## 3 Three-Dimensional Finite-Element Simulation
### Simulation Results and Analyses
#### 3.2.2 Through-Thickness Crack Simulation Results and Analyses  
To investigate the scattering features of nonlinear Lamb waves around a micro-crack to characterize its directivity, two groups of simulations were performed for the 3D aluminum plate with various  
Figure 6: Damage signals collected from sensing path (P\({}_{\text{t1,2}}\)–P\({}_{\text{r5}}\)): (**a**) time-domain signal; (**b**) frequency-domain spectrum. Figure 7: Superposed response signals after pulse inversion: (**a**) time-domain signal; (**b**) frequency-domain spectrum; (**c**) frequency-domain comparison. micro-crack lengths and widths. The simulations of the first group were performed on through-thickness crack model with fixed width (\(w\) = 1 \(\upmu\)m) and various lengths (\(l\) = 2, 4, 6, 8 mm). Figure 8 shows the directivity patterns of the acoustic nonlinear coefficient \(\beta\) for different micro-crack directions, when the mixing frequencies are 450 kHz and 600 kHz. From Figure 8a, the acoustic nonlinear coefficient \(\beta\) at \(\theta\) = 0\({}^{\circ}\) is slightly smaller than \(\beta\) at \(\theta\) = 45\({}^{\circ}\) when the length of the micro-crack is 2 mm. The acoustic nonlinear coefficient \(\beta\) is larger at \(\theta\) = 0\({}^{\circ}\) than in other directions when the micro-crack length increases to 4, 6, and 8 mm, as shown in Figure 8b-d. Moreover, the amplitude of the nonlinear parameter \(\beta\) is the smallest at \(\theta\) = 180\({}^{\circ}\) than in other directions when the micro-crack length varies from 2 mm to 8 mm. In general, the directivity patterns show that the amplitude of \(\beta\) in the forward scattering signal of the crack was the largest, whereas the backscattering signal had a smaller amplitude than the scattering signal for other micro-crack directions. Therefore, the property of the largest forward-scattering and smallest backward-scattering features could be used to characterize micro-crack direction. The micro-crack was perpendicular to the connecting line between the largest forward-scattering point collected by P\({}_{\mathrm{r5}}\) (0\({}^{\circ}\)) and the smallest backward-scattering point collected by P\({}_{\mathrm{r1}}\)(180\({}^{\circ}\)). Moreover, with increasing micro-crack length, the curves of the directivity patterns become more acute, which means that the directivity of long micro-cracks becomes more obvious and the direction of long micro-cracks is easier to identify. The simulations of the second group were performed on through-thickness crack model with a fixed length (\(l\) = 8 mm) and various widths (\(w\) = 1, 5, 8, 10 \(\upmu\)m) at the excited mixing frequencies \(f_{1}\) = 450 kHz and \(f_{2}\) = 600 kHz. The nonlinear parameters \(\beta\) for different micro-crack directions were invested, and the outcomes are shown in Figure 9. The amplitude of the nonlinear parameter \(\beta\) at \(\theta\) = 0\({}^{\circ}\) was larger than in other directions, and the smallest value was obtained at \(\theta\) = 180\({}^{\circ}\). And the figure reflects similar nonlinear characteristics, that the amplitude of the nonlinear parameter \(\beta\) was the largest in the forward-scattering direction and showed the smallest value in the backward scattering  
Figure 8: Directivity patterns of nonlinear parameter \(\beta\) for (**a**) 2 mm; (**b**) 4 mm; (**c**) 6 mm; (**d**) 8 mm length micro-cracks with fixed 1 \(\upmu\)m width. direction. Moreover, with decreasing micro-crack width, the curves of the directivity patterns became more acute and the forward scattering point gradually approaches the boundary of the circle, which means that the directivity of narrow micro-cracks became more obvious and that the direction of narrow micro-cracks was easier to identify. To illustrate further the relationship between acoustic nonlinearity and crack length and width, three kinds of frequency combinations were selected as the incident signal, as shown in Table 2.  
Damage signals for different mixed-frequency combinations were collected.