## 3 Three-Dimensional Finite-Element Simulation
### Simulation Results and Analyses
#### 3.2.2 Through-Thickness Crack Simulation Results and Analyses
To reflect intuitively the directivity of the scattering signals, the relative acoustic nonlinearity \(\beta^{\prime}\) is presented and defined as:  
\[\beta^{\prime}=\frac{\beta_{r_{5}}}{\beta_{r_{6}}} \tag{13}\]  
where \(\beta_{r_{5}}\) and \(\beta_{r6}\) represent the acoustic nonlinear coefficient \(\beta\) of the damage signal as collected by P\({}_{r5}\) and P\({}_{r6}\), respectively. The acoustic relative nonlinearity \(\beta^{\prime}\) can reflect the directivity extent of the scattering signals and can be used for a quantitative evaluation of forward scattering. With the increasing amplitude of \(\beta^{\prime}\), the directivity extent of the scattering signals will become more obvious. The relative variations of the acoustic nonlinearity parameter \(\beta^{\prime}\) with micro-crack lengths (\(l\) = 2, 4, 6, 8 mm) at fixed width (\(w\) = 1 \(\upmu\)m) for three kinds of excited mixed-frequency signal sources \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sequence Number (i)** & **Cycle (n)** & \(f_{1}\) & \(f_{2}\) & \(f_{1}+f_{2}\) & \(f_{2}-f_{1}\) \\ \hline
1 & 22 & 450 kHz & 600 kHz & 1050 kHz & 150 kHz \\
2 & 22 & 470 kHz & 650 kHz & 1120 kHz & 180 kHz \\
3 & 22 & 500 kHz & 700 kHz & 1200 kHz & 200 kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Types of mixed-frequency signals. Figure 9: Directivity patterns of nonlinear parameter \(\beta\) for (**a**) 1 \(\upmu\)m; (**b**) 5 \(\upmu\)m; (**c**) 8 \(\upmu\)m; (**d**) 10 \(\upmu\)m width micro-cracks with fixed 8 mm length. are displayed in Figure 10a. The magnitude of \(\beta^{\prime}\) increased continuously with increasing micro-crack length at the mixing frequencies (\(f_{1}=450\) kHz,\(f_{2}=600\) kHz). As the center frequency of the excitation signal increased, the magnitude of \(\beta^{\prime}\) first increased and then decreased with increasing micro-crack length at both mixing frequencies (\(f_{1}=470\) kHz,\(f_{2}=650\) kHz) and (\(f_{1}=500\) kHz,\(f_{2}=700\) kHz), which means that the forward scattering increased with increasing micro-crack length at the initial stage of micro-crack and then decreased with increasing micro-crack length. A straight explanation is that as the micro-crack became longer, the contact stiffness of the interface increased first and then decreased. From the theory of the contact acoustic nonlinearity, as the contact stiffness of the interface decreased, that is, as the micro-crack became longer, the acoustic nonlinearity increased [31]. Moreover, the variation of the relative acoustic nonlinearity parameter \(\beta^{\prime}\) for micro-cracks of fixed length (\(l=8\) mm) and various widths (\(w=1\), 5, 8, 10 \(\upmu\)m) is shown in Figure 10b. \(\beta^{\prime}\) monotonically decreased with increasing micro-crack width; in the case of three types of mixing frequencies, indications were apparent of a decreasing trend for the forward-scattering degree with increasing micro-crack width. As the micro-cracks become wider, the gap between its two interfaces becomes larger, consequently, parts of the two interfaces are not in contact during the compressional phase of the incident waves, and thus the contact area is reduced. In addition, when the micro-crack was small enough, the scattering features of the nonlinear Lamb wave interacting with the micro-crack began to become inconspicuous. This indicated that the relative nonlinearity coefficient \(\beta^{\prime}\) was less than one as shown in the left lower corner of Figure 10a, which means that the scattering signals no longer had directivity. In order to satisfy the monotonic changing trend of the acoustic nonlinearity parameter \(\beta^{\prime}\), the frequency combination of 450 kHz and 600 kHz should be selected in practice.