Paul Xolin

Christophe Collard

Marc Engels-Deutsch

Ben Zineb

Finite element and experimental structural analysis of endodontic rotary file made of Cu-based single crystal SMA considering a micromechanical behavior model

Keywords: Shape Memory Alloy, Endodontic files, Cu-based single crystal SMA, Finite element analysis, experimental analysis, combined bending-torsion

Shape Memory Alloys (SMAs) are widely used in endodontics as instruments for root canal preparation namely endodontic rotary files. Despite their high performances, compared to stainless steel instruments, it is still possible to enhance their cutting efficiency by acting on their shape and material properties.

As an example, increasing the exhibited reversible martensitic transformation strain makes them more flexible without decreasing their mechanical strength. It becomes possible with single crystal SMAs. Cu-based (Cu-Al-Be, Cu-Zn-Al, Cu-Al-Mn) single crystal SMAs start emerging and can reach about 12% of martensitic transformation strain, in addition to their interesting antimicrobial properties. This study investigates, both numerically and experimentally, the development of endodontic files made of Cu-based single crystal SMAs. The numerical analysis is carried out by the finite element method. The geometry of the instrument is parameterized. Depending on the applied boundary conditions, bending, torsion, or combined bending-torsion loadings are represented. A micromechanical constitutive law is implemented in Abaqus via the UMAT subroutine to describe the thermomechanical behavior of the Cu-based single crystal SMA. Following the thus obtained numerical results, Coltene-Micromega company has manufactured endodontic file prototypes made of Cu-Al-Be sin-

Introduction

Root canal treatment is a dental procedure used to treat inflamed or/and infected tooth pulp. It enables a patient to retain a tooth in a healthy state on his dental arcade. The procedure is often complicated and, once the diagnosis is made, consists in isolating the tooth, opening the crown, shaping and cleaning the root canal, and finally filling it with a biocompatible sealer. To remove the dental pulp and its organo-mineral content, endodontic files are needed. The root canal preparation in endodontics relies on manual then rotary file capacity to remove organic content, microorganisms and predentin and to follow the curved areas while operating without causing any damage and any instrument failure. This capacity was mainly improved during the last decades. Manual stainless steel files were replaced by rotary ones with the same material making the root canal preparation more efficient and with fewer cases of failure. However, stainless steel rotary files had difficulties to follow the relatively curved and conical shape of the root canal for a given tooth. This curvature could have a circular arc shape or a plane/tridimensional "S" shape. This is due to their very weak exhibited reversible deformation induced by elasticity only. That is why endodontic rotary files made of NiTi SMAs emerged naturally as an interesting alternative to stainless steel ones [START_REF] Pedrinha | Influence of File Motion on Shaping, Apical Debris Extrusion and Dentinal Defects: A Critical Review[END_REF]. In fact, SMAs and in particular NiTi alloys are able to deliver much higher reversible deformation thanks to the reversible martensitic transformation strain. It enables the more flexible rotary file to better follow simple or double curved root canals during the preparation process. These properties allow the use of mechanized files driven in a rotation motion from generally 150 rpm to 350-600 rpm [START_REF] Zelada | The effect of rotational speed and the curvature of root canals on the breakage of rotary endodontic instruments[END_REF][START_REF] Padavala | Fatigue in rotary endodontic instruments -A review[END_REF]. In addition, NiTi SMAs are also bio-compatible and show a good resistance to corrosion, a high strength stress and a high resistance to fatigue under bending and torsion loading [START_REF] Shen | Fatigue testing of controlled memory wire nickel-titanium rotary instruments[END_REF][START_REF] Jamleh | Cyclic Fatigue and Torsional Failure of EdgeTaper Platinum Endodontic Files at Simulated Body Temperature[END_REF]. They have two main specific thermomechanical behaviors governed by forward and reverse martensitic phase transformation: superelasticity effect (SE) and shape memory effect (SME), with respectively a self recovering and a thermally assisted recovering of the martensitic transformation strain after unloading. The major available rotary endodontic files are SE even if SME ones, with more flexibility induced by martensite detwinning at quasi-constant buccal temperature, start to emerge and are known in endodontics as heat treated rotary files [START_REF] Prados-Privado | Finite element analysis comparing WaveOne, WaveOne Gold, Reciproc and Reciproc Blue responses with bending and torsion tests[END_REF]. However, continuously rotating NiTi files operate with the inherent risk associated with torque and cyclic fatigue failures. To decrease this risk, files were used in a mechanical reciprocating movement, turning a specific distance clockwise, then rotating counterclockwise [START_REF] Karatas | Effect of movement kinematics on the cyclic fatigue resistance of nickel-titanium instruments[END_REF]. Over the past years, a huge number of carried out research and development works enabled to enhance the cutting performances of these rotary files [START_REF] Schaefer | Cutting Efficiency of Five Different Types of Rotary Nickel-Titanium Instruments[END_REF]. They acted on optimization of material parameters related to (SE) and (SME) mechanisms, [START_REF] Zinelis | The effect of thermal treatment on the resistance of nickel-titanium rotary files in cyclic fatigue[END_REF], on geometrical parameters as cross section shape, tip diameter, taper, pitch, operative length [START_REF] Ahamed | Comparative Evaluation of Stress Distribution in Experimentally Designed Nickeltitanium Rotary Files with Varying Cross Sections: A Finite Element Analysis[END_REF][START_REF] Ha | Stress Generation during Pecking Motion of Rotary Nickel-titanium Instruments with Different Pecking Depth[END_REF][START_REF] De Arruda Santos | Mechanical behavior of three nickel-titanium rotary files: A comparison of numerical simulation with bending and torsion tests[END_REF][START_REF] Arbab Chirani | Comparative analysis of torsional and bending behavior through finite-element models of 5 NiTi endodontic instruments[END_REF][START_REF] Chevalier | Experimental Validation of Numerical Simulations of a New-Generation NiTi Endodontic File Under Bending[END_REF], and also on control parameters as speed rotation for continuous and reciprocating motion and maximum torsion torque. It has been determined that the tip diameter, the taper, and the cross section properties (shape, area and inertia), have in addition to SMA properties, an influence on the bending and torsion flexibilities (or stiffnesses) and on the torsion and fatigue resistance [START_REF] Schäfer | Relationship between design features of endodontic instruments and their properties. Part 2. Instrumentation of curved canals[END_REF][START_REF] Rodrigues | Efficacy of CM-Wire, M-Wire, and Nickel-Titanium Instruments for Removing Filling Material from Curved Root Canals: A Micro-Computed Tomography Study[END_REF][START_REF] Chi | Cyclic fatigue behavior of nickel-titanium dental rotary files in clinical simulated root canals[END_REF]. The fatigue behavior of these NiTi SMA endodontic files and the corresponding induced fracture modes were also investigated by [START_REF] Ferreira | Movement kinematics and cyclic fatigue of NiTi rotary instruments: A systematic review[END_REF][START_REF] Aidasani | Dentinal Defects in Endodontically Treated Teeth using NiTi Rotary vs Reciprocating Endodontic Instruments-A Systematic Review[END_REF] leading to the same conclusions. [START_REF] Kim | Cyclic Fatigue and Torsional Resistance of Two New Nickel-Titanium Instruments Used in Reciprocation Motion: Reciproc Versus WaveOne[END_REF][START_REF] Mcguigan | Endodontic instrument fracture: causes and prevention[END_REF][START_REF] Martins | Fatigue resistance of rotary endodontic files subjected to planar and non-planar curvatures induced by in vitro tooth canals[END_REF] showed that fracture while operating is induced by bending or torsion or both combined loading. Therefore, considering these results, rotating endodontic files performances were highly improved making more efficient, reproductible and safer the canal root preparation and reducing its duration. However, these improvements mainly focus on NiTi rotary files. Recently our research group showed that it is possible to even more improve performances of rotary files by considering Cu-based single crystal SMAs (Cu-SC-SMA) as for example the Cu-Al-Be alloy [START_REF] Vincent | Finite Element Analysis of a Copper Single Crystal Shape Memory Alloy-Based Endodontic Instruments[END_REF][START_REF] Vincent | Experimental and numerical analysis of penetration/removal response of endodontic instrument made of single crystal Cu-based SMA: comparison with NiTi SMA instruments[END_REF].Using this SMA family should also contribute to the reduction of the root canal preparation time thanks to the antimicrobial properties of copper [START_REF] Vincent | Contact killing and antimicrobial properties of copper[END_REF]. Copper is a powerful antimicrobial agent. Microorganisms survive only a few minutes on copper surfaces [START_REF] Grass | Metallic copper as an antimicrobial surface[END_REF][START_REF] Prado | Application of copper bactericidal properties in medical practice[END_REF] especially at 37°C and in conditions of 100% relative humidity (buccal conditions) [START_REF] Ojeil | Evaluation of new invitro efficacy test for antimicrobial surface activity reflecting UK hospital conditions[END_REF]. Associated with hydrogen peroxide, an endodontic irrigant, copper is able to kill bacteria in a very short time through the release of hydroxyl radicals [START_REF] Hartemann | Bactericidal effect of hydrogen peroxide on Escherichia coli[END_REF]. Conversely, sodium hypochlorite, the most widely used irrigation solution in endodontics, requires 60 minutes of contact time to eliminate microorganisms [START_REF] Spratt | An in vitro evaluation of the antimicrobial efficacy of irrigants on biofilms of root canal isolates[END_REF]. Contrary to NiTi single crystals, it is easier to manufacture Cu-SC-SMA wires that are well adapted to process rotary files. Moreover, they have very interesting properties: (i) antimicrobial effect during root canal preparation and (ii) much higher flexibility induced by martensitic transformation strain that can go up to 12%. In addition, it is worth noting that polycrystal Cu-based SMAs present very high incompatibilities between grains inducing high internal stresses and fracture below low reversible transformation strains (about 3 or 4%) which also justifies the choice of Cu-SC-SMA.

However, it has been shown that such Cu-SC-SMA wires, with a [1 0 0] crystallographic orientation in the axis direction, exhibit a very low transformation strain (low volume fraction of martensite) under torsion loading leading to an effective behavior close to the elastic one which is highly anisotropic. Therefore, for given geometrical parameters, the obtained global response of the rotary endodontic instrument under torsion loading is stiffer than under bending loading [START_REF] Sittner | Stress induced martensitic transformations in tension/torsion of CuAlNi single crystal tube[END_REF]. Indeed, when the instrument rotation is hampered by the machined dentin, torsional loading occurs inducing the instrument failure.

A part from it will then remain inside the root canal making it very difficult to eliminate. [START_REF] Vincent | Finite Element Analysis of a Copper Single Crystal Shape Memory Alloy-Based Endodontic Instruments[END_REF] studied by the finite element method the influence of geometrical parameters on responses of Cu-SC-SMA rotary files subjected to bending and torsion loading following the Standard ISO-3630-1 (Int. Org. for Standardization, 2008). This study considers a macroscopic thermomechanical constitutive model proposed by [START_REF] Peultier | Macroscopic constitutive law of shape memory alloy thermomechanical behaviour. Application to structure computation by FEM[END_REF] and improved by [START_REF] Chemisky | Constitutive model for shape memory alloys including phase transformation, martensitic reorientation and twins accommodation[END_REF]. For this preliminary study, the adopted behavior modem cannot accurately predict the specific torsional response of Cu-SC-SMA.

In fact, the experimental analysis of the martensitic transformation process for these alloys, when subjected to torsion loading, shows an activation of very few martensite variants with a very low corresponding volume fraction [START_REF] Tokuda | Experimetnal stuy on performances in Cu-based shape memory alloy under multi-axial loading contions[END_REF][START_REF] Bouvet | A phenomenological model for a Cu-Al-Be shape memory alloy pseudoelasticity under complex loading[END_REF][START_REF] Casciati | Fatigue characterization of a Cubased shape memory alloy[END_REF]. It is related to the high incompatibility between martensite variants favorably oriented with respect to the torsion loading direction. It manifests through the moment-rotation global response with a high slope close to the elastic shear modulus and a very weak hysteresis size. Micromechanical models with crystalline plasticity-like formulation overcome such drawbacks of phenomenological behavior models. They describe the thermomechanical behavior at the grain scale (each grain is assumed as a single crystal) by considering all potentially active variants of martensite.

Each variant is characterized by a set of habit plane-twinning direction, and a volume fraction. Berveiller-Patoor's research group proposed a micromechanical constitutive law based on this approach [START_REF] Patoor | Micromechanical Modelling of Superelasticity in Shape Memory Alloys[END_REF]. They derived a macroscopic model more adapted to polycristalline Cu-based SMAs by applying a self consistent scale transition technique to a Representative Volume Element (RVE) composed of a high number of grains. Each grain has a different crystallographic orientation related to the measured texture [START_REF] Entemeyer | Strain rate sensitivity in superelasticity[END_REF]. Other models considering the same approach are also available in liter-ature and detailed in the review paper [START_REF] Cissé | A review of constitutive models and modeling techniques for shape memory alloys[END_REF]. The single crystal version, initially proposed by [START_REF] Patoor | Micromechanical Modelling of Superelasticity in Shape Memory Alloys[END_REF], improved by [START_REF] Niclaeys | Determination of the interaction energy in the martensitic state[END_REF][START_REF] Merzouki | Coupling between measured kinematic fields and multicrystal SMA finite element calculations[END_REF][START_REF] Collard | Simulation of the effect of elastic precipitates in SMA materials based on a micromechanical model[END_REF] is implemented in the finite element code Abaqus. It well describes the martensite variant activation in a Cu-based single crystal SMA, particularly for torsion loading. It is then more adapted to predict the torsional response of Cu-SC-SMA rotary files.

Therefore, the present paper details the numerical analysis of rotary prototype made of Cu-SC-SMA considering the aforementioned micromechanical constitutive model. It is subjected to respectively bending, torsion and bending-torsion loading. The combined bending-torsion loading occurs on the instrument when it becomes difficult to shape the dentin; in that case, the instrument cannot freely rotate which adds the torsion sollicitation to the bending. The geometry of this rotary file is inspired from already used NiTi files and described with a CAD model developed with Catia and exported to Abaqus . It is meshed with a tetrahedral continuum solid element with quadratic interpolation. The applied boundary conditions and loading respect the recommendation of the ISO Standard 3630-1, (Int. Org. for Standardization, 2008). The identification of the material parameters and the qualification of obtained numerical results rely on experimental results carried out using a specific setup [START_REF] Xolin | Combined bending-torsion testing device for characterization of shape memory alloy endodontic files[END_REF]. It allows to apply bending and torsion loading in combined or separated ways. Prototypes are processed by Coltene-Micromega using Cu-SC-SMA wires manufactured by Nimesis . The obtained numerical results correctly correlate with experimental results and prove the following original features: (i) finite element analysis considering micromechanical constitutive law predicts in a more accurate way the response in torsion of rotary files made of Cu-SC-SMA, (ii) combining bending with torsion for these rotary files reduces their torsional stiffness and then moderates the aforementioned torsional response weakness, (iii) considering the interesting antimicrobial properties of Cu and despite this torsional response weakness, it should be possible to develop new and innovative versions of rotary files made of Cu-SC-SMA. In fact, it shows that responses of Cu-SC-SMA rotary endodontic files are close to the ones of commercial instruments made of NiTi. This is true for both separated and combined bending-torsion loadings. Indeed, obtained values of rotations and their induced moments are in the same range as what is measured for commercial instruments. Furthermore, the analysis of sensitivity to geometrical parameters also leads to the same tendencies for commercial instruments [START_REF] Gavini | Nickel-titanium instruments in endodontics: A concise review of the state of the art[END_REF].

The present paper is organized as follows: after the introduction, section 2 details the finite element numerical model with geometry, meshing, boundary conditions, loading cases and the adopted micromechanical constitutive model and its implementation in the finite element code Abaqus . Section 3 presents the experimental testing device and the Cu-SC-SMA prototypes. Differential Scanning Calorimetry (DSC), tensile, bending, torsion and combined bendingtorsion tests are performed to identify the material parameters of the micromechanical constitutive model describing the Cu-SC-SMA thermomechanical behavior. Bending, torsion and combined bending-torsion loadings are applied to validate the proposed model. Section 4 presents the adopted Taguchi design of experiment (DOE) in order to determine the influence of geometrical parameters on the equivalent response in combined bending-torsion loading. Section 5 presents an overall discussion of the obtained results. The last section gives some conclusions and prospects.

Finite element response analysis of the rotatry endodontic file made of Cu-SC-SMA

Introduction

The proposed finite element analysis aims at predicting the global response of the proposed rotary endodontic file made of Cu-SC-SMA and to analyze the corresponding local state, at each material point, in terms of stresses, strains, activated martensite variants and their corresponding volume fractions. The adopted geometrical model is parameterized allowing to analyze the sensitivity of the rotary file response to the geometrical parameters. Applied boundary conditions are consistent with the ISO Standard 3630-1 (Int. Org. for Standardization, 2008). They induce a bending-torsion loading applied in combined or separated ways. The geometry is meshed with tetrahedral continuum elements and a mesh refinement convergence is carried out. A micromechanical constitutive model describes the thermomechanical behavior of the Cu-based single crystal SMA. This finite element analysis allows to predict the effect of the torsion on the global bending-torsion response of the studied rotary endodontic files.

Geometry, mesh, boundary conditions and loading

The studied Cu-SC-SMA endodontic file presents a parameterized geometry defined by figure 1. It presents a cylindrical part characterized by its diameter and a tapered helical part characterized by its cross-section shape, its length, its taper, its constant or variable pitch and its tip diameter. The variation of these geometrical parameters allows to analyze their effects on the global bending-torsion response. For each set of geometrical parameters, the CAD software CATIA generates the corresponding instrument geometry which is exported into the finite element code Abaqus . Then the meshing process a 3 mm length starting from its tip is clamped. Two rotations corresponding to bending and torsion are applied in two reference points as depicted in figure 3. This leads to three loading cases: torsion, bending and combined torsionbending. As the instrument is generally rotating with a constant speed, the analysis assumes a quasi-static loading. Also note that the boundary conditions can be applied to any wire or prototype instrument with different cross-section shape, diameter, taper and pitch length.

Micromechanical constitutive law for Cu-SC-SMA and its implementation in a finite element code

The thermomechanical behavior of the Cu-SC-SMA is described by a constitutive model, developed by [START_REF] Patoor | Micromechanical Modelling of Superelasticity in Shape Memory Alloys[END_REF], and [START_REF] Niclaeys | Determination of the interaction energy in the martensitic state[END_REF].

Its implementation in Abaqus via the subroutine UMAT, initially undertaken by [START_REF] Merzouki | Coupling between measured kinematic fields and multicrystal SMA finite element calculations[END_REF][START_REF] Collard | Simulation of the effect of elastic precipitates in SMA materials based on a micromechanical model[END_REF], is deeply redrafted in this work leading to an updated version enhancing the convergence process and reducing computation CPU time. It is based on a crystalline plasticity-like approach and describes the activation of martensite variants in the crystalline lattice and its effect on the macroscopic thermomechanical behavior. This

Z Y X RP-1 RP-2 X Y Z Z Y X RP-1 X Y Z Torsion Reference point Bending Reference point Boundary conditions for torsion                U X = f ree U Y = 0 U Z = 0 θ X = θ T θ Y = 0 θ Z = 0               
Boundary conditions for bending

               U X = 0 U Y = 0 U Z = 0 θ X = 0 θ Y = θ B θ Z = 0               
εn ij = 1 2 g(m n i n n j + m n j n n i ) = 1 2 gR n ij (1)
where R n is the symmetric part of the Schmid tensor for the variant (n). In the case of multi-variants simultaneously active, the global transformation strain is the resultant of all active variant transformation strains weighted by their respective volume fractions (f n = Vn V RV E ) as follows:

ε tr ij = n εn ij f n (2)
where V n is the volume occupied by the variant n and V RV E is the total volume of the Representative Volume Element (RVE) characterizing a given material point. The phase transformation process is then characterized by the volume fraction of the 24 variants assumed as internal variables. These volume fractions are constrained by martensitic transformation saturation leading to 0 ≤ f n ≤ 1 and n f n ≤ 1. By considering the defined global transformation strain in addition to the classical elastic and thermal expansion strains, and applying the first principle of thermodynamics, the micromechanical constitutive model derives from the following assumed expression of the Gibbs free energy expressed at the lattice scale:

∆G(σ, T, f n ) = σ ij ε ij -W elas -∆G ch -W inter (3)
where σ, ε and T are respectively the stress and strain tensors and the temperature, in a material point, and where W elas , ∆G ch , W inter are respectively the elastic energy, the chemical energy variation between the austenitic and martensitic phases and the interface energy. This interface energy is assumed negligible due to the flat shape of martensite variants. After some intermediate developments detailed in [START_REF] Siredey | Constitutive equations for polycrystalline thermoelastic shape memory alloys.: Part I. Intragranular interactions and behavior of the grain[END_REF], the final expression of the Gibbs free energy is depicted by equation ( 4), where α, δ ij , T 0 , T I and B are the thermal expansion coefficient, the Krönecker symbol, the thermodynamic temperature equilibrium between austenitic and martensitic phases, the initial temperature of the material and the chemical energy respectively.

∆G(σ, T, f n ) = 1 2 σ ij S ijkl σ kl + σ ij ε tr ij + αδ ij (T -T I )σ ij + B(T -T 0 ) n f n - 1 2 n,m f n H nm f m (4)
The total energy σ : ε is split into an elastic part, a transformation part related to the transformation strain ε tr and a thermal expansion part related to the expansion tensor. The expansion tensor is assumed isotropic and controlled by the scalar α. The elastic energy is expressed as a function of the stress tensor and the fourth order elasticity tensor S. The chemical energy linearly depends on the temperature variation and volume fraction of variants. The last term in equation ( 4) represents the interaction energy related to internal stresses. It is induced by incompatibilities between martensite variants. In order to simplify equations, the interaction between each variant is represented in a matrix H nm given in table 2. It lists the compatibilities and incompatibilities between all variants. Taking into account the constraints on the volume fraction of each variant modifies the expression of Gibbs energy and adds the following terms :

ψ const = n λ n 1 f n + λ(1 - n f n ) (5)
It leads to the following expression of the Lagrangian:

L = σ ij ε ij -∆G ch -W elas -W inter -ψ const = 1 2 σ ij S ijkl σ kl + σ ij ε tr ij + αδ ij (T -T I )σ ij -B(T -T 0 ) n f n - 1 2 n,m f n H nm f m -( n λ n 1 f n + λ(1 - n f n )) (6)
C I C C C C C C I C I C C C C I C I C C C C C C I C I C C C C C C C C C C C C C C C C
where λ n 1 and λ are Lagrangian multipliers allowing to ensure the transformation saturation conditions. The application of the Clausius-Duhem condition integrating the expression of the Lagrangian L leads to the decomposition of strains and expressions of driving forces for martensite variants:

        ∂L ∂σ -ε : σ + ∂L ∂T Ṫ + n ∂L ∂f n ḟ n + ∂L ∂λ n 1 λn 1 + ∂L ∂λ λ L -S RV E Ṫ -q. gradT T         ≥ 0 (7)
where S RV E is the entropy of the RVE, q is the heat flux induced by dissipation and T the current temperature of the RVE. Dot over symbols represent their rate form. Equation ( 7) leads to the following equations:

               S ijkl σ kl + αδ ij (T -T I ) + ε tr ij -ε ij = 0 -B n f n -S RV E = 0 gσ ij R n ij -B(T -T 0 )e n - m H nm f m -λ n 1 + λe n = F n yield (8)
where (e 1 , ..., e k ) is a basis of R k defined as e n i =

0 if i = n 1 if i = n .
The two first driving forces are equal to zero because they correspond to thermoelasticity occurring without dissipation whereas the third driving force is related to phase transformation occurring with dissipation. This justifies the introduction of a yield force F n yield associated to each active variant (n) taking a different value during forward and reverse transformation. The temperature gradient term becomes equal to zero by assuming a uniform temperature in the whole RVE. Combining equations ( 8) and their rate form allows to determine the rate of volume fraction of martensite ḟ n for each active variant (n) which reads:

ḟ n = m Ĥ-1 nm ∼ ε m ij C ijkl εkl - ∼ ε m pq C pqrs α rs + Be m Ṫ (9)
where the fourth order tensor C = S -1 is the elasticity tensor and Ĥnm =

H nm + ∼ ε n ij C ijkl ∼ ε m kl is a k v × k v reversible matrix
where k v is the number of active variants. Combining the rate form of equations ( 8) and ( 9) allows to derive the rate form of the micromechanical based thermomechanical constitutive equations for the Cu-Al-Be single crystal SMA that reads:

σij = l ijkl εkl -m ij Ṫ ( 10
)
where

         l ijkl = C ijkl -n,m C ijrs εn rs Ĥ-1 nm εm pq C pqkl m ij = C ijkl αδ kl -n,m C ijrs εn rs Ĥ-1
nm Be m + εm pq C pqkl αδ kl These constitutive equations are solved with an iterative method based on the return mapping approach. The obtained numerical tool is implemented in Abaqus via the subroutine UMAT. The numerical resolution is detailed in [START_REF] Collard | Simulation of the effect of elastic precipitates in SMA materials based on a micromechanical model[END_REF]. The implementation process in Abaqus via the subroutine UMAT is initially carried out in [START_REF] Merzouki | Coupling between measured kinematic fields and multicrystal SMA finite element calculations[END_REF]. It is redrafted in a deep way in this work enabling to enhance the convergence process and to reduce the computation CPU time. This behavior model presents metallurgical parameters as transformation temperatures and crystallographic characteristics as lattice orientation and habit plane and transformation direction for all variants. It also needs to identify phenomenological parameters as the elastic coefficients, the compatibility and incompatibility coefficients, the transformation amplitude g and the yield forces for each variant n, F n yield . Results issued from this finite element analysis are compared to the ones obtained experimentally. The experimental protocol is detailed in the next section.

Experimental response analysis of the rotary endodontic file made of Cu-SC-SMA

The experimental part of this study has a double objective: (i) the identification of material parameters and (ii) the validation of the FE based numerical approach developed in the previous section. In this section, the testing device for combined bending torsion loadings is briefly presented, then the experimental protocol is detailed.

Experimental setup for combined bending-torsion loading

The experimental testing device is designed to study the bending and torsion responses of endodontic files under bending, torsion or combined bending-torsion loadings. It is based on the standard ISO 3630-1 (Int. Org. for Standardization, 2008). The main goal of these tests on endodontic file is to determine its flexibility and its resistance to torsional and bending loadings. Indeed, a bending loading is applied on the endodontic file when it has to follow the path of a curved root canal, and a torsion loading is applied due to the continuous rotation movement which allows the file to progress inside the narrow root canal and shape it. However, both bending and torsion loadings applied on the endodontic file are combined during the endodontic treatment and represent the most severe loading case. The considered testing device is able to measure the bending, torsion and combined bending-torsion loadings. It is composed as two subsystems: (i) a subsystem for torsion and (ii) a subsystem for bending. The kinematic principle and a detailed picture of the testing device are depicted by figures 5a and 5b respectively. The endodontic file is clamped 3 mm from its tip end and on its rod end 17 mm from the tip. It is loaded in torsion and bending by two rotation actuators. The first rotation is applied around the endodontic file axis whereas the second is applied around an axis perpendicular to this latter. Two corresponding sensors are associated to these actuators in order to measure induced moments. This testing device allows to apply an up to 60°rotation amplitude with 30 N.mm measured torque in bending and 20 N.mm torque with no limitation rotation in torsion. Figure 6 shows the possible applied loading paths of bending or torsion (figure 6a) and combined bendingtorsion (figure 6b). [START_REF] Xolin | Combined bending-torsion testing device for characterization of shape memory alloy endodontic files[END_REF] gives more details about this testing device.

Material parameter identification with Cu-SC-SMA wire

Uniaxial tests at room temperature and DSC tests are carried out to identify Cu-SC-SMA material parameters for the micromechanical behavior model [START_REF] Merzouki | Coupling between measured kinematic fields and multicrystal SMA finite element calculations[END_REF][START_REF] Collard | Simulation of the effect of elastic precipitates in SMA materials based on a micromechanical model[END_REF]. Then, Cu-SC-SMA wires

Time [s] Angle [°]

Load 1 Load 2 ω θ (a) Loading path for simple bending or torsion. Load 1 is equated with bending with an angle θ of 60°and a rotation speed ω of 3°.s -1 for a simple load of bending. It is equated with torsion with an angle θ of 30°and a rotation speed ω of 1.5°.s -1 for a simple load of torsion. Load 2 is the completmentary load, set as 0°.

Time [s] Angle [°] Bending

Torsion

ω 1 ω 2 θ 1 θ 2 (b)
Loading path for combined bending-torsion. Bending load is applied with an angle θ 1 of 60°and a rotation speed ω 1 of 3°.s -1 . Torsion load is applied with an angle θ 2 of 30°and a rotation speed ω 2 of 1.5°.s -1 Figure 7 shows results of DSC tests on Cu-SC-SMA wire. They consist in heating the sample up to 50°C then in cooling it down to -150°C twice and in measuring the heat flux. The obtained results show two peaks, one while cooling and one while heating, which are characteristic of forward and reverse martensitic transformation respectively. The cooling peak is located between -70°C and -78°C which determines M s and M f , the start and finish temperatures for direct martensitic transformation, respectively. The heating peak is visible between -63°C and -57°C which determines A s and A f , the start and finish temperatures for reverse martensitic transformation, respectively. Figure 8 shows the uniaxial tension test for a Cu-SC-SMA wire of 0.79 mm diameter subjected to 10 cycles of 8.5% strain. In addition, experimental tests of 30°350 of bending, 5°of torsion and respectively 30°and 6°of combined bending and torsion are realized on Cu-SC-SMA wires. Applied loadings are selected to show high deformation responses and not overpass range measurement of sensor. Fig-

ure 9 shows responses of these tests. Figure 9a gives the bending moment of a Cu-SC-SMA wire subjected to bending and combined bending-torsion loading.

The elastic domain in both cases is similar until the activation of martensitic transformation. Stiffness during transformation is more important in the case of pure bending. Regarding the torsion moment of wire subjected to torsion and combined bending-torsion loadings (figure 9b), responses to both loadings show a high stiffness. Torsion loading on wire shows a linear response with very low transformation. But, when torsion is combined with bending, the martensitic transformation occurs in wire since the response of the material exhibits a non linear behavior and a hysteresis between loading and unloading path.

Also, it is notable that the elastic domain of responses in pure and combined bending-torsion has different stiffnesses.

In the following, the micromechanical model is used to represent the behavior of Cu-SC-wire under tension, bending, torsion and combined bending-torsion loadings. These tests also provide the maximum strain transformation.

According to the symmetric cubic elasticity approach [START_REF] Belkahla | Elaboration et caractérisation de nouveaux alliages à mémoire de forme basse température type Cu-Al-Be[END_REF], material parameters presented in table 3 are used to identify the material parameters the micromechanical constitutive law describing the behavior of Cu-SC-SMA.

Figure 10 compares the 10th cycle of uniaxial tension of figure 8 with the numerical response of a unique cubic element submitted to 8.5% strain.

To complete material parameters identification, bending, torsion and combined bending-torsion simulations are numerically predicted with previously identified parameter. Numerical simulations are conducted with the boundary conditions presented in section 2.2 to represent experimental loadings. Cu-SC-SMA 22

C A 11 [MPa] C A 12 [MPa] C A 44 [MPa] A f [°C] 123192 94200 110838 - 57
C M 11 [MPa] C M 12 [MPa] C M 44 [MPa] M s [°C] 93456 94200 84084 -70 α [°C -1 ] B [MPa °C-1 ] C [MPa] I [MPa]
1.5e-5 0.14 20-250 2500 wires are represented with 27900 thetraedron elements with quadratic interpolation (codification: C3D10) in Abaqus © software. Figure 11 shows the comparison between experimental and numerical results for these loadings.

g φ 1 [°] Φ [°] φ 2 [°] 0.139 2 0 0

Comparison between numerical and experimental responses of Cu-SC-SMA endodontic file prototypes

Endodontic file prototypes are manufactured from Cu-SC-SMA wire. They are tested under combined bending torsion loading with the testing device described in section 3.1 . In addition, the numerical finite element model previously presented is used with a geometrical modeling of the blade of the endodontic file prototype. Both experimental and numerical responses are compared to estimate the capacity of the model to represent a complex Cu-SC-SMA structure of endodontic file prototypes. The numerical model provides additional information compared to experiments such as stress state and martensitic variant volume fractions into the Cu-SC-SMA. Figure 12 shows the manufactured tested prototypes (figure 12a) and the corresponding finite element model (figure 12b).

They have a two-edged cross section and their geometrical parameters are presented in table 4. The finite element model is composed of 41600 thetrahedron elements with quadratic interpolation and represents only the blade of the prototype to reduce computing time. Three prototypes are tested in bending, torsion

Stress and martensite volume fraction distributions

The presented finite element model allows to predict the local stress and martenistic volume fraction in any material point of the instrument. Figures 14 and 15 depict the numerically obtained response for an endodontic file prototype with a tip diameter of 0.28 mm, a 4% taper and a two-edged cross section, under bending, torsion and combined bending-torsion loadings. Bending loading induces a local stress concentration (figure 14a) in a zone close to the clamped area. Figure 15a shows that martensite appears in this zone subjected to tension and compression. Figure 14b shows the results of torsion loading. A high stress state appears in all blade structures. It corresponds to a high stiffness of numerical and experimental responses. The stress distribution is mainly concentrated in the center of the blade faces, close to the clamped area. Moreover, the numerical model predicts a low value of volume fraction of martensite (figure 15b) for the Cu-SC-SMA endodontic file prototype in torsion. For combined bending-torsion loading (figure 14c), high stress levels concentrate in the bended zone of the instrument. The stress distribution is induced by a superposition of the stresses obtained by bending and by torsion with the addition of the complex geometry effect. Figure 15c shows the volume fraction of martensite for the combined bending-torsion loadings. It follows the stress distribution, visible for the 0 mm, 0.25 mm, 0.5 mm cuts. However, 1 mm and 1.5 mm cuts show differences in the distribution of stress and volume fraction of martensite.

For the 1 mm cut, on figure 15c, volume fraction of martensite is concentrated in inner and outer part of the bended area when the stress distribution (figure 14c) corresponds to torsion loading. Moreover, the numerical model allows to determine the activated martensite variants and their associated volume fraction in each material point. Figure 16 gives, for the main active variants, the zone where each variant occurs with its associated volume fraction. These variants are 7, 13, 14 and 22 defined in table 1. Their distributions show that each martensite variant occurs separatly and partially cohabits with other variants depending on their compatibilities (table 2). Variant 14 occurs in the inner side of the bended area (subjected to a compression stress) next to variants 7 and 22. According to table 2, variants 7-22 are incompatible, but 7-14 and 14-22 are compatible. Variant 13 occurs especially in the outer side of the bended area. Their transformation direction and normal plane are given in table 1 and their interactions in table 2.

Influence of the geometrical parameters on the response of the studied endodontic files

After the identification of material parameters for the Cu-SC-SMA used to manufacture endodontic file prototypes, a Taguchi design of experiment is carried out numerically. It enables to analyse the influence of the geometrical parameters on the global response of these endodontic files during combined bending-torsion loading. It is designed to study multiple factors with a minimum number of simulations on the system and reduces the computation time.

Selected factors and levels

For this study, the selected factors are the cross section shape, the tip diameter at 3 mm from the tip, the taper and the pitch of the blade. Each factor has 3 levels represented in table 5.

1 1 1 1 1 2 1 2 2 2 3 1 3 3 3 4 2 1 2 3 5 2 2 3 1 6 2 3 1 2 7 3 1 3 2 8 3 2 1 3 9 3 3 2 1

Orthogonal table selection

The selected set of factors and their levels are suitable for a Taguchi orthogonal table L9(3 4 ) presented in table 6. Each factor is assigned to a column where its values vary between the previously selected levels. Each line represents a numerical simulation to perform on the system with the levels of the corresponding factor in columns. According to this table, it is possible to model the corresponding blade geometries, which are presented in table 7.

Equivalent combined bending-torsion response

The modeled endodontic file prototype responses are then numerically analyzed in combined bending-torsion. The adopted equivalent moment response of combined bending-torsion M eq is defined by the following relation:

M eq = M 2 bending + 3 × M 2 torsion ( 11
)
where M bending and M torsion are the moment responses of simulated prototypes for bending and torsion respectively. Numerically obtained results are post- According to the mean numerical responses of the factor levels, the best parameters to reduce the global moment response in combined bending torsion loading are: the asymmetric section shape, a diameter at 3 mm from the tip of 0.40 mm, a taper of 4% and a 1 mm pitch length. The evolution of the diameter at 3 mm from the tip and the pitch length shows linear global response variation while the taper seems to minimize this response for a value of 4%. The parameter inducing the widest variation in the response is the cross section, followed by the diameter at 3 mm from the tip , the taper and finally the pitch length.

.

Discussions

Introduction

In this section, experimental and numerical results are detailed and discussed. The first part presents experimental data of uniaxial tension, bending, torsion and combined bending-torsion tests on Cu-SC-SMA wires and endodontic file prototypes with their numerical comparison. The second part presents the analysis of the results of the Taguchi DOE. The third part analyzes distributions of stress and of martensitic variants in endodontic file prototypes loaded in bending, torsion and combined bending-torsion.

Experimental and numerical comparison of bending, torsion and combined bending-torsion responses with Cu-SC-SMA endodontic file prototypes

The comparison between the experimental and numerical responses in bending, torsion and combined bending-torsion loadings depicted in figures 11 for wire and 13 for endodontic file prototype in Cu-SC-SMA, shows satisfactory correlations. The material parameters, identified from DSC measurements, tension and torsion tests, allow to predict the Cu-SC-SMA response under tension and torsion loadings as shown on figures 10 and 11b. Numerical prediction in bending and combined bending-torsion tests on Cu-SC-SMA wires are slightly less correlated with the experimental responses (figure 13). However, numerical and experimental tendencies, such as the decrease of the transformation plateau stiffness, observed between bending and combined bending-torsion is satisfactorily predicted by the numerical model (figure 11a). In addition, elastic stiffness increases and appearance of hysteresis in torsion component for numerical response in combined bending-torsion loading (figure 11b) is well predicted by the model. However, the numerical torsion response in combined bending-torsion loading is under rated compared to experimental results.

The results issued from experimental and numerical predictions of the endodontic instrument response under bending, torsion and combined bendingtorsion loadings are in satisfactory agreement. The differences between the two results need to be solved with enhancement of the adopted finite element model.

On the one hand, elastic stiffness in torsion loading is underestimated (figure 13b). Likewise, in bending loading, the martensitic transformation predicted by the numerical model shows a smaller hysteresis than the experimental response.

Several sources can explain these differences. On the experimental side, the machining process with its imprecision may have an impact on the anticipated shape of the prototypes. Likewise, the clamp of the tip, may generate additional stresses in the area close to the bended one. Moreover, high loading amplitudes are severe to the prototype instrument, which is brought to its mechanical limits and over the clinicaly acceptable work condition. Furthermore, these loading conditions may generate plasticity, which is not taken into account by the adopted thermomechanical constitutive model, and could explain the difference in martensitic transformation between experimentally measured and numerically predicted responses. The aforementioned causes also have an impact on the modeling part. Indeed, the modeled prototype is assumed perfect, without any machining imprecisions or microstrural change, like local recrystallization. In addition, the model represents idealized clamping conditions of contact between a plane and the tapered helicoidal triangular section shape.

On the other hand and despite all these difficulties, the micromechanical model is able to closely reproduce the response of Cu-SC-SMA endontic file prototypes under separated and combined bending-torsion loadings. The comparison of the bending component of the global numerical response between bending and combined bending-torsion (figure 13a) shows that the model correctly predicts the decrease of the response after the start of martensitic tranformation. For the torsion component, the decrease observed between combined and seperated torsion responses, on figure 13b, shows the model capacities to reproduce global behavior of Cu-SC-SMA for complex loading cases.

Distribution of stresses and volume fraction of martensite through the endodontic file

Figures 14 and 15 depict the stress and volume fraction of martensite in endodontic file prototypes made of Cu-SC-SMA. It is visible that bending and torsion loadings, in figures 14a and 14b respectively, generate high stress levels, but only bending loading activates martensitic transformation (figure 15a).

Despite a high shear stress, concentrated in the middle of the blade faces of the Cu-SC-SMA prototypes in torsion loading, a low quantity of martensite is produced (figure 15b). Figure 14c shows how bending and torsion loadings interact in combination and their influence on the martensitic transformation level and its location (figure 15c). The Von Mises equivalent stress distribution for combined bending-torsion is a superposition of both normal and shear stress distribution. The volume fraction of martensite is diffused according to this stress distribution, except for stresses present in the middle of blade faces, close to the 1 mm cut from the clamped area, and beyond (figure 15c). In the case of combined bending-torsion loading, torsion increases the stiffness of the instrument, leading to a more important martensitic transformation induced by bending. Moreover, the complex geometry of the instrument leads to more complex three-dimensional stress state. This directly influences the activation process of martensitic variants (figure 11b and13b). In addition, the flexiblity gain in torsion of numerical responses between pure torsion and combined bending-torsion loadings is visible with the transformation plateau on figure 13b. It is a global effect due to the martensitic transformation initiated by bending.

Moreover, from the combined bending-torsion loading, the actived martensite variants are different for those activated for separated bending and torsion loading. It induces a higher quantity of martensite and thus a higher transformation strain. Therefore, the induced global stiffness in torsion is lower for the combined bending-torsion than in pure torsion. In addition, combining bending and torsion leads to a higher failure level in torsion rotation than in the case of pure torsion.

Accurate geometry according to Taguchi DOE

The Taguchi DOE allows to choose the best set of geometrical parameters, They should be replaced by the exact geometry of the manufactured endodontic files with their imperfection.

Conclusion and prospects

This paper details the experimental and numerical response of a Cu-SC-SMA endodontic file subjected to a bending-torsion loading. This new endodontic 40 instrument makes the preparation of root canal with higher simple or double curvature easier. In addition it has very interesting antimicrobial properties. A specific bending-torsion testing device is designed and manufactured in order to carry out the experimental analysis. The corresponding obtained results enable to identify the material parameters of a micromechanical constitutive law, well adapted to Cu-SC-SMA. This law is coupled with a finite element method to simulate the response of the studied endodontic file prototypes under several loadings. It depicts an increase in the torsion strength of these Cu-SC-SMA endodontic files when subjected to a combined bending-torsion loading in comparison with a response for a pure torsion loading. The parameterized geometrical model enables to analyze the influence of geometrical parameters on the global response of these endodontic files. A Taguchi experimental plan is applied for this analysis. It shows that the best parameter settings to decrease moments in combined bending-torsion loading is an asymmetric section shape, the lowest diameter at 3 mm from the tip of 0.4 mm, a taper of 4% and a pitch length of 1 mm.

In conclusion, the obtained numerical and experimental results show that the different responses of the instruments made of Cu-SC-SMA are close to what was observed for commercial NiTi based instruments [START_REF] Schäfer | Relationship Between Design Features of Endodontic Instruments and Their Properties . Part 3 . Resistance to Bending and Fracture[END_REF][START_REF] Ninan | Torsion and bending properties of shape memory and superelastic nickel-titanium rotary instruments[END_REF][START_REF] Santos | Effects of R-Phase on Mechanical Responses of a Nickel-Titanium Endodontic Instrument: Structural Characterization and Finite Element Analysis[END_REF] and the sensitivity to geometrical parameters is also globally the same. It should be possible to consider endodontic files made of Cu-SC-SMA to maximize micro-organisms elimination while shaping the root canal, and to reduce the risks of failure during endodontic treatment. The main challenges will be the optimization of their shaping performances during the root canal preparation and the carrying out of clinical tests in order to validate such performances in-vivo. Moreover, the manufacturing process of the instrument could modify the Cu-SC-SMA behavior. Additional features should be implemented in the micromechanical model to fully exploit its potential to study microstructure of endodontic files under rigourous constraints, such as plasticity or clamped conditions on the tip which can be improved. Finally, future investigations should explore the shape mem-ory effect of Cu-SC-SMA to investigate the possibilities of an increase of the transformation capacities in torsion of this SMA, which can be favorable in the endodontic domain.

Figure 1 :Figure 2 :

 12 Figure 1: Parametrized geometry of a studied rotary endodontic file.

Figure 3 :

 3 Figure 3: Applied boundary conditions for bending and torsion.

Figure 4 :

 4 Figure 4: Definition of Habit plane and twinning direction where n, m and g are the normal of the habit plane, the twininning direction and the transformation displacement respectively.

  C C C C I I C I C I C I I C I I I I I C C I I I C C C C I I I C I C I C C I I I I I C I I C I I C C C C C I I I C I C I I I I C I C I I I I C I C C C C I C I I I C I C I I C I C I I I I I I C C C C C C I I I I C I C C I I I I C I I C C C C I C I I C I C I I C I I C I I I C C C C I I C I I C I C I I C I I I I C C C C C I I I C C I C I I I I C I I C I C C C C I I I C I I I C C I I I C C C C I I C I I I C I I C I I C C C C I C I I I C I I I I C I C C C C C I I I C I I I I I I C C C C C I C I I C I I I C C C C C I I I I C I I C C C C I I I C I I C I C C C C I I C I I I I C C C C C I

  Figure 5: Presentation of the testing device.

Figure 6 :Figure 7 :

 67 Figure 6: Schematic description of the applied loads on endodontic files.

  are experimentally and numerically tested in bending, torsion and combined bending-torsion. The numerical responses issued from the micromechanical model are comprared to the experimental ones in order to check the ability for the model to corretly predict wire tension response.

Figure 8 :

 8 Figure 8: Uniaxial tension response of Cu-SC-SMA wire of 0.79 mm submitted to 10 cycles of 8.5% strain.

Figure 9 :

 9 Figure 9: Cu-SC-SMA wire response under bending, torsion and combined bending-torsion loadings.

Figure 10 :Figure 11 :

 1011 Figure 10: Experimental and numerical comparison of uniaxial tensile response of Cu-SC-SMA wire of 0.79 mm.

Figure 13 :

 13 Figure 13: Compararison between experimental and numerical responses under bending, torsion and combined bending-torsion loadings of Cu-SC-SMA endodontic file prototypes with a tip diameter of 0.28 mm, 4% taper and two edges cross section.

MoreoverFigure 14 :Figure 15 :

 1415 Figure 14: Stress distribution in an endodontic file prototype with a tip diameter of 0.28 mm, a taper of 4% and two edges cross section made of Cu-SC-SMA submitted to bending, torsion and combined bending-torsion loadings. Clamped area is 3 mm length.

Figure 17 :

 17 Figure 17: Responses in bending and torsion of the 9 finite element model prototypes.

  for the response prediction of endodontic file prototypes in Cu-SC-SMA under combined bending-torsion loading. This parameter set leads to the lowest stiffness in combined bending-torsion. According to figure18, it corresponds to an asymmetric cross-section, with diameter at 3 mm from the tip of 0.4 mm, a taper of 4% taper and 1 mm pitch length. The most influent parameter is the cross-section. The asymmetrical part of the cross section offers the best option acting on the moment-rotation global responses. The taper has the lowest influence on the moment-rotation global response. These results are obtained with the numerical analysis and shown in figure17. Moreover, during canal root preparation, the diameter and the taper of the endodontic file are chosen according to the size of the tooth canal. To go further, experimental investigations should lead to a better prediction of the influence of each geometrical parameter for endodontic files. Numerical simulations consider ideal geometry.

Table 1 :

 1 Values of n and m for Cu-Al-Be single crystal SMA .

	Variant	n 1	n 2	n 3	m 1	m 2	m 3
	1	-0.168 0.688 0.705 -0.121 -0.725 0.678
	2	-0.168 0.705 0.688 -0.121 0.678 -0.725
	3	0.168	0.688 0.705 0.121 -0.725 0.678
	4	0.168	0.705 0.688 0.121	0.678 -0.725
	5	-0.688 -0.168 0.705 0.725 -0.121 0.678
	6	-0.705 -0.168 0.688 -0.678 -0.121 -0.725
	7	-0.688 0.168 0.705 0.725	0.121	0.678
	8	-0.705 0.168 0.688 -0.678 0.121 -0.

Table 2

 2

	: Interaction matrix describing compatibilities (C) and incompatibilities (I) between
	variants.
	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Table 3 :

 3 Material parameters of Cu-SC-SMA wire used with micromechanical model

Table 4 :

 4 Geometrical parameters of the tested prototypes.

	Cross-section Tip diameter	Taper	Pitch
	0.28 mm	4 %	1 mm
	25

Table 5 :

 5 Selected factors with their corresponding levels used in the Taguchi design of experiment.

	Factor	Level 1	Level 2	Level 3
	Cross Section	0.5pt
		Asymmetric
	Tip diameter at 3 mm	0.40 mm	0.45 mm	0.50 mm
	Taper	3 %	4 %	5 %
	Pitch length	0.70 mm	0.85 mm	1.00 mm
	and 14 are compatible but they occur for different loading levels in the bended
	area.

Table 6 :

 6 Taguchi orthogonal table with 4 factors at 3 levels.

	Test	Factor 1	Factor 2	Factor 3	Factor 4
		(Cross	(Tip diameter
				(Taper)	(Pitch)
		Section)	at 3 mm)

Table 7 :

 7 Selected factors with their corresponding level used in the Taguchi design of experiment according to table 6.

Acknowledgements

The Cu-SC-SMA wires were supplied by the Nimesis Technology company,

Mécleuves

. The endodontic files were manufactured by the Coltene-Micromega company, Besançon. The DSC tests were carried out by the research team led by Prof S. ARBAB-CHIRANI of the Laboratory IRDL, Brest, France. The authors want to acknowledge their help for this study. This work was undertaken in the framework of the Phd project of Paul XOLIN, co-advised by Prof. Tarak BEN ZINEB and Prof. Marc ENGELS-DEUTSCH and funded by the French agency ANRT "Association Nationale Recherche Technologie".