Formulation and evaluation of a finite element model for the biphasic model of hydrated soft tissues

Abstract

A finite element formulation of the linear biphasic model for articular cartilage and other hydrated soft tissues consisting of an incompressible, inviscid fluid phase and an incompressible solid phase is presented. The Galerkin weighted residual method is applied to the momentum equation and mechanical boundary conditions of both the solid phase and the fluid phase, and the continuity equation for the intrinsically incompressible binary mixture is introduced via penalty method. The resulting weak form is expressed in terms of the solid phase displacements and fluid phase velocities, which are interpolated for each element in terms of unknown nodal values, producing a system of first order differential equations which are solved using a standard numerical finite difference technique. In the limiting case of steady permeation (no solid displacement) the penalty method is shown to produce L 2 convergence of both the pressure field and the fluid velocity field. In contrast, it is known that the limiting case of an incompressible solid (no fluid flow) yields H 1 convergence of the solid phase displacement. An axisymmetric element of quadrilateral cross-section is developed and applied to the mechanical test problems of a cylindrical specimen of soft tissue in confined and perfectly lubricated unconfined compression, for which independent analytical solutions are available. The effects of mesh size and mesh distortion, solution parameters such as penalty number and time step size, and platen-specimen friction for the unconfined compression problem are evaluated.