Finite element model

Three-dimensional finite element (FE) models were based on those reported by Kasai et al. and consisted of a mandible, natural teeth with periodontal ligaments, and titanium implants with superstructures. All elements were homogenous and isotropic. In the models, eight implants replaced all of the premolars and molars (Fig. 1).

The mass/volume and the shape of the mandible were assumed to be 2 and B, respectively, according to the classification of Lekholm and Zarb. The implant fixtures were 3.75 mm in diameter and 10 mm in length. The dimensions of the natural teeth and periodontal ligaments were based on the literature. The surface area of the periodontal ligament (PDL) corresponded to the anatomical value, and its thickness was 0.25 mm at all sites. The occlusal surfaces of the implants and the teeth were simplified and flattened in agreement with Monson’s sphere. The FE model consisted of approximately 42,000 nodes and 210,000 tetrahedral elements.

The properties of the materials, except for the PDL, were based on previous studies (Table 1). The biphasic properties for the PDL were determined according to the literature. The PDL was assigned two-phase properties. Young’s modulus and Poisson’s ratio were 0.33 MPa and 0.3 for phase 1, respectively. For phase 2, they were 16 MPa and 0.45, respectively. Phase 2 was applied when the von Mises stress exceeded 0.025 MPa. The load-displacement curve of the teeth was verified with the analysis described below (Fig. 2a).