On the application of a viscoelastic model with Rabotnov’s fractional exponential function for assessment of the stress-strain state of the periodontal ligament

Abstract

In the present paper, the mathematical modeling of the viscoelastic periodontal membrane is carried out. Internal surface of the periodontal ligament is adjacent to the outer surface of the tooth root, the geometric shape of which is described by the equation of an elliptic hyperboloid. The external surface of the periodontal membrane is shifted along the normal to the outer surface of the root and fixed on the dental alveolus bone. Relationships between displacements and deformations of the periodontal ligament are formulated with due account for incompressibility of the periodontal tissue. Viscoelastic properties of the periodontium are described by the relaxation kernel with Rabotnov’s fractional exponential function. A system of equations of motion in terms of the translational displacements and rotation angles of the tooth root is obtained. Particular cases of the equations of motion corresponding to the translational motion in the vertical and horizontal directions are formulated.