## 3 Results
As it was proven in Table 2 and Figure 6, after four weeks the orthodontic process can be considered in steady-state regime (the value of the \(y_{00}^{\prime}(t,s)\) signal, for any value of (s) independent variable, reaches a value higher than 95% compared to the steady state value (the value of \(y_{00}^{\prime}(t,s)\) signal reached after seven weeks); practically the output signal, after four weeks, presents value variations which are enclosed between the steady state band of \(\pm 5\%\) near its steady state value). In this context, for the process dynamics, at a first analysis, it is relevant to consider the first four weeks of the treatment. A more complex analysis will be presented next, based on the simulation results. For every \(\mathrm{u}_{0}(\mathrm{t}_{\rm k})=\mathrm{constant}\), the iterative increasing with \(\lambda=(1,2,3,\cdots)\) of the \(\mathrm{s}_{\lambda}=\widetilde{\rm s}+\lambda\cdot\mathrm{As}_{\rm c}\) variable is ensured, where the rotation center testing step (\(\Delta\mathrm{s}_{\rm c}\)) is chosen small enough in order for the numerical integration to be made correctly (for example \(\Delta\mathrm{s}_{\rm c}=s_{\rm f}/1000\)). If for a moment in time (\(\mathrm{t}_{\rm k}\)) and obviously for (\(\mathrm{u}_{0}(\mathrm{t}_{\rm k})\)), as a consequence of the iteratively increase of the variable \(s_{\lambda}=\widetilde{\rm s}+\lambda\cdot\Delta\mathrm{s}_{\rm c}\), the equality between the elastic and the plastic moment is obtained \(\left(M_{\rm E}(\mathrm{t}_{\rm k},s_{\lambda})=M_{\rm P}(s_{\lambda})\right)\), it results that the (\(s_{\lambda}\)) value corresponds with the value of the rotation center (\(s_{c}\)). The rotation center will change its position on the extension in the depth of the tooth's fictional shaft, progressively upward through the top of the tooth's apex, as exemplified in Table 3. We must mention that we have used the same  
As presented in Table 3, the tissue resistance (\(\sigma\)) is measured in grf/mm\({}^{2}\), and it can be interpreted as a measure of the tissue's property to oppose the deformation due to the mechanical strain application (during the orthodontic procedure throughout the elastomer usage); (\(\sigma\)) can be also referred to as the strength coefficient of the plastic environment. As an important remark, the (\(\sigma\)) coefficient is only a weighting coefficient of the tissue resistance, not being the tissue resistance itself. In the presented approach, as it results from Equation (14), due to the fact that the coefficients (\(\delta_{0}\)), (\(\delta_{1}\)), (\(\delta_{2}\)) depend on the position of the rotation center (s\({}_{\rm c}\)) and due to the multiplication of the term from the straight brackets (associated to y (\(\delta_{1}\)), (\(\delta_{2}\))) with (\(\sigma\)), it results that the tissue resistance is given by the value of (s\({}_{\rm c}\)) as well. In this context, the equality between the two members of Equation (14) is verified following this explanation: If the tissue resistance is high, according to Figure 7, the value of (s\({}_{\rm c}\)) is also high, implying a smaller value of the (\(\sigma\)) coefficient; if the tissue resistance is small, according to Figure 8, the value of (s\({}_{\rm c}\)) is low, too, implying a higher value of the (\(\sigma\)) coefficient. The tissue resistance is measured in grf/mm\({}^{2}\), since it results as the multiplication between the (\(\sigma\)) coefficient (expressed in grf/mm\({}^{2}\)) and the multiplicative variation of the term from the bracket introduced by the (s\({}_{\rm c}\)) variation (referring strictly to the value variation of the mentioned term, it is a dimensionless one; for example, if the (s\({}_{\rm c}\)) value increases with 30%, than its new value will be 1.3-s\({}_{\rm c}\), and consequently the introduced multiplicative variation is 1.3). In the transcendent Equation (14), the (\(\sigma\)) coefficient is introduced in order to comparatively determine the plastic environmental resistance in relation to the moment of the elastomeric chain.