## 3 Results
### Stress Distribution and Force Compensation Coefficient for Different Penetration Depths  
The simulation results of the canine are exemplified in Figure 5. The simulation of teeth translation was achieved, as shown in Figure 5a, under 1 N of orthodontic force and the specific moment is shown in Table 3.  
The distal side of the PDL was under tensile stress and the mesial side was compressed. As seen in Figure 5b, the stress distribution is uniform for all subjects. However, with  
Figure 5: Simulation results of the stress distributions for different subjects of canine under a bodily movement. (**a**) The canine executes a bodily movement at the key increment of the simulation. (**b**) The hydrostatic stress distributions along PDL of canines with different penetration depth (\(x\)) during the bodily movement (labial view). PDL is the periodontal ligament. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Penetration Depth (mm)**} & \multicolumn{2}{c}{**M/F for Bodily Movement (mm)**} & \multicolumn{2}{c}{**CR Distance Divide by the Tooth Length, \(y\)/L**} \\ \cline{2-7}  & **Canine** & **Premolar** & **Molar** & **Canine** & **Premolar** & **Molar** \\ \hline
0 & 11.27 & 8.00 & 8.40 & 0.69.41 & 0.6345 & 0.6279 \\
1 & 10.82 & 7.80 & 7.50 & 0.6749 & 0.6244 & 0.5816 \\
2 & 10.52 & 7.60 & 7.20 & 0.6  
the increasing penetration (from 0 mm to 9 mm), the magnitude of hydrostatic stress on the PDL gradually increased (from blue to red). The black area received nearly no stress, because the root apex protruded into the cavity and lost the support of the bone. We quantified the stress values on the PDL for different subjects. The simulation of bodily movement was successfully achieved with the uniform hydrostatic stress distribution. In Figure 6, the averages of the hydrostatic stresses on the PDL versus the penetration depth are shown. Under the same magnitude of the orthodontic force, the hydrostatic stress in the canine PDL was much higher than that in the premolar and molar teeth. For the tooth with a normal periodontal anatomy, the average hydrostatic stress of 2.40 KPa in the canine PDL was nearly five times larger than that in the premolar (0.57 KPa) and molar (0.54 KPa) teeth. As the depth of the root penetration increased, the stress in the PDL increased rapidly. As the penetration depth increased to 9 mm, the stress in the canine PDL increased almost threefold (7.31 KPa) as compared to when the root did not penetrate into the MS (2.40 KPa). The average stress in the premolar PDL increased fourfold from 0.56 KPa to 2.23 KPa. For the molars, the average hydrostatic stress increased from 0.54 KPa to 1.64 KPa. It was found that under the same orthodontic force, as the depth of the penetration increased, the average stress on PDL exhibited a quadratic increase. Similar trends can be observed in all tooth types. When the magnitude of the orthodontic force was constant, the correlation curve between average stresses and penetration depth (\(x\)) could be fitted by a quadratic model as follows:  
\[\sigma_{\text{ave}}(x)=\text{a}x^{2}+\text{b}x+\text{c} \tag{8}\]  
a, b, and c are the parameters. We set b = 0, because the stresses were incremental. Figure 6 shows stresses for each tooth versus penetration depth. The curves were fitted using the quadratic model shown in Equation (8). The fitting parameters are listed in Table 4. The quadratic model was considered a good fit because of the high r-squared values for different teeth. The quadratic term's coefficient (a) indicated the increasing speed of slopes. The stress of the canine increasing much faster than that of the premolar and the molar. Figure 6: Average hydrostatic stresses along the PDL versus the penetration depths into the MSF for different teeth. Experimental data and the quadratic model fitting. According to the hypothesis mentioned above, the optimum force magnitude for the tooth with a low MSF was evaluated based on the average of hydrostatic stresses along the PDL from the FE study.