Finite element formulations for hyperelastic transversely isotropic biphasic soft tissues

This paper presents a finite element model for the three-dimensional (3-D) nonlinear analysis of soft hydrated tissues such as articular cartilage in diarthrodial joints under physiologically relevant loading conditions. A biphasic continuum description is used to represent the soft tissue as a two-phase mixture of incompressible inviscid fluid and a hyperelastic, transversely isotropic solid. Alternate mixed-penalty and velocity-pressure finite element formulations are used to solve the non;nonlinear biphasic governing equations, including the effects of a strain-dependent permeability and a hyperelastic solid phase under finite deformation. The resulting first-order nonlinear system of equations are discretized in time using an implicit finite difference scheme, and solved using the Newton-Raphson method. A significant contribution of this work is the implementation and testing of a biphasic description with a transversely isotropic hyperelastic solid phase. This description considers a Helmholtz free energy function of five invariants of the Cauchy-Green deformation tensor and the preferred direction of the material, allowing for asymmetric behavior in tension and compression. An exponential form is suggested, and a set of material parameters is identified to represent the response of soft tissues in ranges of deformation and stress observed experimentally. After demonstrating the behavior of this constitutive model in simple tension and compression, a sample problem of unconfined compression is used to further validate the finite element implementation.