# Effect of Ultrasonic Vibration on the Surface Adhesive Characteristic of Iced Aluminum Alloy Plate
## 3 Effect of PZT Patch Size on Shear Stress
### The Effect of Side Length  
For exploring the effect of the side length of the PZT patch on shear stress at the adhesive interface, the variation in maximum shear stress at the adhesive interface with frequency was calculated and analyzed by harmonic response analysis for the iced aluminum alloy plate. An aluminum alloy material was selected, whose properties are listed in Table 1. The size of the aluminum alloy plate was 60 mm \(\times\) 60 mm \(\times\) 2 mm. The shape and the side length of the ice layer were the same as those of the aluminum alloy plate, and the thicknesses of the ice and piezoelectric ceramic were both 2 mm. The properties of ice are also listed in Table 1. The shape of the PZT patch was square, and the side length of the PZT patch was in the range of 12 mm\(\sim\)17 mm, which was decided according to the modal analysis results shown in Figure 4. As inferred from Figure 4, the largest vibration amplitude is in the central part of an iced aluminum alloy plate. Therefore, for maximum excitation of the bending deformation, the PZT patch is located in the central part. In this paper, the diameter of the red region was selected as the lower limit, and the diameter of the orange region was selected as the upper limit, which were 12 mm and 17 mm, respectively. For the meshing process, a mapped method was used. The element types of the aluminum alloy plate and the ice were both SOLID186, and that of the PZT patch was SOLID226. The PZT patch was attached to the aluminum alloy plate by command VGLUE, which generated a common plane between the PZT patch and the aluminum alloy plate. By this command, the elements (SOLID226) of the PZT patch coupled with the elements (SOLID186) of the aluminum alloy plate. The parameters of the PZT patch, including elastic matrix \(E\), piezoelectric matrix \(e\), and relative permittivity \(\varepsilon\), were determined [27], and its density is 7.5 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\). The elastic matrix \(E\) of piezoelectric ceramic, which is expressed in stiffness form, is expressed with the following matrix:  
\[E=\left[ \begin{array}{cccccc}13.9&7.43&7.78&0&0&0\\ 0&11.5&7.43&0&0&0\\ 0&0&13.9&0&0&0\\ 0&0&0&2.56&0&0\\ 0&0&0&0&2.56&0\\ 0&0&0&0&0&2.56\\ \end{array} \right]\times 10^{10} \tag{1}\]  
Figure 6: Sketch map of the adhesive position of the piezoelectric ceramic. The piezoelectric matrix \(e\) is expressed with the following matrix:  
\[e=\left[ \begin{array}{ccc}0&0&-5.2\\ 0&0&-5.2\\ 0&0&15.1\\ 0&0&0\\ 0&12.7&0\\ 12.7&0&0\end{array} \right] \tag{2}\]  
The relative permittivity \(\varepsilon\) is expressed with the following matrix:  
\[\varepsilon=\left[ \begin{array}{ccc}1475&0&0\\ 0&1475&0\\ 0&0&1300\end{array} \right] \tag{3}\]  
For electrical boundary conditions, a sinusoidal voltage was applied to the PZT patch. In the process of simulation, the voltage at the interface between the PZT patch and aluminum alloy plate was set to zero, and an alternative voltage was applied on the opposite surface of the PZT patch against the interface. The excitation voltage V\({}_{\text{p-p}}\) loading on the PZT patch (peak to peak) was 400 V. The simulation results are shown in Figure 7.  
As shown in Figure 7, when the side length of the PZT patch is 16 mm \(\times\) 16 mm or 14 mm \(\times\) 14 mm, the maximum values of shear stress at the adhesive interface are high in both cases. Based on the existing research findings, the maximum adhesive shear stress of ice covering on metal substrates is about 1.8 \(\times\) 10\({}^{6}\) Pa [22]. Therefore, in the above simulation cases, there are three kinds of conditions satisfying the deicing condition. The parameters in these conditions and simulation results are listed in Table 2, and the cloud images of the distributions of shear stress at the adhesive interface under these conditions are shown in Figure 8.