# Fourier Transform (FT) Analysis of the Stress as a Tool to Follow the Fatigue Behavior of Metals
## 8 Conclusions  
The fatigue behavior of three metallic materials: cold rolled steel (CRS), aluminum (AL) and brass, was investigated via Fourier transform (FT) analysis of the force (stress) in displacement-controlled tests. Besides the fundamental harmonic at the frequency of the deformation input, higher harmonics at even and odd multiples of the fundamental harmonic were observed. * For the undamaged specimen under large strain amplitudes, beyond the linear regime in a tensile test, the even higher harmonics were found to be within the noise of the machine, while the odd ones quantified the material nonlinearities. * The intensity of the odd higher harmonics, in particular the first odd higher harmonic (\(I_{3/1}\)), was found to scale quadratically with deformation (\(\epsilon_{0}\)), with intensities between \(10^{-3}\) to \(10^{-2}\). * During a fatigue test, the \(I_{3/1}\) intensity was found to linearly increase with the cycle number (regime II) after a transient regime at the beginning of the test (regime I), before increasing rapidly at failure (regime III). * Using the trend of the \(I_{3/1}\) intensity, the fatigue lifetime can be calculated from the constant \(I_{3/1}\) slope (\(dl_{3/1}\)/\(dN\)) and the cumulative nonlinearity (\(Q_{f}\)). The comparison between the power-law correlations for \(N_{f}\) (strain-life curves) revealed lower standard deviations, thus better fatigue predictions using \(I_{3/1}\). * The \(I_{2/1}\) intensity is within the noise level at the beginning, but increased when crack initiation and propagation occurred as cracks create an asymmetry in the stress \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test \#** & \(N_{f}\)**[Cycle]** & \(N_{f}\)**Prediction from \(Q_{f}\)** \\  & \(dl_{3/1}\)/dN [Cycle]** & **[Cycle]** \\ \hline
1 & 42,190 & 37,858 (\(-11\%\)) & 43,651 (4\%) \\
2 & 33,473 & 31,388 (\(-6\%\)) & 24,095 (\(-28\%\)) \\
3 & 25,179 & 22,189 (\(-12\%\)) & 24,665 (\(-2\%\)) \\
4 & 23,993 & 29,092 (\(21\%\)) & 22,735 (\(-5\%\)) \\ Average +/ – one standard deviation & 31,208 \(\pm\) 8448 & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The experimental results for four tests of AL at \(\epsilon_{0}\) = 0.14%, \(\omega_{1}\)/2\(\pi\) = 5 Hz and \(R\) = 0.1, as well as the fatigue life predictions via the nonlinear parameters with their respective deviation in parenthesis. response due to a force reduction at the strain maximum and crack friction at the minimum. * The \(I_{2/1}\) behavior was shown to be a powerful tool to detect and follow crack initiation and growth as it can quantify any asymmetry in the stress response, while \(I_{3/1}\) allowed to quantify the mechanical nonlinearity and its evolution due to fatigue, which can be used to predict failure. Finally, the concept of FT analysis of the stress during fatigue testing was shown to be a powerful tool offering insights into the fatigue process from the time evolution of the mechanical stress response itself (before failure) and allowed better prediction (sensitivity and precision) than standard strain-life curves, while being able to detect cracks with high sensitivity. For future work, the concepts developed here for other materials will be done, in particular for concrete due its wide use. Also, other testing conditions such as load controlled testing or random amplitude/frequency testing would be of high interest. Conceptualization, V.H. and D.R. ; methodology, V.H. ; software, V.H. ; validation, V.H. and D.R. ; formal analysis, V.H. ; investigation, V.H. ; resources, D.R. ; data curation, V.H. ; writing--original draft preparation, V.H. ; writing--review and editing, D.R. ; visualization, V.H. ; supervision, D.R. ; project administration, D.R. ; funding acquisition, D.R. All authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Science and Engineering Research Council of Canada, grant number RGPIN-2016-05958. Not applicable. The data are available upon request.