Elasto-hydrodynamic Lubrication Modelling of Spherical Metal-on-metal Aftificial Hip Joints

A full numerical methodology was developed for the elasto-hydrodynamic lubrication analysis of hip joint implants for the lubrication problem in spherical and conformal contacts. Typical results of a metal-on-metal hip implant were obtained to illustrate the applicability of the numerical methodology developed in the present study. INTRODUCTION The important effect of lubrication on long-term successful performance of artificial hip joints has been increasingly recognized. Elasto-hydrodynamic lubrication (EHL) can play a significant role in the overall tribological performance of artificial hip joints, particularly in reducing wear and wear particles released from the articulating surface, which can cause adverse tissue reactions, osteloysis and ultimately loosening of the prosthesis. Special considerations for EHL analyses of these man-made bearings in a spherical and conformal ball-in-socket configuration are required when compared with a general ballon-plane configuration extensively studied in the engineering literature [1-2]. The main goal of the present study was to develop a complete elasto-hydrodynamic lubrication model for a spherical and conformal contact problem of artificial hip joints, particularly focusing on the numerical methodology. LUBRICATION MODEL AND NUMERICAL APPROACH A typical artificial hip joint employing a metallic acetabular socket against a metallic head was considered. A conformal ball-in-socket configuration was used to represent the articulation between the cup and the head as shown in Fig.1. The conditions employed are shown in table 1 to represent the typical metal-on-metal hip implant. The articulation between the surfaces of the femoral head and the cup generally took place in the presence of a fluid film, which was calculated by considering both rigid separation and elastic deformation due to fluid film pressure. Reynolds equation in the spherical coordinate system was solved for the fluid film lubrication analysis using the finite difference method. The whole solution domain was selected as the hemispherical surface and divided uniformly into two sets of a fine Table 1 Conditions for the hip implant Femoral head radius (R1) 14.0 mm Acetabular cup radius (R2) 14.03mm Cup wall thickness (d) 9.50 mm Elastic modulus for metal (E) 210 GPa Poisson’s ratio for metal (ν) 0.3 Angular velocity of the femoral head (ωx) -2.0 1/s Vertical load applied to the acetabular cup (w) 1500 N grid for 256×256 mesh (∆φ, ∆λ) and a coarse grid for 128×128 mesh (2∆φ, 2∆λ) in spherical coordinate intervals. The elastic deformation of the spherical bearing surface was evaluated by a two dimensional spherical fast Fourier transform (SFFT) method [3]. The corresponding displacement response function for deformation calculation was obtained from the finite element method by applying a unit pressure pulse to an element. For further improving the efficiency of the iteration to the EHL problem, the SFFT technique was combined with numerical transformation of an interpolation for prolongation and a restriction for condensation for the elastic deformation calculations. This allowed the fluid film pressure to be determined on a fine grid, which was subsequently condensed onto the coarse grid by considering the full weight restriction. This was considered to be important for the low frequency components associated with the elastic deformation calculation for a majority of the bearing surfaces and materials for artificial hip joints. The deformation, once calculated on the coarse grid, was prolonged onto the fine grid by the interpolation for the 1 Copyright © 2005 by ASME subsequent pressure iteration. A two dimension cubic interpolation and corresponding full weighting restriction were formulated and used in the present numerical transformation between the fine and coarse grids. Fig.1 Ball-in-cup lubrication models RESULTS Fig.2 and 3 show the predicted film thickness and pressure through the centre of the loading in the entraining and sideleakage directions respectively for a viscosity of 0.001 Pas. The minimum and central film thicknesses for different viscosities were compared separately in Fig.4. DISCUSSION AND CONCLUSIONS Excellent agreements for the predicted film thickness and pressure, as well as the minimum film thickness and the central film thickness were found between results with and without the numerical transformations in the fine-coarse mesh grid. The resultant computational time required for a typical EHL solution was significantly reduced by as much as 3 times and yet the high accuracy required for the solution to the Reynolds equation was kept. The numerical transformation methodology failed to produce convergent results at lower viscosities below 0.01 Pas for such as the numerical transformation between 256×256 and 86×86 meshes. The reason was the numerical R1 R2 Metallic cup Metallic femoral head