Analytical modelling of the elastomeric layer in soft layer hip replacements

A mechanical analysis is carried out for a flat deformable layer, firmly anchored to a rigid substrate, and frictionlessly compressed by a rigid spherical indenter, with reference to the design of hip replacements whose cup is covered by an elastomeric layer. An available perturbation solution of differential character, developed for a layer subject to plane strain conditions, indented by a cylinder approximated as parabolic in shape and valid for high contact widths, is here extended to the axisymmetric problem of a layer compressed by a paraboloidal indenter. In addition to the paraboloidal idealization, an accurate mathematical description of the indenter spherical profile is incorporated into the modelling of this contact problem. Perturbed solutions up to the second order are developed for both compressible and incompressible layers. Selected numerical examples addressing actual joint geometries are included to assess the validity of the proposed analytical modelling, to explore the accuracy in describing the spherical profile and to compare the contact pressure profiles activated by paraboloidal and spherical indenters.