## 2 Methods  
It is well-known that Hertzian theory can also be applied to contacts between elastic bodies whose surfaces are differently curved, one convex and the other concave. Such geometrical relations are present in a contact between a sphere and a spherical socket. If we denote the radii of curvature of the sphere and socket by \(R_{1}\) and \(R_{2}\), the young's moduli by \(E_{1}\) and \(E_{2}\), and the Poisson's ratios by \(v_{1}\) and \(v_{2}\), Hertzian theory gives the following solution:  
\[F_{N}=\frac{4}{3}E^{*}\frac{a^{3}}{R^{*}}, \tag{1}\]  
\[\delta=\frac{a^{2}}{R^{*}}, \tag{2}\]  
where \(F_{N}\) is the applied normal force, \(a\) the contact radius and \(\delta\) the penetration/indentation depth. \(E^{*}\) and \(R^{*}\) are defined by  
\[\frac{1}{E^{*}}=\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}\ \ \text{and}\ \ R^{*}=\frac{R_{1}R_{2}}{R_{2}-R_{1}}=\frac{R_{1}R_{2}}{\Delta R}. \tag{3}\]  
However, this solution is only valid if the contact is non-conformal (incomplete). Among other things, this means that the contact radius must be much smaller than both the radii of curvature of the two surfaces, as well as the dimensions of each body. If the clearance \(\Delta R\) between the sphere and the spherical socket is very small, the size of the contact area increases rapidly when a normal force is applied and can quickly reach the dimensions of the bodies themselves. In this case, an almost conformal contact is present for which the application of the half-space approximation is no longer valid. An indication that the solution according to Equation (1) does not hold for the limiting case \(R_{1}\to R_{2}\) is immediately obvious, as in this case \(R^{*}\) approaches infinity, i.e., the transmission of a finite force would require an infinitely large contact radius. As mentioned above, there is not yet a closed-form analytical solution to this problem. In the following, we present an approximate model, which provides quite simple closed-form analytical solutions. It consists of a rigid sphere indenting a radially directed Winkler foundation, illustrated in Figure 1 on the left, while its kinematics is shown on the right. Straightforward application of the cosine theorem leads to the kinematic relation  
\[R_{1}^{2}=\left(\Delta R+\delta\right)^{2}+\left(R_{2}+u_{r}(\varphi)\right)^{ 2}-2(\Delta R+\delta)(R_{2}+u_{r}(\varphi))\cos\varphi, \tag{4}\]  
where \(u_{r}(\varphi)\) denotes the radial displacement of the foundation at angle \(\varphi\). The quadratic equation in \(R_{2}+u_{r}(\varphi)\) can be solved in an elementary way, yielding the radial displacement as a function of the angle:  
\[u_{r}(\varphi)=\left(\Delta R+\delta\right)\left(\cos\varphi+\sqrt{\left( \frac{R_{1}}{\Delta R+\delta}\right)^{2}-\sin^{2}\varphi}\right)-R_{2}. \tag{5}\]  
Equation (5) was first given by [PERSON] et al. (2016) to represent the kinematics of metal-on-plastic hip implants and has been adopted in other works as well (e.g., ([PERSON] et al., 2018)). As an almost-conformal contact is assumed, allowing for the constraint \(R_{i}\gg\Delta R\) (with \(i=1,2\)), and small displacements and thus a small indentation depth \(R_{i}\gg\delta\) are assumed, \(\sin^{2}\varphi\) under the root can be neglected with respect to the first term. In this way, a very simple, yet sufficiently accurate relationship is obtained for the almost conformal contact  
\[u_{r}(\varphi)=\delta\cos\varphi-\Delta R(1-\cos\varphi). \tag{6}\]  
At the edge of the contact, the radial displacement must be zero, i.e.,  
\[u_{r}(\varepsilon)\overset{!}{=}0, \tag{7}\]  
where \(\varepsilon\) denotes the half-contact angle. From this condition, the half-contact angle can be related to the indentation depth  
\[\cos\varepsilon=\frac{\Delta R}{\Delta R+\delta}. \tag{8}\]  
Figure 1: A rigid ball indenting a radial directed Winkler foundation of a spherical cup: Three-dimensional view (**left**), sectional view including kinematics of contact (**right**).