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Subsubsection 4.1.3.23: Spherical Fiber Distribution Up Subsection 4.1.3: Unconstrained Materials Subsubsection 4.1.3.25: Coupled Transversely Isotropic Mooney-Rivlin

#### 4.1.3.24 Spherical Fiber Distribution from Solid-Bound Molecule

The material type for a spherical (isotropic) continuous fiber distribution with fiber modulus dependent on solid-bound molecule content is “spherical fiber distribution sbm”. 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 [ ] fiber modulus [P] fiber modulus exponent [ ] fiber mass density [M/L] sbm index of solid-bound molecule [ ]
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 . The fiber modulus is dependent on the solid-bound molecule referential density according to the power law relation where is the density at which .
This type of material references a solid-bound molecule that belongs to a multiphasic mixture. Therefore this material may only be used as the solid (or a component of the solid) in a multiphasic mixture (Section 4.10↓). The solid-bound molecule must be defined in the <Globals> section (Section 3.4.3↑) and must be included in the multiphasic mixture using a <solid_bound> tag. The parameter sbm must refer to the global index of that solid-bound molecule. The value of is specified within the <solid_bound> tag. If a chemical reaction is defined within that multiphasic mixture that alters the value of , lower and upper bounds may be specified for this referential density within the <solid_bound> tag to prevent from reducing to zero or achieving excessively elevated values.
Note: In the limit when , the expression for 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:
<solid type="solid mixture">
<solid type="neo-Hookean">
<E>1000.0</E>
<v>0.45</v>
</solid>
<solid type="spherical fiber distribution sbm">
<alpha>0</alpha>
<beta>2.5</beta>
<ksi0>10</ksi0>
<gamma>2</gamma>
<rho0>1</rho0>
<sbm>1</sbm>
</solid>
</solid>


Subsubsection 4.1.3.23: Spherical Fiber Distribution Up Subsection 4.1.3: Unconstrained Materials Subsubsection 4.1.3.25: Coupled Transversely Isotropic Mooney-Rivlin