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 Subsubsection Cubic CLE Up Subsection 4.1.3: Unconstrained Materials Subsubsection Ellipsoidal Fiber Distribution Donnan Equilibrium Swelling

The material type for a Donnan equilibrium swelling pressure is “Donnan equilibrium”. The swelling pressure is described by the equations for ideal Donnan equilibrium, assuming that the material is porous, with a charged solid matrix, and the external bathing environment consists of a salt solution of monovalent counter-ions [55, 37]. Since osmotic swelling must be resisted by a solid matrix, this material is not stable on its own. It must be combined with an elastic material that resists the swelling, using a “solid mixture” container as described in Section↓. The following material parameters need to be defined:
<phiw0> gel porosity (fluid volume fraction) in reference (strain-free) configuration, [ ]
<cF0> fixed-charge density in reference (strain-free) configuration, [n/L ]
<bosm> external bath osmolarity, [n/L ]
<Phi> osmotic coefficient, []
The Cauchy stress for this material is the stress from the Donnan equilibrium response [4]: where is the osmotic pressure, given by and is the fixed-charge density in the current configuration, related to the reference configuration via where is the relative volume. The values of the universal gas constant and absolute temperature must be specified as global constants. For ideal Donnan law, use .
Note that may be negative or positive; the gel porosity is unitless and must be in the range . A self-consistent set of units must be used for this model. For example:
(m, N, s, mol, K) (mm, N, s, nmol, K)
8.314 J/mol K 8.314×10 mJ/nmol K
Eq/m = mEq/L nEq/mm = mEq/L
mol/m = mM nmol/mm = mM
Pa MPa
Pa MPa
Though this material is porous, this is not a full-fledged biphasic material as described in Section 4.8↓ for example. The behavior described by this material is strictly valid only after the transient response of interstitial fluid and ion fluxes has subsided (thus Donnan equilibrium).
Donnan osmotic swelling reduces to zero when either or . Therefore, entering any other values for and at the initial time point of an analysis produces an instantaneous, non-zero swelling pressure. Depending on the magnitude of this pressure relative to the solid matrix stiffness, the nonlinear analysis may not converge due to this sudden swelling. Therefore, it is recommended to prescribe a load curve for either <cF0> or <bosm>, to ease into the initial swelling prior to the application of other loading conditions.
Example (using units of mm, N, s, nmol, K):
<material id="1" type="solid mixture">
  <mat_axis type="local">0,0,0</mat_axis>
  <solid type="Donnan equilibrium">
    <cF0 lc="1">1</cF0>
  <solid type="ellipsoidal fiber distribution">
  <loadcurve id="1">

 Subsubsection Cubic CLE Up Subsection 4.1.3: Unconstrained Materials Subsubsection Ellipsoidal Fiber Distribution