## 4 Discussion  
It is well established theoretically and supported experimentally that typical integer-power-law elastic nonlinearities, i.e., for example quadratic and cubic, cannot induce the temporal shifts  
Figure 6: The dependence of the parameter of hysteretic quadratic nonlinearity on the time of thermal aging of aluminum alloy at 220 \({}^{\circ}\)C. Vertical bars provide a 70% confidence interval for red straight-line fitting in Figure 5.  
of zero points in the acoustic wave velocity profile unless a weak shock front is formed [67; 68]. The experimentally observed shift in the end-point of the leading phase of a longitudinal acoustic pulse is a fingerprint of the hysteretic quadratic nonlinearity, similar to others, such as the linear shift with the vibration amplitude of the resonance frequency of micro-inhomogeneous bars [63; 69] and the generation of the third harmonic with the amplitude proportional to the square of fundamental wave amplitude in some experiments [61]. First, we evaluated the characteristic length scale for a substantial influence of the hysteretic quadratic nonlinearity on the propagation of longitudinal acoustic pulses in our experiments. The so-called nonlinear length, \(x_{NL}\), is defined [57; 58; 65] for the hysteretic quadratic nonlinearity by the same combination of material parameters, as in the case of the elastic quadratic nonlinearity [65; 67], with a replacement of the parameter of the elastic nonlinearity by the parameter of the hysteretic nonlinearity, as follows:  
\[x_{NL}=2c_{L}^{2}\tau/(hV) \tag{2}\]  
where \(\tau\) and \(V\) are the characteristic duration of acoustic transient and the characteristic particle velocity, respectively. Substituting in Equation (2) a half of the maximal particle velocity (\(V\approx 2.5\) m/s) from Figure 5, the characteristic duration of the leading phase (\(\tau\approx 55\) ns) from Figure 5, the maximal measured hysteretic parameter (\(h\approx 300\)) from Figure 6 and the acoustic wave velocity \(c_{L}\approx 6,400\) m/s from Reference [56], we estimate \(x_{NL}\approx 6\ mm\approx 2H\) in our experiments. Thus, Equation (1) can be considered a good approximation for the evaluation of \(h\) and our theoretical approach for the estimation of \(h\) from experimental data is self-consistent. Second, in Figure 7, we compare the dependencies on the duration of aluminum alloy thermal aging of the parameter of the hysteretic quadratic nonlinearity and of the yield strength. The latter was measured for the same samples in Reference [56]. The comparison reveals that aging times for two local maxima of the hysteretic parameter correlate with the position of the minima and the maxima in the variations of yield strength as a function of the thermal aging time of the aluminum alloy (Al6061). A similar correlation was revealed earlier between the maxima of the elastic quadratic nonlinearity in longitudinal waves and the Vickers hardness of another aluminum alloy, Al2024, in Reference [70]. In both cases one of the nonlinearity maxima coincides with the maximum of the yield strength (hardness), while the preceding local minimum in the nonlinearity is achieved at an aging time located between the times of minimum and maximum yield strength/hardness (compare Figure 7 above and Figure 1 in Reference [70]). The fact of the correlation between the local maximum in the parameter of elastic quadratic nonlinear with the absolute maximum in the Vickers hardness in their dependences on thermal aging time was also documented for the steel alloy [23]. There exist theoretical developments [29; 70], based on the analysis of alloy precipitation phases and the effect  
Figure 7: Dependence on the heat-treatment time of the measured parameter of hysteretic quadratic nonlinearity, \(h\), and of the yield strength (adopted from Reference [56]). of precipitate coherency strains on the acoustic harmonic generation, which provide insight in experimentally observed variations of the quadratic elastic nonlinearity with precipitate heat treatment time.