Modeling large-deforming fluid-saturated porous media using an Eulerian incremental formulation

Abstract

The paper deals with modelling fluid saturated porous media subject to large deformation. An Eulerian incremental formulation is derived using the problem imposed in the spatial configuration in terms of the equilibrium equation and the mass conservation. Perturbation of the hyperelastic porous medium is described by the Biot model which involves poroelastic coefficients and the permeability governing the Darcy flow. Using the material derivative with respect to a convection velocity field we obtain the rate formulation which allows for linearization of the residuum function. For a given time discretization with backward finite difference approximation of the time derivatives, two incremental problems are obtained which constitute the predictor and corrector steps of the implicit time-integration scheme. Conforming mixed finite element approximation in space is used. Validation of the numerical model implemented in the SfePy code is reported for an isotropic medium with a hyperelastic solid phase. The proposed linearization scheme is motivated by the two-scale homogenization which will provide the local material poroelastic coefficients involved in the incremental formulation.

Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema

“…4.2] (see also [77,Sect. 4.2] and [49,62]). The boundary conditions in these references permit drainage (fixing a zero fluid pressure) on the whole top lid, setting zero fluid flux elsewhere on the boundary, the bottom edge is clamped (imposing zero displacement) while the vertical walls are on rollers (only the horizontal displacement is set to zero), and the remainder of the top edge is traction-free.…”

“…In this regard, recent works include an enriched Galerkin framework [26], stabilised finite elements [15,77] and hybrid finite elements as well [76]. Although most contributions address the problem in a Lagrangian frame of reference, some alternative formulations include descriptions in Eulerian [62] and ALE [21] coordinates. Besides the stress response, nonlinear models differ from the linear ones in that pressure is not a primary variable, but instead it is given by constitutive modelling [24,30].…”

Homogenization of large deforming fluid-saturated porous structures

“…where f stands for volume forces acting on the porous medium and u is the skeleton (local) velocity, see also [33].…”

“…An alternative approach using the arbitrary Lagrangian Eulerian formulation was published in [11]. Here, we will follow the linearization scheme proposed for the Biot model in our previous work [33], however, we consider quasistatic loading only, so that all inertia effects vanish. At the macro-level, the computational algorithm is consistent with linearization of the residual formulation in the Eulerian framework, such that the incremental scheme uses the updated Lagrangian approach.…”

Viscoelastodynamics of swelling porous solids at large strains by an Eulerian approach

“…This is indeed a classical assumption, especially in connection with swelling in soft materials (like gels) under large strains, see, e.g., [3, 5-7, 10-12, 17] or [2] for the coupling with inelastic strain (reflecting plasticity or creep). The swelling distortion can be modeled, alternatively to the multiplicative decomposition of the total strain, by adopting the Biot model [4] for the large strains, as used in [15,25] for neo-Hookean material.…”