Fig。 1 – The four 3D finite element models of the edentulous mandible and prosthetic components: (A) represents model A (single-implant model); (B) represents model B (two-implant model); (C) represents model C (three-implant model); and (D) represents  model D (four-implant model)。

and titanium cap。 The abutment and cap were made of Ti6Al4V titanium alloy, as was the implant。 The material properties of the cortical and cancellous bone, mucosa and prosthetic components were determined from values obtained from the literature (Table 2)。 All materials were assumed to be isotropic, homogeneous and linearly  elastic。

2。3。 Contact management and loading conditions

Implants were considered totally osseointegrated。 Therefore, a   mechanically   perfect   interface   was   presumed   to exist

between implant and bone。 However, the interface between the overdenture and the mucosa was not fixed when function- ing。 Instead, the overdenture was able to rotate and slide on the mucosa in different directions。 To simulate this displacement, we assumed that sliding friction existed between the over- denture and mucosa。 The coefficient of sliding friction between the overdenture and mucosa was set to be 0。334 in accordance with previous experiments carried out by our team。27

The models were constrained at the nodes on the mesial and distal bone in all degrees of freedom。 Three types of load were applied to the overdenture in each model to simulate functional loading, namely 100 N vertical and inclined loads on the left first molar and 100 N vertical load on the lower incisors。 To facilitate discussion, the three loading conditions have been abbreviated as VM, IM and VI for vertical load on the left first molar, inclined load on the left first molar and vertical load on the lower incisors, respectively。 IM refers to a 45-8 angled force buccolin- gually applied at the centre of the left first molar。

3。 Results

3。1。 Strain distribution in peri-implant cortical bone

Maximum equivalent strains in the cortical bone around implant under three types of load for each model is shown in Table 3。 Strain distributions in the peri-implant cortical bone of each model under three loading conditions are illustrated in Figs。 2–5。 Under all three loading conditions, the maximum strain values were below 2500 me in all models。 In models A, C and D, the peak strain values in the cortical bone showed an increasing trend as the number of implants increased, and the

Loading condition Model A Model B Model C Model D

VM 474。5 535。9 843。3 835。4

IM 1320 1180 1609 2082

VI 606。6 1340 992。3 1323

Fig。 2 – Equivalent strain distribution in the cortical bone of model A under three loading conditions ((A) VM, (B) IM and (C) VI)。 Colours indicate level of strain from dark blue (lowest) to red (highest)。 The arrows show the sites at which peak strain values occur。

Fig。 3 – Equivalent strain distribution in the cortical bone of model B under three loading conditions ((A) VM, (B) IM and (C) VI)。 Colours indicate level of strain from dark blue (lowest) to red (highest)。 The arrows show the sites at which peak strain values occur。