## 3 Application to Total Hip Arthroplasty with UHMWPE Liner  
As an application example, we consider the contact in total hip arthroplasty (THA) with a stiff steel head and a soft plastic liner in the acetabular cup. As the clearance between head and inner cup radius is very small, the contact can be described as almost conformal. Typically, the liner sits inside a metallic acetabular shell that is fixated to the hip bones. Thus, the maximum outer cup radius is limited by individual anatomy. Table 1 summarizes the available head radii and liner thicknesses as well as the commonly used radial clearances and outer cup radii. The traditionally used plastic material for the plastic liner is ultra-high molecular weight polyethylene (UHMWPE). Recently, radiation cross-linking of UHMWPE has been used to increase the wear resistance. Even though the radiation changes all mechanical properties slightly, the elastic modulus is measured in the range 0.5-1 GPa for both conventional and cross-linked UHMWPE [13; 28]. For the Poisson's ratio, values of 0.4-0.46 are commonly assumed [13; 29]. The material properties used for the application example are shown in Table 2.  
A convenient way to configure the new model for mapping the normal contact in total hip arthroplasty is to adopt the foundation modulus from already established models. One possibility would be to take the foundation modulus from the famous model of [PERSON] et al. [16], which reads  
\[K_{\text{B}}=\frac{2(1-2\
u)+(1+\
u)\kappa^{3}}{(1+\
u)(1-2\
u)(1-\kappa^{3})} \cdot\frac{E}{R_{2}}\quad\text{with}\quad\kappa:=\frac{R_{2}}{R_{2}+L}. \tag{14}\]  
In contrast to the original model of [PERSON] et al., which must be evaluated numerically, very simple closed-form analytical solutions are obtained. In the range of small radial clearances (almost conformal contact), both models give almost the same results. However, it is well-known that the stiffness of the model of [PERSON] et al. originates from the elastic spherical vessel under uniform (internal and external) pressure and thus is somewhat larger than that of the liner. Noticeable deviations occur, especially for larger Poisson's ratios [17; 18; 30]. This is already evident from Equation (14), as for incompressible material, \(\
u\to 1/2\), the foundation modulus becomes unbounded. The same holds for the foundation modulus of the constrained column model, which in the following is also called bonded-layer-model. It is defined by  
\[K_{\text{BL}}=\frac{(1-\
u)}{(1+\
u)(1-2\
u)}\cdot\frac{E}{L}. \tag{15}\]  
and originates from the indentation into a very thin plane compressible layer bonded to a rigid substrate. In addition to the constraint \(\
u<0.45\), it is assumed that the linear dimensions of the contact area are much larger than the layer thickness to justify neglecting in-plane deformations and to ensure constrained uniaxial compression [24]. The model \begin{table}
\begin{tabular}{c c c c} \hline
**Head Radius** & **Liner Thickness** & **Radial Clearance** & **Outer Cup Radius** \\ \(R_{1}\) & \(L\) & \(\Delta R\) & \(R_{2}+L\) \\ \hline
11–23 mm & 3–15 mm & 0.05–0.2 mm & 16–26 mm \\ \hline \end{tabular}
\end{table} 
Table 1: Geometrical properties in THA [13; 26; 27]. \begin{table}
\begin{tabular}{c c c} \hline
**Material** & **Elastic Modulus (GPa)** & **Poisson’s Ratio** \\ \hline Steel & 230 & 0.3 \\ UHMWPE & 1 & 0.4, 0.45 \\ \hline \end{tabular}
\end{table} 
Table 2: Material properties. is widely applied to map the indentation into thin cartilage layers [19; 20; 21; 22]. However, its application to ball-in-socket contact is questionable, at least for larger liner thicknesses that reach the order of linear dimensions of the contact area. For the above reasons, we avoid adopting the foundation modulus from other models here. To determine the foundation modulus (stiffness of the foundation per unit area) and to validate the predictions of the new model, an axisymmetric finite element analysis was conducted with the software Abaqus FEA.