# Energy Concepts and Critical Plane for Fatigue Assessment of Ti-6 Al-4V Notched Specimens
## 3 Criterion Validation  
In order to check the accuracy of the criterion presented in Section 2, some experimental data are selected from the technical literature [27; 33]. Such data are related to uniaxial and multiaxial fatigue tests (with nominal loading ratio equal to \(-1\)) carried out on circumferentially V-notched cylindrical specimens characterized by (Figure 3):  
* V-notch with a depth of 6 mm;
* Opening angle of 90\({}^{\circ}\);
* Notch root radius of 0.1 mm. Figure 3: Geometrical sizes of the titanium alloy V-notched specimens subjected to tension and/or torsion fatigue loading. The above specimens were made of grade 5 titanium alloy (Ti-6 Al-4V), commonly used in aerospace and naval applications. The above titanium alloy is characterized by very good static and fatigue properties (Table 1) with a high strength-to-mass ratio. The Manson-Coffin parameters of both tensile and torsional equations are reported in Reference [22]. By taking full advantage of \(v_{e}\), \(\Delta K_{1A}\), \(\Delta K_{3A}\), \(\Delta\sigma_{1A}\) and \(\Delta\tau_{3A}\), the control volume radii \(R_{1}\) and \(R_{3}\) are equal to 0.051 mm and 0.837 mm, respectively. Before being fatigue tested, the specimens have been polished in order to remove surface scratches and marks due to machine tools. Fatigue tests have been performed by means of an MTS 809 servo-hydraulic axial-torsional testing system with a 100 kN axial cell and a 1100 Nm torsion cell. Moreover, all tests have been carried out under load control, with a frequency between 10 and 15 Hz. Details of the loading conditions related to the experimental fatigue tests being examined are reported in Reference [27; 33]. In particular, we consider four different loading conditions characterized by experimental fatigue life, \(N_{f,\text{exp}}\), between \(10^{3}\) and \(6\cdot 10^{5}\) loading cycles, and more precisely:  
1. Pure tension fatigue loading;
2. Pure torsion fatigue loading;
3. Combined in-phase (\(\Phi=0^{\circ}\)) tension and torsion fatigue loading;
4. Combined out-of-phase (\(\Phi=90^{\circ}\)) tension and torsion fatigue loading. The biaxiality ratio \(\lambda\) related to multiaxial loading conditions is equal to 2. According to the above loading conditions, the value of the distance \(r\) (Equation (1)) turns out to be:  
1. \(r=1.9\cdot R_{m}\) for pure tension fatigue loading (\(\lambda=0\));
2. \(r=11.3\cdot R_{m}\) for pure torsion fatigue loading (\(\lambda=\infty\));
3. \(r=10.8\cdot R_{m}\) for combined tension and torsion fatigue loading (\(\lambda=2.0\)). Figure 4 shows experimental fatigue life, \(N_{f,\text{exp}}\), plotted against the theoretical one, \(N_{f}\). In particular, 79% of results is conservative and 63% is included into 3\(\times\) band. Moreover, we can remark that better estimations are obtained by considering only the in-phase multiaxial fatigue data (Figure 4b) since all the results fall within 3\(\times\) band, whereas almost all the results related to out-of-phase data are outside the above band (Figure 4c). In any case, when estimations do not fall within the reference bands, the errors made by the present criterion are, in general, on the conservative side. This strongly supports the idea that the [PERSON] et al. criterion, applied together with the control volume concept, can be used successfully to assess notched components in situations of practical interest, always allowing an adequate margin of safety to be reached. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(E\) & \(\
u_{e}\) & \(\sigma_{u}\) & \(\sigma_{y}\) & \(\Delta K_{1A}\) & \(\Delta K_{3A}\) & \(\Delta\sigma_{1A}\) & \(\Delta\tau_{3A}\) \\
**[GPa]** & **[-]** & **[MPa]** & **[MPa]** & **[MPa\(\cdot\)mm\({}^{0.445}\)]** & **[MPa\(\cdot\)mm\({}^{0.333}\)]** & **[MPa]** & **[MPa]** \\ \hline
**110.0** & **0.3** & **978.0** & **894.0** & **452.0** & **1216.0** & **950.0** & **776.0** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Static and fatigue properties of material examined [27; 33].