Large Strain Poroelasticity
When a porous body experiences large deformations (large strains or large rotations), the infinitesimal theory cannot be used. In a large deformation context, the momentum equation, Equation 8-6, reads
(8-9)
where the gradient ∇ = ∂/∂X corresponds to the material gradient with respect to the undeformed configuration, and P is the (effective) first Piola–Kirchhoff stress tensor. See Equation of Motion in the Structural Mechanics Theory chapter for details.
The pore pressure contribution enters as a load in Equation 8-9. The first Piola–Kirchhoff stress generated from the pore pressure reads
(8-10)
where F is the deformation gradient and J is the volume ratio. In biphasic poroelasticity, the total stress in the mixture is often of interest. The corresponding Cauchy stress is computed as
(8-11)
where σ is the (effective) Cauchy stress computed in the Solid Mechanics interface.
Darcy’s law is also written with respect to the undeformed configuration. The material velocity field reads
where κ is the material permeability tensor of the porous matrix, μ is the fluid viscosity, and pf is the pore pressure. The symbol ∇ = ∂/∂X corresponds to the gradient with respect to the undeformed configuration.
Darcy’s velocity in the current (deformed) configuration is then computed with the so-called Piola transformation (Ref. 2)
For numerical reasons, and also since the solid mechanics equations are solved in a total Lagrangian formulation (see Lagrangian Formulation for details), it is more efficient to write the mass conservation equation for the fluid in the undeformed configuration. It is then given by (Ref. 2)
(8-12)
where ϕ = Jεp is the Lagrangian porosity and Qm is an external mass source or sink per unit reference volume. The material time derivative can be expanded as
(8-13)
The current porosity is derived from the mass conservation equation of the solid. For Biot poroelasticity, it is expressed in terms of the volume change J and the pore pressure pf as
For biphasic poroelasticity, the current porosity has the simple closed-form expression
where ε0 stands for the initial (undeformed) porosity.
Equation 8-13 can now be expressed as
The poroelastic storage (Equation 8-3) in a large strain context still reads
(8-14)
with the important difference that the current porosity in the deformed matrix, εp, is used. The fluid mass conservation Equation 8-12 is then formulated as
(8-15)
Similarly, the biphasic poroelasticity formulation reads
(8-16)
since the first term on the left hand side vanishes (Sp = 0) and the Biot–Willis coefficient equals 1.
The choice between the large strain and small strain formulation is normally governed by the Include geometric nonlinearity checkbox in the study step. If some feature in the model forces geometric nonlinearity, such as Contact, but the strains and rotations are expected to be small, it is still possible to let the Poroelasticity node use the small strain formulation. For this, select Geometrically linear in the Formulation list under the Linear Elastic Material node within the Solid Mechanics interface to enforce a small strain formulation.
Then, Darcy’s law is implemented according to the small strain formulation described by Equation 8-8, the poroelastic storage coefficient Sp follows the small strain formulation described in Equation 8-3, and the initial porosity ε0 is used instead of the current porosity εp.