A penalty finite element analysis for nonlinear mechanics of biphasic hydrated soft tissue under large deformation

Abstract

AbstractThe non‐linear response of soft hydrated tissues under physiologically relevant levels of mechanical loading can be represented by a two‐phase continuum model based on the theory of mixtures. The governing equations for a biphasic soft tissue, consisting of an incompressible solid and an incompressible, inviscid fluid, under finite deformation are presented and a finite element formulation of this highly non‐linear problem is developed. The solid phase is assumed to be hyperelastic, and the stress‐strain relations for the solid phase are defined in terms of the free energy function. A finite element model is formulated via the Galerkin weighted residual method coupled with a penalty treatment of the continuity equation for the mixture. Using a total Lagrangian formulation, the non‐linear weighted residual statement, expressed with respect to the reference configuration, leads to a coupled non‐linear system of first order differential equations. The non‐linear constitutive equation for the solid phase elasticity is incrementally linearized in terms of the second Piola‐Kirchhoff stress and the corresponding Lagrangian strain. A tangent stiffness matrix is defined in terms of the free energy function; this matrix definition can be applied to any free energy function, and will yield a symmetric matrix when the free energy function is convex. An unconditionally stable implicit predictor‐corrector algorithm is used to obtain the temporal response histories. The confined compression mechanical test of soft tissue in stress relaxation is used as an example problem. Results are presented for moderate and rapid rates of loading, as well as small and large applied strains. Comparison of the finite element solution with an independent finite difference solution demonstrates the accuracy of the formulation.