## 3 Results
### Quantitative Evaluation of the Hysteretic Quadratic Nonlinearity of Aluminum Alloy  
Theoretical predictions and experimental observations for acoustic pulse propagation in the media with hysteretic quadratic nonlinearity indicate that the end-time of the first (compressional) phase of the particle velocity profile in an acoustic pulse exhibits more pronounced variations as a function of both pulse amplitude and propagation distance [57; 58] than the time of the particle velocity peak arrival. This fact could be appreciated by the examination of the acoustic pulse profile transformation depicted in Figures 5, 7 and 8 in Reference [57] and Figure 4 in Reference [58]. That is why, for the determination of the hysteretic parameter of specimens, we measured variations in the end-time of the detected leading phase of the particle velocity profile as a function of detected peak velocity amplitude. The results of these measurements for all samples subjected to different heat treatment durations are presented in Figure 5. Note that not only the differences in hysteretic acoustic nonlinearity of differently aged samples but also the differences in sample thicknesses contribute to the variations of the arrival times in Figure 5. However, independently of the sample's thickness, the results in Figure 5 explicitly demonstrate the tendency that in all characterized samples the growth of the amplitude of acoustic pulses results in a delayed end-time of the leading phase. Figure 5: Variations in the end-time of the leading phase of the particle velocity profile in detected acoustic pulses as a function of the detected amplitude of this phase. Straight red lines provide the best linear fits for experimental points. In theory [57] the local shift in end-time of the leading phase, \(d\tau_{0}\), is proportional to the parameter of the hysteretic quadratic nonlinearity, \(h\), to the local amplitude of the leading phase of propagating acoustic pulse, \(V(x)\), and to the increase in propagation distance of the acoustic pulse, \(dx\) (see, as an example, Equation (31) in Reference [57]), as follows: \(d\tau_{0}(x)=\left[h/\left(2c_{L}^{2}\right)\right]V(x)dx\). Here \(c_{L}\) denotes linear longitudinal acoustic velocity in the medium. In the case of weak manifestations of the hysteretic quadratic nonlinearity, as it is observed in Figure 5, the integration of the above relation along the propagation distance of an acoustic pulse leads in the first leading approximation to the following: \(\tau_{0}(H)=\left[h/\left(2c_{L}^{2}\right)\right]V(H)H\), where \(H\) denotes the thickness of the aluminum alloy plate in our experiments. Consequently, the parameter of the hysteretic quadratic nonlinearity of the material can be found in our experiments by evaluating the slope, in linear dependence, of the measured end-time of the leading phase on the measured amplitude, \(V_{0}(H)\), of the following phase:  
\[h=(4c_{L}^{2}/H)[\tau_{0}(H)/V_{0}(H)] \tag{1}\]  
In deriving Equation (1) we took into account that the particle velocity at the mechanically free surface of our samples is twice larger than in propagating acoustic wave, \(V(H)=V_{0}(H)/2\). To evaluate the hysteretic parameter in the frame of Equation (1) we used sound velocities documented for our samples in Reference [56], our measurements of the aluminum alloy plate thickness, and the slopes, \(\tau_{0}(H)/V_{0}(H)\), of the best linear fit to experimental data presented in Figure 5. The evaluated dependence of the hysteretic quadratic nonlinearity on the aging time of the aluminum alloy is presented in Figure 6.  
Experimental results presented in Figure 6 indicate positive values of the hysteretic parameter, \(h\), as it is expected theoretically [63; 64; 65] for all possible materials. They reveal values of the parameter of hysteretic quadratic nonlinearity, which are at least an order of magnitude larger than the values of the parameter of elastic quadratic nonlinearity in homogeneous solids [66].