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Subsection 4.2.2: Uncoupled Fiber Models Up Subsection 4.2.2: Uncoupled Fiber Models Subsubsection 4.2.2.2: Fiber with Neo-Hookean Law Uncoupled

#### 4.2.2.1 Fiber with Exponential-Power Law, Uncoupled Formulation

The material type for a single fiber with an exponential-power law, in an uncoupled strain energy formulation, is “fiber-exp-pow-uncoupled”. Since fibers can only sustain tension, this material is not stable on its own. It must be combined with a stable uncoupled material that acts as a ground matrix, using a “uncoupled solid mixture” container as described in Section 4.1.2.14↑. The following material parameters need to be defined:
 representing a measure of the fiber modulus [P] coefficient of exponential argument [ ] power of exponential argument [ ]
The fiber is oriented along the unit vector , where are orthonormal basis vectors representing the local element coordinate system when specified (Section 4.1.1↑), or else the global Cartesian coordinate system. The stress for this fibrous material is given by where is the square of the fiber stretch, , and is the unit step function that enforces the tension-only contribution.. 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:
Single fiber oriented along , embedded in a Mooney-Rivlin ground matrix.
<material id="1" type="uncoupled solid mixture">
<mat_axis type="local">0,0,0</mat_axis>
<k>10e3</k>
<solid type="Mooney-Rivlin">
<c1>10.0</c1>
<c2>0</c2>
</solid>
<solid type="fiber-exp-pow-uncoupled">
<ksi>5</ksi>
<alpha>20</alpha>
<beta>3</beta>
<mat_axis type="angles">
<theta>0</theta>
<phi>90</phi>
</mat_axis>
</solid>
</material>

Example:
Two fibers in the plane orthogonal to , oriented at ±25 degrees relative to , embedded in a Mooney-Rivlin ground matrix.
<material id="1" type="uncoupled solid mixture">
<mat_axis type="local">0,0,0</mat_axis>
<k>10e3</k>
<solid type="Mooney-Rivlin">
<c1>10.0</c1>
<c2>0</c2>
</solid>
<solid type="fiber-exp-pow-uncoupled">
<ksi>5</ksi>
<alpha>20</alpha>
<beta>3</beta>
<mat_axis type="angles">
<theta>90</theta>
<phi>25</phi>
</mat_axis>
</solid>
<solid type="fiber-exp-pow-uncoupled">
<ksi>5</ksi>
<alpha>20</alpha>
<beta>3</beta>
<mat_axis type="angles">
<theta>-90</theta>
<phi>25</phi>
</mat_axis>
</solid>
</material>


Subsection 4.2.2: Uncoupled Fiber Models Up Subsection 4.2.2: Uncoupled Fiber Models Subsubsection 4.2.2.2: Fiber with Neo-Hookean Law Uncoupled