# Fourier Transform (FT) Analysis of the Stress as a Tool to Follow the Fatigue Behavior of Metals
## 7 Behavior of the Nonlinear Parameters during Fatigue
### The Rate of Change of I\({}_{3/1}\) (\(Al_{3/1}\)/dN) and the Cumulative Nonlinearity (Q/)  
As seen in Figure 3, the I\({}_{3/1}\) intensity quantifies the mechanical nonlinearity and increases linearly in regime II with ongoing fatigue with the cycle number as shown in Figure 4, so a constant slope can be determined. This slope (\(dI_{3/1}\)/\(dN\)) was found to be correlated with the fatigue life time (\(N\)/) for polymers [26]. The same analysis was performed for the metallic specimens and the results are presented in Figure 9a. It can be seen that a strong correlation exists between both parameters for all three materials.  
The results of Figure 9a can be used to extract a power-law relation as:  
\[N_{f}=A_{I_{3/1}}\left(\frac{dI_{3/1}}{dN}\right)^{-1.3} \tag{13}\]  
with the proportionality factor \(A_{I3/1}\). Here, the exponents are the same (\(-1.3\)) for the three materials and the values of \(A_{I3/1}\) are reported in Table 1.  
The second nonlinear fatigue criterion investigated is the cumulative nonlinearity (Equations (10) and (11)) as reported in Figure 9b. A strong correlation between \(N_{f}\) and \(Q_{f}\) is again observed as:  
\[Q_{f}=A_{Q}\;N_{f} \tag{14}\]  
with a proportionality factor of \(A_{Q}\). Interestingly, the same exponent is obtained for all the materials investigated. CRS has the lowest \(Q_{f}\) for a given fatigue lifetime compared with AL and brass, which also has shorter fatigue lifetimes under the same measurement conditions; i.e., shorter fatigue lifetime corresponds to a higher mechanical nonlinearity, which is quantified by \(A_{Q}\) as reported in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Metal**} & \(A_{\textit{strain-life}}\) & **Standard Deviation** & \(A_{\textit{I3/1}}\) & **Standard Deviation** & \(A_{\textit{Q}}\) & **Standard Deviation** \\  & **[Cycle]** & **[\%]** & **[10\({}^{-6}\)]** & **[\%]** & **[-]** & **[\%]** \\ \hline AL & \(282\pm 86\) & 31 & 49 \(\pm\) 9 & 19 & 2.63 \(\pm\) 0.49 & **19** \\ CRS & \(3632\pm 1661\) & 46 & 1.4 \(\pm\) 0.3 & 22 & 3.99 \(\pm\) 0.42 & **11** \\ Brass & \(2237\pm 636\) & 28 & 71 \(\pm\) 8 & 11 & 0.96 \(\pm\) 0.09 & **9** \\ \hline \hline \end{tabular}
\end{table}
Table 1: The proportionality factors (\(A_{\textit{strain-life}}\), \(A_{\textit{I3/1}}\) and \(A_{Q}\)) for the materials investigated, as well as their standard deviations as absolute and relative values for easier comparison. The lowest standard deviations are marked in bold.  
Figure 9: **(a)** The number of cycles to failure (\(N\)/) as a function of the rate of change of \(I_{3/1}\) (\(dI_{3/1}\)/\(dN\)) in regime II for CRS (black square), AL (red circle) and brass (blue triangle), as well as their regression showing the same slope for all materials. **(b)** The correlation between the fatigue lifetime and the cumulative nonlinearity (\(Q_{f}\)) also showing a high correlation with similar slope for the materials investigated.  
Comparison between the Nonlinear Parameters (\(dl_{3/1}\)/\(dN\) and \(Q_{f}\)) with the Strain-Life Curves--Standard Deviations Analysis  
The prefactor \(A_{strain-life}\), \(A_{13/1}\) and \(A_{Q}\), as well as their standard deviations are listed and compared in Table 1. The lowest standard deviation (the best prediction) is marked in bold. While the standard deviation of the strain-life curve is in the range of 30-50%, the ones of the nonlinear criteria are only around 20%, revealing the lowest standard deviation for the two nonlinear criteria \(dl_{3/1}\)/\(dN\) and \(Q_{f}\), with an even lower value for the \(Q_{f}\) vs. \(N_{f}\) correlation. Similar results were observed for polymers in our recent studies [31].