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

[1] 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 [1] for the mixture stress,

[2] 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

[3] 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 [3]) should be prescribed as a nodal boundary condition. t (which should be evaluated from equation [2] 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.