## 1 Introduction  
The superelasticity of nickel-titanium (Ni-Ti) shape memory alloys is considered the reason for the positive mechanical behavior (consisting of a nonlinear elastic behavior) that allows the material to undergo high deformation without plastic residual-strain after stress removal, during transforming from the austenite to martensitic phase [1]. The NiTi is only composed of two elements: titanium and nickel, however, the physical and mechanical properties of NiTi alloys are significantly different from pure titanium and nickel base alloys [2]. In dental practice, the phenomenon of NiTi endodontic instruments fracturing as a consequence of torsional overloading or cyclic fatigue is continually reported [3; 4; 5]. This usually happens when the tip of the instrument binds to the canal while the shaft continues to rotate [6]. On the other hand, flexural and fatigue failure occurs when the instrument rotates in the curved canal, before undergoing a great number of tensile-compressive strain cycles in the zone with maximum curvature. Instrument durability applies an important role during the management of endodontic compromised teeth to avoid mechanical complications during dental treatment [7]. In this sense, bending and torsion are considered critical conditions that must be investigated in order to understand the mechanical behavior of the endodontic instruments. Pitch length, taper and conicity, alloy chemical composition and thermomechanical processing are also main factors that influence the mechanical response of the alloy [8]. A close correlation has also been demonstrated between the cross-sectional shape and the stress distribution pattern of an endodontic instrument subjected to bending or torsion [9]. Taking into account the cross-sectional geometry, the bending orientation may have an important influence on the mechanical responses of the instrument. Recently, thermomechanical treatment of NiTi alloys allowed for the design of a new generation of endodontic instruments characterized by a change in the phase composition, with the appearance of the martensite and/or R phase under clinical conditions [9; 10]. This allowed for the introduction of new heat-treated NiTi endodontic instruments with different mechanical behaviors in comparison to the conventional ones [10; 11]. Several laboratory tests were performed to study the flexibility and torsional stiffness of NiTi instruments during the conventional kinematics of rotary instrumentation [12]. Theoretical studies based on mathematical and numerical analyses have also been carried out in dentistry, by means of a finite element analysis (FEA), to calculate new mechanical and structural aspects of the behaviors of materials in endodontics, restorative or implant dentistry [13; 14; 15; 16; 17; 18; 19]. This tool can efficiently illustrate the stress distribution pattern through an analysis of finite small elements, the behavior of which can be summarized and calculated [20]. Moreover, FEA can easily provide the stress magnitude and distribution, as well as the influence of various materials which cannot be investigated by traditional in vitro tests [21]. The simulation of kinematics (rotation) occurs differently from traditional in vitro tests, such as in cyclic fatigue resistance tests [22; 23; 24; 25; 26], Weibull analyses [27], and metallurgical and micromorphological analyses [22; 23; 28]. In all these studies, the friction of NiTi alloys is performed with devices simulating an artificial metallic canal (with a higher elastic modulus and hardness than in human dentine) with different curvature radii [22; 23; 24; 25; 26]. Currently, there is no standardized methodology to analyze the cyclic and torsional aspects involved in the fatigue test of NiTi instruments [29]. FEA has been successfully used in endodontics, revealing that it is an efficient tool to investigate, for example, the instrument mechanical responses in root canal simulations.