# Propagation of Non-Linear Lamb Waves in Adhesive Joint with Micro-Cracks Distributing Randomly
## 1 Introduction
Quite a few studies have shown that the nondestructive ultrasonic evaluation method employing non-linear Lamb waves is capable of characterizing early material degradations and micro-cracks initiation by examining the non-linear effect which is exhibited from the ultrasonic waves [25, 26, 27, 28, 29, 30]. Interaction between the elastic wave and micro-cracks with the clapping mechanism has been adopted to explain the generation of higher harmonics of non-linear Lamb waves. As the propagating elastic wave meets a micro-crack within the material, the micro-crack may open or close under the action of tension or compression stress [31,32]. The Lamb wave can propagate through the micro-crack during its compressive period, but not during the tensile period. In this way, the non-linear wave with higher harmonic components is thus generated because of the bi-linear response of the micro-crack under the action of compressive and tensile stresses. In application it is quite a challenge, however, to detect the higher harmonic components of the Lamb wave because of its dispersive property. Theoretical and experimental studies have been conducted extensively based on the generation of higher harmonics of the non-linear Lamb-wave. For example, [PERSON] et al. [33] analyzed the selection of the primary Rayleigh-Lamb ultrasonic wave mode that generated cumulative second harmonics in a homogeneous isotropic plate using theoretical modeling. [PERSON] et al. [34] experimentally investigated the accumulative effect of the generated second harmonic wave, and their results verified that the generated second harmonic wave increased in a linear way with the increasing propagation distances. [PERSON] et al. [35] used the non-linear guided waves to evaluate the plasticity driven material damage in a metal plate, and proved that the accumulated plasticity, which caused an increase in the generation of second harmonics, can also be measured with Lamb waves. [PERSON] et al. [36] investigated the symmetry properties of second harmonic Lamb waves using an experimental method and proved that the generation of second harmonics will not happen in anti-symmetric Lamb wave modes. [PERSON] et al. [37] investigated the second harmonics generation of guided wave propagation in an isotropic and stress-free elastic pipe, and verified that the second harmonics in the pipe have a cumulative effect with the propagation distance. [PERSON] et al. [38] investigated the influences of the interfacial properties on the second harmonic generation of the Lamb waves in layered planar structures. Their results indicated that the efficiency of second harmonic generation is closely dependent on the interfacial properties. [PERSON] et al. [39] investigated the second harmonic generation in composites and indicated that the phase matching and nonzero power flux criteria are necessary conditions for cumulative second harmonic existence, and only symmetric modes can be generated as cumulative second harmonics for symmetric composite laminates, regardless of the propagation direction of the primary mode. Because of the presence of interfaces, the waves in the multilayered structures will be refracted and reflected many times, making the propagation characteristic of the Lamb waves more complicated. The numerical simulation method, therefore, is also an effective tool to investigate the process and mechanism of the higher harmonic generation of the Lamb waves for materials with damage. [PERSON] et al. [40] investigated the generation of non-linear Lamb wave propagation in a thin plate with micro-cracks distributing in a random way. The results showed that the micro-cracks, which lead to the generation of second harmonics, can be applied to simulate the early micro damages in structures. [PERSON] et al. [41] examined the strongly cumulative second harmonic due to the Lamb and SH wave mode interactions using finite element modeling, and the results showed that only the symmetric Lamb mode of the secondary wave field can be cumulative. [PERSON] et al.