# Propagation of Non-Linear Lamb Waves in Adhesive Joint with Micro-Cracks Distributing Randomly
## 4 Results and Discussions
(**a**) 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). Figure 9: Magnitudes of (**a**) fundamental harmonics and (**b**) second harmonics (collected at position 1) versus the propagation distance. ANPs increase when the distances of the Lamb wave propagating in the micro-cracks damage zone increase, except for the signals collected at position 1 in the \(z\)-direction. It demonstrates that the second harmonic Lamb waves show a property of accumulation with the propagation distance. Note that there is an exception for the ANP collected at position 1 in the \(z\)-direction. Investigations have demonstrated that the magnitude of the second harmonics would increase in a linear way with the propagating distances of non-linear Lamb waves when the phase velocities of the fundamental and second components are equal [49]. If they are not equal to each other, the magnitude of the second harmonics will then vary in the form of a sinusoidal curve with increasing propagation distances. Although we try to excite the fundamental and second harmonics of Lamb waves with the same phase velocity, the existence of the micro-crack, however, will lead to small deviations of the phase velocities when the non-linear Lamb waves propagate in the region with micro-cracks. The ANPs, therefore, do not always exhibit a linear relationship with the propagation distance, particularly in the region of micro-cracks. This effect is more obvious for the ANP collected at position 1 in the \(z\)-direction. Furthermore, Figures 10 and 11 show that the ANPs in the \(z\)-direction are bigger than those in the \(x\)-direction, which indicates that the non-linear effect in the \(z\)-direction is stronger. Particularly, the ANPs in the \(z\)-direction from the signals collected at position 1 are much bigger than that at position 2. This phenomenon can be explained using the displacement field of the excited second harmonic Lamb waves, as shown in Figure 12. It is clear that the normalized displacement in \(z\)-direction is bigger at the surface (position 1) than that at the adhesive layer (position 2). This fact indicates that it is more effective to evaluate the non-linearity of the adhesive structure just by employing the \(z\)- component of the Lamb waves at the surface, which is easy to realize in practice. Figure 11: ANP (collected at position 2) versus the propagation distance. Figure 10: ANP (collected at position 1) versus the propagation distance. As a quantitative indicator for the micro-cracks, the crack density, as defined in Equation (7), is closely related to the number and length of the cracks within the adhesive layer. Figure 13 shows the ANP as a function the micro-crack number when other factors, such as the crack length and friction coefficient, remain unchanged. It is demonstrated that there is a linear relationship between the ANP and the micro-crack number, both in the \(x\)- and \(z\)-directions. Because the ANP based on the signals collected at position 1 is more sensitive to the micro-crack related damage, here we only consider the signals collected at position 1. Figure 14 plots the relationship between the ANP and micro-crack length. It is clear that there is a linear relationship between them. Furthermore, it also shows that the ANPs are more sensitive to the change of the micro-crack length when other factors are kept constant. The ANP versus the micro-crack density with increasing micro-crack number and increasing radius is shown in Figure 6:0 Considering that the micro-cracks are distributing randomly for the finite element modeling, it is still demonstrated that the results based on these two cases are basically consistent. At the same time, when such factors as the frequency of the exciting signals and friction coefficient of the micro-crack surface keep constant, the ANPs also increase in a linear way with the increasing micro-crack density.