## 2 Subsurface Crack Forms and Their Distribution in the UVAG Process
### Variety of Subsurface Crack Forms and Their Proportions  
Figure 4 shows the observed subsurface cracks with various shapes and sizes appearing on the cross-section surface (observed surface). Among them, three typical subsurface crack forms (i.e., arc-shaped, bifurcated, and straight median and lateral cracks) can be distinguished, and a few cracks with irregular shapes (other cracks) can also be found. By examining the entire cross-section surface, it is observed that the cracks of the three typical forms constitute a significant proportion of all subsurface cracks.  
In order to further investigate the proportion of each crack form, the crack propagation depths were examined by an image measurement software. To filter out cracks of small depth, a threshold must be selected. In ultrasonic-vibration-assisted grinding of optical glasses, the subsurface cracks with maximum propagation depth always originated from median cracks, which means that the lateral cracks do not have a significant effect on the subsurface damage. According to the previous studies [16; 17], the ratio between the propagation depth of lateral and median subsurface cracks is in the range 0.2-0.3; thus, the threshold value adopted in this work (\(h_{c}\)) was as follows:  
\[h_{c}=0.2\times MSSCD_{i} \tag{1}\]  
Figure 4: Subsurface cracks observed in experiments.  
Figure 3: Profile measurement for subsurface crack depth determination.  
where \(MSSCD_{i}\) (\(i=1\), \(2\),..., \(40\)) is the maximum subsurface depth of all the cracks in \(i\)th experiment.  
By using the above measuring method, the propagation depths of all selected cracks were obtained; thus, the proportions of the three typical subsurface cracks and other cracks were determined, as shown in Figure 5.  
According to the statistics of the number of subsurface cracks obtained from 40 groups of experiments, the average number of linear and lateral cracks in each group is about 3000, the average number of arc cracks is about 4000, and the average number of bifurcation cracks is about 3600. The average number of other cracks is about 750.  
From Figure 5, the proportion of the three typical subsurface cracks is over 85% of the total, which means the subsurface cracks in the 40 experiments are mainly straight median and lateral, arc-shaped, and bifurcation cracks. The proportion of other cracks is the smallest, while that of the arc-shaped cracks is the biggest. The average proportions of arc-shaped, bifurcation, and straight median cracks in the 40 experimental sets are 34.5%, 30.6%, and 27.4%, respectively. This means that more than 70% of the cracks did not follow the standard paths of median and lateral cracks. It can also be seen from Figure 5 that with the change in the ultrasonic vibration amplitude, the proportions of arc-shaped and bifurcated cracks increased, while those of the straight median and lateral cracks and other cracks decreased.  
By examining the morphologies of all the maximum depth cracks in the 40 sets of experiments, a summary of maximum-depth subsurface crack form distribution was obtained, as shown in Table 2.  
It is clear from Table 2 that in the 40 sets of UVAG of optical glass experiments, almost 78% (31/40) of the subsurface cracks with maximum propagation depth were not straight median cracks, which means the cracks with maximum depth were more likely to evolve from the arc-shaped and bifurcated cracks.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **Arc-Shaped** & **Bifurcation** & **Straight Median** & **Other Cracks** \\  & **Cracks** & **Cracks** & **and Lateral Cracks** & **Other Cracks** \\ \hline Times of the maximum & & & & \\ depth cracks & 17 & 12 & 9 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Proportion of cracks with maximum depth.  
Figure 5: Proportions of subsurface cracks.