Viscoelasticity of periodontal ligament: an analytical model

Abstract

BackgroundUnderstanding 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.MethodsIn 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.ResultsTranslational 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.ConclusionsThe 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.

The effects of a periodontal ligament analogue on occlusal contact forces

“…1,2 Moreover, as are most biological tissues, the PDL is characterised as a viscoelastic (VE) material. 3,4 Viscoelastic materials exhibit time-dependent mechanical behaviours, with 'creep' and 'recovery' specifically relevant to this study. Typically, while a constant force is applied to a VE material, it tends to continuously deform (creep).…”

Modelling and evaluating periodontal ligament mechanical behaviour and properties: A scoping review of current approaches and limitations

“…Constitutive modelling of the PDL refers to the material response to various loading, where stress conditions in the periodontium may be expressed as a system of differential equation 9 and has been used to develop non-linear, porous, viscous and fibrous PDL material models. For example, the PDL has been modelled 31 Similarly, another material model 23 describes the porous and fibrous nature of the PDL under various loading conditions. In addition, we see comprehensive numerical study for the PDL stress relaxation behaviour based on different simpler models, such as Burgers model, Five-Parameter model, Schapery's method and modified superposition concludes that the modified superposition model is a better choice in predicting the PDL stress relaxation or viscoelastic characteristics for a variety of load applications.…”

Frequency-related viscoelastic properties of the human incisor periodontal ligament under dynamic compressive loading

“…The PDL consists of collagen fibers and a liquefied ground substance of proteoglycans and glycoproteins, which convey its unique mechanical viscoelastic properties [ 2 ]. In fact, the PDL composite structure is the basis for the manifestation of a nonlinear and time-dependent mechanical behavior in this tissue [ 3 – 5 ].…”