Materials Science and Engineering C 32 (2012) 1594–1600

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An analytical mechanical model to describe the response of NiTi rotary endodontic files in a curved root canal Agnès Marie Françoise Leroy a, b, Maria Guiomar de Azevedo Bahia c, Alain Ehrlacher b, Vicente Tadeu Lopes Buono a,⁎ a b c

Department of Metallurgical and Materials Engineering, School of Engineering, Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, MG, Brazil Department of Mechanical and Materials Engineering, École des Ponts Paristech (ENPC), Champs-sur-Marne, France Department of Restorative Dentistry, Faculty of Dentistry, Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, MG, Brazil

a r t i c l e

i n f o

Article history: Received 10 September 2011 Received in revised form 26 February 2012 Accepted 22 April 2012 Available online 28 April 2012 Keywords: Large transformations mechanics Rotary endodontic files Nickel–titanium alloys Superelasticity

a b s t r a c t Aim: To build a mathematical model describing the mechanical behavior of NiTi rotary files while they are rotating in a root canal. Methodology: The file was seen as a beam undergoing large transformations. The instrument was assumed to be rotating steadily in the root canal, and the geometry of the canal was considered as a known parameter of the problem. The formulae of large transformations mechanics then allowed the calculation of the Green–Lagrange strain field in the file. The non-linear mechanical behavior of NiTi was modeled as a continuous piecewise linear function, assuming that the material did not reach plastic deformation. Criteria locating the changes of behavior of NiTi were established and the tension field in the file, and the external efforts applied on it were calculated. The unknown variable of torsion was deduced from the equilibrium equation system using a Coulomb contact law which solved the problem on a cycle of rotation. Results: In order to verify that the model described well reality, three-point bending experiments were managed on superelastic NiTi wires, whose results were compared to the theoretical ones. It appeared that the model gave a good mentoring of the empirical results in the range of bending angles that interested us. Conclusions: Knowing the geometry of the root canal, one is now able to write the equations of the strain and stress fields in the endodontic instrument, and to quantify the impact of each macroscopic parameter of the problem on its response. This should be useful to predict failure of the files under rotating bending fatigue, and to optimize the geometry of the files. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Endodontic treatment is the clinical intervention that aims to cure a tooth considered so threatened that an infection is very probable and can lead to the necrosis of the tooth. It includes a step of shaping the root canal that is made by the dentist using endodontic instruments or files, which can be operated manually or driven by an electric engine. Rotary files are made of superelastic NiTi, an alloy that owes its uncommon properties to a reversible first order phase transition: the martensitic transformation [1]. The properties of NiTi depend strongly on their chemical composition and thermo-mechanical history. It was shown that the use of NiTi rotary files improved the quality of canal shaping and thus the success rate of the treatment [2], but the instruments still have their limitations: they can occasionally break inside the root

⁎ Corresponding author at: Department of Metallurgical and Materials Engineering, Universidade Federal de Minas Gerais, Av. Antonio Carlos, 6627, Campus Pampulha, Escola de Engenharia, Bloco 2, sala 2640, 31270-901 Belo Horizonte, MG, Brazil. Tel.: + 55 31 3409 1859; fax: + 55 31 3409 1815. E-mail address: [email protected] (V.T.L. Buono). 0928-4931/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.msec.2012.04.049

canal. This failure can jeopardize the treatment success and can be caused by two mechanisms: fatigue, due to rotating bending in a curved root canal, or fracture under torsional overload, which occurs mainly when the file locks in the canal walls [3]. Fatigue of NiTi rotary instruments is a mechanical problem that can be tackled through three complementary approaches: empirical study, computational analysis and analytical modeling. The empirical study is indispensable and has already led to various publications [4–6] but is fastidious and expensive, which partly explains the development of the other two approaches. Though computational analysis has been much studied over the last few years [7], the construction of an analytical mathematical model of an endodontic instrument appeared later as being a possibly efficient method, as shown by Zhang et al. in 2011 [8]. Until now, few works of numerical or analytical simulation of the files succeeded in taking into account the non-linear behavior of NiTi, even less the hysteresis and the asymmetry between tension and compression of the material's mechanical response that were observed through experiments [1,9]. Moreover, most of the mathematical studies aimed to improve the geometry of the cross-section of the instrument, but few aimed to forecast its lifetime [10].

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Unfortunately, endodontic instruments cannot be dimensioned so as to have an infinite lifetime. Besides that, assessing the damage done in an instrument between two treatments is exceedingly difficult. On the one hand, it depends on the geometry of the file and on the properties of its constituting material. The geometry of the files varies much according to the manufacturer, the model, the caliber, etc. Besides, the mechanical behavior of NiTi highly depends on the histories of temperature and stresses, so that it evolves during the life of the instrument. The use of a lubricant irrigant fluid allows a control of the temperature of the file during the treatment, so it can also have an influence on NiTi's mechanical behavior. It can also be mentioned that sterilization procedures seem to increase the fatigue resistance of the files, though they were not shown to have an influence on other mechanical properties of the files [11]. On the other hand, the damage done in an instrument depends on the root canal shape, which is inherent to the canal considered and can be irregular. Indeed, the amount of strain in the file depends directly on the geometry of the root canal. Finally, it is influenced by the contact between the file and the root canal walls, which induces a torsional loading of the instrument. The friction coefficient associated to this contact is not known, and its value is very probably influenced by the use of a lubricant irrigant fluid. The way the file is in contact with the root canal also depends on the geometry of the latter. For instance, round files only touch two parts of the walls of oval canals. Another issue is the presence of imperfections at the walls of the root canal, which can lead to a locking of the file. The latter then suffers a high amount of torsion, due to the fact that the engines for endodontics impose a torque on the files, which can lead directly to its failure. If the instrument does not break due to torsion, it can be assumed that this high solicitation will have an influence on its remaining lifetime and on the mechanical behavior of NiTi. Some engines offer a control of the torque imposed on the file, so that when the latter suffers too high a torsion, the engine stops applying the torque. This may help to avoid failure of the files due to torsion, but one does not know exactly the effect such a solicitation has on their future behavior. One should also consider that flexural fatigue will decrease the torsional resistance of the instrument so that torque control can be ineffective with used files [12]. In spite of these difficulties, the only solution to avoid file breakage seems to be the calculation of the damage they endure at each treatment. The aim of this work was to build a mathematical model to describe the response of endodontic instruments when submitted to both torsion and rotating bending in the root canal. Such a model would allow the study of the influence of the geometry of the instrument on its resistance and to forecast the remaining lifetime of a file after treatment of a given root canal. As it is difficult to measure the intensity of forces applied on the instruments in service, the model was built starting from the root canal geometry, measurable on a radiograph. In order to make a general study of the mechanical behavior of NiTi rotary files, it was necessary to introduce a concept of “standard root canal”. Indeed, it is a known fact that the shape of root canals varies according to the patient and to the kind of tooth considered [13]. Such a standard root canal is easily represented analytically in two dimensions by writing the equation of the angle of bending along the file. Various studies already allow one to make an accurate description of a standard canal's geometry in two dimensions [12,14–17]. This leads to a definition of the standard root canal in which the curvature radius is equal to 5 mm at 3 mm from the apex. To represent such a root canal in three dimensions, one will have to introduce an additional degree of freedom in the equation, which can be easily accomplished. The same thing can be done in the model so as to receive three-dimensional entries regarding the geometry of the root-canal. Though, as will be commented later, it is difficult to obtain an accurate three-dimensional description of the root canal geometry. To build the model, the initial and deformed geometries of the file were expressed, then the strain field was calculated and the stress

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field was deduced from it, using the equations of the material's behavior. The temperature was assumed to be constant during all experiments and endodontic treatments. The different steps needed to build the general model, which are part of three main fields: kinematics, mechanical behavior of NiTi and external efforts, are described here. The results obtained are then discussed aiming to identify the limits of the model. 2. General mechanical model 2.1. Kinematics The assumptions made concerning kinematics are that the variations of length of the file are negligible and that its cross-section remains plane and perpendicular to the neutral axis, without change of shape. The initial geometry is the one of the straight file. Let E z be the longitudinal axis of the file, composed of the centers of gravity of the cross-sections. Its origin, O, is placed at the center of gravity of the largest cross-section of the file, the base cross-section (Fig. 1). Let s be the curvilinear abscissa along the file and L its length. Working in cylindrical coordinates, whose axis is EZ, a particle X of the file is located by the coordinates (r,θ,s): X ¼ r cosðθÞE X þ r sinðθÞE Y þ sE Z

ð1Þ

The particles of the base section at t = 0 belong to
Si ð0Þ ¼ X ¼ r cosðθÞE X þ r sinðθÞE Y jr ∈ ½0; Rb ðθÞ; θ ∈ ½0; 2πg

ð2Þ

where Rb(θ) defines its geometry. All the sections can be deduced from one another using a homothety and a rotation, so that the particles belonging to the section placed at the abscissa s at the initial belong to

Si ðsÞ ¼

8 > < > :

X ¼ r cosðθÞE X þ r sinðθÞE Y þ sE Z jr ∈ ½0; f ðsÞRb ðθ þ βðsÞÞ; θ ∈ ½0; 2π

9 > = > ;

ð3Þ

Where β(s) represents the rotation of the sections around the longitudinal axis (the pitch length of the file, which may vary along its length), and f(s) describes the taper of the instrument. As many instruments present variable pitch and taper, such characteristics were represented in the model. The description of the geometry of the file was made considering that its cross-sections are perfect equilateral triangles, which is not the case in reality, though one can perform measurements on the instruments so as to know the mean discrete repartition of pitch lengths and cross-section diameters along the file, as was done in the work by Peixoto et al., 2010 or Câmara et al., 2009 [18,19]. In these works, measurements were made on groups of ten instruments of different kinds. A mean value of the pitch length for each pitch number, and a mean value of the diameter of the cross-section at each millimeter from the tip were estimated. On the one hand, the values of the mean pitch lengths allow one to find the angle of rotation of the cross-section, β(s), as a function of its longitudinal abscissa. Depending on the cross-section of the file, the mean pitch lengths correspond to a different amount of rotation of the cross-section. Indeed, let us consider a file composed of n pitches of mean lengths L1,...,Ln and whose cross-sections present three

Fig. 1. Referential system in which the initial geometry of the instrument is expressed [20].

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corners, such as a triangular file. It is generated by the translation, rotation and homothety of the base cross-section guided by three curves, which are the geometrical places of the corners of the cross-sections. These three curves can be deduced from one another by a rotation of 2π/3 around the longitudinal axis of the file. The measurement of the pitch length of the file is made on a projection of these three curves on a plane, as shown in Fig. 2. Since these three curves present the same geometry (considering that the cross-sections are perpendicular to the longitudinal axis), one can observe in Fig. 2 that one pitch length measured on such a file corresponds to a rotation of 2π/3 of the cross-section. In this manner, the function describing the rotation of the cross section along the longitudinal axis was discretized in intervals of rotation of 2π/3 of the cross-section, which allows an accurate description of the variation of the pitch along the file. Then, to find the value of β(s) one can proceed as described in Table 1. In the case of a file whose cross-sections present only two corners, as for example the Mtwo files, each measured pitch length corresponds to a rotation of π of the cross-section, so that the formulation in Table 1 should be adapted. On the other hand, if the mean values of the diameters of the crosssections at each millimeter from the tip are known, a linear interpolation can be done to estimate the diameter of every section along the file, and so to know the value of the upper bound of Rb(θ) for any s. Assuming that the root canal is contained in a plane, let (O′,ex,ey,ez) be an orthonormal referential system belonging to the root canal. The file is positioned in the root canal so that its longitudinal axis belongs to the plane (O′,ex,ez), its base section belongs to (O′,ex,ey) and O and O′ are coincident. The bending is thus applied around ey (Fig. 3a). The longitudinal axis of the file in the deformed configuration corresponds to the longitudinal axis of the root canal, so its geometry, which is invariant in time in the referential system associated to the tooth, is given by n o A ¼ g ¼ g x ðsÞe x þ g z ðsÞe z ð6Þ

Table 1 Algorithm for obtaining the value of the angle of rotation of the cross-section located at the abscissa s, β(s), for a file composed of n pitches of lengths Li (i = 1,...,n) and which cross-section presents three tops. 1: 2:

pitch = vector(n, [L1,...,Ln]) for i = 1,2,...,n do i P Lk then if sb

3:

k¼1

4: 5: 6: 7: 8: 9: 10:

number of pitch = i break end if end for if number of pitch = 1 then s β ðsÞ ¼ 2π 3 pitchð1Þ else i−1 P

11: 12:

k¼1 β ðsÞ ¼ 2π 3 pitchðnumber of pitchÞ end if

s−

Lk

The configuration at the instant t is given by: Ωt ¼

      L−sð1−α Þ x ðr; θ; s; t Þr ∈ 0; Rðθ þ βsÞ ; θ ∈ ½0; 2π; s ∈ ½0; L ð10Þ L

The vector t (s) is introduced so that (t (s),ey n (s)) is an orthonormal referential system along the longitudinal axis of the file (Fig. 3b). Using this referential system, the gradient of the transformation F (X, t) can be calculated using the formulae of derivation of composite functions. The Green–Lagrange deformation field e (X, t) is then given by [21] e ðX ; t Þ ¼

1 t F ðX ; t Þ:F ðX ; t Þ−1 2

ð11Þ

Assuming that the rotation is uniform in every section, the rotation of the section located around the longitudinal axis at s can be written as

where 1 is the identity tensor. This gives the strain field expressed in the referential system belonging to the tooth. The second order terms appear to be despicable because of the very small dimensions of the instrument. Neglecting them makes appear the strain field as sum of two tensors corresponding to bending and torsion. The calculation of the stress field requires the application of equations describing the material's behavior. It is thus necessary to express the strain field in the current referential system, adopting Euler's point of view. This referential system belongs to the sections and is obtained by applying two rotations to the initial cylindrical referential system, corresponding to the bending and to the rotation of the section around n(s), including both terms of speed and torsion. The strain field expressed in the current referential system e * is then

ωðs; t Þ ¼ τðsÞ þ Ωt

e ðr; θ; sÞ ¼ e bending ðr; θ; sÞ þ e torsion ðr; θ; sÞ

Since variations in the length of the file were neglected, s is the same abscissa than at the initial state. Let n(s) be the unitary vector tangent to the longitudinal axis at s, and φ(s) the angle between n(s) and ez (Fig. 3b). Then, s

s

g x ðsÞ ¼ ∫ sinðφðuÞÞdu ; g z ðsÞ ¼ ∫ cosðφðuÞÞdu 0

ð7Þ

0

ð8Þ



with:

where Ω is the rotation speed applied to the instrument and τ(s) is an unknown term of angular torsion modeling the effect of friction between the file and the root canal. The particle that was at X at the initial state is located at the time t at

þΩtÞ sinðφðsÞÞÞe z

ð9Þ



0

0 B e ðr; θ; sÞ ≈@ 0 bending 0 

0 0 0

e

B B B ðr; θ; sÞ ≈B B torsion @ 

ð12Þ

1 0 C 0 A dφ cosðθÞ −r ds

0

x ðr; θ; s; t Þ ¼ ðg x ðsÞ þ r cosðθ þ τ ðsÞ þ Ωt Þ cosðφðsÞÞÞe x þr sinðθ þ τ ðsÞ þ Ωt Þe y þ ðg z ðsÞ−r cosÞθ þ τ ðsÞ



0

0

0

0

1 dτ 1 dτ sinðθÞ r cosðθÞ − r 2 ds 2 ds

Fig. 2. Schematic representation of the rotation of the cross-section of a file.

ð13Þ 1 1 dτ sinðθÞ C − r 2 ds C 1 dτ C r cosðθÞ C ð14Þ C 2 ds A 0

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2.2. Mechanical behavior of NiTi The behavior of superelastic NiTi in uniaxial tension at room temperature can be divided into four stages (Fig. 4). The first stage corresponds to the elastic deformation of austenite (the crystalline phase present at room temperature in superelastic NiTi), the second one to the stress induced martensitic transformation, the third to the elastic deformation of martensite and the last one to the plastic deformation of martensite [1]. Since endodontic instruments should not reach plasticity during root canal shaping (they should always recover their original shape after use, otherwise they should be immediately discarded), only the behavior of NiTi until the third phase needed to be modeled here. This was done using results obtained in uniaxial tension to write the constitutive equations. The mechanical behavior was assumed to be the sum of the torsion and bending terms. Another assumption was that the unknown angular torsion would not exceed 40° (which was confirmed by the results), so that torsion would only cause elastic deformation of austenite: 

σ ðr; θ; sÞ ¼ σ



bending ðr; θ; sÞ

þσ



torsion ðr; θ; sÞ

ð15Þ

One was to monitor the principal strain: when it reached the value corresponding to a change of behavior in uniaxial tension, the change of behavior occurred under bending. The other approach was to monitor the strain energy. Let eA, eTM and eM be the values of the principal strain corresponding to a change of behavior in the case of uniaxial tension (Fig. 4). For instance let us consider the approach based on the principal strain. Then the interval of elastic deformation of austenite in a cross-section of the instrument is the range of r.cos(θ) belonging   −1  −1  to −eA dφ ; eA dφ . In this case the behavior is elastic: ds ds σ

 bending

ðr; θ; sÞ ¼

 torsion

ðr; θ; sÞ ¼

EA  e ðr; θ; sÞ 1 þ ν torsion

ð16Þ

where σ torsion is the stress tensor related to e torsion , e bending is  the stress tensor related to e bending in the current referential system, EA is the Young's modulus of austenite and ν the Poisson coefficient of NiTi, which was taken as equal to 0.33. The different stages of the macroscopic mechanical behavior and the transition between them were described in three-dimensions, similarly to what was done by Auricchio et al. [22]. The differences compared to their work were that in the present one, the microscopic behavior of NiTi was not taken into account and the stress field was deduced from the strain field. Each stage was described by a linear stress–strain relationship. The martensitic transformation was described as occurring at constant stress. Criteria applied to the strain field allowed to define intervals of behavior along t(s), which appeared to depend on the bending gradient along the file. The model describes qualitatively the heterogeneity of behavior in the cross-sections under bending: for instance, martensitic transformation occurs more easily close to the surface of the instrument along the t(s) axis, where stress levels are the highest. Since the asymmetry between tension and compression of NiTi was not quantified for the material used in this work, it was not taken into account. Two approaches were tested to describe the transition between two stages of behavior of the material.

  νEA EA  1þ e :Trace e ð1 þ ν Þð1−2ν Þ 2ð1 þ ν Þ bending bending

ð17Þ

The same method was applied to describe the behavior in the intervals of martensitic transition and of elastic deformation of martensite, differentiating the cases of tension and compression. For instance, the behavior on the interval of elastic deformation of martensite under compression was written as σ

bending

∀ðr; θ; sÞ ∈ Ωt ; σ

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ðr; θ; sÞ ¼ −

EA eA νEA eA ðn ðsÞ⊗n ðsÞÞ ⋅1 − ð1 þ ν Þð1−2ν Þ 2ð

1 þ νÞ

 νEM  :Trace e þ eM ðn ðsÞ⊗n ðsÞÞ ⋅1 ð1 þ νÞð1−2ν Þ bending

 EM  þ e þ eM ðn ðsÞ⊗n ðsÞÞ ð18Þ 2ð1 þ ν Þ bending

þ

where EM is the Young's modulus of martensite. 2.3. External efforts The external bending and torsion torques were obtained by integration on the cross-sections [23] 

−Bending torque : MðsÞ ¼ −∫SðsÞ r cosðθÞ σ 33 ðr; θ; sÞ dS

ð19Þ

−Torsion torque : T ðsÞ

  ¼ −∫SðsÞ r sinðθÞσ 13 ðr; θ; sÞ−r cosðθÞσ 23 ðr; θ; sÞ dS

ð20Þ

This torsion torque is due to friction with the root canal walls. Writing a Coulomb contact law and matching the two expressions of the torsion torque made appear a differential equation that allowed to calculate the unknown angle of torsion. In the case of a wire of diameter 1 mm and of length 15 mm undergoing a rotary bending until 85°, it was found that if the friction coefficient was equal to 0.2, which seemed to be a high value considering that the canal is irrigated during treatment [24], the angular deflection due to torsion at the tip of the wire would be of about 15°. Considering that a high value of bending was applied compared to the usual solicitation on a file, this confirmed the assumption that torsion would not provoke martensitic transition. 3. Results and discussion

Fig. 3. (a) Orthonormal referential system belonging to the tooth, which corresponds to the referential of the base section of the file. (b) Cylindrical referential system belonging to the file in the deformed configuration.

The model was compared to empirical results obtained in threepoints bending on NiTi wires. One should keep in mind that endodontic files have complex cross-sectional shapes, and so testing the model on a wire does not represent accurately what happens in reality. Though, it appeared as a simple way in a first approach to test the accuracy of the model for a simple geometry, and being able to perform reproducible experiments. Various superelastic NiTi wires presenting different characteristic transformation temperatures were tested in order to confirm that the model was able to describe the behavior of all of them. The transformation temperature considered was Af, defined as the end temperature of transformation from martensite to austenite upon heating

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a martensitic NiTi alloy. The superelastic behavior at room temperature takes place in NiTi alloys with Af in the range −10 °C to 25 °C [1]. Wires presenting temperatures Af equal to 13 °C, 15 °C, 19 °C and 32 °C were tested, in order to see if the model would describe the behavior of superelastic NiTi for various characteristic Af temperatures. Three uniaxial tension experiments were managed on every kind of wire and extracted the mean value of each parameter required in the model. Three three-point bending experiments were then managed on each kind of wire, and the empirical and theoretical curves of the bending moment at the middle of the wire in function of the vertical distance traveled by the middle of the wire were plotted. The experiments were led on a specially designed device, which allowed a precision of 10 μm and 0.01 N, in the measure of the displacement and of the force respectively. The distance between the supports was chosen to be equal to 26 mm, so as to reach 4% of maximum deformation in the wire when a vertical distance of 5 mm was traveled by its center-point. This corresponds approximately to the maximum amount of deformation suffered by a file of caliber 25/0.06 at 3 mm from its tip, when introduced in a standard root canal, according to the equations of kinematics presented in this work. The expression of φ(s) was estimated as a third order polynomial of the curvilinear abscissa s, so that the equation of the deformed wire was a second order polynomial of s. A good mentoring of the empirical result was obtained for all wires. The wire which characteristic temperature Af was equal to 19 °C was the less well represented (see Fig. 5 and Table 1). NiTi mechanical behavior was modeled as being symmetric in tension and compression, and the martensitic transformation as occurring at a constant stress. These two choices should reduce the theoretical bending moment one has to apply to reach a given angular deflection. It appeared that when the theoretical changes of behavior rely on criteria based on the strain energy, the model represented the wire as being more flexible than it is in reality, as it ought to be (Fig. 5), though the use of criteria based on the principal strain introduced an artificial rigidity in the theoretical response since the model then described the martensitic transformation as starting later. This allowed a better description of the experiment. For various wires, as shown in Table 2, a good mentoring of the empirical results was only obtained considering that the Poisson ratio was inferior to 0.3. This consideration was only made with regards to the elastic part of the behavior of the material. At the beginning of the experiment, the material behaves elastically and the model is linear, so that it should represent well the experiment, which did not occur for all wires. This can be due to the fact that the Poisson ratio of the material is known to vary around 0.3 depending on its metallurgical

Fig. 4. Uniaxial tensile test on three superelastic NiTi wires with Af = 6 °C. (1) Elastic deformation of the austenite; (2) martensitic transformation; (3) elastic deformation of the martensite; (4) plastic deformation.

Table 2 Mechanical properties of the NiTi wires used in the model (the Poisson coefficient was adjusted in function of the elastic part of the behavior in bending) and the mean error between the results given by the model in its two variants and the empirical data in three-point bending tests. Af

13 °C

15 °C

16 °C

19 °C

32 °C

EA (GPa) eA (%) ν Mean error (%)—criteria based on the principal strain Mean error (%)—criteria based on the strain energy

43.1 1.04 0.33 5.33

44.3 1.08 0.32 2.66

53.2 0.98 0.30 3.59

49.3 0.94 0.29 9.49

39.8 1.26 0.29 5.81

16.43

12.25

11.38

14.62

12.62

properties [25] and also to the fact that the model did not describe the asymmetry between tension and compression in NiTi mechanical behavior. In consequence, these results do not invalidate the model, though they highlight the fact that the mechanical behavior of the material is not exactly known. Managing three-point bending experiments on NiTi wires and comparing the results to the model's results in the elastic part of the experiment could allow an estimation of an apparent Poisson coefficient of NiTi, which would account for the asymmetry between tension and compression. The model was applied to a wire undergoing rotating bending and the strain–stress curve was plotted on a cycle at the point farthest from the center of the cross-section (Fig. 6). In the case presented in Fig. 6 with a motor rotating at 300 rpm, a particle located at the periphery of the wire undergoes five cycles in 1 second, the amplitude of tension strain being of 5%. The amplitude of compression strain is smaller in reality than in the results presented here [9], but the compression phase of the cycle should not to provoke the opening or growth of cracks so the amplitude that interests us most is actually that of tension. The calculations were made on a wire, but they are applicable to an endodontic instrument undergoing bending associated to a small torsion. To calculate the response of the instrument undergoing a high torsion, it would be necessary to introduce in the model components of warping of the sections: the angular torsion would depend on the three coordinates of a particle and the cross-sections would not remain plane and perpendicular to the neutral axis of the instrument. Such components should make appear the repartition of stress described by Zhang et al. in their work [8] which would lead to a more accurate study of the impact of the cross-section geometry on the resistance to failure under an excess of torsion and to fatigue under rotating bending. The equation of a root canal or of a deformed wire in the case of an experiment can be approached by a polynomial function and the expression of φ(s) deduced from it, so that the model can be applied

Fig. 5. Comparison between empirical and theoretical results with regard to three-points bending on a superelastic NiTi wire with Af= 12.6 °C; d is the vertical distance traveled by the middle of the wire and M(L/2) is the bending moment at the middle of the wire.

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To model three-dimensional root canals analytically, one will have to introduce an additional degree of freedom in the canal's geometry equation (and so in the file's deformed geometry), but this is not a difficult task. However, the actual three-dimensional geometry of the root-canal is hard to obtain, since it is currently impossible to measure it accurately on a radiograph. Not all curvatures are detectable on a radiograph, and a two-dimensional image does not provide all the relevant features of a three-dimensional curved root canal [27]. In such conditions, one can conclude that one limitation of the mathematical model is the lack of accurate knowledge concerning root canals' geometries. Moreover, finite elements analysis or another kind of numerical analysis should prove to be a better tool than an analytical model to accurately describe the three-dimensional geometries of root canals and also files presenting complex cross-sections, as well as to model the non-steady part of a treatment. However, one should keep in mind that the same limitations will hold with respect to the lack of some mechanical parameters such as friction coefficient or fatigue law and to obtaining an accurate three-dimensional numerical image of the root canal. The analytical model developed in this work provides a complementary approach to describe the steady part of the treatment and to understand better the behavior of the instruments. 4. Conclusions Fig. 6. Theoretical strain–stress curve in the direction of the normal to the section corresponding to a cycle of rotating bending on a superelastic NiTi wire with Af= 6 °C, 1 mm in diameter, 15 mm length, and 43° of angular deflection in pure bending, which corresponds to a constant bending gradient equal to 0.1.

in the case of an endodontic treatment. The expressions of the strain and stress fields show that the mechanical response of a file depends linearly on the bending gradient. The latter being equal to the inverse of the radius of curvature of the longitudinal axis of the file which corresponds to the one of the root canal, the response of an instrument is proportional to the inverse of the radius of curvature of the root canal. Such dependence has already been observed through experiments [26]. The advantages of this analytical model compared to finite elements analysis are that it permits to describe more easily the contact with the root canal and to take into account every characteristic of the material's behavior. It also makes appear explicitly the equations of the stress and strain fields in the file which could allow for instance an optimization of the cross-section geometry, or a prediction of the files' fatigue life, considering only the steady part of the behavior. To achieve such a prediction, it would be necessary to build a model of the nucleation and propagation of cracks in NiTi and to set criteria of failure, so as to estimate the lifetime of a file based on the values of the local strain field over a cycle of rotation, which can be obtained from the model presented in this work. An accurate empirical study of the behavior of superelastic NiTi undergoing rotating bending would be required: the material is not known to follow the classical laws of fatigue life [4]. The model also has its limits: it does not allow a calculation of the instrument's mechanical response when submitted to high rates of torsion. Still, the construction of a model describing high rates of torsion is feasible by applying the method described in this work. Another limitation of the model is that it does not include a representation of the evolution of the canal's geometry, or what happens when the file is introduced in the canal. It only represents the steady behavior of a file rotating in a non-changing root canal. Besides, some mechanical characteristics of the problem are not accurately known, such as the friction coefficient between the file and the root canal. Some points are to be deepened in order to improve the model: introducing a warping of the sections, quantifying the friction coefficient and the asymmetry between tension and compression existing in the mechanical behavior of NiTi alloy.

Building an analytical mathematical model to study the response of endodontic instruments during a treatment proves to be an efficient method. Using the mathematical description of the problem presented here, one is now able to express the mechanical response of a NiTi rotary file in function of its characteristics and of the ones of the root canal: this could prove to be useful to assess the damage made on the instrument at each treatment and to optimize the geometry of the instruments. Acknowledgements This work was partially supported by the Fundação de Amparo à Pesquisa do Estado de Minas Gerais—FAPEMIG, Belo Horizonte, MG, Brazil, and Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brasília, DF, Brazil. References [1] K. Otsuka, C.M. Wayman, Shape Memory Materials, Cambridge University Press, 1998. [2] C.R. Glosson, R.H. Haller, S.B. Dove, C. Del Rio, J. Endod. 21 (1995) 146–151. [3] J. Camps, W.J. Pertot, J. Endod. 20 (1994) 395–398. [4] A.M.G. Figueiredo, P.J. Modenesi, V.T.L. Buono, Int. J. Fatigue 31 (2009) 751–758. [5] M.G.A. Bahia, R.F. Dias, V.T.L. Buono, Int. J. Fatigue 28 (2006) 1087–1091. [6] J.M. Young, K.J. Van Vliet, J. Biomed. Mater. Res. B: Appl. Biomater. 72B (2005) 17–26. [7] S. Necchi, L. Petrini, S. Taschieri, F. Migliavacca, J. Endod. 36 (2010) 1380–1384. [8] E.W. Zhang, G.S.P. Cheung, Y.F. Zheng, Int. Endod. J. 44 (2011) 72–76. [9] K. Gall, H. Sehitoglu, Y.I. Chumlyakov, I.V. Kireeva, Acta Mater. 47 (1999) 1203–1217. [10] G.S.P. Cheung, E.W. Zhang, Y.F. Zheng, Int. Endod. J. 44 (2011) 357–361. [11] A.C.D. Viana, B.M. Gonzalez, V.T.L. Buono, M.G.A. Bahia, Int. Endod. J. 39 (2006) 709–715. [12] E.P. Vieira, R.K.L. Nakagawa, V.T.L. Buono, M.G.A. Bahia, Int. J. Endod. 42 (2009) 947–953. [13] O.A. Peters, C.I. Peters, K. Schonenberger, F. Barbakow, Int. J. Endod. 36 (2003) 93–99. [14] J.P. Pruett, D.J. Clement, D.L. Carnes, J. Endod. 23 (1997) 77–85. [15] M.G.A. Bahia, V.T.L. Buono, Oral Surg. Oral Med. Oral Pathol. Oral Radiol. Endod. 100 (2005) 249–255. [16] R.C. Martins, M.G.A. Bahia, V.T.L. Buono, Oral Surg. Oral Med. Oral Pathol. Oral Radiol. Endod. 100 (2006) 99–105. [17] E.P. Vieira, E.C. França, R.C. Martins, V.T.L. Buono, M.G.A. Bahia, Int. Endod. J. 41 (2008) 163–172. [18] I.F.C. Peixoto, E.S.J. Pereira, J.G. Silva, A.C.D. Viana, V.T.L. Buono, M.G.A. Bahia, J. Endod. 36 (2010) 741–744. [19] A.S. Câmara, R.C. Martins, A.C.D. Viana, R.T. Leonardo, V.T.L. Buono, M.G.A. Bahia, J. Endod. (2009) 113–116.

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[20] Dentsply Maillefer, ProTaper Universal, http://www2.dentsplymaillefer.com/ #/218x624/218x7718/line_218x7727/product_218x7740/ [last accessed on 04/19/2011]. [21] A. Ehrlacher, Lesson of the Department of Mechanical Engineering of the École des Ponts ParisTech, 2009, pp. 76–96. [22] F. Auricchio, R.L. Taylor, J. Lubliner, Comput. Methods Appl. Mech. Eng. 146 (1977) 281–312. [23] G. Cailletaud, Mécanique des Matériaux Solides : IAE Poutres Planes, ENS Mines de Paris, http://mms2.ensmp.fr/mms_paris/poutre/transparents/f_Beam.pdf [last accessed on 04/28/2011]. [24] Roymech, Coefficients Of Friction, RoyMech Index page, http://www.roymech.co. uk/Useful_Tables/Tribology/co_of_frict.htm [last accessed on 04/14/2011]. [25] T. Duerig, A. Pelton, C. Trepanier, Nitinol: Monotonic Mechanical Properties, ASM International: The Material Information Society, http://www.asminternational. org/content/News/Duerig_1010_forWeb.pdf (last accessed on 03/17/2011). [26] J.P. Pruett, D.J. Clément, D.L. Carnes, J. Endod. 23 (1997) 77–85. [27] C.J. Cunningham, E.S. Senia, J. Endod. 18 (1992) 294–300.

Agnès Marie Françoise Leroy graduated in 2012 in Mechanics and Materials Engineering at the École des Ponts ParisTech, France, and in Civil Engineering at the Universidade Federal de Minas Gerais, Brazil. She worked at the Department of Metallurgy and Materials Engineering of the Universidade Federal de Minas Gerais researching NiTi instruments for one year and a half during her undergraduate studies. She has now begun a PhD in computational fluid dynamics at Electricité de France (EDF) R&D, France, in the Laboratoire d'Hydraulique Saint-Venant.

Maria Guiomar de Azevedo Bahia received an MSc degree in Dentistry from the Universidade Federal de Minas Gerais (UFMG) in 1987 and a PhD in Metallurgical and Materials Engineering from the same university in 2004. She is associate professor of Endodontics at UFMG where she works on mechanical and clinical behavior of NiTi instruments, being the author of many papers on this subject.

Alain Ehrlacher was born in 1952. He graduated from the École Polytechnique in 1976 and the École des Ponts, ParisTech in 1978, and completed the “Doctorat d'Etat” at the University of Paris VI in 1985. He is currently the Head of the Mechanical and Material Engineering Department at École des Ponts, ParisTech. Professor Ehrlacher has published many papers in Fracture Mechanics, Damage Mechanics, Concrete and Thermo-Hydro-Chemical Modeling of Porous Medium, Composite Materials, Adhesive and Reinforcement of Structures. He has also worked on biomechanics.

Vicente Tadeu Lopes Buono received an MSc degree in Physics from the Catholic University of Leuven in 1982 and a PhD in Physical Metallurgy from the Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil, in 1995. He is currently an associate professor of Physical Metallurgy at this university and works on phase transitions in metals and alloys, physical and mechanical properties of materials, biomaterials and surface coated materials. He also works as consultant for various companies on thermomechanical treatment of steels and related products.