## 1 Introduction  
Nonlinear ultrasonic behaviors, such as harmonics, mix frequencies, and the resonance frequency shift, have been proven to be sensitive to structure imperfections and early degradation of materials [1; 2; 3; 4; 5; 6]. In the early stage of damage, material fatigue can induce a number of micro-cracks with a typical length of 1-100 \(\upmu\)m by continuous loading cycles, then the micro-cracks further grow, coalesce with other micro-cracks and eventually form macro-cracks [7]. Since the fatigue cracks are localized [8], not uniformly distributed in the structure, the generated nonlinear response is basically dependent on the configuration of the crack area related to the localized hysteretic deformation. Clapping between the contacting surface and dissipative mechanism due to frictional sliding and so on [7; 8], leads to a much stronger nonlinearity than the surrounding material behaving linearly [8]. However, non-linear effects induced by the localized cracks in the materials are not obvious enough to conveniently be measured and analyzed [9; 10]. In addition, the use of power amplifier, transducers, and coupling media in ultrasonic testing system also bring about external non-linear change. Since it is difficult to separate structure-induced non-linearity in materials from external non-linearity, non-linear ultrasonic technology is not applied widely to accurately evaluate and locate the structure imperfections in practical applications [11]. In terms of the uncertainties in real-life testing conditions, \"big data\" sets, collected from acoustic echo-signals of vast amounts of damaged material, show different but distinguishable statistical characteristics compared with intact material [12; 13; 14; 15; 16; 17]. Some statistical models, also called data-driven models, relate the degree of damage to the probability of detection (PoD). They were first introduced by the National Aeronautics and Space Administration (NASA) in 1973 and were soon accepted as a standard method [18; 19]. Later [PERSON] and [PERSON] further developed statistical methods to estimate a time-to-failure distribution for a broad class of degradation structures [16]. [PERSON] employed [PERSON] to update a method to develop a closed-form residual-life distribution for the monitored device [20]. [PERSON] et al. proposed a framework to meet the challenge by systematically evaluating material damage based on large data sets collected by using acoustic emission (AE) [12]. [PERSON] et al. investigated AE relative energy, amplitude distribution as well as amplitude spectrum to discern the delamination damage mechanism of the composites [21]. [PERSON] et al. employed the model-based Clustering (Hellinger divergence) method to classify certain attributes of the original pure data obtained directly from the acoustic emission signals and form normed frequency spectra to perform physical separation tasks of AE random signals [22]. However, most data-driven methods are probabilistic and obtained historical degradation data or empirical knowledge [23; 24], which require a large amount of experimental data to construct the reference curves or the preset feature threshold. Hence, in our previous work [25], we first introduced consensus self-organizing model (COSMO), neither any domain knowledge nor supervision to extract useful features, detecting a single flaw located at a fixed positon of steel specimens based on acoustical echo-signals. Nevertheless, as the contacting surfaces of the flaw produced by an electrical discharge machine were totally separated in [25], evaluating the actual fatigue cracks by applying COMSO method would potentially be problematic. In this paper, four cracks distributed along the length direction of a steel specimen were investigated, and the crack's growth (produced using fatigue testing [26]) was further discussed in details, which is probably accord with the mechanisms involved in contact acoustic nonlinearity and hysteretic nonlinearity.