## 3 Illustration of the Proposed Approach: Results of Case Studies
### Tensile Test  
Figure 13 shows the true stress-strain curve and the expended mechanical power density \(p_{M}\) (estimated by using DIC mapping and averaged over the entire specimen gauge part) and its variance as a function of strain. The strain hardening rate \(\theta(\varepsilon_{T})=\frac{d\sigma_{T}}{d\varepsilon_{T}}\) was calculated and its fragment is shown in the figure to highlight the crossover with the \(\sigma_{T}(\varepsilon_{T})\) indicating the onset of necking according to the Considere criterion (cf. [120; 121; 122] for other forms of instability criteria based on the evolution of the total dislocation density)  
\[\frac{d\sigma_{T}}{d\varepsilon_{T}}=\sigma_{T} \tag{22}\]  
The alloy demonstrates an excellent hardenability resulting in the true strength exceeding 1.3 GPa and uniform strain up to 0.35 before the Considere condition is met. One can notice that the mechanical power exhibits interesting behaviour, the details of which unveil several deformation stages, as will be discussed in what follows. The distribution of locally measured \(p_{M}\) values is of its own interest, and is shown in Figure 14 representing three typical situations marked in Figure 13: (a,b) the onset of plastic yielding when the Luders band forms at the upper edge of the gauge part of the specimen; (c,d) the uniform deformation stage; (e,f) the developed necking stage. Dur  
Figure 13: True stress vs. true strain, strain hardening rate, the mechanical power and its variance evolution during tensile testing on the Inconel-625 alloy. The vertical dashed line indicates the Considère point. The solid vertical lines correspond to the moments of interest (MOI) shown in Figure 14.  
the \(p_{M}\) distribution has a symmetric Gaussian shape, which is reasonably expected for this stage. Not only does the distribution of \(p_{M}\) become significantly skewed towards the left hand-side of the histograms (i.e., the most probable values lie in the low power domain, indicating that most parts of the specimen do not deform plastically) when the strain localisation occurs, but also its variance can serve as a measure of strain localisation. Indeed, Figure 13 illustrates it explicitly--the variance of the \(p_{M}\) distribution increases dramatically when necking sets in under plane stress domination at the necking strain \(\varepsilon_{N}\) = 0.35 (it should be, however, noticed, that the accelerated increase in the \(p_{M}\) variance becomes appreciable even before the macroscopic loss of stability commences according to the mechanistic criterion from Equation (22). Outside of the neck region, the material, however, slightly unloads since the global reaction force decreases in response to the shrinking cross-sectional area within the neck [123], thus resulting in some negative values in the mechanical power distribution shown in Figure 14e. Figure 14: _Cont._  
Figure 15 compiles the results from all of the methods in a single plot: (i) the average expended mechanical power density is plotted in parallel with the dissipated and stored power, and all curves are synchronised with the stress-strain data. The strain hardening rate is also plotted for comparison. The typical evolution of the AE signal in the Inconel-625 alloy is illustrated in terms of the dissipated AE power \(P_{AE}\) and the median frequency \(f_{m}\) of the PSD function synchronised with the tension loading curves and strain hardening rate. Analysis of Figure 15 reveals that, typically for ductile materials, AE commences already at the relatively low applied stress and peak when the Luders band nucleates and propagates. Then, the AE level decreases continually until fracture, but never vanishes. Overall, AE exhibits the behaviour typical of metals and alloys with dislocation-mediated plasticity [38; 124]. While the power (or energy) reduces with strain in response to strain hardening and the concomitant shortening of the dislocation mean free path, the median frequency \(f_{m}\) of the PSD increases steadily for the same reason in parallel with the flow stress.