Analytical Modeling of the Viscoelastic Behavior of Periodontal Ligament with Using Rabotnov’s Fractional Exponential Function

  • Sergei BosiakovEmail author
  • Sergei Rogosin
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 343)


The mathematical modeling of a stress-strain state of the viscoelastic periodontal membrane is carried out. Internal and external surfaces of the periodontal ligament are described by a symmetrical two-sheeted hyperboloid. Tooth root is assumed to be a rigid body. Displacements of points on the internal surface of the periodontal ligament coincide with the displacements of the corresponding points of the external surface of the tooth root. The relationships between the displacements and strains for periodontal ligaments are formulated as an assumption that the periodontal tissue approaches to incompressible materials. Viscoelastic constitutive law with a fractional exponential kernel for periodontal ligament was used. The equations of motion for the periodontal ligament relative to translational displacements and rotation angles of its points are derived. In the particular case the vertical translational motion of the tooth root, as well as corresponding displacements are analyzed. Constants of the fractional viscoelastic function were assessed on the basis of the experimental data about the behavior of the periodontal ligament. The obtained results can be used to determine a load for orthodontic tooth movement corresponding to the optimal stresses, as well as to simulate bone remodeling on the basis of changes of stresses and strains in the periodontal ligament during orthodontic movement.


Periodontal ligament Tooth root Viscoelastic model Fractional exponential function Translational displacement 



The research is supported by the FP7 IRSES Marie Curie grant TAMER No 610547. The authors are thankful to professor Francesco Mainardi and to professor Ivan Argatov for valuable discussions of the results of the paper.


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Theoretical and Applied MechanicsBelarusian State UniversityMinskBelarus
  2. 2.Institute of Mathematics, Physics and Computer SciencesAberystwyth UniversityPenglais, Aberystwyth CeredigionUK
  3. 3.Department of EconomicsBelarusian State UniversityMinskBelarus

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