## 4 Discussion
This probably indicates slight differences in the structure and/or elemental/phase composition of the material in the apex area in comparison to the main body. According to the Hook's law, a higher modulus of elasticity imposes higher mechanical stresses for the same elastic deformation, which corelates well with the occurrence of the instrument rupture points (=distance from the apex) during fatigue tests. In materials science, not only hardness and modulus values but also their ratios are very often used for description of the material response to external loading [55]. For example, in tribological practice, the \(H/E\) (defined as the elastic strain to failure) or \(H^{3}/E^{2}\) ratios (defined as resistance to plastic deformation) are often considered more relevant parameters for description of wear behavior than mere hardness. In addition, the \(H/E\) ratio appears in many equations describing critical stress intensity, Kc, representing or indicating the indentation fracture toughness. Based on these equations, a material's resistance to crack formation and propagation is proportional to \((E/H)^{1/2}\)[56,57]. Another important material property with direct practical relevance is contact yield pressure, or in other words, resistance to plastic deformation, which scales with \(H^{3}/E^{2}\)[58]. It is usually observed that critical load for non-elastic deformation grows with the value of \(H^{3}/E^{2}\) and in turn postpones the fracture initiation during quasi-static indentation tests. On the other hand, damage tolerance was proved to scale reciprocally to \(H^{3}/E^{2}\) (i.e., to \(E^{2}/H^{3}\)) during dynamic impact mechanical testing, as shown by [PERSON] [59]. This clearly reflects the complexity of tribo/mechanical testing and emphasizes the peculiarities of different loading conditions (static, quasistatic, dynamic). This means that material elasticity, as well as ductility, plays an important role in a material's erosive, abrasive and impact wear, and must be considered when overall tribo-mechanical performance is evaluated. Comparison of the performance of individual instruments through the probability of fracture vs. time to failure (Figure 2) and hardness to elastic modulus ratios (Figure 6) implies a relationship. The increased operation time, i.e., longer time to fracture, seems to scale with the decrease of \(H/E\), which predicts higher fracture toughness, as can be seen from a comparison of Unicone and the other instruments (WaveOne Gold, Unicone Plusand Reciproc Blue). This study also shows that \(H^{3}/E^{2}\) values are lowest for the Unicone Plus and Reciproc Blue, with the longest operation time, and highest for Unicone, with the shortest operation time; compare especially the values obtained at the distance of 6.4 mm from the apex of the instrument (Figure 5d). A possible explanation might be based on more effective mechanisms of energy dissipation and strain accommodation. However, it should also be notated that influences other than material-related parameters might be responsible for the best performance, such as the shape and design of the instrument, the cross-sectional area, and/or the surface roughness. A similar correlation between \(H^{3}/E^{2}\) and the results of a dynamic fatigue test was also reported by [PERSON], who reported that resistance to crack propagation during repetitive impact increases with the decrease of \(H^{3}/E^{2}\) in the case of optical coatings [59]. It should be noted that the multi-method approach used in this study follows current trends in endodontics research. Similar multi-method approaches combining cyclic fatigue, torsion resistance, flexibility and cutting efficiency were recently reported by [PERSON] et al. [60]. It is believed that this approach is necessary for the further development of understanding of the mechanical strength and limits of endodontic instruments.