## 2 Materials and Methods
### Energy and Damage Quantification  
During the fatigue process, the energy in the volume of material consisting of a prismatic base cross-section of the specimen and height being the separation between the reference points of the extensometer, within a cycle and between two instants of time \(t_{i}\) and \(t_{i}\)+1 is obtained from Equation (2).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Step.**} & \(\mathbf{F_{max}}\) & \(\mathbf{F_{min}}\) & \(\mathbf{\sigma_{max}}\) & \(\mathbf{\sigma_{min}}\) & \(\mathbf{\Delta\sigma}\) \\  & **(N)** & **(N)** & **(MPa)** & **(MPa)** & **(MPa)** \\ \hline
1 & 1200 & 120 & 30 & 3 & 27 \\ \hline
2 & 1300 & 130 & 32.5 & 3.25 & 29.25 \\ \hline
3 & 1400 & 140 & 35 & 3.5 & 31.5 \\ \hline
4 & 1500 & 150 & 37.5 & 3.75 & 33.75 \\ \hline
5 & 1600 & 160 & 40 & 4 & 36 \\ \hline
6 & 1700 & 170 & 42.5 & 4.25 & 38.25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Load values during the Locati test steps.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Stage.**} & \begin{tabular}{c} **Break?** \\ **(Yes/No)** \\ \end{tabular} & \
\begin{tabular}{c} **Cycles/Step** \\ **(Cycles)** \\ \end{tabular} &
\begin{tabular}{c} **\(\mathbf{\Delta F_{max}}\)** \\ **(N)** \\ \end{tabular} \\ \hline Locati 1 & No & 25,000 & 100 \\ \hline Locati 2 & Yes & 25,000 & 100 \\ \hline Locati 3 & No & 25,000 & 50 \\ \hline Locati 4 & No & 50,000 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Locati test variants.  
Figure 2: Descriptive loading scheme of the Locati fatigue test.  
\[E_{t_{1}/t_{i+1}}=\int_{t_{i}}^{t_{i+1}}dE=\int_{t_{i}}^{t_{i+1}}F\cdot d\Delta \tag{2}\]  
For a whole cycle, this total energy can be decomposed into: dissipated energy represented by the hysteresis loop (E\({}_{\text{d}}\)) and accumulated energy (E\({}_{\text{a}}\)), which is recovered by the element at the end of the cycle, as shown in Figure 3.  
Therefore, as shown in Equation (3), the total energy per cycle, \(E_{T}\), is the total of the two previous ones. This \(E_{T}\) will be the energy used to determine the damage generated in the fatigue process.  
\[E_{T}=E_{a}+E_{d} \tag{3}\]  
A parameter called total damage, \(T_{D}\), is defined to quantify the damage suffered by the material during the fatigue process. The definition of this parameter will be made from the variation in energy parameters with respect to their initial value and will be obtained from the expression:  
\[T_{D}=\left(1-\frac{E_{T0}}{E_{T1}}\right)\cdot 100\ [\%] \tag{4}\]  
where \(E_{T0}\) will be the total energy measured in the first of the fatigue cycles and \(E_{T1}\) the total energy measured in cycle \(i\).  
Since the stress level throughout the Locati test is changing, a new parameter is defined, which will be the individual net damage (\(D_{I}\)) and will consider the variation in the total energy with respect to the initial one of each of the steps. See Figure 4.  
\[D_{NI_{j}}=D_{I_{j}}-\left(D_{0_{j}}-D_{NIf_{j-1}}\right) \tag{5}\]  
where \(D_{NIf_{j-1}}\) is the individual net damage value of the previous step.  
Figure 3: Descriptive diagram of the energy dissipated and accumulated during a dynamic load cycle.