# 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
### Crack Detection  
As described before, the complex modulus is not a very sensitive parameter for the detection of a crack. After crack initiation, the newly created surfaces touch/compress each other as crack friction, resulting in an additional force, superposing with the material force response, so the force response of the material becomes asymmetric. If the stress signal is decomposed via FT, this leads to even higher harmonics, so the \(I_{2/1}\) intensity increases above the noise level (initial value). A similar behavior was observed for polymers in T/T [23,31]. As reported using logarithmic axes for the \(I_{n/1}\), Figures 6-8 show that the \(I_{2/1}\) intensity is constant (noise level) at the beginning of a test. Then, the \(I_{2/1}\) intensity increases at a specific time (crack initiation) until failure, after \(N\) = 75,000 and 90,000 in Figure 6, after \(N\) = 90,000 and 280,000 cycles in Figure 7 and after \(N\) = 220,000 and 45,000 cycles in Figure 8 respectively, as a larger crack will result in higher crack friction, i.e., an asymmetric force from the newly created crack surfaces. As a consequence, the force response between the strain maximum and strain minimum becomes asymmetric, which can be quantified by the even higher harmonics, in particular the \(I_{2/1}\) intensity. This can be observed for lower strain amplitudes and longer lifetimes, when crack growth is a significant part of the fatigue lifetime, thus crack propagation occurs before the sudden and complete brittle fracture when the material can no longer sustain the stress concentration at the crack tip. The visual analysis of the fracture surface (fractography) shows for the six tests presented in Figures 6-8 a smooth and a rough surface, which are separated by red lines as a guide for the eye. Cracks are generated and grow slowly in the smooth surface area [33], before fast crack growth occurs when the remaining cross-section area can no longer withstand the applied deformation. Therefore, it can be concluded that crack growth occurs, which can be followed in the mechanical stress response by the \(I_{2/1}\) intensity. The decrease in the \(I_{2/1}\) intensity in Figure 6b, instead of the typical \(I_{2/1}\) increase with crack onset, can be explained by a vector compensation in a complex plane of the \(I_{2}\) contribution caused by the noise of the machine and the asymmetry in the force response caused by the crack initiation. When vectors of the noise of the machine and of the material response have a phase shift of 180\({}^{\circ}\), they compensate each other resulting in an \(I_{2/1}\) decrease to a value close to zero (<10\({}^{-4}\)) followed by an intensity increase when the asymmetry of the material response, and consequently its vector in a complex plane, exceeds the noise level. Consequently, FT of the stress can detect cracks as soon as the \(I_{2/1}\) intensity significantly decreases or increases from the noise level. Figure 6: Mechanical parameters (\(1\,E^{*}\)\(1\), \(I_{3/1}\) and \(I_{2/1}\)) for a fatigue test of AL at \(\varepsilon_{0}=0.08\%\) (**a**) and \(\varepsilon_{0}=0.072\%\) (**b**) at \(5\,\mathrm{Hz}\), revealing a change of the \(I_{2/1}\) intensity from the noise level after \(N=75,\!000\) and \(90,\!000\) cycles, respectively. On the right side the fracture surface is presented, using \(40\times\) magnification, revealing a smooth and a rough surface area, correlating with crack initiation and brittle failure, respectively. Compared with the rough area, the smooth area is small matching with the very small \(1\,E^{*}\)\(\downarrow\) change before failure. Figure 7: \(Cont.\)  
Figure 8: Mechanical parameters (\(1E^{*}\)!, \(I_{3/1}\) and \(I_{2/1}\)) for a fatigue test of brass at \(\varepsilon_{0}\) = 0.1% (**a**) and \(\varepsilon_{0}\) = 0.14% (**b**) at 5 Hz, revealing the increase of the \(I_{2/1}\) intensity above the noise level after \(N\) = 220,000 and 45,000 cycles, respectively.