# Ultrasonic Deicing Efficiency Prediction and Validation for a Flat Deicing System
## 4 Results and Discussion
### Stress Distribution and Deicing Affect Analysis  
The following study revolved around configuration 2. Stress distribution of ice layer was mainly investigated in this section. The ice/substrate interface shear stress nephogram at XZ plane and YZ plane at frequency 213 kHz were studied. In order to show these results more intuitively, 13 paths at the interface of the ice layer including left and right boundaries were established along the long side of the ice layer, which were parallel to the Y axis. Similarly, 22 paths at the interface of the ice layer including two boundaries were established along the short side of ice layer, which were parallel to the X axis. Based on these, the variations of shear stress with paths can be obtained. Then, the shear stress amplitude of 13 paths parallel to the Y axis and 22 paths parallel to the X axis were averaged  
Figure 9: Energy density needed for deicing and deicing efficiency factor (DEF) of seven different configurations. separately. Average the shear stress amplitude of 13 paths parallel to the Y axis, the average shear stress \(\tau_{13}\) and \(\tau_{23}\) vary along the Y axis can be calculated. Similarly, the results of average shear stress amplitude vary along the X axis can also be estimated. The results are shown in Figure 11.  
From the simulation results of Figure 11a,b, the conclusion can be drawn that the average shear stress \(\tau_{13}\) and \(\tau_{23}\) amplitudes are symmetric about the neutral axis. Combining these two figures results with shear stress nephogram, it can be found that \(\tau_{13}\) is symmetric about the neutral axis, while \(\tau_{23}\) is antisymmetric about the neutral axis. From the results of Figure 11, it can be seen that the average  
Figure 11: Average shear stress amplitude varies along the axis direction: (**a**) \(\tau_{13}\) amplitude varies along the Y axis; (**b**) \(\tau_{23}\) amplitude varies along the Y axis; (**c**) \(\tau_{13}\) amplitude varies along the X axis; (**d**) \(\tau_{23}\) amplitude varies along the X axis. Figure 10: (**a**) Ice/substrate interface shear stress nephogram for four different icing cases; (**b**) The total shear stress for delamination and deicing efficiency factor for the four icing cases. shear stresses are higher at edges and corners of the ice layer which means ice debonding starts from the edges and corners of the ice layer. Meanwhile, the shear stress distributions of deicing efficiency top three, configuration 2 (\(r\) = 5 mm), configuration 6 (\(r\) = 9 mm), and configuration 7 (\(r\) = 10 mm) were compared. In order to find the principle of stress distributions of these three configurations, the shear stress \(\tau_{13}\) and \(\tau_{23}\) were normalized. The results are shown in Figures 12 and 13. Figures 12 and 13 reveal the relationship between shear stress distribution and deicing efficiency. The normalized shear stress distribution can reflect the stress gradient from edges to the center of the ice layer. In order to describe this type of stress gradient, a coefficient by referring to the definition of variance was defined. The coefficient named variance of maximum shear stress (VMSS) represents the degree of dispersion between shear stress at the edges and middle of the ice layer. For the average shear stress at the edges of the ice layer at maximum, the average shear stress at the edges would be equal to one after normalization. Thus, this coefficient can be described as Formula (13):  
\[\text{VMSS}=\frac{1}{N}\sum_{i=1}^{N}\left(\tau_{i}-\tau_{max}\right)^{2}=\frac {1}{N}\sum_{i=1}^{N}\left(\tau_{i}-1\right)^{2}, \tag{13}\]  
where \(N\), \(\tau_{i}\) represent the number of node and shear stress for each node, respectively, \(\tau_{max}\) means shears stress at edges, which equals to one after normalization. The larger the value of VMSS, the larger the degree of dispersion. That presents the shear stresses at the edges of the ice layer reach the deicing stress threshold, the shear stresses at the rest of the part of the ice layer are still relatively small.