title: Viscoelasticity of periodontal ligament: an analytical model.
abstract: BACKGROUND
  Understanding of viscoelastic behaviour of a periodontal membrane under physiological conditions is important for many orthodontic problems. A new analytic model of a nearly incompressible viscoelastic periodontal ligament is suggested, employing symmetrical paraboloids to describe its internal and external surfaces.
  
  
  METHODS
  In the model, a tooth root is assumed to be a rigid body, with perfect bonding between its external surface and an internal surface of the ligament. An assumption of almost incompressible material is used to formulate kinematic relationships for a periodontal ligament; a viscoelastic constitutive equation with a fractional exponential kernel is suggested for its description.
  
  
  RESULTS
  Translational and rotational equations of motion are derived for ligament's points and special cases of translational displacements of the tooth root are analysed. Material parameters of the fractional viscoelastic function are assessed on the basis of experimental data for response of the periodontal ligament to tooth translation. A character of distribution of hydrostatic stresses in the ligament caused by vertical and horizontal translations of the tooth root is defined.
  
  
  CONCLUSIONS
  The proposed model allows generalization of the known analytical models of the viscoelastic periodontal ligament by introduction of instantaneous and relaxed elastic moduli, as well as the fractional parameter. The latter makes it possible to take into account different behaviours of the periodontal tissue under short- and long-term loads. The obtained results can be used to determine loads required for orthodontic tooth movements corresponding to optimal stresses, as well as to simulate bone remodelling on the basis of changes in stresses and strains in the periodontal ligament caused by such movements.
authors: Department of Mechanics and Mathematics, Belarusian State University, 4, Nezavisimosti Avenue, Minsk, 220030 Belarus.
authors: Sergei M
authors: 0000-0003-4524-9304
authors: SM
authors: Bosiakov
authors: Department of Economics, Belarusian State University, 31, K. Marx, Minsk, 22030 Belarus.
authors: Anna A
authors: AA
authors: Koroleva
authors: Institute of Mathematics, Physics and Computer Science, Department of Mathematics, Aberystwyth University, Penglais, Aberystwyth Ceredigion, SY23 3BZ UK.
authors: Sergei V
authors: SV
authors: Rogosin
authors: Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Leicestershire, LE11 3TU UK.
authors: Vadim V
authors: VV
authors: Silberschmidt
keywords: Fractional exponential function; Periodontal ligament; Tooth root; Translational displacement; Viscoelastic model