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Abstract
A hybrid finite element method has been developed to solve the equations governing the linear biphasic behavior of soft tissues. The biphasic model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, one solid and one fluid, and employs mixture theory to derive equations governing its mechanical behavior. These equations are time dependent, and are solved by a spatial finite element and a temporal finite difference approximation. The first step in derivation of this hybrid method is application of the finite difference rule to the solid phase velocity and displacement, thus obtaining equations with only velocities at discrete times as primary variables. A weighted residual statement of the temporally discretized governing equations, employing C$\sp0$ continuous interpolations of the solid and fluid phase velocities and interpolations of the pore pressure and elastic stress which may be discontinuous, is then derived. The stress and pressure functions are chosen to satisfy the total momentum equation of the mixture, yielding what is referred to as an equilibrated stress and pressure field. The weighting functions are interpolated using the same functions as the velocity and stress/pressure fields. The matrix equations resulting from substitution of the interpolations into the weak form of the weighted residual statement are symmetric. This formulation is first applied to plane strain problems using six noded triangular elements with several different stress and pressure fields in element local coordinates. These elements are tested against analytical solutions and a mixed-penalty finite element formulation of the same equations. The hybrid method is found to be robust and produce excellent results; preferred 2-D elements are identified on the basis of these results. Ten noded tetrahedral elements implementing two different stress and pressure fields, as well as the mixed-penalty method, are developed and applied to several 3-D test problems. The capabilities to handle materials with transversely isotropic mechanical properties and geometry with boundary conditions in tangent/normal coordinates are also implemented, and such problems solved.
