## 1 Introduction  
To reduce the risk of catastrophic failure and prolong the lifespan of structures, structural health monitoring (SHM) sensitive to defect-related changes in a real-time manner has been widely applied in aerospace, transportation, civil and mechanical structures, etc. Among various SHM techniques, ultrasonic Lamb wave has been considered as a promising way to detect damages in plate-like structures due to its appealing advantages of high sensitivity to different types of defects and its large inspection range with low attenuation [1]. The damage localization of linear cracks and through holes was achieved by the ellipse method, based on the time-of-flight of damage-scattered waves with a spatially distributed array of sensors attached to a homogeneous plate structure [2, 3]. Moreover, efforts have been dedicated to characterizing the features of damage regions, such as the size and shape through various characterization algorithms [4, 5]. Qualitative or quantitative detection of multiple damage cases [6, 7] in engineering structures has become an increasing concern, although it seems to be a challenging task. In general, the above-mentioned methods based on the linear features of Lamb waves (e.g., velocity and attenuation) are commonly sensitive to gross defects or open-type ones (e.g., circle holes, notches and cracks), and insensitive to small variations in microstructures (e.g., precipitates, inclusions and micro-voids) or contact-type damages (e.g., disbands, delamination and micro-cracks). If we want to improve the sensitivity of the linear Lambwave to the small defects, an effective solution is to increase the excitation frequency, which will bring an additional complexity to signal processing because of the multimodal and dispersive properties of Lamb waves at higher frequencies. Recently, an alternative way to overcome the shortcoming is to employ the nonlinear feature of Lamb waves [8]. There are basically two types of acoustic nonlinearities, classical and non-classical, introduced by the coupling of nonlinear ultrasound and damages. The classical nonlinearity is mostly associated with intrinsic nonlinearity of materials such as anharmonicity and imperfection of atomic lattices, whereas the non-classical nonlinearity mainly arises from contact-type damages in structures through various mechanisms such as stress-strain hysteresis, contact nonlinearity, rough surfaces contact, [PERSON] effect, etc. [9]. As a typical nonlinear Lamb wave, higher harmonic generation has a close tie with the behavior of the material nonlinearity. In the past two decades, the cumulative second harmonics of Lamb waves have attracted considerable attention because of their high sensitivity to material nonlinearities. Phase-velocity matching and non-zero power flux are regarded as the necessary conditions for cumulative second harmonic of Lamb waves generation [10; 11]. Mode pairs that satisfy these conditions (e.g., S1-s2, A2-s4, and S2-s4) have been used for nonlinear ultrasonic detection of plasticity [12], fatigue [13], creep [14], pitting [15], and microstructural damage [16]. Recently, mode pair S0-s0 in a low-frequency range satisfying approximate phase-velocity matching was proposed to quantitatively evaluate the incipient stage of material degradation [17; 18; 19]. In addition, mode pair S0-s0 was also used to detect closed defects or imperfect bonding interfaces that open and close as the waves pass across them. [PERSON] et al. [20] explored nonlinear interaction of S0 mode Lamb wave with an imperfect joint of plates by perturbation analysis and numerical simulation. [PERSON] et al. [21] performed numerical simulations on the propagation of S0 Lamb waves in thin plates with randomly distributed micro-cracks to study the behavior of nonlinear Lamb waves. They found that the acoustic nonlinear parameter increased linearly with the micro-crack density and the size of the micro-crack zone. [PERSON] et al.