A note on an asymptotic solution for the contact of two biphasic cartilage layers in a loaded synovial joint at rest

Abstract

Retaining the first terms of asymptotic expansions and assuming zero gradient of the contact pressure at the contact edge in the perpendicular direction to the edge, Ateshian et al. (1994, J. Biomechanics 27, 1347–1360; 1992, Adv. Bioeng. ASME 22, 191–194) have presented an asymptotic axially symmetric and plane strain solutions to the contact problems of the two thin biphasic cartilage layers in the synovial joint (with the synovial fluid film in between already depleted), subjected to a sudden normal load. Both the immediate and an early time response to a step loading have been analysed. The present note shows that the contact width thus obtained immediately after the load application differs from the numerical values for a dry frictionless contact of incompressible single-phase elastic layers available in the literature. The difference increases with the decreasing contact width-to-layer thickness ratio. It is proposed to improve the above solutions by releasing zero pressure gradient condition at the contact edge and calculating the instantaneous contact widths from the equations proposed for the corresponding case by Matthewson (1981, J. Mech. Phys. Solids 29, 89–113) and Meijers (1967, Appl. Sci. Res. 18, 353–383) that approximate better the numerical values. The instantaneous rate of change in the contact width is then obtained for a step load varying in time.