## 1 Introduction
From contact mechanics point of view, the hip joint implant can be considered by a ball-in-socket model, where the head and the cup have approximately but not exactly the same radius of curvature, so that a small radial clearance remains (almost conformal contact). [PERSON] [15] applied the usual procedure for non-conformal contacts and approximates the ball-in-socket model by an equivalent ball-on-plane model, which is solved numerically using the fundamental solution for the indentation into a linear elastic bonded layer. However, while the contact radius is insensitive to the mapping procedure, significant differences in the maximum pressure arises compared to FE calculations. The most frequently adopted model for a hard-on-soft implant such as a metal head on (metal backed) plastic was developed by [PERSON] et al. [16]. It is based on the solution of the elastic spherical vessel under uniform (internal and external) pressure, which implies that the tangential displacements present in the plastic layer are neglected. Several important findings emerged from the work of [PERSON] et al., e.g., bonding of the plastic layer on the metal avoids higher tensile stresses at the edge of the contact zone or the minimum thickness of the plastic layer (liner) should not be less than 4 mm. One problem of the model is that it does not provide a closed-form analytical solution for the force as a function of indentation depth; instead, a numerical solution is required. In addition, it is not applicable to liner materials with a Poisson's ratio close to 0.5 (incompressible materials). [PERSON] et al. [17] have demonstrated that already at a Poisson's ratio of 0.46 the results from [PERSON]'s model clearly deviate from FE results. Thus, they introduced a correction factor, which is a linear function of the ratio of inner to outer radius of the cup, and was obtained by fitting on the maximum contact pressure of corresponding FE simulations. As the correction factor is always smaller than 0.883, it is concluded that the model of [PERSON] et al. considerably overestimates the stiffness. [PERSON] et al. [18] performed a parametric analysis of the contact stress in ultra-high molecular weight polyethylene (UHMWPE) acetabular cups by means of a simplified model which uses the constrained column concept in radial direction (radial acting Winkler foundation) with stiffness taken from the model of [PERSON] et al. They studied the effects of layer thickness, radial clearance, radii of femoral head and cup, as well as contact area on maximum pressure. However, they also noted that values obtained from FEM were 10 to 20% higher than predicted by the simplified model. It is also common to adopt the original constrained column model that is often used to map contacts with thin cartilage layers attached to bones. In this model, the thin (plane) compressible elastic layer bonded to a rigid substrate acts similar to a linear Winkler foundation and its stiffness is specified by the modulus of constrained uniaxial compression [19; 20; 21; 22; 23]. However, the application of this bonded-layer-model generally requires certain assumptions to be satisfied. The Poisson's ratio should not exceed 0.45 as the stiffness becomes unbounded in the limit of an incompressible material. Furthermore, the layer thickness must be much smaller than the linear dimensions of the contact area [18; 24]. The latter condition is fulfilled for some but by no means all common geometric pairings of femoral head and acetabular cup in total hip arthroplasty. In the present work, an approximate model for the almost conformal elastic contact between a sphere and a spherical cup is proposed, which leads to simple closed-form analytical solutions. As an application example, the frictionless normal contact of artificial hip joints is considered. It is demonstrated that the new model is able to predict the local and global contact mechanical quantities with great accuracy for all commonly used hard-on-soft hip implants.