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4.2.1.1 Fiber with Exponential-Power Law
The material type for a single fiber with an exponential-power law is “fiber-exp-pow”. 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:
| <ksi> | representing a measure of the fiber modulus | [P] |
| <alpha> | coefficient of exponential argument | [ ] |
| <beta> | 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 Cauchy 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 neo-Hookean ground matrix.
<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="fiber-exp-pow">
<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 neo-Hookean ground matrix.
<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="fiber-exp-pow">
<ksi>5</ksi>
<alpha>20</alpha>
<beta>3</beta>
<fiber type="angles">
<theta>90</theta>
<phi>25</phi>
</fiber>
</solid>
<solid type="fiber-exp-pow">
<ksi>5</ksi>
<alpha>20</alpha>
<beta>3</beta>
<fiber type="angles">
<theta>-90</theta>
<phi>25</phi>
</fiber>
</solid>
</material>