A micromechanically-based interface model for the periodontal ligament

Abstract

An interface, 3D finite element is presented, specifically designed to simulate the mechanical behavior of the periodontal ligament (hereafter shortened into PDL). The PDL is a thin layer (about 0.25 mm in adult humans) of soft tissue that connects the root of a tooth to the surrounding alveolar bone. From the mechanical viewpoint, it can be considered as a thin interface made by a solid phase, consisting mainly of collagen fibers, immersed into a so-called ground substance. Previous work [1] attempted its finite element modeling in terms of a nonlinear interface description, based, however, on the simple fitting of experimental results. Here we attempt to substantially improve on what done in [1], by developing an interface element whose constitutive behavior is obtained from a simple, yet reasonably complete micromechanical model. This is based on an idea described in [2], but further extended to cover the case of a 3D fiber arrangement. The model is defined by a 2 cable, X-shaped structure, each cable representing a large number of collagen fibers, each of a different length, and presenting a crimped geometry at rest. Until a fiber is not fully uncrimped, it offers no stiffness contribution. The cables, in the model, are inclined, both with respect to the direction of the root main axis, and with respect to the direction orthogonal to the root’s axis, so as to be able to catch all the stiffness contributions provided in reality by the PDL. In each cable, a single collagen fiber is governed by a local (microscopic) non-linear stress-strain relationship, which includes a failure strain in tension. Simple statistical integrations over the length of each fiber, as described in [3], allow us to obtain a macroscopic behavior, coupled in tension and shear, including a toe stiffness, the subsequent quasi-linear stiffness, a peak macroscopic stress and strain, and a post-peak behavior until the complete failure of the element. The behavior in compression, much simpler, is governed by the same phenomenological equations as described in [1]. We can thus take into account, in a complex FE model of the tooth-bone system, several features of the PDL at the microscopic level, as well as study their influence on the overall response, on the basis of a relatively simple model, whose parameters have a physical meaning.