Knee Prostheses Modelling: Deformation of the Low Rigidity Component

Synovial fluid is usually assumed to be Newtonian despite its viscoelastic behaviour. This paper presents a joint replacement lubrication model, based on the Reynolds equation and Phan–Thien and Tanner rheological law (PTT). The model, of ellipsoid-on-plane type, contemplates the curvature of articular surfaces and calculates deformation of the tibial component—typically made of ultra-high molecular weight polyethylene (UHMWPE)—either by solving full 3D linear elasticity equations (EoPE) or by using a simplified column model (EoPEc). Applying a numerical procedure based on the Finite Element Method, equations governing the non-Newtonian lubrication flow were simultaneously solved to find the component of the total stress tensor for the PTT model, in perpendicular direction to the flow and lubricating film thickness. From these results, we calculated friction forces and coefficients on articular surfaces. Simulations were carried out corresponding to the squeeze film lubrication mode, varying the relaxation constant λ of the fluid. Small differences in friction coefficients are observed when comparing the EoPE and EoPEc models. However, EoPEc model overestimates the lubricant film thickness, regardless of the value of λ considered; therefore, quantitative predictions derived from the EoPE model are more conservative, especially when considering the height of the lubrication channel.