## 3 Application to Total Hip Arthroplasty with UHMWPE Liner
### Results and Discussion
Figure 4: Comparison of the FEM solutions with Hertzian theory, [PERSON]’s analysis, results from the bonded-layer-model, and the new proposed model, respectively, for a Poisson’s ratio of 0.45, inner cup radius 18 mm, clearance 0.1 mm, and liner thickness 6.75 mm: normal force as a function of indentation depth (**left**), and pressure distribution as a function of angular coordinate for a normal force of 3000 N (**right**). Figure 5: Comparison of the normal force \(F_{N}\) against normal displacement \(\delta\) and contact pressure against angular coordinate \(\varphi\) or half arc length \(R_{1}\varphi\) relations between FEM solution and analytical model. (**a,b**) different radial clearances \(\Delta R\) with \(R_{1}=16\) mm and \(L=6\) mm, (**c,d**) different head radii \(R_{1}\) with \(\Delta R=0.1\)mm and \(L=6\) mm, and (**e,f**) different liner thicknesses \(L\) with \(\Delta R=0.1\) mm and \(R_{1}=16\) mm. As shown in Figure 5a,b, increasing the clearance decreases the contact stiffness and contact angle while increasing the maximum contact pressure. Figure 5c,e show that the overall contact stiffness is increased by increasing the head radius at constant liner thickness and decreasing the liner thickness at constant head radius. In Figure 6, the trends with respect to the maximum contact pressure and the contact area can be studied more thoroughly. Both were calculated at a normal force of 3000 N for geometries within the limitations listed in Table 1. The contact area can be determined from the head radius and the half-contact angle by  
\[A=2\pi R_{1}^{2}(1-\cos\varepsilon). \tag{19}\]  
The lowest contact pressures are achieved for large head radii with a moderate liner thickness. The opposite trend is found for the contact area, where smaller head radii in combination with thin liners give the lowest values. In the design of the contact pairs in THA with soft UHMWPE cups, various factors must be taken into account. In addition to low wear rates, the avoidance of edge loading due to the inclination of the cup, the risk of dislocation, and the risk of cup fracture are some important criteria [31; 32]. With the proposed analytical model, it is possible to investigate significant influencing factors especially regarding the wear rate. Recent experimental investigations of polyethylene wear indicate that minimizing the contact area is more important for low wear rates than minimizing the contact pressure [31; 32; 33]. As shown in Figures 5 and 6, this can be achieved either by using small head radii and small liner thicknesses or increasing the radial clearance. However, the latter option may be disadvantageous, as another major influence factors for wear, the lubrication film thickness, is reducing significantly for increasing clearances [13]. Of course, the proposed analytical model in the current form can only serve as a first assessment in the dimensioning for THA with respect to a given wear law. For a thorough design, important factors such as the inclination of the cup, a lubrication regime, and explicit wear calculations must be considered as well.