# Propagation of Non-Linear Lamb Waves in Adhesive Joint with Micro-Cracks Distributing Randomly
## 1 Introduction  
Adhesively-bonded structures are widely employed in industries because of their inherent nature of providing uniform stress transfer, high fatigue resistance, and the reduction in structural weight [1]. Under in-service conditions, defects such as voids, micro-cracks, disbonding and kissing bonds often occur in adhesive joints due to fatigue loading and environmental corrosion. For example, micro-cracks are always the precursor to structural failures, which can lead to serious consequences and huge costs in engineering without timely and efficient detection. Thus, the methods of evaluating the quality of bonded joints have been broadly investigated for the purpose of structural safety, which include non-destructive evaluation (NDE) techniques-based ultrasonic testing, acoustic emission, and acoustic elasticity technology [2]. As one of the most powerful non-destructive techniques, the ultrasonic detection method has been applied extensively in detecting such defects as voids, cracks and fractures in structural materials. It is based on the principle of linear acoustic physics, including transmission, reflection, scattering, and absorption of the acoustic energy. For example, [PERSON] et al. [3] analyzed the propagation of guided waves in a plate consisting of an isotropic aluminium layer bonded to an anisotropic laminate made of carbon fibre reinforced plastics. Their results show that most of the wave energy can be concentrated in a certain layer depending on the mode used, and damage presenting in this layer can be detected. However, it is not effective for linear ultrasonic evaluation methods to characterize the early material degradation when the above-mentioned defects are tiny. Particularly, it is maybe impossible to detect the micro-cracks in closing when the ultrasonic testing technology based on linear acoustic physics is employed. Transmissions, reflections and scatterings could not be discerned as the ultrasonic wave propagates through the micro-cracks in closing [4]. For adhesive bonded structures, studies also show that linear Lamb wave is not sensitive to the mechanical property of the adhesive layer, although it is strongly sensitive to the thicknesses and mechanical properties of other plates in the adhesive structure [5,6]. In recent years, the non-linear ultrasonic technique has received attention for the reason that it is highly sensitive to material micro-structures whose dimensions are much smaller than the wavelength of an ultrasonic wave [7, 8, 9, 10, 11, 12]. [PERSON] et al. [13,14] conducted studies on monitoring the thermal aging of aluminum alloy and detecting the delamination of lined anti-corrosion pipes using non-linear acoustics, which demonstrate the sensitivity of the non-linear ultrasonic technique to the damage. Compared with the linear ultrasonic technique, it is becoming a quite promising tool in the nondestructive characterization of the degrees of damages which are related to the micro-cracks [15, 16, 17, 18]. Non-linear ultrasonic technique is based on the non-linear interactions of the ultrasonic wave with the microstructures of materials, such as dislocation, precipitation, and micro-cracks. According to different non-linear effects, sub-harmonic technology, acoustoelasticity, harmonic generation, and wave mixing are common methods for measuring the non-linearity in solids. Both experimental investigations and theoretical studies show that these techniques can detect and evaluate micro damages effectively, and the sensitivity is much higher than traditional linear ultrasonic techniques [19, 20, 21, 22]. Non-linear Lamb waves are widely acknowledged as one of the most promising means for quantitatively evaluating the material damages because of its long propagation distance in plate-like structures [23,24].