## 2 Methods
The contact is considered as frictionless, so that no tangential stresses occur in the spherical contact area. The spring stresses of the Winkler foundation sketched on the left in Figure 1 are proportional to the spring displacements in the radial direction  
\[p_{r}(\varphi)=K\,u_{r}(\varphi), \tag{9}\]  
where \(K\) represents the foundation modulus. Note, that it is also called \"subgrade reaction modulus\" and has the unit of measure \(\text{N}/\text{m}^{3}\). It represents the elastic modulus of the foundation divided by the thickness of the foundation. The incremental contact force \(dF_{\text{C}}\) in _vertical direction_ produced by the pressure acting on an annular ring element of the spherical surface corresponding to an angle \(\varphi\) reads  
\[\delta=\frac{a^{2}}{R^{*}}, \tag{2}\]0  
The whole contact force and thus the normal force result from the sum of the contributions of all loaded annular elements  
\[\delta=\frac{a^{2}}{R^{*}}, \tag{2}\]1  
After substituting Equation (9) into Equation (11) and taking Equation (6) into account, the integration yields  
\[\delta=\frac{a^{2}}{R^{*}}, \tag{2}\]2  
Thus, the normal force as a function of the half-contact angle is determined. Equation (8) can then be used to substitute the half-contact angle, which results in the normal force as a function of the indentation depth  
\[\delta=\frac{a^{2}}{R^{*}}, \tag{2}\]3  
Equations (8) and (12) provide an extremely simple closed-form analytical approximate solution for the almost conformal contact between a sphere and a spherical cup. Hence, the relationships between the macroscopic quantities normal force, indentation depth, and half-contact angle are completely defined. Moreover, the pressure distribution is given in a simple way via Equation (9) by taking Equation (6) into account. The proposed model can be applied to approximate the almost conformal contact of ball joints in mechanical engineering as well as ball-and-socket joints in biomechanics, e.g., (artificial) hip and shoulder joints. One comparably simple model that also provides closed-form analytical solutions was developed by [PERSON] et al. [25]. Unfortunately, without any physical explanation, they mapped the contact within a spherical joint with clearance onto an equivalent contact of a rigid cone and a plane Winkler foundation whose thickness is given by the radius of the original spherical cavity. They have even presumed a priori that the contact stress distribution in vertical direction along the profile of the rigid cone is ellipsoidal, as given by Hertzian theory. After fitting the maximum pressure by an introduced parameter, the force as a function of indentation depth obtained with their model agrees well with the results of an FE simulation, especially for small clearances. However, although they investigate the influence of the radial clearance, they do not vary the thickness of the spherical shell. In addition, it is explicitly stated that identical linear elastic materials are assumed. In the following, the proposed model will be adapted to adequately simulate the frictionless normal contact for commonly used hard-on-soft hip implants. For this purpose, the foundation modulus must be determined.