Finite Element Methods For Articular Cartilage Of Biphasic Porous Medium Nonlinear Model

The mechanical behavior of articular cartilage is essential for the research of movement mechanism, load carrying capacity and pathology of diarthrosis and has great significances and practical values for the treatment and diagnosis of arthropathy as well as the development of artificial articular materials.Biphasic porous medium model based on mixture theory can be used to accurately depict the mechanical behavior of articular cartilage. Experimental data show that the permeability of cartilage tissue is related with the volume strain of solid phase, which cannot be ignored. Based on biphasic porous medium model of articular cartilage, this thesis regards the solid phase as isotropic linear elastic medium, the fluid phase as ideal fluid, the deformation as small, and the permeability depending on the volume strain of solid phase, and deduce out the finite element method for this nonlinear model. First, the penalty finite element balance equation for quasi-static problem is obtained by the application of Galerkin weighted residual method and the introduction of penalty parameter. The system equation is nonlinear due to the dependence of permeability on the volume strain of solid phase. Differential method is used for the discretion of time zone, and a direct solution method and an incremental solution method are given out for the solution of the non-linear system equation.Second, a finite element program with four nodes - rectangle element is coded. A one-dimensional problem is analyzed with this program, and in turn, the creep and stress relaxation behaviors of articular cartilage under confined compression are numerically simulated. The numerical responses at different depths are consistent with those obtained by theories. In addition, the comparisons of displacements, velocities, solid effective stresses and pressure, obtained respectively by nonlinear model and linear model, display that the differences of the results become more obvious as the increase of the applied load, which demonstrates that the non-linear dependence of permeability on volume strain of solid phase is important as the deformation is not too small, and therefore cannot be ignored.The consistency between the numerical and the theoretical results illustrates that the finite element method given out in this thesis is correct and the designed program is reliable. The non-linear numerical analysis method obtained lays a foundation for the further researching of the mechanical behavior of articaular cartilage.