Triphasic finite element model for swelling porous media

The equations describing the mechanical behaviour of intervertebral disc tissue and other swelling porous media are three coupled partial differential equations in which geometric and physical non‐linearities occur. The boundary conditions are deformation‐dependent. To solve the equations for an arbitrary geometry and arbitrary boundary conditions, we use the finite element (FE) method. The differential equations are rewritten in an integral form by means of the weighted residual method. The domain of the integral is defined via a set of shape functions (i.e. finite elements). By applying the Gauss theorem and rewriting with respect to the reference state (total Lagrange), non‐linear equations are obtained. These are solved by means of the Newton‐Raphson technique. In order to get a finite set of equations, the weighted residual equations are discretized. The shape functions are chose as weighting functions (Galerkin method). This discretization results in a non‐symmetric stiffness matrix. A general description is given for the elements implemented into the commercial FE package DIANA (DIANA Analysis B.V., Delft, Netherlands). The numerical results of unconfined compression of a schematic intervertebral disc with varying proteoglycan concentration are given.