Three-dimensional contact of TITH cartilage layers, a closed-form solution

TL;DR

This paper presents a closed-form solution for the three-dimensional contact problem of inhomogeneous, anisotropic TITH cartilage layers, providing insights into contact pressure distribution and bone approach relevant for in vivo experimental interpretation.

Contribution

It extends previous models by deriving a closed-form solution for TITH cartilage contact, accounting for inhomogeneity and anisotropy in a thin-layer approximation.

Findings

Solution in closed-form for TITH cartilage contact

Contact pressure distribution influenced by inhomogeneity

Bone approach proportional to contact area

Abstract

Inhomogeneity and anisotropy play a crucial role in attributing articular cartilage its properties. The frictionless contact model constructed here consists in two thin biphasic transversely isotropic transversely homogeneous (TITH) cartilage layers firmly attached onto rigid substrates and shaped as elliptic paraboloids of different radii. Using asymptotic techniques, a solution to the deformation problem of such material has been recently obtained extending previous ones referred to homogeneous materials. The layer itself is thin in comparison with the size of the contact area and the observed time is shorter than the hydrogel characteristic time. The emerging three-dimensional contact problem is solved in closed-form and numerical benchmarks for constant and oscillating loads are given. The results are shown in terms of contact pressure and approach of the bones. The latter is derived to be directly proportional to the contact area. Existing experimental data are reinterpreted in view of the current model formulation. Comparisons are made with existing solutions for homogeneous biphasic materials in order to underline the functional importance of inhomogeneity in spreading the contact pressure distribution across the contact area. Particular attention is paid to the applicability of the retrieved formulas for interpreting measurements of in vivo experiments. Future directions are also prospected.