## 1 Introduction  
In modern endodontics NiTi rotary instruments are used universally for root canal shaping. Compared to conventional stainless-steel files, they are more elastic and have a higher cutting quality. In addition, they allow us to achieve the tapered root canal shape with less canal transportation because of their super-elasticity [1]. [PERSON] et al. first introduced NiTi root canal files to solve the issue of the rigidity of stainless-steel instruments and thus, to improve the preparation of curved root canals [2]. However, unexpected fracture of NiTi rotary instruments during their use (i.e., inside the tooth) remains a current problem and has a negative influence on the treatment outcome [3; 4; 5; 6; 7]. Three different ways have been described in the literature regarding the causes for which NiTi instruments tend to fracture: due to torsion, to flexural fatigue [8; 9; 10], or to a combined mechanism of flexural and torsional stress [11]. Flexural fatigue appears to be more common in curved root canals, while torsional failure can occur in a straight root canal [12; 13; 14]. An ideal mechanical preparation of the endodontic space has a continuous tapered shape along the entire length of the root canal, from the coronary level to the apical third, respecting the anatomical shape and the multi-planar curves of the canal, while mantaining the size of the foramen as small as possible [15]. This desideratum is still a major challenge in current clinical practice, in terms of narrow and curved root canals [16]. When the tip or another component of the instrument gets stuck in the root canal as the shank continues to rotate, an overload torsional fracture occurs [8]. If the torque generated by the handpiece reaches the elastic limit of the metal, fracturing of the tip becomes unavoidable [17]. Instruments fractured due to torsional loads show elements of plastic deformation that occur prior to failure [8]. These issues have led to the introduction of M-wire alloys as revolutionary systems that have higher resistance to cyclic fatigue compared to conventional NiTi instruments [18; 19]. Thus, in 2007 a patented thermomechanical manufacturing process was created by (Sportswire LLC, Langley, OK, USA), aimed to produce a more flexible NiTi alloy with improved cyclic fatigue resistance [20]. This resistance is linked to several characteristics of endodontic files, including diameter, flexibility of the file, metal density, regressive surface area, cross-sectional shape, and presence of transitional zones in these files [21]. How to chose between the various types of NiTi endodontic instruments on the market today, each with its own cross-sectional shape that behaves differently on stress distribution and fracture tendency? This is a question at which this study aims to contribute for an answer. Thus, four cross-sectional shapes can be found: convex triangle, triangle, rectangular, and parallelogram [22]. They are important (as studied in this work, as well), because it has been shown for example that stress accumulation under bending conditions of rotary files during preparation of curved root canals can be minimized by using designed rotary NiTi instruments with rectangle centered cross-section configuration, a low pitch, and reduced taper [23]. Studies regarding torsional stress and cyclic fatigue have been performed on many types of rotary instruments, but especially on those made from M-wire alloys, as they are part of the most used systems in current endodontic practice. Two such types of instruments, made from an M-wire alloy are considered in this study: Reciproc R25 (VDW, Munich, Germany) and Protaper Next X2 (Dentsply Maillefer, Ballaigues, Switzerland). Their performances are compared with the Endostar E3 instrument (Endostar, Poldent Co. Ltd., Warsaw, Poland), made out of conventional (i.e., untreated) NiTi.