Analytical modelling of the elastomeric layer in soft layer hip replacements

A mechanical analysis is carried out for a flat deformable layer, firmly an chored to a rigid substrate, and frictionlessly compressed by a rigid spher ical indenter, with reference to the design of hip replacements whose cup i s covered by an elastomeric layer. An available perturbation solution of di fferential character, developed for a layer subject to plane strain conditi ons, indented by a cylinder approximated as parabolic in shape and valid fo r high contact widths, is here extended to the axisymmetric problem of a la yer compressed by a paraboloidal indenter. In addition to the paraboloidal idealization, an accurate mathematical description of the indenter spherica l profile is incorporated into the modelling of this contact problem. Pertu rbed solutions up to the second order are developed for both compressible a nd incompressible layers. Selected numerical examples addressing actual joi nt geometries are included to assess the validity of the proposed analytica l modelling, to explore the accuracy in describing the spherical profile an d to compare the contact pressure profiles activated by paraboloidal and sp herical indenters.