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Subsubsection 4.1.3.22: Solid Mixture Up Subsection 4.1.3: Unconstrained Materials Subsubsection 4.1.3.24: Spherical Fiber Distribution from Solid-Bound Molecule

4.1.3.23 Spherical Fiber Distribution

The material type for a spherical (isotropic) continuous fiber distribution is “spherical 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:
 parameter [ ] parameter [ ] 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 fiber strain energy density is given by where , , and .
Note: In the limit when , this expressions produces a power law, Note: When , the fiber modulus is zero at the strain origin (. Therefore, use when a smooth transition in the stress is desired from compression to tension.
Example:
<material id="1" type="solid mixture">
<solid type="neo-Hookean">
<E>1000.0</E>
<v>0.45</v>
</solid>
<solid type="spherical fiber distribution">
<ksi>10</ksi>
<alpha>0</alpha>
<beta>2.5</beta>
</solid>
</material>


Subsubsection 4.1.3.22: Solid Mixture Up Subsection 4.1.3: Unconstrained Materials Subsubsection 4.1.3.24: Spherical Fiber Distribution from Solid-Bound Molecule