## 3 Illustration of the Proposed Approach: Results of Case Studies
### Fatigue Crack Growth Test  
In the fracture mechanics approach to the stable fatigue crack propagation, in the vast majority of structural materials, the crack length \(a\) increases under cyclic load according to the empiric Paris-Erdogan equation  
\[\frac{da}{dN}=C\Delta K^{m} \tag{23}\]  
where \(\frac{da}{dN}\) represents the crack growth rate (the crack length increment per cycle), and \(C\) and \(m\) are experimentally determined material properties known as Paris coefficient and Paris exponent, respectively (both dependent also on the environment, temperature and loading conditions). Figure 16 shows the Paris plots for the crack growth rate \(\frac{da}{dN}\) vs. the stress intensity factor range \(\Delta K\), which are obtained independently from the IRT (a) or RVI (b) data used to find the crack tip position and estimate the crack length. The vertical lines indicate the validity of the linear elastic fracture mechanics (LEFM) and the \(\Delta K\) concept according to the ASTM standard [90], which is verified by the criterion:  
\[(W-a)\geq\frac{4}{\pi}\bigg{(}\frac{K_{max}}{\sigma_{\text{sys}}}\bigg{)}^{2} \tag{24}\]  
where \(W\) is the effective width of the specimen, \(a\) is crack length, \(K_{max}\) is the highest stress intensity factor occurring within a cycle, and \(\sigma_{\text{sys}}\) is the materials yield stress. The stress intensity factor needs to be calculated within the small scale yielding (SSY) approximation of an elastic-plastic body with a Mode I crack in plane strain. In Figure 17, the results from both techniques are compared. Not only an excellent agreement between both methods is seen, but also the scatter of experimental data for \(\frac{da}{dN}\) for each technique is reasonably  
Figure 15: Summary of results of the tensile test, representing the behaviour of the expended and stored mechanical power—\(p_{M}\) and \(p_{S}\), respectively, dissipated heat \(p_{D}\), and the strain hardening rate \(\theta\), synchronised with the stress-strain curve and the key descriptive AE variables—power \(P_{AE}\) and median frequency of the PSD \(f_{m}\). small. Both methods yield practically the same Paris parameters \(C\) and \(m\) of \(3.12\pm 0.03\). Additionally, for both methods the slight deviation from the LEFM prediction according to the Paris' law becomes visible precisely at \(\Delta K\) expected from the SSY criterion of \(62\,\mathrm{MPa}\sqrt{\mathrm{m}}\) corresponding to the crack length \(a\) of \(24.8\,\mathrm{mm}\). The reversible thermoelastic effect and the irreversible heat dissipation due to plastic deformation in the cyclic plastic zone are the two main contributors to the heat source function \(\dot{q}\) in Equation (10). Since the thermoelastic effect assumes the linearity between the stress and temperature increment, one can make use of the cyclic loading to eliminate this effect from the analysis by performing the integration of the \(\dot{q}\) over each loading cycle. An example of the heat dissipation behaviour per cycle is illustrated in Figure 18, where the heat power density \(\dot{q}\) at the crack tip is plotted versus the applied load. An example of the map of the energy dissipated during one cycle \(w_{D}\) is shown in Figure 18c, highlighting the cyclic plastic zone ahead of the crack tip. Figure 16: Crack growth curves and their linear fit for IRT (**a**) and DIC data (**b**). The vertical line represents the calculated linear elastic fracture mechanics limit according to the small scale yielding criterion. Figure 17: Paris plot obtained from IRT and DIC measurements. Both datasets are plotted for comparison. The vertical line represents the LEFM limit according to the small yielding criterion. The plot compiling the main findings from all three in situ methods is represented in Figure 19, where the average (per cycle) AE energy and median frequency are plotted together with the dissipated heat energy at the crack tip as a function of the crack length, which serves as a damage parameter for FCG.