Finite element formulations for biological soft hydrated tissues under finite deformation

This thesis addresses finite element based computational models for the 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. The theoretical foundations of this theory are reviewed with emphasis on constitutive modeling. Alternate mixed-penalty and velocity-pressure finite element formulations are used to solve the 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. Both formulations are used to develop quadrilateral and triangular elements in 2-D and hexahedral and tetrahedral elements in 3-D. Numerical examples, including those representative of soft tissue material testing and simple human joints, are used to validate the formulations and to illustrate their applications. While not the central objective of this work, important insights have been gained from selected linear biphasic analyses. A focus of this work is the comparison of the alternate formulations for nonlinear problems. While it is demonstrated that both formulations produce a range of converging elements, the velocity-pressure formulation is found to be more efficient computationally.

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.