Analytical Modelling of the Tooth Translational Motions: Comparative Analysis for Different Shapes of Root

Abstract

The orthodontic treatment planning may be carried out based on the finite element and analytical models of periodontal ligament (PDL). For analytical modelling of the PDL behavior the shape of the tooth root mainly was approximated by circular or elliptical paraboloid. Another shape of the tooth root is the elliptical two-sheeted hyperboloid. Semi-axes of the tooth root cross-section in the shape of the elliptical paraboloid and two-sheeted hyperboloid on the alveolar crest level are the same, but the shape of a two-sheeted hyperboloid allows employing the additional parameter for describing the root apex rounding. The aim of this study is the comparative analysis of the hydrostatic stresses patterns during the tooth root translational displacements in the almost incompressible PDL for the root in the shape of the elliptical paraboloid and two-sheeted hyperboloid. As a result, patterns of the hydrostatic stresses in the PDL during translational displacement are nearly identical for the tooth root in the shape of a paraboloid and the tooth root in the shape of a two-sheeted hyperboloid with the rounded apex of the tooth root. The translational movement of the tooth root with a pointed apex leads to the higher hydrostatic stresses in the PDL compared with the tooth root with a rounded apex. The obtained results indicated that the rounding of the tooth root should be considered during planning of orthodontic treatment.

Appendix

$$\begin{aligned} \begin{aligned} a_{11}^{(k)} = -A \iint \limits _{F_k} (\sin (\alpha _k) (G_k (2 \nu -1) n_z^{(k)} + 2 H_k (\nu -1) n_x^{(k)})+\cos (\alpha _k) n_y^{(k)} (1-2 \nu )) dF_k, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} a_{16}^{(k)} = a_{61}^{(k)} = A \iint \limits _{F_k} (\sin (\alpha _k ) (G_k^3 (2 \nu -1) n_z^{(k)} y + H_k (2 G_k^2 (\nu -1) n_x^{(k)} y+ n_y^{(k)} x (1-2 \nu ))+ \\ \nonumber + G_k H_k^2 (2 \nu -1) n_z^{(k)} y+2 H_k^3 (\nu -1) n_x^{(k)} y) +\cos (\alpha _k) (n_y^{(k)} y-2 \nu (n_x^{(k)} x+n_y^{(k)} y))) dF_k, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} a_{22}^{(k)} = A \iint \limits _{F_k} (2 (\nu -1) n_y^{(k)} \cos (\alpha _k) -(2 \nu -1) \sin (\alpha _k) (G_k n_z^{(k)} +H_k n_x^{(k)})) dF_k, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} a_{34}^{(k)} = a_{43}^{(k)} = - A \iint \limits _{F_k} (\cos (\alpha _k) (G_k^2 (1-2 \nu ) n_y^{(k)} y+ \\ + H_k^2 (1-2 \nu ) n_y^{(k)} y-2 \nu n_z^{(k)} z)+\sin (\alpha _k) (2 G_k^3 (\nu -1) n_z^{(k)} y + \\ +G_k^2 H_k (2 \nu -1) n_x^{(k)} y+G_k (2 H_k^2 (\nu -1) n_z^{(k)} y +(1-2 \nu ) n_y^{(k)} z)+H_k^3 (2 \nu -1) n_x^{(k)} y)) dF_k, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} a_{44}^{(k)} = A \iint \limits _{F_k} (\cos (\alpha _k) (G_k^2 (2 \nu -1) y (n_y^{(k)} y-n_z^{(k)} z)+ \\ \nonumber +H_k^2 (2 \nu -1) y (n_y^{(k)} y-n_z^{(k)} z)+2 z (\nu n_y^{(k)} z+\nu n_z^{(k)} y-n_y^{(k)} z))+ \\ +\sin (\alpha _k) (-2 G_k^3 y (\nu n_y^{(k)} z+(\nu -1) n_z^{(k)} y)+G_k^2 H_k (1-2 \nu ) n_x^{(k)} y^2+\\+G_k ((2 \nu -1) z (n_y^{(k)} y-n_z^{(k)} z)-2 H_k^2 y (\nu n_y^{(k)} z+(\nu -1) n_z^{(k)} y))-\\ -H_k (2 \nu -1) n_x^{(k)} (H_k^2 y^2+z^2))) dF_k, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} a_{55}^{(k)} = - A \iint \limits _{F_k} (\sin (\alpha _k) (G_k (n_z^{(k)} (2 (\nu -1) x^2+\\ \nonumber +(2 \nu -1) z^2)+n_x^{(k)} x z)+H_k (n_x^{(k)} ((2 \nu -1) x^2+ \\ +2 (\nu -1) z^2)+n_z^{(k)} x z))-(2 \nu -1) n_y^{(k)} \cos (\alpha _k) (x^2+z^2)) dF_k, \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} a_{66}^{(k)} = A \iint \limits _{F_k} (\cos (\alpha _k) (n_x^{(k)} x y +n_y^{(k)} (2 (\nu -1) x^2+(2 \nu -1) y^2))+\\ \nonumber +\sin (\alpha _k) (H_k (n_x^{(k)} ((1-2 \nu ) x^2-2 G_k^2 (\nu -1) y^2)+\\+n_y^{(k)} x y (-2 (G_k^2-1) \nu -1))-G_k (2 \nu -1) n_z^{(k)} (G_k^2 y^2+x^2)+\\+G_k H_k^2 (1-2 \nu ) n_z^{(k)} y^2-2 H_k^3 y (\nu n_x^{(k)} y+\nu n_y^{(k)} x-n_x^{(k)} y))) dF_k, \end{aligned} \end{aligned}$$

$$\begin{aligned} A=\frac{E}{2\delta (1+\nu )(1-2\nu )}, G_k=\frac{\partial F_k}{\partial x}, H_k=\frac{\partial F_k}{\partial z}, k=1, 2. \end{aligned}$$