## 3 Three-Dimensional Finite-Element Simulation
### 3D Models of an Aluminum Plate
#### 3.1.1 Through-Thickness Micro-Crack Model  
The research object of the model was an aluminum plate. The size and parameters of the aluminum plate are shown in Table 1.  
For convenience in discussion, a coordinate system was defined with its origin at the center of the plate, as shown in Figure 2a. Two piezoelectric chips (P\({}_{t1}\) and P\({}_{t2}\)) were used to excite the incident Lamb wave signal; they were located at (-70 mm, 0 mm) and were symmetrically distributed on the upper and lower surfaces of the plates. Another eight piezoelectric chips were bound to the upper surface of the aluminum plate to collect Lamb wave structural response signals, which were labeled as P\({}_{t1}\)-P\({}_{t8}\), forming an 80 mm diameter circular transducer network. These eight piezoelectric chips were located at \(r\) = 40 mm and \(\theta\) = 0\({}^{\circ}\) (P\({}_{r5}\)), 45\({}^{\circ}\) (P\({}_{r6}\)), 90\({}^{\circ}\) (P\({}_{r7}\)), 135\({}^{\circ}\) (P\({}_{r8}\)), 180\({}^{\circ}\) (P\({}_{r1}\)), 225\({}^{\circ}\) (P\({}_{r2}\)), 270\({}^{\circ}\) (P\({}_{r3}\)), and 315\({}^{\circ}\) (P\({}_{r4}\)) [29]. The through-thickness micro-crack was modeled by an extruded entity, with the in-plane midpoint of the micro-crack at (0 mm, 0 mm) and the shape of the 3D micro-crack being an elliptical cylinder. The crack profile was elliptical, and the major and minor axes of the ellipse determined the length (\(l\)) and width (\(w\)) of the micro-crack. The height (\(l\)) of the micro-crack was perpendicular to the upper and lower surfaces of the plate as shown in the enlarged micro-crack in damaged part in Figure 3. It was located at the center of the aluminum plate in the \(y\)-axis direction. The orientation of the micro-crack is the direction of micro-crack expanding, in the models the crack direction is along the micro-crack's length \(l\), that is, the \(y\)-axis direction. Three-dimensional eight-node brick elements in Abaqus (C3D8R) were used to mesh the plane. To ensure the computational efficiency and accuracy of the model, it was essential to adopt the appropriate grid cell edge length (\(l_{e}\)) and the integration time step (\(\Delta t\)) calculated by Equations (10) and (11) [16]. For sidebands at the sum frequency (1200 kHz), the calculated mesh element size was 0.259 mm, and the time step was 41.667 ns. Because the model thickness was 2 mm, the grid cell edge length was selected as 0.2 mm to ensure that the grid along the direction of plate thickness was divided into an integer number of cells. A time step of 40 ns was deemed sufficient to ensure accuracy [30]:  
\[L_{e}\leq\frac{\lambda_{min}}{10} \tag{10}\]  
\[\Delta t\leq\frac{1}{20f_{max}}. \tag{11}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Dimensions (Length \(\times\) Width \(\times\) Height)** & **Density \(\rho\)** & **Elasticity Modulus \(E\)** & **Poisson’s Ratio \(A\)** \\ \hline \(300\times 300\times 2\) mm & 2700 kg/m\({}^{3}\) & 70 GPa & 0.33 \\ \hline \end{tabular}
\end{table}
Table 1: Model dimensions and aluminum plate material parameters.  
Figure 2: Damage model of aluminum plate: (**a**) schematic diagram; (**b**) three-dimensional (3D) model.  
According to the results of the analysis in Section 2, an exciting single S\({}_{0}\) mode Lamb wave signal could facilitate the extraction and analysis of the response signal. To obtain a single S\({}_{0}\) mode, two piezoelectric chips (P\({}_{t1}\) and P\({}_{t2}\)) were used to excite the Lamb wave satisfying the following conditions: (i) P\({}_{t1}\) and P\({}_{t2}\) vertically applied stress at the same time on the upper and lower surfaces of the plate; (ii) P\({}_{t1}\) and P\({}_{t2}\) applied stress in opposite directions, as shown in Figure 2b.