# Propagation of Non-Linear Lamb Waves in Adhesive Joint with Micro-Cracks Distributing Randomly
## 4 Results and Discussions  
Figure 6 shows the displacements contour of the Lamb wave which propagates in the tri-layer adhesive structure with micro-cracks distributing in a random way within the adhesive layer. U\({}_{1}\) and U\({}_{2}\) are the displacements in the \(x\)- and \(z\)- directions, respectively. It is clear that the length of the micro-crack is much shorter than the wavelength of the Lamb wave. In addition, it also shows that there is no big change in the waveform of the Lamb waves after they interact with the micro-cracks when passing through the region with micro-cracks. States of opening and closing can also be observed when the deformed shape is magnified 5000 times for the micro-cracks, as shown in Figure 6a. The Lamb waves collected at position 1 (as shown in Figure 3) in the \(x\)- and \(z\)-directions are shown in Figures 7 and 8, respectively. Note that, plots shown in Figures 7b and 8b are based on the spectrum graph sliced at the fundamental and second harmonic frequency, respectively. In both directions, the S\({}_{0}\) mode was also generated besides S\({}_{1}\) and S\({}_{2}\) modes because the thickness profile of the S\({}_{0}\) mode is a bit similar to that of the S\({}_{2}\) mode at the fundamental primary frequency. The solid line represents the waves for the adhesive layer without cracks, while the dashed line represents those with randomly distributed micro-cracks. As we can see, the time-domain signals for both cases, with and without micro-cracks, are almost overlapped, as shown in Figures 7a and 8a. Magnitudes of fundamental harmonics and second harmonics (collected at position 1) versus the propagation distance  
Figure 5: The average Acoustic Non-linearity ParameterANP versus the average number of simulations based on the displacement collected in the \(x\)-direction and \(z\)-direction using the models. Figure 6: The Lamb waves propagating in the tri-layer adhesive structure. (**a**) The displacement contour in the \(x\)-direction and the 5000-times magnified deformed shapes; (**b**) the displacement contour in the \(z\)-direction. are shown in Figure 9. We can see that magnitudes of fundamental harmonics vary slightly with the propagation distance, while there is accumulative increase of the magnitude of second harmonics with the propagation distance. It means that we can neglect the influence of the micro-crack on the propagation of fundamental Lamb waves. According to the frequency domain plot shown in Figures 7b and 8b, however, the dashed line contains second harmonic waves for time-domain signals in either the \(x\)- or \(z\)-direction when there are micro-cracks within the adhesive layer, which demonstrates that the micro-cracks lead to the generation of second harmonic Lamb waves. The minimum propagation distance in our simulation is 4 mm; we can see that there are second harmonics at this distance. Figure 8: _Cont._  
Figure 7: Lamb waves collected at position 1 in the \(x\)-direction. (**a**) In the time domain; (**b**) the spectrum graph sliced at the fundamental frequency 0.52 MHz and second harmonic frequency 1.04 MHz (\(c=0.002\), \(\mu=0.3\)). Figures 10 and 11 show the ANPs versus the propagation distances of the Lamb wave signals collected at position 1 and position 2, respectively. To validate the numerical simulation, ANPs without randomly distributed micro-cracks are also provided. It is clear that the ANPs in the healthy state are not only much smaller than with randomly distributed micro-cracks, but also change little with the propagation distances. This indicates that it is the randomly distributed micro-cracks that lead to the generation of second harmonic waves. In Figure 3, the length of the damage zone with micro-cracks varies from 4-16 mm. Note that the signals are collected at the end of the micro-crack zone; the length of micro-crack zone, therefore, is the propagation distance of the Lamb waves. As can be seen, the  
Figure 8: Lamb waves collected at position 1 in the \(z\)-direction.