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Subsubsection 4.1.3.4: Donnan Equilibrium Swelling Up Subsection 4.1.3: Unconstrained Materials Subsubsection 4.1.3.6: Ellipsoidal Fiber Distribution Neo-Hookean

#### 4.1.3.5 Ellipsoidal Fiber Distribution

The material type for an ellipsoidal continuous fiber distribution is “ellipsoidal fiber distribution”. Since fibers can only sustain tension, this material is not stable on its own. It must be combined with a stable compressible material that acts as a ground matrix, using a “solid mixture” container as described in Section 4.1.3.22↓. The following material parameters need to be defined:
 parameters [ ] parameters [P]
The Cauchy stress for this fibrous material is given by [38, 7, 4]: Here, is the square of the fiber stretch , is the unit vector along the fiber direction, in the reference configuration, which in spherical angles is directed along , , and is the unit step function that enforces the tension-only contribution.
The fiber stress is determined from a fiber strain energy function, where in this material, The materials parameters and are assumed to vary ellipsoidally with , according to The orientation of the material axis can be defined as explained in detail in Section 4.1.1↑.
Example:
<material id="1" type="solid mixture">
<mat_axis type="local">0,0,0</mat_axis>
<solid type="neo-Hookean">
<E>1000.0</E>
<v>0.45</v>
</solid>
<solid type="ellipsoidal fiber distribution">
<ksi>10, 12, 15</ksi>
<beta>2.5, 3, 3</beta>
</solid>
</material>


Subsubsection 4.1.3.4: Donnan Equilibrium Swelling Up Subsection 4.1.3: Unconstrained Materials Subsubsection 4.1.3.6: Ellipsoidal Fiber Distribution Neo-Hookean