Viscoelastic behavior of periodontal ligament: Stresses relaxation at translational displacement of a tooth root

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.