# Fourier Transform (FT) Analysis of the Stress as a Tool to Follow the Fatigue Behavior of Metals
## 1 Introduction
### Nonlinear Material Behavior during Fatigue
Consequently, the modulus becomes a function of the applied deformation as \(E^{*}=E(\epsilon^{*})\) and a Taylor series expansion of the strain dependent modulus can be developed containing both even and odd powers of the strain as [18]:  
\[E^{*}(\epsilon)=E_{0}+E_{1}\;\epsilon^{*1}+E_{2}\;\epsilon^{*2}+E_{3}\;\epsilon^{* 3}+\ldots \tag{4}\]  
The stress response of the simplest nonlinear 1D scalar model for an elastic solid body (spring) can be described by [PERSON]'s law as:  
\[\sigma^{*}(t)=E^{*}(\epsilon^{*})\epsilon^{*}(t) \tag{5}\]  
Inserting Equation (3) into Equation (5) and simplifying the notation gives:  
\[\sigma^{*}(t)=E_{0}\;\epsilon_{s}+(E_{0}+\epsilon_{s}E_{1})\epsilon_{0}\;e^{ i\omega_{1}t}\;+(E_{1}+\epsilon_{s}E_{2})\epsilon_{0}^{2}\;e^{2i\omega_{1}t}\;+(E_{2}+ \epsilon_{s}E_{3})\epsilon_{0}^{3}\;e^{3i\omega_{1}t}+\ldots \tag{6}\]  
In general, for viscoelastic materials, Equation (6) can be rewritten including phase angles to show that an asymmetric nonlinear stress response includes a constant value and a fundamental harmonic, as well as even and odd higher harmonics as [12; 18; 19]:  
\[\sigma^{*}(t)=\sum_{n=0}^{\infty}I_{n}\;e^{i(n\omega t+\delta_{n})} \tag{7}\]  
A nonlinear, but symmetric, stress response occurs if the stress response is independent of the deformation direction; i.e., it is similar during loading and unloading: \(\sigma^{*}(t)(\epsilon+)\)\(-\)\(\sigma_{man}=-(\sigma^{*}(t)(\epsilon-)\)\(-\)\(\sigma_{man})\). In this case, the Taylor series of the stress only contains odd higher harmonics due to geometrical symmetry arguments, underlining the different geometric meanings of the even and odd higher harmonics [12]. This is why both contributions need to be analyzed separately to understand the physical origin of mechanical nonlinearity. The magnitude of the generated higher harmonics (\(I_{n}\), \(n>1\)) are typically normalized by the fundamental (\(I_{1}\)) as \(I_{n}/I_{1}=I_{n/1}\). For example, the relative third harmonic intensity is presented as \(I_{3/1}\). Equation (6) shows that the normalized intensities of the higher harmonics should scale as power-law functions of \(\epsilon_{0}\) with exponents of \(n-1\)[20]. For example, \(I_{2/1}\) and \(I_{3/1}\) should scale linearly and quadratically with respect to \(\epsilon_{0}\). This was experimentally observed and mathematically derived for several materials [20; 21; 22; 23]. To quantify the materials nonlinearity while being independent of the measurement conditions, and to obtain a nonlinear material parameter, the proportionality factor between \(I_{3/1}\) and \(\epsilon_{0}\) is defined as the \(Q\) parameter [20; 22]:  
\[Q=\frac{I_{3/1}}{\epsilon_{0}^{2}} \tag{8}\]  
In a strain controlled fatigue test, the applied strain amplitude (\(\epsilon_{0}\)), as well as the angular frequency (\(\omega_{1}\)), are constant, while the stress response changes due to ongoing fatigue [24]. On the one hand, the stress amplitude decreases and its nonlinearity (deviation from a sinusoidal waveform with the fundamental frequency \(\omega_{1}\)) increases [25]. Both phenomena result in decreasing \(E^{\prime}\) and \(E^{\prime\prime}\) combined with increasing mechanical nonlinearity quantified via the \(I_{3/1}\) intensity. Recently, a framework was developed to analyze, quantify and predict the fatigue behavior via Fourier transform of the stress response by individually analyzing each higher harmonic for brittle polymers under T/T. Three concepts based on the higher harmonics were introduced: (I) the detection of macroscopic cracks via the \(I_{2/1}\) intensity [19], (II) the prediction of fatigue failure via the rate of change of the \(I_{3/1}\) intensity [26], and (III) the cumulative nonlinearity \(Q_{f}\) as a fatigue criterion [27]. With ongoing fatigue, the nonlinear mechanical parameter (\(I_{3/1}\) and \(Q\)) were found to monotonously increase for rectangular notched and unnotched samples [26; 27]. Based on previous results, it was found that during testing \(I_{3/1}\) goes through three different regimes until failure.