## 3 Application to Total Hip Arthroplasty with UHMWPE Liner
### Results and Discussion  
In this section, the solutions of the new analytical model are compared with the ones of the previously discussed models and finite element analysis. Furthermore, a parametric study is shown. All results presented comply with the geometrical and material properties listed in Tables 1 and 2, respectively. In a first example, an inner cup radius of \(R_{2}=18\) mm, a liner thickness of \(L=6.75\) mm, and a radial clearance of \(\Delta R=0.1\) mm is chosen to clarify the influence of [PERSON]'s ratio. Figure 3 shows the normal force in terms of normal displacement on the left for different models in comparison with the FEM solution for \(\
u=0.4\). Figure 2: Dimensionless foundation modulus \(p_{0}R_{2}/\delta E\) as a function of normal force \(F_{N}\) (**left**), radial clearance \(\Delta R\) (**middle**), and ratio of inner cup radius to liner thickness \(R_{2}/L\) (**right**) for different Poisson’s ratios as determined by FEM. Dotted lines show the fitted dimensionless foundation modulus \(\widetilde{K}\) used in the model with the constants listed in Table 3.  
As expected, owing to the high degree of conformity, the Hertzian solution according to Equations (1)-(3) is not adequate for the investigated contact problem. For both the original model of [PERSON] et al. [16] and the bonded-layer-model (Note, that the analytical solution from the new model was used but with consideration of the bonded-layer-stiffness according to Equation (15)), deviations from the results of the finite element analysis are observable, but they are still within acceptable limits. The same applies to the pressure distribution as a function of angle illustrated on the right in Figure 3. The maximum pressure for both models is slightly higher than predicted by the FE simulations. On the other hand, the results with the new model considering the foundation modulus according to Equation (18) show excellent agreement with the FE results. Unfortunately, such an excellent agreement does not hold for a Poisson's ratio of 0.45. As already discussed in the last section, this is a consequence of a small error accepting stiffness adjustment for higher Poisson's ratios. Nevertheless, this leads to only minor deviations compared to the FE results, as demonstrated in Figure 4, which is why the model is well suited to represent the conformal contact even in the range of higher Poisson's ratios. It is also evident from Figure 4 that both the model of [PERSON] et al. and the bonded-layer-model produce results that differ significantly from the FE results. Both models react considerably stiffer and overestimate the maximum pressure by 17% and 25%, respectively. In the following, the results obtained from the new model are compared with results of the FE analysis for common geometries of hip joint implants summarized in Table 1. A Poisson's ratio of 0.4 is assumed. Figure 5 shows the normal force in terms of displacement, as well as the contact pressure distribution at 3000 N in comparison with FEM results for a variety of geometries. In all cases, and for both normal force and pressure distribution, the agreement with the FEM solution is very good. Only for the contact angle is a mentionable difference found as the model predictions are generally 3-8% larger than the FEM results. Figure 3: 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 0.4, 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**).