## 3 Illustration of the Proposed Approach: Results of Case Studies
### Fatigue Crack Growth Test
For the material under investigation, the mechanisms related to mechanical twinning and fracture of hard particles can be ruled out due to Inconel's high stacking fault energy. Therefore, the primary energy dissipation mechanisms are supposed to be attributed to the crack advance and the dislocation activity in the plastic zone. One can note that, in line with many previous observations, the average (per cycle) AE power increases exponentially as the crack advances. Even though the observed behaviour of the AE power \(P_{AE}\) presented in Figure 20 as a function of \(\Delta K\) can be approximated by a straight line in log-log scale, the linear fit is applicable only for the low \(\Delta K\) domain. It is interesting that the power relation (25) holds reasonably only up to the critical \(\Delta K\) value corresponding to the LEFM limit, whereas beyond this point, the AE power tends to increase rapidly with the crack advance, which agrees with the development of the excess plasticity ahead of the crack tip. However, the observed behaviour contradicts the predictions by [PERSON] et al. (129) since the \(n_{AE}\) value of 2.2, which is obtained from the least square fit in Figure 20, is far from being equal to \(m+2\) at relatively small \(\Delta K\), as anticipated from the cited work for the AE count rate (129). Thus, even though there seem to be a reasonable agreement between the AE power and the stress intensity factor range indeed, at least within small scale yielding, there is still no consensus about the \(n_{AE}\) index, and there is still a big deal to establish the dependence (if any) of \(n_{AE}\) on the microstructure and testing conditions. Figure 19 unveils another interesting correlation between the AE power, the heat dissipation power, and the strain range. The latter quantity assessed by DIC as the difference between maximum and minimum local strain per cycle serves as a measure of the expended mechanical power in the plastic zone. One can notice that the measured strains appeared to be substantially plastic (see the blue curve in Figure 19, which represents the strain range measured by DIC). Therefore, obtaining the expended mechanical power from DIC is not as straightforward as that in the case of uniaxial tension. Nonetheless, the observed correlation is noteworthy and the expended mechanical power can be obtained in a more elaborated approach. Although the AE power (energy or count rate) has long been recognised to be generally related to the fatigue crack length or the stress intensity factor range, the behaviour of the AE spectral density during fatigue crack propagation has not been explored so far, to the authors best knowledge (that is at sharp contrast to the monotonic tensile testing). In the present work, we illustrate the evolution of the AE PSD function by the behaviour of its median frequency (compare to that in the tensile test, Figure 15). As cycling proceeds and the crack advances, \(f_{m}\) shows a general trend to decrease (Figure 19). This apparent trend opposes that in monotonic tension, where the \(f_{m}\) value increases with strain hardening, as has been discussed above. According to the statistical interpretation of the median frequency as a property reciprocal to the correlation time in the ensemble of emitting defects proposed by [PERSON] et al. (2015) (see also ([PERSON] and [PERSON], 2016)), the increase in the \(f_{m}\) value signifies the increase in randomness of AE sources, while the opposite trend (decreasing \(f_{m}\)) heralds the increase in their temporal correlation or spatial scale. This interpretation appears to be in harmony with the observed behaviour of \(f_{m}\), since during FCG the plastic zone expands progressively at the expense of the undeformed material, resulting in longer and longer crack jumps. Thus, AE is emitted not from the most heavily deformed plastic zone, but primarily from its expanding periphery, thus involving greater and greater undeformed volume into the process of cyclic plastic zone growth.