# Fatigue Analysis of NiTi Rotary Endodontic Files through Finite Element Simulation: Effect of Root Canal Geometry on Fatigue Life

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## Abstract

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## 1. Introduction

- The strain–life relationship is similar to that observed in low-cycle fatigue for metals, with a decrease in NCF for higher strain amplitudes, corresponding to highly curved canals [14].
- Apart from the ‘structural fatigue’ resulting in the final fracture, NiTi exhibits ‘functional fatigue’, a significant and asymptotic change in the stress–strain curve and the phase transformation stresses during the first 100–140 cycles, resulting in a reduction in hysteresis cycle area and an increase in residual permanent strains after cycling [20].

## 2. Materials and Methods

- Segment AB has a length of $A{B}_{RC}=10\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$, and it is perpendicular to the external surface, as indicated in Figure 2a. Line ${L}_{1}$ passes through point B, and it is inclined at angle ${\theta}_{RC}$ with respect to segment AB.
- A fillet, for which its radius is given by ${r}_{RC}$, is defined between segment AB and line ${L}_{1}$, as illustrated in Figure 2b. The tangency points of the fillet with the existing segments are denoted by D and E.
- Point F is located over line ${L}_{1}$ in such a manner that the total length from point A to point F is ${L}_{RC}=16\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$. By performing this, the entire active part of the endodontic rotary files can be inserted within the canal. The resulting curve ADEF is the neutral axis of the root canal.
- Finally, a conic surface is created by sweeping a circumference along the neutral axis of the root canal, as illustrated in Figure 2c. At the entrance of the canal, the diameter of this circumference is ${D}_{RC}=1.26\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$, and at the end of the canal it is ${d}_{RC}=0.26\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$.

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

- Insertion step: In the first step, the endodontic rotary file is inserted into the root canal. This is performed by prescribing a displacement at its reference node, which takes place along the y axis and has a magnitude equal to the length of the active part of the endodontic rotary file (${L}_{ap}$). The rest of the movements of the reference node (displacements in x and y directions and all the rotations) are restricted in this step.
- Rotation step: In the second step, after the active part of the endodontic rotary file is inserted in the root canal, the endodontic rotary file performs a complete revolution along its axis of rotation. This is performed by prescribing a ${360}^{\circ}$ rotation along the y axis, while the rest of the movements of the reference node are restricted (rotations along x and z axes and all the displacements). The rotated angle is denoted by $\phi $.

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

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Bürklein, S.; Schäfer, E. Critical evaluation of root canal transportation by instrumentation. Endod. Top.
**2013**, 29, 110–124. [Google Scholar] [CrossRef] - Iandolo, A.; Amato, A.; Martina, S.; Latif, D.A.; Pantaleo, G. Management of severe curvatures in root canal treatment with the new generation of rotating files using a safe and predictable protocol. Open Dent. J.
**2020**, 14, 421–425. [Google Scholar] [CrossRef] - Iandolo, A.; Abdellatif, D.; Pantaleo, G.; Sammartino, P.; Amato, A. Conservative shaping combined with three-dimensional cleaning can be a powerful tool: Case series. J. Conserv. Dent.
**2020**, 23, 648–652. [Google Scholar] [CrossRef] - Kuzekanani, M. Nickel-titanium rotary instruments: Development of the single-file systems. J. Int. Soc. Prev. Community Dent.
**2018**, 8, 386–390. [Google Scholar] [CrossRef] - Chi, C.W.; Li, C.C.; Lin, C.P.; Shin, C.S. Cyclic fatigue behavior of nickel–titanium dental rotary files in clinical simulated root canals. J. Formos. Med. Assoc.
**2017**, 116, 306–312. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Plotino, G.; Grande, N.M.; Cordaro, M.; Testarelli, L.; Gambarini, G. A Review of Cyclic Fatigue Testing of Nickel-Titanium Rotary Instruments. J. Endod.
**2009**, 35, 1469–1476. [Google Scholar] [CrossRef] - Scattina, A.; Alovisi, M.; Paolino, D.S.; Pasqualini, D.; Scotti, N.; Chiandussi, G.; Berutti, E. Prediction of cyclic fatigue life of nickel-titanium rotary files by virtual modeling and finite elements analysis. J. Endod.
**2015**, 41, 1867–1870. [Google Scholar] [CrossRef] [Green Version] - Lee, M.H.; Versluis, A.; Kim, B.M.; Lee, C.J.; Hur, B.; Kim, H.C. Correlation between experimental cyclic fatigue resistance and numerical stress analysis for nickel-titanium rotary files. J. Endod.
**2011**, 37, 1152–1157. [Google Scholar] [CrossRef] [PubMed] - Sattapan, B.; Nervo, G.J.; Palamara, J.E.; Messer, H.H. Defects in rotary nickel-titanium files after clinical use. J. Endod.
**2000**, 26, 161–165. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Peng, B.; Shen, Y.; Cheung, G.S.; Xia, T.J. Defects in ProTaper S1 instruments after clinical use: Longitudinal examination. Int. Endod. J.
**2005**, 38, 550–557. [Google Scholar] [CrossRef] [PubMed] - Wei, X.; Ling, J.; Jiang, J.; Huang, X.; Liu, L. Modes of Failure of ProTaper Nickel-Titanium Rotary Instruments after Clinical Use. J. Endod.
**2007**, 33, 276–279. [Google Scholar] [CrossRef] - McGuigan, M.B.; Louca, C.; Duncan, H.F. Endodontic instrument fracture: Causes and prevention. Br. Dent. J.
**2013**, 214, 341–348. [Google Scholar] [CrossRef] [PubMed] - Lo Savio, F.; Rosa, G.L.; Bonfanti, M.; Alizzio, D.; Rapisarda, E.; Pedullà, E. Novel Cyclic Fatigue Testing Machine for Endodontic Files. Exp. Tech.
**2020**, 44, 649–665. [Google Scholar] [CrossRef] - Cheung, G.S.; Darvell, B.W. Fatigue testing of a NiTi rotary instrument. Part 1: Strain-life relationship. Int. Endod. J.
**2007**, 40, 612–618. [Google Scholar] [CrossRef] - Tabassum, S.; Zafar, K.; Umer, F. Nickel-titanium rotary file systems: What’s new? Eur. Endod. J.
**2019**, 4, 111–117. [Google Scholar] [CrossRef] - Figueiredo, A.M.; Modenesi, P.; Buono, V. Low-cycle fatigue life of superelastic NiTi wires. Int. J. Fatigue
**2009**, 31, 751–758. [Google Scholar] [CrossRef] - Eggeler, G.; Hornbogen, E.; Yawny, A.; Heckmann, A.; Wagner, M. Structural and functional fatigue of NiTi shape memory alloys. Mater. Sci. Eng. A
**2004**, 378, 24–33. [Google Scholar] [CrossRef] - Vilaverde Correia, S.; Nogueira, M.T.; Silva, R.J.C.; Pires Lopes, L.; Braz Fernandes, F.M. Phase Transformations in NiTi Endodontic Files and Fatigue Resistance. In European Symposium on Martensitic Transformations; EDP Sciences: Les Ulis, France, 2009; p. 07004. [Google Scholar] [CrossRef] [Green Version]
- Dornelas Silva, J.; Lopes Buono, V.T. Effect of the initial phase constitution in the low-cycle fatigue of NiTi wires. SN Appl. Sci.
**2019**, 1, 1591. [Google Scholar] [CrossRef] [Green Version] - Maletta, C.; Sgambitterra, E.; Furgiuele, F.; Casati, R.; Tuissi, A. Fatigue properties of a pseudoelastic NiTi alloy: Strain ratcheting and hysteresis under cyclic tensile loading. Int. J. Fatigue
**2014**, 66, 78–85. [Google Scholar] [CrossRef] - Chien, P.Y.; Walsh, L.J.; Peters, O.A. Finite element analysis of rotary nickel-titanium endodontic instruments: A critical review of the methodology. Eur. J. Oral Sci.
**2021**, 129, e12802. [Google Scholar] [CrossRef] - Ha, J.H.; Lee, C.J.; Kwak, S.W.; El Abed, R.; Ha, D.; Kim, H.C. Geometric optimization for development of glide path preparation nickel-titanium rotary instrument. J. Endod.
**2015**, 41, 916–919. [Google Scholar] [CrossRef] [PubMed] - Xu, X.; Eng, M.; Zheng, Y.; Eng, D. Comparative study of torsional and bending properties for six models of nickel-titanium root canal instruments with different cross-sections. J. Endod.
**2006**, 32, 372–375. [Google Scholar] [CrossRef] [PubMed] - Carpinteri, A.; Spagnoli, A.; Vantadori, S.; Bagni, C. Structural integrity assessment of metallic components under multiaxial fatigue: The C-S criterion and its evolution. Fatigue Fract. Eng. Mater. Struct.
**2013**, 36, 870–883. [Google Scholar] [CrossRef] - Cheung, G.S.; Zhang, E.W.; Zheng, Y.F. A numerical method for predicting the bending fatigue life of NiTi and stainless steel root canal instruments. Int. Endod. J.
**2011**, 44, 357–361. [Google Scholar] [CrossRef] [PubMed] - Pruett, J.P.; Clement, D.J.; Carnes, D.L. Cyclic fatigue testing of nickel-titanium endodontic instruments. J. Endod.
**1997**, 23, 77–85. [Google Scholar] [CrossRef] - Roda-Casanova, V.; Zubizarreta-Macho, A.; Sanchez-Marin, F.; Alonso Ezpeleta, O.; Albaladejo Martinez, A.; Galparsoro Catalan, A. Computerized Generation and Finite Element Stress Analysis of Endodontic Rotary Files. Appl. Sci.
**2021**, 11, 4329. [Google Scholar] [CrossRef] - Auricchio, F.; Petrini, L. A three-dimensional model describing stress-temperature induced solid phase transformations: Solution algorithm and boundary value problems. Int. J. Numer. Methods Eng.
**2004**, 61, 807–836. [Google Scholar] [CrossRef] - de Arruda Santos, L.; López, J.B.; de Las Casas, E.B.; de Azevedo Bahia, M.G.; Buono, V.T.L. Mechanical behavior of three nickel-titanium rotary files: A comparison of numerical simulation with bending and torsion tests. Mater. Sci. Eng. C
**2014**, 37, 258–263. [Google Scholar] [CrossRef] - Ha, J.H.; Cheung, G.S.; Versluis, A.; Lee, C.J.; Kwak, S.W.; Kim, H.C. ‘Screw-in’ tendency of rotary nickel-titanium files due to design geometry. Int. Endod. J.
**2015**, 48, 666–672. [Google Scholar] [CrossRef] - Karolczuk, A.; Macha, E. Selection of the critical plane orientation in two-parameter multiaxial fatigue failure criterion under combined bending and torsion. Eng. Fract. Mech.
**2008**, 75, 389–403. [Google Scholar] [CrossRef] - Kang, G.; Song, D. Review on structural fatigue of NiTi shape memory alloys: Pure mechanical and thermo-mechanical ones. Theor. Appl. Mech. Lett.
**2015**, 5, 245–254. [Google Scholar] [CrossRef] [Green Version] - Gavini, G.; dos Santos, M.; Caldeira, C.L.; Machado, M.E.d.L.; Freire, L.G.; Iglecias, E.F.; Peters, O.A.; Candeiro, G.T.d.M. Nickel-titanium instruments in endodontics: A concise review of the state of the art. Braz. Oral Res.
**2018**, 32, e67. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Geometry of the endodontic files P2 (

**a**) and P3 (

**b**) and normalised transversal cross section for both of them (

**c**).

**Figure 2.**Parametrisation of the geometry of the root canal: (

**a**) definition of the segment $A{B}_{RC}$ and line ${L}_{1}$, (

**b**) definition of the fillet and (

**c**) definition of the root canal surface.

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

**Figure 6.**Coffin–Manson relation between strain amplitude and number of cycles to failure (NCF). Parameters for NiTi from [25]: ${\epsilon}_{F}^{\prime}=0.68$, ${\sigma}_{F}^{\prime}=705\phantom{\rule{0.277778em}{0ex}}\mathrm{MPa}$, $E=42.5\phantom{\rule{0.277778em}{0ex}}\mathrm{GPa}$, $c=-0.6$, $b=-0.06$.

**Figure 10.**Maximum bending strain range as a function of the geometry of the root canal. (

**a**) P2 and (

**b**) P3.

**Figure 11.**Expected life in number of fatigue cycles ${N}_{f}$ as a function of the geometry of the root canal. (

**a**) P2 and (

**b**) P3.

**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.

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**MDPI and ACS Style**

Roda-Casanova, V.; Pérez-González, A.; Zubizarreta-Macho, Á.; 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

**AMA Style**

Roda-Casanova V, Pérez-González A, Zubizarreta-Macho Á, Faus-Matoses V.
Fatigue Analysis of NiTi Rotary Endodontic Files through Finite Element Simulation: Effect of Root Canal Geometry on Fatigue Life. *Journal of Clinical Medicine*. 2021; 10(23):5692.
https://doi.org/10.3390/jcm10235692

**Chicago/Turabian Style**

Roda-Casanova, Victor, Antonio Pérez-González, Álvaro Zubizarreta-Macho, and Vicente Faus-Matoses.
2021. "Fatigue Analysis of NiTi Rotary Endodontic Files through Finite Element Simulation: Effect of Root Canal Geometry on Fatigue Life" *Journal of Clinical Medicine* 10, no. 23: 5692.
https://doi.org/10.3390/jcm10235692