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. 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. 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. 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.

中文翻译：

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牙周膜的粘弹性：一种分析模型。

了解牙周膜在生理条件下的粘弹性行为对于许多正畸问题很重要。提出了一种几乎不可压缩的粘弹性牙周膜的新分析模型，该模型采用对称的抛物面来描述其内表面和外表面。在模型中，假定牙根是刚体，在其外表面和韧带的内表面之间具有完美的结合。几乎不可压缩的材料的假设被用来建立牙周膜的运动学关系。建议使用具有分数指数核的粘弹性本构方程进行描述。得出了韧带点的运动平移和旋转方程，并分析了牙根的平移位移的特殊情况。基于对牙周膜对牙齿平移的响应的实验数据来评估分数粘弹性函数的材料参数。定义了由牙根的垂直和水平平移引起的韧带中的静水应力分布特征。通过引入瞬时弹性模量和松弛弹性模量以及分数参数，提出的模型可以对粘弹性牙周膜的已知分析模型进行推广。后者使得考虑短期和长期负荷下牙周组织的不同行为成为可能。获得的结果可用于确定与最佳应力相对应的正畸牙齿运动所需的载荷，