Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Finite element modeling of superelastic nickel–titanium orthodontic wires$ Ines Ben Naceur a,d,n, Amin Charfi b,e, Tarak Bouraoui c,f, khaled Elleuch a,d a

Laboratoire de Génie des Matériaux et Environnement LGME, ENIS, Université de Sfax, BPW 3038 Sfax, Tunisia Laboratoire des Systèmes Electromécaniques, ENIS, Université de Sfax, BPW 3038 Sfax, Tunisia c Laboratoire de Génie Mécanique LGM, ENIM, Avenue Ibn El Jazzar, Université de Monastir, 5019 Monastir, Tunisia d Ecole National D'Ingénieurs de Sfax, département des Génie des Matériaux, BPW 3038 Sfax, Tunisia e Institut Supérieur des Systèmes Industriels de Gabès Rue, SlahEddine Ayoubi, 6011 Gabès, Tunisia f Ecole National D'Ingénieurs de Monastir, Avenue Ibn El Jazzar, 5019 Monastir, Tunisia b

art ic l e i nf o

a b s t r a c t

Article history: Accepted 5 October 2014

Thanks to its good corrosion resistance and biocompatibility, superelastic Ni–Ti wire alloys have been successfully used in orthodontic treatment. Therefore, it is important to quantify and evaluate the level of orthodontic force applied to the bracket and teeth in order to achieve tooth movement. In this study, three dimensional finite element models with a Gibbs-potential-based-formulation and thermodynamic principles were used. The aim was to evaluate the influence of possible intraoral temperature differences on the forces exerted by NiTi orthodontic arch wires with different cross sectional shapes and sizes. The prediction made by this phenomenological model, for superelastic tensile and bending tests, shows good agreement with the experimental data. A bending test is simulated to study the force variation of an orthodontic NiTi arch wire when it loaded up to the deflection of 3 mm, for this task one half of the arch wire and the 3 adjacent brackets were modeled. The results showed that the stress required for the martensite transformation increases with the increase of cross-sectional dimensions and temperature. Associated with this increase in stress, the plateau of this transformation becomes steeper. In addition, the area of the mechanical hysteresis, measured as the difference between the forces of the upper and lower plateau, increases. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Orthodontic wire Nitinol Superelastic behavior Finite element method

1. Introduction Superelastic (SE), nickel–titanium-alloy (NiTi) wires have become the wires of choice for orthodontic aligning and leveling mechanics due to their good mechanical properties, biocompatibility and resistance to corrosion, greater strength and lower elastic modulus compared with stainless steel alloys. These wires readily sustain large deflections without exceeding the elastic limit of the alloy (Es-Souni and Brandies., 2001). As an SE NiTi wire is deflected, it is firstly deformed in an elastic manner in the austenitic state. As the stress induced in the wire increases, a phase transformation begins from austenitic toward martensitic phase. In practice, the transformation is likely incomplete at wire engagement, when the wire is allowed to

☆ Department or institution to which the work should be attributed: Laboratoire de Génie des Matériaux et Environnement LGME, ENIS, Université de Sfax, BPW 3038 Sfax, Tunisia. n Correspondence to: Ecole National D'Ingénieurs de Sfax, département des Génie des Matériaux, BPW 3038 Sfax, Tunisia. Tel.: þ216 58 310 883: fax: þ216 74 228 781. E-mail address: [email protected] (I.B. Naceur).

unload, hysteresis occurs, thus causing the alloy to return to its austenite phase by delivering light continuous forces over a wider range of deformation which is claimed to allow dental displacements. In orthodontics, the normal force is the component which acts perpendicularly to the direction of the desired movement. Two categories of normal force are present in orthodontic leveling. Firstly, slot-wire contact(s) exist due to wire-curvatures at/through the slot. Secondly, the ligation securing the wire in the slot can create normal force (Trenton, 2008). It has been demonstrated that elastomeric ligation produces the highest normal force, followed by stainless steel ties and self-ligating mechanisms (Khambay et al., 2005). For this reason, manufacturers continue to develop self-ligating designs to decrease the normal force exerted on the arch wire in order to reduce friction at the bracket arch wire interface that might prevent attaining optimal orthodontic force levels in the supporting tissues (Krishnan et al., 2009). Normal force induced by wire curvatures is responsible for correcting malpositioned teeth and simultaneously exerts pressure on anchorage teeth. Actually, it can be directly related to the stiffness of the activated arch wire. As the flexural stiffness(es) of a wire increase(s), the amplitude(s) of normal forces which press

http://dx.doi.org/10.1016/j.jbiomech.2014.10.007 0021-9290/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

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against the walls increase(s), it is/they are also influenced by interbracket distance and cross-sectional shape and size (Queiroz et al., 2012). During recent years, the area of constitutive modeling of shape memory alloys (SMA) has been the point of interest of many researchers (Arghavani, 2010; Auricchio et al., 1997; Dumoulin and Cochelin, 2000; Etave et al., 2001; Huang et al., 2000; Gao and Brinson, 2002; Marketz and Fischer, 1996; Majo et al., 2004). The main difficulty is the modeling of the temperature–stress dependence and a nonlinear behavior of SMA alloy. Therefore, it is necessary to adopt constitutive laws able to be implemented in commercial Finite Element codes and to predict the SMAs response. In this study, one of the famous phenomenological constitutive models of shape memory alloys based on Boyd and Lagoudas (1996) formulation is proposed and implemented as a user defined subroutine using the finite element analysis software, ABAQUS (Abaqus, Analysis User's Manual, 2009). The aims were to compare and evaluate the level of the normal force generated by SE NiTi orthodontics wires subjected to 3-mm deflection with different cross-sectional shapes, sizes and temperatures. 2. Methods 2.1. Shape memory alloy constitutive model

In fact, it is a three-dimensional model based on small deformation assumptions. Besides, it is a thermodynamic model based upon the expression of the Gibbs free energy which is defined as the portion of enthalpy available for doing work at constant temperature (Lagoudas and Bo, 1999). Moreover, such potential considers the martensitic volume fraction (ξ) and the transformation strain (ε) as the internal state variables since both of them play an important role in characterizing the phase transformation and the observable thermomechanical response of SMAs. In this constitutive model, the total specific Gibbs free energy, of a polycrystalline SMA is assumed to be equal to the mass weighted sum of the free energy of each phase (martensitic and austenitic) plus the free energy of mixing (Boyd and Lagoudas, 1996).       11 1  T G σ; T; ξ; εt ¼  σ : S : σ  σ : αðT  T 0 Þ þ εt þ c ðT  T 0 Þ  T In 2ρ ρ T0  s0 T þ υ0 þ f ðξÞ;

where σ, ξ, εt, T and T0 are defined as the Cauchy stress tensor, martensitic volume fraction, transformation strain tensor, current temperature and reference temperature, respectively. Other material constants α, S, ρ, c, u0 and S0 are the effective thermal expansion tensor, effective compliance tensor, density, effective specific heat, and effective specific internal energy at reference state and effective entropy at reference state. In (1), the hardening function f(ξ), is responsible for the transformation induced strain hardening in the SMA material. This function is given by 8   < 1 ρbM ξ2 þ μ1 þ μ2 ξ; ξ_ 4 0; 2 f ðξÞ ¼   A 2 1 : ρb ξ þ μ1  μ2 ξ; ξ_ 40: 2 A

The constitutive model used in this paper is based on the formulation proposed by Boyd and Lagoudas (1996). In this section, a brief description of this model is proposed.

ð1Þ

ð2Þ

M

where ρb , ρb , μ1 and μ2 are material constants for transformation strain hardening. The first condition in (2) represents the forward phase transformation (A-M) and the second one represents the reverse phase transformation (M-A). The constitutive relation of a shape memory material can be obtained by the total

70 (110)B2 NiTi

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Wire Samples Heating element

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Fig. 2. Tensile test setup machine for superelastic NiTi wire.

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

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Fig.3. Modeled and experimental superelastic behavior of NiTi orthodontic wires: (a) at room temperature, (b) at body temperature, and (c) at 50 1C.

Table 1 UMAT input material parameters for constitutive model implemented in Abaqus (T ¼ 37 1C).

Fig. 4. The nitinol DSC diagram. Gibbs free energy as ε ¼ ρ

∂G S : σ þ αðT  T 0 Þ þ εt ∂σ

ð3Þ

The implementation of the SMA constitutive equations proposed in this work was done as a user defined material UMAT. Numerical simulations was carried out using the commercially available nonlinear finite element code ABAQUS. 2.2. Experimental investigation: material parameters identification We made use of Ti50 Ni50 (SE) alloy with rectangular cross section (0.46  0.64 mm2), produced by Forestadent company. To determine the different states present in material, X-ray diffraction (XRD) measurements were recorded using a Bruker D8 Advance diffractometer operating with copper radiation Kα¼ 1.5418 Å.

Parameter Description

Value

EA EM ν ƐL As Af Ms Mf ρΔSA ρΔSM αA αM

44 GPa 30 GPa 0.33 0.052 284.5 K 296.5 K 288 K 280 K  0.27 MPa K  1  0.82 MPa K  1 11  10  6 K  1 6  10  6 K  1

Austenite elasticity modulus Martensite elasticity modulus Poissons ration Maximum transformation strain Austenite start temperature Austenite finish temperature Martensite start temperature Martensite finish temperature Austenite stress influence coefficient Martensite stress influence coefficient Coefficient of thermal expansion for the austenite Coefficient of thermal expansion for the austenite

The result of this test is depicted in Fig. 1, we can note that austenitic cubic phase (B2) dominates the microstructure of NiTi orthodontic wire at room temperature, XRD also show the presence of little amounts of Ni3Ti precipitates. The material constants required for the constitutive model were obtained from the isothermal tensile tests and from differential scanning calorimetry (DSC) measurement. Experimental stress–strain characterization was carried out with an imposed strain rate of 10  4 s  1 using an Instron tensile machine type 5966 with a load cell of 10 kN and equipped with a temperature controlled heating element. Fig. 2 shows the setup for tensile wire. From the inspection of the obtained experimental stress–strain curve (Fig. 3), the material parameters for a specific reference temperature (Young modulus of the austenite and martensite phases (EA, EM), the different values of phase transition AM MA MA stresses (σ AM s ; σ f ; σ s ; σ f Þ, maximum transformation strain H and austenite and martensite stress influence coefficients, ρΔsM and ρΔsA) of the constitutive model used in this study to simulate the superelastic behavior of NiTi wires are identified and used in subsequent simulations. The transformation temperatures were determined from a Differential Scanning Calorimetry (DSC) test. This technique measures the energy

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

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quantity adsorbed or regained by a sample, not stressed, when it is heated or cooled in its transformation range. The exothermic and endothermic higher and lower values, corresponding to the energy variations, can be easily evaluated (Iijima et al., 2002). Therefore, the sample underwent a cooling cycle of 100 1C to  100 1C and then a heating cycle of  100 1C to 100 1C with a rate of 710 1C/min. The result of DSC test is shown in Fig. 4, with the indication of the transformation temperatures determined by applying the method of the tangents to the various peaks; the obtained parameters are listed in Table 1. 2.3. Finite element simulation To show the model capability of reproducing the macroscopic behavior of SMA material, the generated output finite element results were compared with the experimental tensile and bending tests. 2.3.1. Tensile test In this study, the simulated stress–strain curves from the proposed constitutive model for superelastic NiTi orthodontics wires were compared with the experimental

data at 27 1C, 37 1C and 50 1C. The NiTi SE orthodontics wires, with 0.46  0.64 mm2 of rectangular cross section and 40 mm in length, were subjected to a tensile testing loading–unloading history inducing a full conversion of austenite into martensite and getting a maximum strain of 7%. Both experimental and numerical tensile tests were conducted with an applied displacement rate of 6.5  10  3 mm s  1 in 409 s at one end while the other end was fixed. A total of 390 linear hexahedral elements C3D8 were used in the finite element model. Fig. 4a–c show the numerical and experimental stress–strain curves of the orthodontic wire at 25, 37 and 50 1C, respectively. As can be seen, being at a temperature above Af, the pseudo-elastic effect is perfectly reproduced and a complete strain recovery is obtained as the stress is driven to zero. A qualitatively good agreement between experiments and simulations is obtained for the proposed model, which is able to reproduce the main characteristics of the experimentally observed behaviors.

2.3.2. Bending test The model was confirmed in bending against a previously published experiment conducted by Lombardo et al. (2012) (Fig. 5a) who used NiTi SE orthodontic wires produced by Forestadent company. The used wire had a round cross section of 0.35 mm,

Bracket modeled as a discrete rigid shell

Orthodontic wire modeled with solid element

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Fig. 5. Validation in bending against experimental results by Luca Lombardo et al. (2012): Deflection, with 1 mm blade, of mounted arch wire kept in a water bath at a constant temperature of 37 1C (Luca Lombardo et al., 2012) (a), stress distribution on the beam (b). Force versus the displacement (c).

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

I.B. Naceur et al. / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎ a length of 55 mm and an interbracket distance of 7 mm. A complete loading–unloading test was simulated at 37 1C. The simulation was performed using quadratic hexahedral elements with reduced integration (C3D20R). It is worth noting that for bending, the shear locking phenomenon occurs with the full integration of 8 node brick elements and, as a result, the simulation predicts well the actual stiffness of the wire (Eshghinejad., 2012). A convergence analysis was performed for choosing the appropriate elements size by considering the normal load–displacement response as the convergence criteria. It was considered to be converged when the maximum difference was smaller than 0.2 N. A total of 624 quadratic hexahedral elements were used for modeling the orthodontic wires by choosing an element size of 0.7 mm. In order to simplify the model, a symmetry condition case was considered. The brackets and the metal blade were modeled like a rigid non-deformable body (Fig. 5b). A 4-mm ramp vertical displacement was applied at the central rigid body and the brackets were blocked in their reference points. Contact elements permit the interaction between the parts and no boundary conditions were applied between them allowing the orthodontic wire to slide freely through the brackets. The purpose was to reproduce the real clinical case in which the wire was sought to slide through the slot of the bracket as freely as possible with the minimum of friction (Fercec et al., 2012). A friction coefficient of 0.3 was taken from the literature (Noda et al., 1993). The comparison between the numerical and the experimental results is presented in terms of the applied normal force versus the midspan inflection. Both experimental and numerical responses at 37 1C are reported in Fig. 5c, from which it can be seen that the numerical results are quite in agreement with the experimental data, respecting the load levels. The ability of the model to completely recover the deformation during the unloading cycle, i.e. to reproduce the pseudo-elastic effect in bending test as well, is also interesting to note.

5

3. Result

Iijima et al., 2002; Lombardo et al., 2012; Sun and Hwang, 1993; Thériault et al., 2006; Trenton, 2008), therefore, a bending test is simulated to study the force variation of an orthodontic arch wire when it loaded up to a deflection of 3 mm. The thermomechanical loading–unloading test were undertaken at 25, 37 and 50 1C. The properties of the materials used were the same as those used in previous tests (Table 1). The tested wires had a cross section of 0.46  0.64 mm2 and 0.41  0.56 mm2 for the rectangular wires, and a diameter of 0.46 and 0.35 mm for the circular wires, sliding in 0.55  0.71 mm2 slots, which is a common clinical situation (Lombardo et al., 2012). Initially, the SE wire was drawn in their initial arch form (Fig. 6a), and then a vertical displacement was applied at the center of the wire until 3 mm at constant temperature. To simplify the problem one half of the arch wire and the surrounding 3 brackets were modeled. In this study, the bracket was modeled as a rigid body. Since the nitinol orthodontic wires are used in bending, C3D20R elements were used. That's why a mesh consisting of 790 quadratic hexahedral elements for the large section (Fig. 6b) was built up. Contact properties, element size, boundary and initial conditions were the same as the simulation of the previous bending test. The inter-bracket distance of 7 mm was chosen as an average clinical inter-bracket distance.

3.1. SMA orthodontic NiTi wire analysis: FE method

3.2. FE result

The bending test actually simulates the application of the wire pressure on the teeth in the oral cavity (Auricchio and Sacco., 1999;

The bending curves for each cross-sectional dimension and at different temperatures are shown in Fig. 7. Indeed, since the

Fig. 6. (a) Boundary conditions and load applied to the orthodontic wire (a). (b) Three-dimensional orthodontic wire mesh.

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

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Fig. 7. Force/deflection curves at 25, 37 and 50 1C for different cross sections: (a) 0.46  0.64 mm2, (b) 0.41  0.56 mm2, (c) 0.46 mm and (d) 0.35 mm.

tested temperatures are greater than Af, the large deformation is fully recovered upon unloading but with different loading and unloading levels, thus accurately reflecting the pseudoelastic behavior (Auricchio and Sacco, 1999). The activation and deactivation forces of the orthodontic wires measured at different deflections are shown graphically in Fig. 8 to observe the evolution of curves when the total bending is achieved at 3 mm. The numerical results also provide information about stress levels and martensitic volume fraction within the orthodontic arch wire. The contour plots of the stress and martensitic volume fraction distributions at the loading phase obtained from the finite element method are depicted respectively in Fig. 9a and b. A concentration of stress is located at both upper and lower surfaces of the orthodontic arch wire. The maximum stress is obtained when the wire is fully deflected. The distribution of the martensitic fraction is symmetric according to the middle line of the wire. The interfaces between the austenite and the martensite are clearly observed, and these interfaces move with the increase in loading. The resulting axial stress–strain response and the corresponding martensitic volume fraction for the largest rectangular cross section during the loading and unloading processes at 37 1C at the more solicited zone are depicted in Fig. 9c and d, respectively. Actually this region is where the contact will occur between the wire and the center bracket. From these two figures, the martensitic transformation can be seen to begin in the first loading segment at a stress level of 380 MPa, the transformation is complete at 500 MPa and the material is transformed into pure martensitic state, upon unloading, the strain is fully recovered and the material then elastically unloads to the austenite.

4. Discussion 4.1. Overall observations and comparisons to existing results The use of SMA superelastic properties for the correction of teeth malocclusions in orthodontics is a very successful application, since it allows the obtention of an optimal teeth movement as well as the control and drastic shortening of therapy (Auricchio and Sacco, 1999). In this study, we have used a three dimensional thermomechanical model of shape memory alloys, based on Boyd and Lagoudas (1996) and coded in Abaqus UMAT subroutine. The material parameters, determined by tensile tests at different temperatures and DSC measurement on superelastic NiTi orthodontic wires, are shown in Table 1. The method of extracting the critical points from the experimental data is more explained in the research paper by Boyd and Lagoudas (1996). The performance of the finite element numerical code was verified with experimental tensile and bending tests. The greatest advantage of performing this simulation is the acquirement of the normal force applied by the orthodontic arch wire on the bracket to be subsequently able to apply in future tribological tests between bracket and wire. It also allows us to know the maximal stress and martensitic volume fraction generated by deflecting the wire until 3 mm. From Fig. 7, it can be noticed that at small deformation the alloy shows a linear elasticity, thus, Hook's law can be used to describe the stress and strain relationship. Above a certain force, which depends on temperature, the martensitic transformation occurs and the elasticity behavior becomes nonlinear.

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

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Furthermore, two plateau regions are visible. In fact, the upper plateau represents the force needed to engage the wire in the bracket (Elayyan et al., 2010), during this process, nickel– titanium arch wire undergoes a phase transformation from austenite to a preferentially-oriented stress-induced martensite(SIM) plates (Garrec and Jordan, 2004). The stress required for the martensite transformation increases with the increase in cross-sectional dimensions. Associated with this increase in stress tends to be a change in the gradient of the SIM plateau, i.e. the gradient gradually becomes steeper with the sample size. This behavior is more pronounced in the 0.46  0.64 mm2 arch wire (Fig. 7a). This phenomenon corresponds to the increase in elastic strain energy stored in the alloy during the forward martensite transformation. In other words, the ratios of martensitic and frictional energy transformations lost because of interfaces movement are higher. Therefore, the applied stress required to transform austenite to martensite gradually increases with the volume fraction of the transformed martensite (Morgan and Friend., 2001; Garrec and Jordan., 2004; Ahmadabadi et al., 2009). The position of the variants in the deformed state is not stable when the stress is removed. The stored elastic energy obviously contributes to the driving force which assists to the reverse transformation and causes the return of variants to their original positions (Otsuka and Ren, 1999). The force level of the unloading process, which represents the forces delivered to teeth during the leveling and aligning stage of treatment (Elayyan et al., 2010), is considerably less than that of the loading plateau (Fig. 8). This phenomenon is amplified with higher cross-sectional dimensions because the volume fraction of the transformed martensite is more important (Garrec et al., 2005).

Comparing the different cross-sectional dimensions, the 0.46  0.64 mm2 arch wire was found to exhibit a greater load than the others for all the deflections (Fig. 8a). At 1.5 mm deflection, the difference in the force levels on the upper plateau between 0.46  0.64/0.41  0.56, 0.41  0.56/0.46 and 0.46/0.35 mm arch wires was about 37%, 59% and 64% respectively at 37 1C. This difference is clearly less important in the unloading process. Therefore, the area of the mechanical hysteresis, measured as the difference between the forces of the upper and lower plateau, increases, because at the same maximum deformation, the SIM volume increases with the cross-sectional dimension. This leads to the increase in the stored elastic energy with the same proportion, hence facilitating the reverse transformation (Garrec et al., 2005). For example, at 2-mm deflection, the difference for the 0.46  0.64-mm2 wire was approximately 36%, 71% and 88% greater than that for the 0.41  0.56, 0.46 and 0.35 mm arch wires, respectively at 37 1C. The critical stresses for direct and inverse phase transformation are significantly affected by temperature changes. Regardless of the cross section, the orthodontic wires revealed the same pronounced temperature dependency: the force level had a tendency to be higher with the increase in temperature. These results can be explained by the Glausius–Clapayron model (Miyazaki and Otsuka, 1989), which gives the relationships between stress and temperature for shape memory and superelastic alloy. According to this model, the critical stress to induce martensite would increase on heating, therefore, the load increases with the increase in temperature when it is higher than Af (25–50 1C) (Iijima et al., 2002). The force values for the loading plateau are steeper for the highest temperature. This confirms the results reported by Sakima et al. (2006) who also recorded a steeper plateau as well as temperature increase.

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

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Fig. 9. Evolution of stress and martensitic volume fraction in the rectangular orthodontic wire (0.46  0.64 mm2) at 37 1C: (a) contour plot of the tensile stress and (b) the martensite volume fraction distributions. (c) Tensile stress versus tensile strain and (d) versus martensitic volume fraction at the more solicited zone.

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

I.B. Naceur et al. / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4.2. Summary It is important to quantify and evaluate the orthodontic force applied by NiTi arch wire to the bracket and teeth in order to achieve tooth motion. Therefore, based on a three-dimensional material model for shape memory alloys, the implementation of the constitutive equations describing the pseudoelastic behavior of SMA in a finite element computer code has been presented. To ensure the validity of the proposed model, the response of the orthodontic wire at tensile and bending test were simulated and validated with the experimental study. From the simulation results, the nickel–titanium wires were found to display temperature and cross sectional size and shape sensitivity. Besides, the numerical results successfully reproduce the superelastic effect of SMA. The rectangular cross section showed much higher force than the circular one. In addition, the model can predict well the reduction in hysteresis area, and the increased slope of the loading transformation plateau with the increase in cross sectional size and temperature. To this end, the proposed constitutive model offers a useful tool for the simulation of superelastic SMA-based devices in biomedical applications. Conflict of interest statement None. References Abaqus, Analysis User's Manual, 2009. Dassault Systemes of America Corp., Woodlands Hills, CA, USA. Ahmadabadi, M.N., Shahhoseini, T., Parsa, M.H., Haj-Fathalian, M., Nik, T.H., Ghadiria, H., 2009. Static and cyclic load-deflection characteristics of NiTi orthodontic archwires using modified bending tests. J. Mater. Eng. Perform. 18, 793–796. Auricchio, F., Taylor, R.L., Lubliner, J., 1997. Shape-memory alloys: macromodelling and numerical simulations of the superelastic behavior. Comput. Methods Appl. Mech. Eng. 146, 281–312. Auricchio, F., Sacco, E., 1999. A temperature-dependent beam for shape-memory alloys: constitutive modeling, finite-element implementation and numerical simulations. Comput. Methods Appl. Mech. Eng. 174, 171–190. Arghavani, J., 2010. Thermo-mechanical Behavior of Shape Memory Alloys Under Multiaxial Loadings: Constitutive Modeling and Numerical Implementation at Small and Finite Strains (Ph.D. thesis). Sharif University of Technology, Iran. Boyd, J.G., Lagoudas, D.C., 1996. A thermodynamical constitutive model for shape memory materials. Part 1. The monolithic shape memory alloy. Int. J. Plast. 6, 805–842. Dumoulin, C., Cochelin, B., 2000. Mechanical behavior modeling of balloonexpandable stents. J. Biomech. 33, 1461–1470. Elayyan, F., Silikas, N., Bearn, D., 2010. Mechanical properties of coated superelastic archwires in conventional and self-ligating orthodontic brackets. Am. J. Orthod. Dentofac. Orthop. 137 (2), 213–217. Eshghinejad, A., 2012. Finite Element Study of a Shape Memory Alloy Bone Implant (Ph.D. thesis). The University of Toledo, États-Unis.

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Es-Souni, M., Brandies, H.F., 2001. On the transformation behaviour, mechanical properties and biocompatibility of two NiTi-based shape memory alloys: NiTi42 and NiTi42Cu7. Biomaterials 22, 2153–2161. Etave, F., Finet, G., Boivin, M., Boyer, J.-C., Rioufol, G., Thollet, G., 2001. Mechanical properties of coronary stents determined by using finite element analysis. J. Biomech. 34, 1065–1075. Fercec, J., Glisic, B., Scpan, I., Markovic, E., Stamenkovic, D., Anzel, I., Flasker, J., Rudolf, R., 2012. Determination of stresses and forces on the orthodontic system by using numerical simulation of the finite elements method. Acta Phys. Pol. 122, 659–665. Gao, X., Brinson, L., 2002. A simplied multivariant sma model based on invariant plane nature of martensitic transformation. J. Intell. Mater. Syst. Struct. 13, 795–810. Garrec, P., Jordan, L., 2004. Stiffness in bending of a superelastic Ni–Ti orthodontic wire as a function of cross-sectional dimension. Angle Orthod. 74, 691–696. Garrec, P., Tavernier, B., Jordan, L., 2005. Evolution of flexural rigidity according to the cross-sectional dimension of a superelastic nickel titanium orthodontic wire. Eur. J. Orthod. 27, 402–407. Huang, M., Gao, X., Brinson, L.C., 2000. A multivariant micromechanical model for smas part 2. Polycrystal model. Int. J. Plast. 16, 1371–1390. Iijima, M., Ohno, H., Kawashima, I., Endo, K., Mizoguchi, I., 2002. Mechanical behavior at different temperatures and stresses for superelastic nickel–titanium orthodontic wires having different transformation temperatures. Dent. Mater. 18, 88–93. Khambay, B., Millett, D., McHugh, S., 2005. Archwire seating forces produced by different ligation methods and their effect on frictional resistance. Eur. J. Orthod. 27, 302–308. Krishnan, M., Kalathil, S., Abraham, K.M., 2009. Comparative evaluation of frictional forces in active and passive self-ligating brackets with various archwire alloys. Am. J. Orthod. Dentofac. Orthop. 136, 675–682. Lagoudas, D., Bo, Z., 1999. Thermomechanical modeling of polycrystalline smas under cyclic loading. Int. J. Eng. Sci. 14, 1089–1204. Lombardo, L., Marafioti, M., Stefanoni, F., Mollica, F., Siciliani, G., 2012. Load deflection characteristics and force level of nickel titanium initial archwires. Angle Orthod. 82, 507–521. Majo, D.G., Paterson, R.J., Curtis, R.V., Saidb, R., Wood, R.D., Bonet, J., 2004. Optimisation of the superplastic forming of a dental implant for bone augmentation using finite element simulations. Dent. Mater. 20, 409–418. Marketz, F., Fischer, F.D., 1996. Modelling the mechanical behavior of shape memory alloys under variant coalescence. Comput. Mater. Sci. 5, 210–226. Morgan, N.B., Friend, C.M., 2001. A review of shape memory stability in NiTi alloys. J. Phys. IV 11, 325–332. Miyazaki, S., Otsuka, K., 1989. Development of shape memory alloys. ISIJ Int. 29, 353–377. Noda, T., Okamoto, Y., Hamanaka, H., 1993. Frictional property of orthodontic wires —evaluation by static frictional coefficients. J. Jpn. Orthod. Soc. 52, 154–160. Otsuka, K., Ren, X., 1999. Martensitic transformations in non ferrous shape memory alloys. Mater. Sci. Eng. 273–275, 89–105. Queiroz, G.V., Ballester, R.Y., De Paiva, J.B., Neto, J.R., Galon, G.M., 2012. Comparative study of frictional forces generated by NiTi archwire deformation in different orthodontic brackets: in vitro evaluation. Dent. Press J. Orthod. 17, 45–50. Sakima, M.T., Dalstra, M., Melsen, B., 2006. How does temperature influence the properties of rectangular nickel–titanium wires. Eur. J. Orthod. 28, 282–291. Sun, Q.P., Hwang, K.C., 1993. Micromechanics modelling for the constitutive behavior of polycrystalline shape memory alloys I. Derivation of general relations. J. Mech. Phys. Solids 41, 1–17. Thériault, P., Brailovski, V., Gallo, R., 2006. Finite element modeling of a progressively expanding shape memory stent. J. Biomech. 39, 2837–2844. Trenton, D., 2008. Unloading Behavior and Potential Binding of Superelastic Orthodontic Leveling Wires: a Gingivally Malposed Cuspid Model (Master's Thesis). Saint Louis University, Madrid.

Please cite this article as: Naceur, I.B., et al., Finite element modeling of superelastic nickel–titanium orthodontic wires. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.10.007i

Finite element modeling of superelastic nickel-titanium orthodontic wires.

Thanks to its good corrosion resistance and biocompatibility, superelastic Ni–Ti wire alloys have been successfully used in orthodontic treatment. The...
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