Viscoelastic Behavior of Periodontal Ligament: Stresses Relaxation at Translational Displacement of a Tooth Root

  • S. Bosiakov
  • G. Mikhasev
  • S. RogosinEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


Understanding of viscoelastic response of a periodontal membrane under the action of short-term and long-term loadings is important for many orthodontic problems. A new analytic model describing behavior of the viscoelastic periodontal ligament after the tooth root translational displacement based on Maxwell approach is suggested. In the model, a tooth root and alveolar bone are assumed to be a rigid bodies. The system of differential equations for the plane-strain state of the viscoelastic periodontal ligament is used as the governing one. The boundary conditions corresponding to the initial small displacement of the root and fixed outer surface of the periodontal ligament in the dental alveolus are utilized. A solution is found numerically for fractional viscoelasticity model assuming that the stress relaxation in the periodontal ligament after the continuing displacement of the tooth root occurs approximately within five hours. The character of stress distribution in the ligament over time caused by the tooth root translational displacement is evaluated. Effect of Poisson’s ratio on the stresses in the viscoelastic periodontal ligament is considered. The obtained results can be used for simulation of the bone remodelling process during orthodontic treatment and for assessment of optimal conditions of the orthodontic load application.


Viscoelastic periodontal ligament Translational displacement Root of the tooth Stress relaxation 

Mathematics Subject Classification (2010)

74L15 35Q92 45E10 



This paper is the result of project implementation: TAMER “Trans-Atlantic Micromechanics Evolving Research: Materials containing inhomogeneities of diverse physical properties, shapes and orientations” supported by FP7-PEOPLE-2013-IRSES Marie Curie Action “International Research Staff Exchange Scheme”. It is also supported by Belarusian Fund for Fundamental Scientific Research (grant F17MS-002).


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Belarusian State UniversityMinskBelarus

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