# Influence of Cross-Section and Pitch on the Mechanical Response of NiTi Endodontic Files under Bending and Torsional Conditions—A Finite Element Analysis

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

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

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

- In the bending analysis, an increasing displacement was imposed at reference node B in the negative direction of the y-axis, until the maximum von Mises stress along the endodontic file ${\sigma}_{max}$ reached the end of the martensitic elastic regime. As the results of the bending analyses are sensitive to the orientation of the endodontic file with respect to the bending direction, the analysis was conducted in 24 different angular positions, given by a rotation ${\phi}_{z}=\{{0}^{\circ},{15}^{\circ},{30}^{\circ},\dots ,{360}^{\circ}\}$ of the endodontic file with respect to the z-axis.
- In the torsional analysis, an increasing rotation was imposed at reference node B along the positive direction of z-axis, until the maximum von Mises stress along the endodontic file ${\sigma}_{max}$ reached the end of the martensitic elastic regime. Here, the results of the analysis do not depend on the orientation of the file.

## 3. Results

#### 3.1. Bending Analysis

#### 3.2. Torsional Analysis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts 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

- Let $i\in [1,{n}_{e}]$ refer to each of the ${n}_{e}$ tetrahedral finite elements in the model, and $j\in [1,4]$ refer to the integration points in each tetrahedral element. The von Mises stress at a given element and integration point is denoted as ${\sigma}_{ij}$, and the volume associated with each integration point is denoted as ${V}_{ij}={V}_{i}/4$ (where ${V}_{i}$ is the volume of element i).
- The von Mises stress ${\sigma}_{ij}$ and the volume ${V}_{ij}$ at each integration point of the model are retrieved and stored in an array $\mathsf{\Sigma}$ with ${n}_{i}=4\xb7{n}_{e}$ rows. Each row m in $\mathsf{\Sigma}$ contains the von Mises stress and the volume associated with a given integration point, with the shape$$\mathsf{\Sigma}\left[m\right]=[{\sigma}_{ij},{V}_{ij}].$$
- The rows in $\mathsf{\Sigma}$ 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 ${\sigma}_{ij}$ stress in the analysis frame after the array $\mathsf{\Sigma}$ is created.

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

- On one hand, the Coffin–Manson relation is based on a uni-axial strain, but the strain results obtained from the finite element model correspond to a multi-axial strain state. Thus, a criterion to reduce the obtained multi-axial strain state to an equivalent uni-axial strain condition is required.
- On the other hand, the bending analysis conducted using the proposed finite element model does not represent the actual strain history of the endodontic file when it is rotating inside the root canal, as bending is applied in just one direction (uni-directional fatigue). Thus, a conversion method must be proposed to convert the obtained strains into a purely reversed fatigue phenomenon.

## Appendix B. Literature Review

**Table A1.**Previous FE studies considering the effects of cross-section and pitch on rotary endodontic files.

Source | Section Type | Tip Diameter; Taper | Pitch (mm) | Material Model and Parameters | FE Code; Model Type | Boundary Conditions; Number of Nodes/Elements | Conclusions | Limitations |
---|---|---|---|---|---|---|---|---|

Xu et al., 2006 [32] | 6 shapes (ProTaper, Hero642, Mtwo, ProFile, Quantec, NiTiflex) | 0.4 mm, $4\%$ | 3.6 | Multi-linear kinematic hardening plastic model ${E}_{A}$ = 34.3 GPa, ${\nu}_{A}$ = 0.33, ${\sigma}_{L}^{S}$ = 480 MPa, ${\sigma}_{L}^{E}$ = 755 MPa | N/A, Static | Loads: progressive 0–2.5 Nmm torsion in shank, Constraints: fixed at tip, # nodes: Not available. # elements: Not available. | (1) Sections with higher core area show lower stresses for the same torque | (1) Sections analyzed have different total areas |

Kim et al., 2009 [34] | 4 shapes (ProFile, HeroShaper, Mtwo, NRT) | 0.3 mm; $6\%$ | Several, N/A | ${E}_{A}$ = 36 GPa, ${\nu}_{A}$ = 0.3, ${\sigma}_{L}^{E}$ = 504 MPa, ${\sigma}_{L}^{S}$ = 755 MPa | ABAQUS; Static (cases I to IV) Dynamic (case V): Simulated shaping | Case I (or II), Load: 1 N (or 2 mm) bending in tip Constraint: shank fixed Case II (or III), Load: 2.5 Nmm (or 10°) torsion in shank Constraint: fixed at 4 mm from tip Case V, Constraint: shank rotation 240 rpm, file introduction in simulated root canal; # nodes: 7018–18,214 # elements: 5300–9440 | (1) Rectangle-based sections have lower expected fatigue life than triangle-based sections | (1) Material model not clearly defined |

Baek et al., 2011 [36] | 4 theoretical shapes (triangle, slender rectangle, rectangle, square) | 0.3 mm; $4.4\%$ | 3.2, 1.6, 1.1 | ${E}_{A}$ = 36 GPa, ${\nu}_{A}$ = 0.3 | ABAQUS; Static | Load: 20° torsion in shank Constraint: fixed at 4 mm from tip; # nodes: Not available. # elements: Not available. | (1) Rectangle-based sections, even with smaller areas, have higher torsional stiffness than triangular section; (2) Reduction in pitch increases torsional stiffness | (1) Linear material model; (2) Mesh quality not provided |

Arbab-Chirani et al., 2011 [35] | 5 shapes (Hero, Hero Shaper, Mtwo, ProFile, ProTaper F1) | 0.2mm; $6\%$ | Several, N/A | SMA material model, ${E}_{A}$ = 47 GPa, ${\nu}_{A}$ = 0.3 ${\sigma}_{L}^{S}$ = 505 MPa | Cast3M; Static | Case 1: Load: bending at tip 3.8 mm, Constraint: shank fixed Case 2: Load: torsion at tip 22°, Constraint: shank fixed; # nodes: 66,023–73,561 # elements: 14,100–16,200 | (1) ProTaper F1, Hero Shaper, and Hero are stiffer than Mtwo and ProFile; (2) Maximum stresses near the tip for both cases and similar for all the files | (1) Different pitch among files; (2) Deformations applied are low to extend martensitic transformation to a significant part of the file |

Versluis et al., 2012 [33] | 4 theoretical shapes (triangle, slender rectangle, rectangle, square) | 0.3 mm; $4\%$ | 3.2, 1.6, 1.1 | SMA material model, ${E}_{A}$ = 36 GPa, ${\nu}_{A}$ = 0.3 ${\sigma}_{L}^{S}$ = 504 MPa, ${\sigma}_{L}^{E}$ = 600 MPa | MSC.Marc; Static | Load: bending at tip 5 mm (all possible orientations with respect to the cross-section), Constraint: shank axis orientation and shank end location fixed; # nodes: Not available. # elements: Not available. | (1) Flexural stiffness and stress decreases with decreasing pitch; (2) Decreasing the pitch reduces the oscillation of stress when the file rotates; (3) Flexural stiffness and stress correlates with center-core area; (4) Effect of section greater than that of pitch; (5) Maximum stress is affected by bending orientation for rectangular section | (1) Deformations applied are low to extend martensitic transformation to a significant part of the file (max. stresses below 504 MPa) |

De Arruda et al., 2014 [10] | 3 shapes (Mtwo, RaCe, PTU F1) | 0.25 mm; $6\%$ | Several (Not available) | Shape-memory alloy material model implemented as ABAQUS sub-routine, ${E}_{A}$ = 42.53 GPa, ${\nu}_{A}$ = 0.33 ${\sigma}_{L}^{S}$ = 492 MPa, ${\sigma}_{L}^{E}$ = 630 MPa | ABAQUS; Static | Case 1: Load: bending in shank from 0° to 45° (two perpendicular orientations), Constraint: fixed at 3 mm from tip Case 2: Load: 3 Nmm torsion in shank, Constraint: fixed at 3 mm from tip; # nodes: 84,126–91,372 # elements: 48,460–55,009 | (1) Finite element analysis results agree with experimental results; (2) RaCe and Mtwo are more flexible than PTU F1 in bending and torsion; (3) Shape of the section affects the maximum stress and the variation in stress with bending orientation | (1) Only three section geometries and two orientations for bending considered; (2) Different pitch among files |

Ha et al., 2015 [37] | 4 theoretical shapes (triangle, slender rectangle, rectangle, square) | 0.3 mm; $4.4\%$ | 3.2, 1.6, 1.1 | ${E}_{A}$ = 26 GPa, ${\nu}_{A}$ = 0.3 | ABAQUS; Not available. | Load: Prescribed rotation inside the root canal, Constraint: Contact with friction in 3 simulated root canal (15°, 30°, 45° curvature), shank axis orientation & shank end location fixed; # nodes: 10,230–18,042 # elements: 8325–15,540 | (1) The square cross-section shows the highest ‘screw-in’ force and reaction torque; (2) ‘Screw-in’ force and reaction torque are higher for greater pitch and higher root canal curvature | (1) Linear material model; (2) Very low friction coefficient (0.1) considered between file and root canal; (3) Solid surface used as root canal model; (4) Only 3 root canal geometries considered |

Basser-Ahamed et al., 2018 [31] | 5 theoretical shapes (triangle T, convex triangle C, concave triangle U, combined CTU, combinedUTC) | 0.25 mm; $6\%$ | 1.6 | File: ${E}_{A}$ = 36 GPa, ${\nu}_{A}$ = 0.3 Root canal: E = 18.6 GPa, ${\nu}_{A}$ = 0.3 | ANSYS; Not available. | Load: Torque 2 Nm, Constraint: Contact with simulated root canal (45° curvature), shank axis orientation and shank end location fixed, rotation of 180° at 240 rpm; File: # nodes: 16,750–42,785 # elements: 75,430–152,432 Root canal: # nodes: 3000 # elements: 3500 | (1) A combined section CTU (C coronal third, T middle third, U apical third) presents lower stresses than constant section | (1) Geometry of the root canal not clearly defined defined; (2) Contact and friction conditions undefined; (3) Does not consider changes in dentin properties within the root canal; (4) Effect of pitch not analyzed |

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**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**).

**Figure 5.**The von Mises stress plots for the bending analysis of endodontic files with ${p}_{z}=4\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ and $\phi ={0}^{\circ}$.

**Figure 6.**Bending moment–rotation relationships for the bending analysis of endodontic files with ${p}_{z}=4\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$: squared cross-section (

**a**) and triangular cross-section (

**b**).

**Figure 7.**Bending analysis: effect of the pitch on the maximum rotation (

**a**) and maximum applied torque (

**b**) when the end of the martensitic elastic regime is reached.

**Figure 8.**Bending analysis: bending stiffness of the endodontic rotary files with (

**a**) square and (

**b**) triangular cross-section.

**Figure 9.**Bending analysis: effect of the pitch on the maximum principal strain (

**a**) and the expected number of cycles (

**b**) when the rotated angle is ${\theta}_{x}={20}^{\circ}$.

**Figure 10.**The von Mises stress plots for the torsional analysis of endodontic files with ${p}_{z}=4\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$.

**Figure 11.**Torque–rotation relationships for the torsional analysis of endodontic files with different axial pitch: squared cross-section (

**a**) and triangular cross-section (

**b**).

**Figure 12.**Torsional analysis: effect of the pitch on the applied torque (

**a**) and rotation (

**b**) when the end of the martensitic elastic regime is reached.

**Figure 13.**Torsional analysis: torsion stiffness of the endodontic rotary files with (

**a**) square and (

**b**) triangular cross-section.

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

Parameter | Variable | Magnitude |
---|---|---|

Young’s modulus of austenite | ${E}_{A}$ | 42,530 MPa |

Austenite Poisson’s ratio | ${\nu}_{A}$ | 0.33 |

Young’s modulus of martensite | ${E}_{M}$ | 12,828 MPa |

Martensite Poisson’s ratio | ${\nu}_{M}$ | $0.33$ |

Uni-axial transformation strain | ${\epsilon}_{L}$ | $6\%$ |

Slope of the stress–temperature curve for loading | ${(\delta \sigma /\delta T)}_{L}$ | $6.7$ |

Start of transformation loading | ${\sigma}_{L}^{S}$ | $492\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa}$ |

End of transformation loading | ${\sigma}_{L}^{E}$ | $630\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa}$ |

Reference temperature | ${T}_{0}$ | 22 ${}^{\circ}$C |

Slope of the stress–temperature curve for unloading | ${(\delta \sigma /\delta T)}_{U}$ | $6.7$ |

Start of transformation unloading | ${\sigma}_{U}^{S}$ | $192\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa}$ |

End of transformation unloading | ${\sigma}_{U}^{E}$ | $97\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa}$ |

End of martensitic elastic regime | ${\sigma}_{ME}^{E}$ | $1200\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa}$ |

Parameter | Variable | Magnitude |
---|---|---|

Fatigue ductility coefficient | ${\epsilon}_{F}^{{}^{\prime}}$ | $0.68$ |

Fatigue strength coefficient | ${\sigma}_{F}^{{}^{\prime}}$ | $705\phantom{\rule{3.33333pt}{0ex}}\mathrm{MPa}$ |

Fatigue ductility exponent | c | $-0.6$ |

Fatigue strength exponent | b | $-0.06$ |

Modulus of elasticity | E | $42.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{GPa}$ |

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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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. *Journal of Clinical Medicine*. 2022; 11(9):2642.
https://doi.org/10.3390/jcm11092642

**Chicago/Turabian Style**

Roda-Casanova, Victor, Antonio Pérez-González, Alvaro Zubizarreta-Macho, and Vicente Faus-Matoses.
2022. "Influence of Cross-Section and Pitch on the Mechanical Response of NiTi Endodontic Files under Bending and Torsional Conditions—A Finite Element Analysis" *Journal of Clinical Medicine* 11, no. 9: 2642.
https://doi.org/10.3390/jcm11092642