## 3 Three-Dimensional Finite-Element Simulation
### Simulation Results and Analyses
#### 3.2.3 Buried Crack Model Simulation Results and Analyses  
To confirm the validity of nonlinear Lamb wave-mixing technique in identifying the direction of micro-crack, the acoustic nonlinear coefficient \(\beta\) for buried crack was investigated. The simulations of the first group were performed on buried crack model with fixed height (\(h=1.8\) mm) and various lengths (\(l=2\), 4, 6, 8 mm) when the mixing frequencies are 450 kHz and 600 kHz and the results are shown in Figure 11. The results reflected the similar nonlinear characteristics as for the through-thickness crack that the amplitude of \(\beta\) in the forward-scattering direction of the crack was the largest, whereas the backscattering signal had a smaller amplitude than the scattering signal of the other micro-crack directions. The direction pattern of nonlinear parameter show that 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 \(P_{\mathrm{r1}}\)(\(180^{\circ}\)) and with increasing micro-crack length, the  
Figure 10: Relative nonlinear parameter \(\beta^{\prime}\) for (**a**) crack-length variation; (**b**) crack-width variation with three types of mixed-frequency incident signals. curves of the directivity patterns become more acute, which means that the directivity of long buried micro-cracks was more obvious and easier to identify like the through-thickness crack. For buried micro-crack, the simulations of the second group were performed on model with fixed length (\(l=8\) mm) and various heights (\(h=1.4\), \(1.6\), \(1.8\), \(2.0\)\(\upmu\)m) at the excited mixing frequencies \(f_{1}=450\) kHz and \(f_{2}=600\) kHz. The results of nonlinear parameter \(\beta\) for different micro-crack dimensions are displayed in Figure 12 and which show that the amplitude of the nonlinear parameter \(\beta\) at \(\theta=0^{\circ}\) was larger than in other directions and the smallest value was obtained at \(\theta=180^{\circ}\). The direction patterns of nonlinear parameter \(\beta\) mean that the largest and smallest values were obtained in the forward and backward scattering directions, respectively. Moreover, with increasing micro-crack height, the curves of the directivity patterns became more acute, which means that the directivity of higher buried micro-cracks became more obvious and its direction was easier to identify. Figure 11: Direction patterns of nonlinear parameter \(\beta\) for (**a**) 2 mm; (**b**) 4 mm; (**c**) 6 mm; (**d**) 8 mm length micro-cracks with fixed 1.8 mm height. The relative variations of the acoustic nonlinearity parameter \(\beta^{\prime}\) with micro-crack lengths (\(l\) = 2, 4, 6, 8 mm) at fixed height (\(l\) = 1.8 mm) for three kinds of excited mixed-frequency signal sources were investigated for the buried micro-crack with characteristic of zero width. The results are displayed in Figure 13a, which show that at the mixing frequencies (\(f_{1}\) = 450 kHz, \(f_{2}\) = 600 kHz) the magnitude of \(\beta^{\prime}\) monotonically increased with increasing micro-crack length, and at both mixing frequencies (\(f_{1}\) = 470 kHz, \(f_{2}\) = 650 kHz) and at (\(f_{1}\) = 500 kHz, \(f_{2}\) = 700 kHz) the magnitude of \(\beta^{\prime}\) first increased and then decreased with increasing micro-crack length which means that the forward scattering degree of the micro-crack increased with increasing buried micro-crack length at the initial stage of micro-cracking and then decreased with increasing length of micro-crack. These variation tendency means that the change property of acoustic nonlinearity parameter \(\beta^{\prime}\) for the buried crack is consistent to the through-thickness crack. Because of the buried micro-crack was embedded in different depths in the plate, the directivity of micro-crack could be carried out by changing the height of micro-crack (the buried depth of micro-crack).