Finite Element Formulation of Poro-Elasticity Suitable for Large Deformation Dynamic Analysis

A finite element model based on mixture theory is presented for the analysis of a mechanical phenomenon involving dynamic expulsion of fluids from a fully saturated porous solid matrix in the regime of both infinitesimal and finite deformation. The governing equations are obtained by applying the conservation laws of momentum and mass to each phase and the solid-fluid mixture. A complete formulation based on the motion of the solid and fluid phases is first presented; then approximations are made with respect to the relative acceleration vector to arrive at a so-called u-p formulation, which is subsequently implemented in a finite element model. The variational forms and matrix formulations are presented. The matrix equations are consistently linearized. The Newmark method is chosen as the global solution algorithm for solving the general finite element matrix equations.

In the u-p formulation for the finite deformation analysis, a modified compressible neo-Hookean hyperelastic model with a Kelvin solid viscous enhancement for the solid matrix is implemented as a test function for the nonlinear constitutive model. The constitutive model for fluid flow is represented by a generalized Darcy’s law formulated with respect to the current configuration. Fluid compressibility is also considered in terms of volumetric logarithmic strain.

Numerical examples in 1-D and 2-D are presented to validate the finite element model. Results of the small and finite deformation analyses are compared at different strain levels. For the 1-D case the numerical simulation was also compared with the analytical solution. These examples demonstrate the significance of large deformation effects on the transient responses of porous structures, as well as the strong convergence profile exhibited by the iterative algorithm.