 mm for P2 and *pz* = 3 mm for P3.

**Figure 1.** Geometry of the endodontic files P2 (**a**) and P3 (**b**) and normalised transversal cross section for both of them (**c**).

On the other hand, the geometry of the root canal was constructed as follows (Figure 2):


The resulting geometry of the root canal depends on two parameters: the angle of curvature *θRC* and the radius of curvature *rRC*, according to the method proposed by Pruett [26]. The variation of these parameters allows us to consider different geometries for the root canal. In this study, three different values for the angle of curvature *θRC* = [30◦, 45◦, 60◦] and three different radii of curvature *rRC* = [5 mm, 10 mm, 15 mm]

were considered. Combining these variables, 9 different geometries of the root canal were obtained, which are shown in Figure 3 (denoted as RC1, RC2, ..., RC9).

**Figure 2.** Parametrisation of the geometry of the root canal: (**a**) definition of the segmen<sup>t</sup> *ABRC* and line *L*1, (**b**) definition of the fillet and (**c**) definition of the root canal surface.

**Figure 3.** Geometries of root canal considered for the study.

#### *2.1. Definition of the Finite Element Model*

Figure 4 shows the finite element model used in this study, which consists of an endodontic rotary file and a root canal. The root canal is modelled as a rigid surface under the assumption that its deformations are so small compared to the deformations of the endodontic rotary file that they can be neglected. The root canal remains immovable during the analysis.

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

The geometry of the endodontic file is generated and then discretised into quadratic finite element tetrahedrons following the ideas provided in [27]. The average element size has been set to 0.1 mm, which has proven to provide a good compromise between accuracy and computational cost. The resulting finite element model has 103,609 nodes and 68,367 elements.

The top surface of the endodontic rotary file is defined as a rigid surface (shaded in grey in Figure 4), and its movements are coupled to the movements of a reference node that is used to define the boundary conditions of the endodontic rotary file.

The superelastic behaviour of the NiTi alloy was modelled by using the material model developed by Auricchio [28], which is summarized in Figure 5. Here, *EA* and *EM* represent the Young's modulus of austenite and martensite, respectively. The beginning and the end of the loading transformation phase are given by *σSL* and *σEL* , respectively, whereas the beginning and the end of the unloading transformation phase are given by *σSU* and *<sup>σ</sup>EU*. Finally, *εL* represents the uniaxial transformation strain, and *σEME* indicates the end of the martensitic elastic regime. In this work, the material properties that characterise this material model were extracted from [29].

**Figure 5.** Definition of the stress–strain curve for the constitutive model of the superelastic NiTi alloy.

The mechanical interaction between the root canal and the endodontic rotary file was considered by using a node-to-surface contact. A penalty-based constraint enforcement method was selected in order to enhance the convergency of the numerical solution. The tangential behaviour of the contact was also taken into account in the finite element model, with a constant coefficient of friction *μ* = 0.1 [30].

The finite element model was solved by using transient analysis, which was conducted by using a large displacement formulation, and performed in two sequential steps:


#### *2.2. Fatigue Life Estimation from the Results of the Finite Element Analysis*

The objective of this study was to predict the fatigue life of NiTi rotary files as they are rotating inside the root canal. For such a purpose, the strain results obtained from the rotation step of the finite element analysis were used in combination with the Coffin– Manson relation. This relation is conveniently expressed by the following equation:

$$\frac{\Delta \varepsilon}{2} = \varepsilon\_F^{\prime} \cdot \mathcal{N}\_f^{\varepsilon} + \frac{\sigma\_F^{\prime}}{E} \cdot \mathcal{N}\_f^{b} \tag{1}$$

where *Nf* is equivalent to NCF, *εF* is the fatigue ductility coefficient, *σF* is the fatigue strength coefficient, *c* is the fatigue ductility exponent, *b* is the fatigue strength exponent, Δ*ε* is the total strain range and Δ*ε*/2 is the strain amplitude. The prime in the equation indicates that the properties correspond to the cyclic properties, i.e., those after the initial 100–140 cycles.

In this equation, the first addend of the right side corresponds to the plastic strain amplitude Δ*ε p*/2 and the second one to the elastic strain amplitude Δ*εe*/2. Figure 6 shows a logarithmic plot of the Equation (1), showing the contribution of these two terms, with the parameters for NiTi used in the present study, taken from [25]. The exponents *b* and *c* in the equation are negative, because the number of cycles correlates negatively with the strain amplitude. For high strain amplitudes, the plastic strain is much higher than the elastic strain, and the number of cycles to failure is low (low-cycle fatigue, LCF); for very low strain amplitudes, the second term of the equation is dominant because there is no significant plastic strain, and the number of cycles to failure is high (high-cycle fatigue, HCF). The transition between LCF and HCF can be observed as a change in the slope of the curve, which is typically located close to 103–104 cycles.

Since the Coffin–Manson relation is based on a uniaxial strain, a criterion to reduce the obtained multiaxial strain state to an equivalent uniaxial strain condition is required. The critical plane concept has been extensively used for such a purpose, with successful results both for high and low cycle fatigue [31]. In the critical plane approach, the assessment of the fatigue failure is carried out in the material plane where the amplitude of some stress/strain components (or a combination of them) exhibits a maximum [24]. In the discretised finite element model of the endodontic rotary file, each node *i* on the surface will have an associated critical plane Π*i* characterised by its normal direction *ni*.

**Figure 6.** Coffin–Manson relation between strain amplitude and number of cycles to failure (NCF). Parameters for NiTi from [25]: *εF* = 0.68, *σF* = 705 MPa, *E* = 42.5 GPa, *c* = −0.6, *b* = −0.06.

In this study, the critical plane Π*i* was defined in such a manner that its normal direction *ni* is parallel to the direction of the maximum principal strain produced in node *i*, when the amplitude of this maximum principal strain reaches its maximum value. The direction *ni* could be determined by observing the maximum principal strain at each frame of the analysis. However, in order to speed up the calculations, in this study it will be assumed that this maximum principal strain is normal to the plane that contains the trajectory of the observed node, as illustrated in Figure 7. Hence, *ni* will be normal to the plane of rotation of node *i*.

After the critical plane Π*i* is determined for node *i*, a bending strain value *<sup>ε</sup>i*,*j* can be then obtained for that node at each analysis frame *j* by transforming the strain tensor and selecting the strain component in the direction of *ni*. Finally, the total strain range Δ*εi* for node *i* is defined as follows.

$$
\Delta \varepsilon\_i = \max\_{j=1\ldots n} (\varepsilon\_{i,j}) - \min\_{j=1\ldots n} (\varepsilon\_{i,j}) \tag{2}
$$

This total strain range Δ*εi* can be used in the Coffin–Manson relation to assess the fatigue life associated to node *i*. The material parameters considered for the application of the Coffin–Manson relation were extracted from [25].

**Figure 7.** Determination of the critical plane and bending strain for node *i*.

The fatigue life of the endodontic rotary file was defined by the minimum value of fatigue life considering all the nodes in the surface of the endodontic rotary file. The node where Δ*εi* reaches a maximum value is the critical node, and it is denoted as *i* = *crit*.
