# Thread: Prescribing a Fluid Pressure on a Free Surface of a Triphasic Material

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## Prescribing a Fluid Pressure on a Free Surface of a Triphasic Material

In triphasic analyses, it is sometimes necessary to apply a fluid pressure on a free surface (a surface whose motion is not constrained by displacement boundary conditions). For example, in a permeation problem where a triphasic material has a free upstream surface and a constrained downstream surface, we may wish to prescribe a fluid pressure on the unconstrained upstream side, which will result in a compaction of the triphasic material. This example explains how to properly prescribe this type of boundary condition.

Recall that in a triphasic (or multiphasic) mixture, the total (or mixture) Cauchy stress is

 T=-p I + Ts

where p is the fluid pressure, I is the identity tensor and Ts is the stress resulting from the state of strain in the solid matrix. As explained in the Theory Manual, the fluid pressure p is not continuous across the boundary of a triphasic material (since p may include a Donnan osmotic pressure contribution).

Parameters that do satisfy continuity across the boundaries of triphasic materials are the mixture traction and the effective fluid pressure. Let the normal component of the mixture traction be denoted by t. According to the above relation  for the mixture stress,

 t = -p + ts

where ts is the normal component of traction resulting from Ts. In a fluid bath, ts=0 (since there is no solid there). Therefore, the value of t = -p in the bath.

Let the effective fluid pressure be denoted by pe. In general, pe is related to p according to

 pe = p - R*T*Phi*osmolarity

where R=universal gas constant, T=absolute temperature, Phi=osmotic coefficient (=1 for ideal solutions), and osmolarity is the sum of concentrations of all solutes.

We now have all the necessary relations to prescribe a fluid pressure p on a free triphasic surface, using a fluid bath whose osmolarity is given. In PreView, pe (which the user must evaluate from equation ) should be prescribed as a nodal boundary condition. t (which should be evaluated from equation  with ts=0) should be prescribed as a "biphasic normal mixture traction".

The attached example file, PermeationUnclamped.feb, illustrates this type of boundary condition. In this steady-state one-dimensional permeation analysis, a triphasic material is first allowed to swell (from t=0 to t=1) by decreasing the fixed charge density from 0 to -200 mEq/L, while the upstream and downstream bath pressures are set to p=0, with a bath osmolarity of 300 mM. Given R=8.314 mJ/nmol.K and T=293 K, it follows that pe=-0.731 MPa (upstream and downstream) and t=0 MPa (prescribed upstream only, since the downstream face is fixed). Then, from t=1 to t=2, p is raised from 0 to 0.5 MPa, which is equivalent to raising pe from -0.731 MPa to -0.231 MPa and decreasing t from 0 to -0.5 MPa over that time interval.
Last edited by ateshian; 07-29-2012 at 01:00 PM. Reason: This post was modified to show a simpler alternative for this problem.  Reply With Quote

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## I am wondering whether the triphasic material takes electric potential and FCD gradients into account when modelling fluid flux? Quadriphasic (Huyghe and Janssen, 1997) theory suggests that these gradients may play an important role in fluid flow in incompressible porous media. I have looked at the governing equations employed in the triphasic/multiphasic material but am not too sure if these gradients are accounted for in the flux relations.

Can you also tell me upon which theory the multiphasic material is based?

Mark  Reply With Quote

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## Hi Mark,

The foundation for the triphasic material is the paper by Lai et al.,

Lai, W. M., Hou, J. S., and Mow, V. C., 1991. A triphasic theory for the swelling and deformation behaviors of articular cartilage. J Biomech Eng 113, 245-258

The generalization to multiphasic materials may be found in

Gu, W. Y., Lai, W. M., and Mow, V. C., 1998. A mixture theory for charged-hydrated soft tissues containing multi-electrolytes: passive transport and swelling behaviors. J Biomech Eng 120, 169-180

The implementation in FEBio generalizes these frameworks under finite deformation and also accounts for the solute frictional interactions with the solid (in addition to the traditional diffusion models that account for frictional interactions with the solvent) and for the solute solubility in the porous solid matrix (the exclusion of the solute from some of the pore space) as described in

Mauck, R. L., Hung, C. T., and Ateshian, G. A., 2003. Modeling of neutral solute transport in a dynamically loaded porous permeable gel: implications for articular cartilage biosynthesis and tissue engineering. J Biomech Eng 125, 602-614

The summary of governing equations used in FEBio may be found in the Theory Manual . In equation (2.125) you can see that the electric potential appears in the mechano-electrochemical potential of the solutes. Based on this dependency, it also appears in the partition coefficient in equation (2.133), which is used in the definition of the effective solute concentration in equation (2.132). The solvent and solute fluxes in (2.134)-(2.135) are dependent on the gradients of the effective solute concentrations, implying that they are directly dependent on the gradient of the electric potential. These dependencies are all consistent with the framework of Huyghe's quadriphasic theory.

The FCD appears in the electroneutrality condition, eq.(2.126), which is used to derive the divergence-free current condition of eq.(2.130). Equation (2.126) is used to solve for the electric potential once the effective concentrations are known at a given step in the iterative solution process. Though the gradient in the FCD does not appear explicitly in the governing equations, inhomogeneous distributions of FCD will produce inhomogeneous electric potentials accordingly.

Best,

Gerard  Reply With Quote

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## Attached please find an update of this tutorial, which uses the current version of FEBio.  Reply With Quote

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