## 3 Results
If 0 < \(\sigma\) < 1 grf/mm\({}^{2}\), then the plastic environmental (tissue) resistance is high, and if (\(\sigma\)) is greater than 1, the plastic environmental resistance becomes progressively lower. From Equation (14) it can be easily deduced that \(\sigma=\sigma(t,s)\):  
\[\sigma=\sigma(t,s)=\frac{u_{0}(t)\cdot s_{\rm c}}{\delta_{0}\cdot s_{\rm f}+ \frac{\delta_{1}}{2}\cdot(s_{\rm f}{}^{2}+2\cdot s_{\rm f}\cdot\widehat{s})+ \frac{\delta_{2}}{3}\cdot(s_{\rm f}{}^{3}+3\cdot s_{\rm f}{}^{2}\cdot\widehat{ s}+3\cdot s_{\rm f}\cdot\widehat{s}{}^{2})} \tag{15}\]  
This approach presented in Equation (15) could lead to the possibility of developing an interesting study on the localization of the plastic environmental resistance, in relation to the depth variation (s) and in relation to the evolution in time (_t_) of the orthodontic treatment. As previously mentioned, the simulations results presented in Table 3 and Figures 9 and 10 are made for the particular constant value of the tissue resistance coefficient \(\sigma=2\) grf/mm\({}^{2}\). From the practical perspective (perspective which the aim of the paper addresses), the approximation that (\(\sigma\)) is constant stands with accuracy. This aspect is due to the fact that, in this paper, we do not study the tissue resistance on intervals of (s) independent variable,  
Figure 10: The evolution of the rotation center (s\({}_{\rm c}\)) in relation to time (t), for \(\sigma=2\) grf/mm\({}^{2}\). our focus being to determine the total value of the plastic moment of the tissue-reacting force (in this case, the integration limits from Equation (14) remain the same). Moreover, the (\(\sigma\)) value presents small variations in relation to time, variations which can be neglected from the practical point of view (even if (\(\sigma\)) would present consistent variations in relation to time (t), its most important value, for the process dynamics, is the initial one). The value \(\sigma=2\) grf/mm\({}^{2}\), being higher than 1, highlights a small resistance of the tissue. This value was chosen for the simulation since it implies approximately the same variation of \(y^{\prime}_{\alpha}(t)\) as in the case of the simulation presented in Table 2 and in Figure 6, but in contrast to the previous simulation, it generates a negative value of the \(y_{\beta}(t)\) signal, in steady-state regime. Using the evolution of the rotation center (s\({}_{\mathrm{C}}\)) obtained by applying the previously presented procedure, we could compute using Equation (12) the tooth's apex movement (\(y_{\beta}\)), which being smaller than zero (excepting the first 1.35 weeks from the orthodontic treatment procedure start, as it results from Table 3 and Figure 9) shows that the orthodontic treatment regime corresponds to the rotation from Figure 8. Although (\(y^{\prime}_{\alpha}\)) increases progressively, it is remarked (in Table 3 and Figure 9) that after a period \(t=0.45\) weeks the (\(y_{\beta}\)) movement begins to decrease. The explanation of this phenomenon is based on the low value of the tissue resistance coefficient (\(\sigma=2\) grf/mm\({}^{2}\)), the initial applied force \(\mathrm{u}_{0}=100\) grf having a too high value for the corresponding treatment. In this case, as a result of the force action, the incisal edge of the tooth will be quickly accelerated, and the tooth root cannot follow its movement with a comparable acceleration. As consequence, the rotation regime occurs. In some particular cases, in which the tissues have high resistance (small values of (\(\sigma\))), the decrease of the (\(y_{\beta}\)) signal can occur, too. The medical explanation for this phenomenon would be that the elastomer used in order to generate the movement begins to decrease its force in a few days after appliance, so the force initially applied represents only a stimulus (mathematically interpreted as an impulse).